Testing Alternative Theories of Quantum

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Testing Alternative Theories of Quantum Mechanics with Optomechanics, and Effective Modes for Gaussian Linear Optomechanics
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Helou, Bassam Mohamad
(2019)
Testing Alternative Theories of Quantum Mechanics with Optomechanics, and Effective Modes for Gaussian Linear Optomechanics.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/KJ1K-9268.
Abstract
Optomechanics has made great strides in theory and experiments over the past decade, which culminated in the first direct detection of gravitational waves in 2015 by LIGO. This thesis explores how optomechanics can be used to test fundamental physics other than the theory of general relativity. Our emphasis will be on falsifiable theories (ultimately, only experiments can decide whether a theory is correct) that address two outstanding issues in quantum mechanics: the measurement problem, and reconciling quantum mechanics with the theory of general relativity. In particular, we show that the space experiment LISA pathfinder places aggressive bounds on two objective collapse models, which are non-linear stochastic modifications of the Schroedinger equation that can resolve the measurement problem. Moreover, we show that state-of-the-art torsion pendulum experiments can test the Schroedinger-Newton theory, which is the non-relativistic limit of a non-linear theory combining quantum mechanics with a fundamentally classical spacetime.
Along the way, we propose how to resolve two major difficulties with determining the predictions of non-linear quantum mechanics in an actual experiment. First, we cannot use the density matrix formalism in non-linear quantum mechanics and so we have to suggest and justify a particular ensemble for the thermal bath. Separating out quantum and classical fluctuations helped us propose a reasonable ensemble. Second, most researchers believe that deterministic non-linear quantum mechanics must violate the no-signaling condition. We show this isn't necessarily the case because different interpretations of quantum mechanics make different predictions in non-linear quantum mechanics. We propose an interpretation, the causal-conditional prescription, that doesn't violate causality by noticing that once we fix an initial state, the evolution of a system under many non-linear theories is equivalent to evolution under a linear Hamiltonian with feedback. The mapping allows us to leverage the tools of quantum control, and it tells us that if the non-linear parameters of a non-linear Hamiltonian respond causally (i.e. with an appropriate delay) to measurement results, then the theory can be made causal.
We also contribute to the theory of quantum optomechanics. We introduce two new bases that one can view environment modes with. In linear optomechanics a system interacts with an infinite number of bath modes. We show that the interaction can be reduced to one with finite degrees of freedom. Moreover, at any particular time, the system is correlated with only a finite number of bath modes. We show that if we make the assumption that we can measure any commuting environment modes, then this basis allows us to understand the one-shot quantum Cramer-Rao bound in a simple way, and allows us to sweep large parameter regimes and so find promising optomechanics topologies for quantum state preparation tasks that we can then analyze without the assumption of being able to measure any observable of the environment. We also use this basis to show that when we are interested in the conditional dynamics of a test mass, we can only adiabatically eliminate a lossy cavity when we measure the optomechanical system at a slow enough rate. Finally, we develop an analytic filter for obtaining the state of a generic optomechanical system that interacts linearly with its environment and is driven by Gaussian states, and where the outgoing light is measured with a non-linear photon-counting measurement. We hope that our work will help researchers explore optomechanics topologies that make use of photon counters.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Quantum Mechanics, Optomechanics, Schroedinger-Newton theory, collapse models, non-linear quantum mechanics
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Chen, Yanbei
Group:
TAPIR, Astronomy Department
Thesis Committee:
Schwab, Keith C. (chair)
Painter, Oskar J.
Miao, Hiaoxing
Chen, Yanbei
Defense Date:
11 December 2018
Record Number:
CaltechTHESIS:12182018-142547647
Persistent URL:
DOI:
10.7907/KJ1K-9268
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DOI
PRD article adapted for Ch. 2
DOI
PRD article adapted for Ch. 3
arXiv
arXiv article adapted for Ch. 4
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Author
ORCID
Helou, Bassam Mohamad
0000-0003-2760-7622
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No commercial reproduction, distribution, display or performance rights in this work are provided.
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11321
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CaltechTHESIS
Deposited By:
Bassam Helou
Deposited On:
27 Mar 2019 20:51
Last Modified:
04 Mar 2020 22:06
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Testing alternative theories of Quantum Mechanics with
Optomechanics, and effective modes for Gaussian linear
Optomechanics

Thesis by

Bassam Helou

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2019
Defended 12/11/2018

Bassam Helou
ORCID: 0000-0003-2760-7622

iii

ACKNOWLEDGEMENTS
My time at Caltech wasn’t just an academic journey. I would like to first thank the
people who made a lasting impact on my self. Haixing Miao was a great mentor,
and his outlook on life continues to inspire me. William H. Hurt welcomed me to his
family, re-ignited a love of reading, and shared his worldview with me. Robert Wills
cultivated a love of the outdoors. Karthik Seetharam cultivated a love of working out.
Zhewei Chen cultivated a love of climbing and other outdoor adventures. Locatelli
Rao gave me some tools and understanding of physical therapy. Lee Coleman
introduced me to mindfulness meditation.
I am also thankful for friends who have me kept me sane over the years: Anandh
Swaminathan, Stephan Zheng, Rose Yu, Li Zhiquin, Tejas Deshpande, Zachary
Mark, Linhan Shen, Eric Wolff, Min-feng Tu, Nicole Yunger-Halpern, Pavan Bilgi,
Evan Miyazano, Ivan Papusha, Jenia Mozgunov, Howard Hui, Daniel Naftalovich,
Stephen Perry, Krzysztof Chalupka, Xiaoyu Shi, Becky Schwantes, Jacklyn Phu,
Corwin Shiu, Tung Wongwaitayakornkul, Mark Harfouche, Rebecca Herrera, Luis
Goncalves, Vipul Singhal, Tobias Bischoff, Chris Rollins, Julius Su, Reed Scharff,
Elena Murchikova and Tal Einav. I’d also like to thank the Alpine club for organizing
many amazing trips. I’d also like to thank ’the drunkest table’ for staying in touch
and organizing many events. Specifically, I’d like to thank Tony Liao, Utkarth Gaur,
Mike Wang, Andrew Ng, Velora Chan, the two Jon Ngs, Jimmy Lu, Hecheng Wang,
Roshny Francis, Adrienne Wang and Patricia Ko.
I am thankful for many stimulating discussions with other researchers that made me
feel like I was part of a community. I would like to thank Maaneli Derakhshani,
Antoine Tilloy, Huan Yang, Yiqiu Ma, P. C. E. Stamp, Dan Carney, Belinda Pang
and Xiang Li.
I’d like to thank my family for their unwavering support: Hikmat Helou, Mohamad
Helou, Akram Helou, Randa Helou, Rym Helou, Randa Kabbara, Malek Kabbara,
Khaled Helou, my grandfather Akram Helou and Houda Helou. I’d like to also
thank extended family for their support: Amer Mikkewi, Mo Mikkewi, Kareem
Mikkewi, Jihad Mikkewi, Vivian Cunanan, Wilfredo Cunanan, Bernadette Glen,
Douglas Murray, Alex Murray, Molly Purnell, Kelley Purnell and Mark Purnell.
I’d like to thank the ISP office, JoAnn Boyd, Jennifer Blankenship, Alfrida King
and Christy Jenstad for their kindness and amazing administrative support.
I’d like to thank CTLO and the Caltech library staff for being supportive of my ideas.

iv
I’d also like to thank Kerry Vahala for allowing me to experiment with TA ideas.
Finally, I’d like to thank Yanbei Chen for his patience, support and the freedom he
gave me. His boundless creativity and curiosity is inspirational. I am a much better
researcher and problem solver because of him!

ABSTRACT
Optomechanics has made great strides in theory and experiments over the past
decade, which culminated in the first direct detection of gravitational waves in 2015
by LIGO. This thesis explores how optomechanics can be used to test fundamental
physics other than the theory of general relativity. Our emphasis will be on falsifiable
theories (ultimately, only experiments can decide whether a theory is correct) that
address two outstanding issues in quantum mechanics: the measurement problem,
and reconciling quantum mechanics with the theory of general relativity. In particular, we show that the space experiment LISA pathfinder places aggressive bounds
on two objective collapse models, which are non-linear stochastic modifications of
the Schroedinger equation that can resolve the measurement problem. Moreover, we
show that state-of-the-art torsion pendulum experiments can test the SchroedingerNewton theory, which is the non-relativistic limit of a non-linear theory combining
quantum mechanics with a fundamentally classical spacetime.
Along the way, we propose how to resolve two major difficulties with determining
the predictions of non-linear quantum mechanics in an actual experiment. First, we
cannot use the density matrix formalism in non-linear quantum mechanics and so
we have to suggest and justify a particular ensemble for the thermal bath. Separating
out quantum and classical fluctuations helped us propose a reasonable ensemble.
Second, most researchers believe that deterministic non-linear quantum mechanics
must violate the no-signaling condition. We show this isn’t necessarily the case
because different interpretations of quantum mechanics make different predictions in
non-linear quantum mechanics. We propose an interpretation, the causal-conditional
prescription, that doesn’t violate causality by noticing that once we fix an initial state,
the evolution of a system under many non-linear theories is equivalent to evolution
under a linear Hamiltonian with feedback. The mapping allows us to leverage the
tools of quantum control, and it tells us that if the non-linear parameters of a nonlinear Hamiltonian respond causally (i.e. with an appropriate delay) to measurement
results, then the theory can be made causal.
We also contribute to the theory of quantum optomechanics. We introduce two
new bases that one can view environment modes with. In linear optomechanics a
system interacts with an infinite number of bath modes. We show that the interaction
can be reduced to one with finite degrees of freedom. Moreover, at any particular
time, the system is correlated with only a finite number of bath modes. We show
that if we make the assumption that we can measure any commuting environment
modes, then this basis allows us to understand the one-shot quantum Cramer-Rao
bound in a simple way, and allows us to sweep large parameter regimes and so find
promising optomechanics topologies for quantum state preparation tasks that we
can then analyze without the assumption of being able to measure any observable
of the environment. We also use this basis to show that when we are interested
in the conditional dynamics of a test mass, we can only adiabatically eliminate

vi
a lossy cavity when we measure the optomechanical system at a slow enough
rate. Finally, we develop an analytic filter for obtaining the state of a generic
optomechanical system that interacts linearly with its environment and is driven
by Gaussian states, and where the outgoing light is measured with a non-linear
photon-counting measurement. We hope that our work will help researchers explore
optomechanics topologies that make use of photon counters.

vii

PUBLISHED CONTENT AND CONTRIBUTIONS

B. Helou, J. Luo, H. Yeh, C. Shao, B. J. J. Slagmolen, D. E. McClelland, and Y. Chen
(2017). “Measurable signatures of quantum mechanics in a classical spacetime”. In:
PRD 96, 044008. doi: 10.1103/PhysRevD.96.044008.
BH participated in the conception of the project, co-wrote the manuscript, co-carried
the calculations, carried out the simulations and co-analyzed the results.
B. Helou, B J. J. Slagmolen, D. E. McClelland, and Y. Chen (2017). “LISA pathfinder
appreciably constrains collapse models”. In: PRD 95, 084054. doi: 10.1103/PhysRevD.95.084054.
BH participated in the conception of the project, co-wrote the manuscript, co-carried
the calculations, carried out the simulations and co-analyzed the results.
B. Helou, and Y. Chen (2017). “Different interpretations of quantum mechanics make
different predictions in non-linear quantum mechanics, and some do not violate the
no-signaling condition”. In: arXiv:1709.06639.
BH conceived of the project, co-developed the results, and wrote the manuscript

viii

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . vii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Introduction to collapse models . . . . . . . . . . . . . . . . . . . . 3
1.3 Alternatives to quantum gravity . . . . . . . . . . . . . . . . . . . . 5
1.4 Overview of contributions to the theory of optomechanics . . . . . . 11
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter II: LISA pathfinder appreciably constrains collapse models . . . . . . 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Constraining the collapse models . . . . . . . . . . . . . . . . . . . 23
2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Chapter III: Measurable signatures of quantum mechanics in a classical spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Free dynamics of an optomechanical setup under the SchroedingerNewton theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Nonlinear quantum optomechanics with classical noise . . . . . . . . 39
3.4 Measurements in nonlinear quantum optomechanics . . . . . . . . . 50
3.5 Signatures of classical gravity . . . . . . . . . . . . . . . . . . . . . 54
3.6 Feasibility analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.8 Appendix: Conservation of energy in the SN theory . . . . . . . . . 72
3.9 Appendix: Derivation of p0 → ξ and p0 ← ξ . . . . . . . . . . . . 74
3.10 Appendix: More details on calculating B̂(ω) ξ . . . . . . . . . . . 76
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Chapter IV: Different interpretations of quantum mechanics make different
predictions in non-linear quantum mechanics, and some do not violate the
no-signaling condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 Multiple measurements in sQM and the no-signaling condition . . . 84
4.3 Ambiguity of Born’s rule in NLQM . . . . . . . . . . . . . . . . . . 87
4.4 The no-signaling condition in NLQM . . . . . . . . . . . . . . . . . 91

ix
4.5 Causal-conditional: A sensible prescription that doesn’t violate the
no-signaling condition . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Chapter V: Measurable signatures of a causal theory of quantum mechanics
in a classical spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2 NLQM is formally equivalent to quantum feedback . . . . . . . . . . 109
5.3 An example of continuously monitored optomechanical systems . . . 117
5.4 Signature of SN with the causal-conditional prescription . . . . . . . 123
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Chapter VI: Effective modes for linear Gaussian optomechanics. I. Simplifying the dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2 Effective modes for an optomechanical setup driven by pulsed bluedetuned light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.3 Effective modes for general setups . . . . . . . . . . . . . . . . . . . 144
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.5 Appendix: Constructing effective modes that simplify the dynamics
of a cavity optomechanical setup interacting with a single sideband
of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.6 Appendix: An alternative proof based on group theory . . . . . . . 155
6.7 Appendix: Constructing effective modes that simplify the dynamics
of a general optomechanical setup . . . . . . . . . . . . . . . . . . . 155
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Chapter VII: Effective modes for linear Gaussian optomechanics. II. Simplifying the entanglement structure between a system and its environment . . 166
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.2 Setup and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.3 Entanglement structure . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.5 Correlation structure of a system with its optical bath . . . . . . . . . 185
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Chapter VIII: Adiabatically eliminating a lossy cavity can result in gross
underestimations of the conditional variances of an optomechanical setup 195
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.2 Unconditional dynamics of a cavity optomechanical setup . . . . . . 196
8.3 Numerics showing the breakdown of adiabatic elimination in describing conditional dynamics . . . . . . . . . . . . . . . . . . . . . 202
8.4 Insights from a simplified version of the problem . . . . . . . . . . . 203
8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.6 Appendix: Introduction to quantum state preparation in optomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

8.7 Appendix: The unconditional covariance matrix for the setup in Sec.
8.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Chapter IX: The conditional state of a linear optomechanical system that is
being monitored by a non-linear, photon-counting, measurement . . . . . 218
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
9.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
9.3 Switching to the Wigner function . . . . . . . . . . . . . . . . . . . 221
9.4 The Projection operator in terms of the Wigner function . . . . . . . 221
9.5 Calculation of the conditional state . . . . . . . . . . . . . . . . . . 225
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Chapter X: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

LIST OF ILLUSTRATIONS

Number
Page
1.1 An optomechanical system with n degrees of freedom interacts with
(n)
(1)
, which evolve into n effecthrough Âin
n effective input modes, Âin
(1)
(n)
tive output modes, Âout through Âout . The operators b̂1 through b̂n
represent system degrees of freedom. The remainder of the effective
environment modes scatter into themselves and nothing else. . . . . . 14
2.1 Upper and lower bounds on the CSL collapse rate λCSL obtained
from laboratory experiments operating at different frequencies. Blue,
green, black and gray regions: exclusion regions obtained from LISA
pathfinder, LIGO, a millikelvin-cooled nanocantilever [21] and spontaneous emission from Ge [14, 9], respectively. Our calculation of
the bounds obtained from LIGO follow that of [8]. The dashed blue
line is the upper bound limit obtained from the LISA pathfinder repos
sults if Sa were used instead of Sa . The red and orange domains
are regions in which the collapse rate is too slow to explain the lack
of macroscopic superpositions and measurements, respectively. The
red region is below the lower bound of 10−17 s−1 proposed by Ghirardi, Pearle and Rimini [16]. The orange region’s boundary is the
Adler lower bound 10−8±2 s−1 , below which latent image formation
on a photographic emulsion consisting of silver halide suspended in
gelatine wouldn’t occur fast enough [1]. The orange error bars reflect
the uncertainty in this lower bound. . . . . . . . . . . . . . . . . . . 26
3.1 Left Panel: according to standard quantum mechanics, both the vector
(h x̂i, h p̂i) and the uncertainty ellipse of a Gaussian state for the center
of mass of a macroscopic object rotate clockwise in phase space, at
the same frequency ω = ωCM . Right panel: according to Eq.(3.7),
(hxi, hpi)
q still rotates at ωcm , but the uncertainty ellipse rotates at
2 > ω . (Figure taken from [29]). . . . . . . . . 34
ωq ≡ ωcm 2 + ωSN
cm
3.2 The proposed low-frequency optomechanical experiment. . . . . . . 39

xii
3.3

3.4

3.5

3.6

3.7

Two ways of forming the same Gaussian density matrix. In the left
panel, we have an ensemble of coherent states parameterized by a
complex amplitude α, which is Gaussian distributed. The red circle
depicts the noise ellipse, in phase space, of one such state. The
green ellipse depicts the total noise ellipse of the density matrix.
In the right panel, we have an ensemble of squeezed states with
amplitudes ε, which achieves the same density matrix with a fixed
squeeze amplitude and a uniform distribution of squeeze angles. . .
The two prescriptions, pre-selection (top) and post-selection (bottom), that can be used to calculate measurement probabilities. Both
prescriptions are equivalent in linear quantum mechanics, but become
different under non-linear quantum mechanics. . . . . . . . . . . . .
A depiction of the predicted signatures of semi-classical gravity. The
pre-selection measurement prescription’s signature is a narrow and
tall Lorentzian peak, while the post-selection measurement prescription’s signature is a shallow but wide Lorentzian dip. Both prescriptions predict a Lorentzian peak of thermal noise at ωcm . Note that
the figure is not to scale and throughout this article, we follow the
convention of 2-sided spectra. . . . . . . . . . . . . . . . . . . . . .
A histogram showing the distribution of two sets of 105 realizations of
ξ˜c (t) over a period of 200/γ (with γ set to 1), and a time discretization
of dt = 0.14/γ. In one set, ξ˜c (t) is chosen to have a spectrum of
Sd with d = 0.62, and in the second set, ξ˜c (t) has a spectrum of 1.
yth , which is chosen to be 2 in this example, allows us to construct
a decision criterion: if the collected measurement data’s estimator
satisfies Y < −yth , we decide that its noise power spectrum is Sd , if
Y > yth , white noise and if −yth ≤ Y ≤ yth , no decision is made. . .
Simulation results showing the minimum measurement time, τmin ,
required to distinguish between a Lorentzian spectrum and a flat
background in such a way that the probabilities of indecision and of
making an error are both below 10%. Plot (a) shows results for a
Lorentzian peak, while plot (b) is for a Lorentzian dip. The coherence
time is given by the inverse of the half width at half maximum of the
Lorentzian. Note that both plots are log-log plots. . . . . . . . . . . .

xiii
Simulation results showing the minimum measurement time, τmin ,
required to distinguish between the Schroedinger-Newton theory with
the pre-selection measurement prescription (which has the signature
of a Lorentzian with depth h) and standard quantum mechanics in
such a way that the probabilities of indecision and of making an error
are both below p%. The coherence time is given by the inverse of the
half width at half maximum of the Lorentzian. Note that the y-axis is
on a log scale. Moreover, the dashed lines are only to guide the eye
(and are fits of the form a ln (p) + b). . . . . . . . . . . . . . . . . .
3.9 Simulation results showing the minimum measurement time, τmin ,
required to distinguish between the Schroedinger-Newton theory with
the post-selection measurement prescription (which has the signature
of a Lorentzian with depth d) and standard quantum mechanics in
such a way that the probabilities of indecision and of making an error
are both below p%. The coherence time is given by the inverse of the
half width at half maximum of the Lorentzian. Note that the x-axis
is scaled by the inverse of the complimentary error function, er f c−1 ,
and the y-axis is on a log scale. Moreover, the dashed lines are to
2
guide the eye and are fits of the form a − b × er f c−1 (p/100) . . .
3.10 Minimum measurement time required to distinguish between the
Schroedinger-Newton theory with the post-selection measurement
prescription and standard quantum mechanics in such a way that the
probabilities of indecision and of making an error are both below
10%. Note that we interpolated the data given in Fig. 3.7 to create
this figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 A spacetime diagram showing multiple measurement events. Event C
describes the preparation of an ensemble of identical 2-particle states
|Ψini i by Charlie. Event A (B) describes Alice (Bob) measuring her
(his) particles. The dashed lines show the light cone centered around
each event. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8

xiv
4.2

Assignment of boundary conditions after two measurements according to the causal-conditional prescription. M1 and M2 are two measurement events at spacetime locations (t1, x1 ) and (t2, x2 ), respectively, and the dashed lines show the light cones centered around
each of them. To keep the figure uncluttered, we work with a onedimensional quantum field, and we have discretized space and time
into 10 points each. Each degree of freedom of the field is represented
by a dot on the figure. How we fill the dot depends on what boundary
condition (B.C.), which is indicated on the legend at the top of the
figure, is assigned to the time-evolution of the wavefunction that the
non-linear Hamiltonian at the spatial location of the dot depends on
(see section 3.3 for more details). Note that the initial state of the
field is |Ψini i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 A setup similar to that described by Fig. 4.1, but more elaborate.
Event D is Dylan preparing the state Ψini , Event C is Charlie measuring the eigenstate |Ψini i. Event A (B) describes Alice (Bob) measuring her (his) particles. Bob then sends his particle to be measured
by Eve at event E. The dashed lines show the light cone centered
around each event. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Partioning of spacetime into different regions according to which
boundary state is associated with time evolution. There are 4 measurement events: C, A, B and E, that we’ve arranged identically as
in Fig. 4.3. We didn’t include Event D to limit clutter. The 4 events
result in 6 regions. The boundary state associated with the non-linear
time-evolution operator of each region is the time-evolved initial state
of the experiment conditioned on measurement events presented in
the legend at the top of the figure. . . . . . . . . . . . . . . . . . . . 99
4.5 A general configuration of measurements, labeled by M j where 1 ≤
j ≤ n, occurring before an event B, which describes Bob performing
a measurement. The dashed lines shows the past light cone of event B. 102

xv
5.1

5.2
6.1

6.2

6.3
6.4

7.1
7.2

Showing how causal NLQM and causal feedback are equivalent in
a simple example. At time t1 , Alice performs a measurement. The
corresponding measurement event is denoted byM A. The result of the
measurement is broadcast along M A’s future light cone. It reaches
Bob at time t2 = t1 + |xB − x A | /c. In the NLQM picture, at time t2 ,
the classical field at xB suddenly changes to incorporate information
about Alice’s measurement result. In the quantum feedback picture,
Bob switches his feedback strategy at t2 to incorporate information
about Alice’s measurement result. |λ1 (t)i and |λ0 (t)i are obtained
from solving a non-linear Schroedinger equation with initial conditions given by (5.30) and (5.31), respectively. . . . . . . . . . . . . . 116
Alice and Bob’s optomechanical setups. Note that i = A, B. . . . . . 119
An optomechanical system with n degrees of freedom interacts with
(1)
(n)
n effective input modes, Âin
through Âin
, which evolve into n effec(1)
(n)
tive output modes, Âout through Âout . The operators b̂1 through b̂n
represent system degrees of freedom. . . . . . . . . . . . . . . . . . 132
The setup proposed by Hofer et al. in Ref. [5]. The cavity has a
single movable mirror with a center of mass mode denoted by âm .
The incoming light pulse, shown in dashed blue, is blue-detuned
and of length τ. Note that the ingoing and outgoing light modes,
âin (t) and âout (t) respectively, form a continuum but we show them
as discrete modes for simplicity. . . . . . . . . . . . . . . . . . . . . 134
The transformation of input degrees of freedom to output degrees of
freedom under the matrix M. . . . . . . . . . . . . . . . . . . . . . . 136
A diagram showing a hypothetical beam-splitter interaction that
swaps the quantum states of the system degrees of freedom, b̂1
(1)
through b̂n , with that of the effective modes it interacts with: Âin
(n)
through Âin
. The remainder of the effective modes are assumed to
have an arbitrary interaction amongst themselves. . . . . . . . . . . 151
General optomechanical setup with n degrees of freedom interacting
with m bosonic environment fields. . . . . . . . . . . . . . . . . . . 168
A cavity optomechanical setup. A test mass’ center of mass position
with corresponding quadratures ( x̂1, p̂1 ) is driven by the thermal bath
field operators b̂in (t). The cavity field with corresponding quadratures ( x̂2, p̂2 ) is driven by the optical field operators âin (t). âin (t)
and b̂in (t) then evolve into âout (t) and b̂out (t), respectively. . . . . . . 170

xvi
7.3

7.4

7.5

7.6

7.7

8.1
8.2

Two-mode squeezing between the modes ŝi and êi for 1 ≤ i ≤ n. The
bottom two graphs show the phase space distribution of both of these
modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Quantifying the entanglement between the cavity optomechanical
setup shown in Fig. 7.2 with its environment. N1 (N2 ) is the logarithmic negativity of the first (second) diagonalizing symplectic
system mode of Eq. (7.102) with the effective environment mode it
is correlated with. We set the equilibrium thermal occupation of the
test mass to 1/2 and the quality factor to 106 . . . . . . . . . . . . . . 178
Optimal squeezing of x̃1 for Q = 106 , n = 1/2 and different values
of Γm and ΓΘ for the cavity optomechanical setup discussed in Sec.
7.3.3. We remind the reader that x̃1 = 2 x̂1 /∆xzp , where x̂1 is the
center of mass position of the test mass, and ∆xzp = ~/2mωm is its
zero-point fluctuations. Moreover, Q is the test mass’ quality factor,
n is the test mass’ thermal occupation number, Γm = ωm /γ, where
γ is the cavity decay rate, and ΓΘ is the dimensionless measurement
3 . . . . . . . . . . . . . . . . . . . . . . . . 182
strength ΓΘ3 = ~g 2 / mωm
The logarithmic negativity between the cavity and the test mass for
the setup discussed in Sec. 7.3.3, if we were to measure the two
environment modes êθ1 and êθ2 given by Eqs. (7.75-7.76). . . . . . . 186
The correlation structure, at its simplest, of a general optomechanical
system with its optical bath. The optomechanical system consists of
n degrees of freedom, and is correlated with only n effective optical
modes. These optical modes are also only correlated with n effective
optical modes. This correlation ’chain’ extends for the rest of the
optical bath modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
The cavity optomechanical setup we examine in this article. y (t) is
the measurement record collected by the experimentalist. . . . . . . . 196
Predictions of the squeezing of x̃ after measuring the phase quadrature
âout,2 of the outgoing light and at different measurement strengths ΓΘ ,
when the cavity is adiabatically eliminated and when it isn’t. When
phase
âout,2 is measured, Vx̆ x̆
is the steady state conditional variance of
x̃ if the cavity were adiabatically eliminated, and Vx̃min
x̃ the steady state
conditional variance of x̃ under the full dynamics. The inset shows
phase
Vx̆ x̆
at different measurement strengths. We chose the thermal
occupation number n to be 1. . . . . . . . . . . . . . . . . . . . . . 203

xvii
8.3

8.4

8.5

8.6

9.1

Simulation results of the critical measurement strength, ΓΘc , when the
optimal squeezing variance for x̃ according to the exact dynamics
is half of that as when the cavity is adiabatically eliminated. Each
black dot represents ΓΘc for a different choice of Γγ and Q in the
range 10−1 ≤ Γγ ≤ 10−6 and 104 ≤ Q ≤ 109 . We chose the thermal
occupation number n to be 1. The gray surface is fit to guide the eye,
and is equal to −0.155 − 0.09 log10 Q − 0.97 log10 Γγ . . . . . . . . . 204
Predictions of the optimal squeezing of x̃ at different measurement
strengths when the cavity is adiabatically eliminated and when it
isn’t. Vx̆min
x̆ is the minimum achievable conditional variance of x̃
if the cavity were adiabatically eliminated, and Vx̃min
x̃ the minimum
achievable conditional variance of x̃ under the full dynamics. We
chose the thermal occupation number n to be 1. . . . . . . . . . . . 205
Eq. (8.61) is an accurate approximation of Vx̃, x̃ in the parameter
regime of interest. We generated 105 different possible values of
the triplet Q, ΓΘ Γγ, Γγ , 3 × 104 of which are shown in the inset to
demonstrate that they cover most of the regime 104 ≤ Q ≤ 1011 ,
10−5 ≤ Γγ ≤ 10−1 , and 10−5 ≤ ΓΘ Γγ ≤ 10−1 . We evaluated Vc,x
exactly and Eq. (8.61) over them, and obtained that for 98.8% of the
triplets δ = Vx̃c exact − Vx̃c approx /Vx̃c exact ≤ 1%, and for 1.2% of
the triplets δ is between 1 and 3%. . . . . . . . . . . . . . . . . . . . 210
Testing the accuracy of Eq. (8.65). The simulation results are the
same as those in Fig. 8.3. The plane shows the value of ΓΘ for which
α̃12 /4ν1 = α̃22 /4ν2 for different values of Q and Γγ . . . . . . . . . . . 211
A simple example optomechanical setup. A free test mass is driven
by an incoming light continuum, labeled by âin . âin then interacts
with the test mass’ center of mass motion position and momentum
operators, x̂ and p̂. The reflected light forms an outgoing light light
continuum, labeled by âout . âout is continuously monitored by a
photon counter. We denote the resultant measurement record by n (t). 220

xviii

LIST OF TABLES
Number
Page
2.1 LISA pathfinder test mass parameters (Ref. [3]). We estimated ρ
and a with weighted averages of the densities and lattice constants,
respectively, of the materials in the alloy that the test masses are made
out of. The composition of this alloy is 73% Au and 27% Pt. . . . . . 23
3.1 Characteristic Schroedinger-Newton angular frequency ωSN for several elemental crystals. Density is approximated by values at room
temperature, and the Debye-Waller factor B (at 1 K) is provided by
Ref. [20]. *: Note that Osmium’s Debye-Waller factor is solely
obtained from theoretical calculations. . . . . . . . . . . . . . . . . . 35
3.2 The probabilities of the different outcomes of the likelihood ratio test
on a particular measurement data stream with an estimator following
either of the two distributions shown in Fig. 3.6. The three possible
outcomes are (1) deciding that the data has a spectrum of Sd , (2)
deciding that it has a white noise spectrum (S = 1) or (3) making
no decisions at all. P (correct) stands for the probability of deciding
(1) or (2) correctly, P (wrong) is the probability of making the wrong
decision on what spectrum explains the data, and P (indecision) is the
probability of outcome 3. Note that a different table would have been
generated if a different threshold, yth , had been chosen in Fig. 3.6. . 63

Chapter 1

INTRODUCTION
1.1

Overview

By directly detecting gravitational waves, LIGO validated the use of optomechanics for testing fundamental physics. The discovery also spurred research in how
gravitational wave astronomy can test whether the theory of general relativity will
break down in the strong gravity regime. Concurrently, researchers have been looking at whether optomechanics can be used to test alternative theories of Quantum
Mechanics (QM). The search for alternatives of QM is fueled by two issues: the
measurement problem, and reconciling QM with the theory of general relativity.
The measurement problem describes a fundamental conceptual difficulty with QM.
The Schroedinger equation deterministically predicts the dynamics of a particle’s
wavefunction. However, if we wish to probe the particle’s state with a measurement
device’ (a concept in QM with no concrete definition), we no longer use deterministic
linear evolution equations. Instead, we have to switch to a non-deterministic nonlinear formalism. The non-determinism is conceptually unsatisfying, but isn’t the
main issue. Since the Schroedinger’s equation can predict the dynamics of any
collection of particles, it should be able to predict the dynamics of the measurement
device. However, that doesn’t seem to be the case.
The measurement problem can also be understood in a different way. Quantum mechanics has been spectacularly successful at predicting the behavior of microscopic
particles, such as electrons and atoms, but the macroscopic world doesn’t exhibit
any of the weirdness of quantum mechanics. Nothing around us seems to be in a
superposition of different states, and no large object is entangled with any other large
object. We’d like to understand why the macroscopic world is so well described by
classical mechanics.
Theoretical solutions to these issues exist. String theory and loop quantum gravity
promote gravity to the quantum realm. However, the signatures of quantum gravity
are extremely weak and cannot be detected with current state-of-the-art technology.
As a result, these theories are outside this thesis’ scope. We are only concerned with
theories that are falsifiable in the near future. The Everett interpretation of QM,
Stochastic mechanics [13], and Bohmian Mechanics all resolve the measurement
problem. With the possible exception of Stochastic Mechanics, they make the exact

same predictions as standard QM, and so are not falsifiable and are outside this
thesis’ scope. Ultimately, only experiments can decide whether a theory is correct.
Other solutions exist, but they are not as popular because they suffer from some conceptual issues and their formalism isn’t as appealing (Occam’s razor would choose
other solutions over them). Nonetheless, researchers have recently demonstrated
that these theories are falsifiable with current optomechanics technology. This thesis builds upon their work. We show how current optomechanics experiments place
bounds on some of the parameters of some of these theories. We also discuss how
low-frequency optomechanics experiments can test the idea that gravity is fundamentally classical. Moreover, we show that fundamentally classical gravity doesn’t
necessarily violate causality.
Along the way, we make contributions to the fundamental theory of optomechanics.
We introduce two new bases that one can view environment modes with. The first
basis reduces the interaction between a linear optomechanical system (with a finite
number of internal modes) and an environment with an infinite number of internal
modes to an interaction with finite degrees of freedom. The second basis simplifies
the correlation structure between an optomechanical system and its environment at
any particular time: the system is only correlated with a finite number of effective
environment modes. This simplified entanglement structure allowed us to derive the
one-shot quantum Cramer-Rao bound in a simple way. We also use this structure to
explain why adiabatically eliminating a lossy cavity can grossly underestimate the
conditional variances of a strongly-monitored cavity-optomechanical setup. When
such a setup is strongly driven, a small amount of information becomes trapped in the
cavity’s state, and a large amount of information about the test mass escapes to the
environment. We can recover the latter information by measuring the outgoing light,
thereby reducing our uncertainty about the test mass’ state. However, eventually we
become limited by the information that is locked in the cavity and which we cannot
recover. Finally, an important goal of optomechanics is to prepare a test mass in
a state with a negative Wigner function (which is an indisputable sign of quantum
behavior). A promising proposal, which has been partially realized in experiments,
is to use a photon counter and an appropriately detuned laser. Photon counting
is a non-linear measurement scheme, which greatly complicates the theory as it is
no longer linear. We developed an analytic filter to obtain the conditional state of
such a system. We hope that our work will help researchers explore optomechanics
topologies that make use of photon counters.
The rest of this chapter briefly reviews objective collapse models, and the theory

that gravity is fundamentally classical. This chapter also provides short summaries
of my projects. The subsequent chapters contain my works on these projects.
1.2

Introduction to collapse models

This section briefly reviews objective collapse models. We refer the interested reader
to Refs. [5, 4, 16] for an exhaustive review.
Objective collapse models modify Schroedinger’s equation to solve the measurement problem. Such a modification has to satisfy five constraints. First, it has
to be non-linear, because measurements seem to break the linearity of quantum
mechanics. Second, the modification is stochastic because non-linear modifications
of quantum mechanics violate the no-signaling condition [35]. Although we show
this isn’t always the case (see Chapter 4), making collapse models stochastic greatly
simplifies them. Otherwise, one would have to determine how a deterministic theory could agree with Born’s rule. Third, the predictions of collapse models have to
match Born’s rule. There is evidence, but no definitive proof, that collapse models
agree with Born’s rule for any measurement topology [6, 5]. Fourth, the predictions
have to match existing experimental data. This means that the modifications to
Schroedinger’s equation should negligibly affect the behavior of microscopic particles, while strongly affecting the dynamics of macroscopic bodies to the extent that
they’d behave classically. Fifth, the theory has to be causal.
One could argue that spontaneous collapse models should also be consistent with
relativistic quantum field theories. Relativistic extensions of collapse models is an
ongoing research project. It is a difficult endeavor because the Bell test experiments
imply that the wavefunction collapses instantaneously (or at least faster than the
speed of light).
Objective collapse models seem daunting to learn because there are so many of
them. As in Refs. [40, 38, 14], we will present a unified view of most collapse
models, and then specialize to two of the most popular models: the Continuous
Spontaneous Localization (CSL) model, and the Diosi-Penrose (DP) model.
Ironically, even though objective collapse models were proposed to resolve the
measurement problem, they can be stated in terms of the formalism of continuous
measurements in standard quantum mechanics. In fact, continuous and Markovian
modifications that satisfy the constraints we’ve indicated above have to correspond
to the continuous measurement of a particular operator [3, 15]. For CSL and DP,

that particular operator is a smeared version of the mass density operator:
Φ̂ (x) =

Φ̂ j (x) =

m j Φ̂ j (x) ,

(1.1)

d 3 y g (y − x) ψ †j (y) ψ (y) ,

(1.2)

where g is the smearing function, j labels different matter fields, and ψ †j (y) creates
a particle from the jth matter field at location y. It is no surprise that CSL and DP
pick the mass density operator at different locations to be the degree of freedom that
is monitored, because collapse models need to prevent superpositions of massive
bodies over large distances. In the non-relativistic limit, Φ̂ (x) in first quantized
form is (assuming just one type of particle)
f ir st
Φ̂ (x) −−−−−−−−−→ m d 3 y g (y − x) |yi hy| .
(1.3)
quantization

Φ̂ (x) will be monitored everywhere, resulting in the master equation (~ = 1)

d 3 xd 3 yΓ (x, y) Φ̂ (x) , Φ̂ (y) , ρ̂ (t) dt
d ρ̂ (t) = −i Ĥ, ρ̂ (t) dt −
d 3 xd 3 yΓ (x, y) H Φ̂ (x) ( ρ̂ (t)) dW (y, t) ,
(1.4)
where Ĥ is the total Hamiltonian for the system, and the superoperator H is
H Φ̂ (x) ( ρ̂) = Φ̂ (x) , ρ̂ − 2Tr Φ̂ (x) ρ̂ ρ̂.

(1.5)

dW (y, t) is a Brownian motion process and its correlation function is
E (dW (y, t) dW (x, t 0)) = Γ−1 (x, y) δ (t − t 0) dt 2 .

(1.6)

It is important to note that Eq. (1.4) is just a way to present and interpret collapse
models, which are phenomenological and were created to address the measurement
problem. Collapse models were not derived from first principles and they have
no ontologies associated with them. Nonetheless, Eq. (1.4) makes a substantial
claim: such an evolution equation describes all measurements (even non-position
measurements, such as measurements of spin). Only one evolution equation is
enough to describe all quantum phenomena and we no longer have to interrupt
unitary evolution with projection operators.

For CSL and DP, g is a Gaussian (it’s width is traditionally denoted as rCSL for
CSL, and a ’regularization parameter’ for DP). CSL simply chooses uncorrelated
measurement results:
ΓCSL (x, y) = γδ (x − y) ,
(1.7)
while DP chooses
ΓDP (x, y) =

|x − y|

(1.8)

For the DP model, γ has the same dimensions as the gravitational constant G.
This isn’t surprising because DP is motivated (by heuristic arguments) to gravity
introducing noise in the Schroedinger equation because the time derivative in this
equation is ambiguous when spacetime is in a superposition state.
In Eq. (1.4), we don’t have access to the measurement record and so we have to
take the expectation value of d ρ̂ with respect to the stochastic variables dW. The
last term in Eq. (1.4) averages to 0, and we are left with the predictions of standard
quantum mechanics (the first term), and decoherence (the second term) which will
be the signature of objective collapse models. In optomechanics, the decoherence
results in an additional white noise force on the test mass’ center of mass motion
(see supplemental material of [24]).
Various experiments have placed upper bounds on CSL and DP’s parameters. For
example, we showed that the space experiment LISA pathfinder places the best
bounds on CSL and DP at low frequencies. Theorists have also placed rough lower
bounds by imposing that the CSL and DP models have to be strong enough to match
the fast rate at which measurement devices collapse a wavefunction. The upper and
lower bounds are still separated by orders of magnitude. It might be a while before
we can decisively rule out collapse models.
Finally, we end with some hope. Although collapse models are justifiably ad-hoc
and non-elegant extensions of quantum mechanics, a recent experiment has found
evidence for a non-thermal force of unknown origin [41]. The experiment doesn’t
offer conclusive evidence for collapse models, but warrants further investigation.
The European Commission has recently awarded a €4.4 million grant for an experiment that monitors the position of small micrometer-scale levitated glass spheres at
an unprecedented accuracy.
1.3

Alternatives to quantum gravity

Theorists have faced enormous difficulties in quantizing the theory of general relativity (GR). Part of the difficulty is that quantum theory and GR treat time and space

(i.e. spacetime) differently. In quantum mechanics, spacetime is an absolute background but is a dynamical object in GR. Nonetheless, researchers have succeeded
in constructing elegant theories that quantize gravity. There is no experimental
evidence for any of these theories, nor do we expect that in the near future any
laboratory experiment will detect any of the very weak signatures of these theories.
Because of this, some researchers have moved away from quantum gravity, and are
looking at other models that reconcile quantum mechanics with gravity.
1.3.1

Overview of alternative models of quantum gravity, and outlook

The simplest alternative model first appeared nearly half a century ago by Møller
and Rosenfeld [22, 32]. Specifically, they proposed that spacetime is sourced by the
quantum expectation value of the stress energy tensor:
G µν = 8π Φ|T̂µν |Φ ,

(1.9)

(1.10)

where V (®
r ) is the non-gravitational potential energy at r® and U (t, r®) is the Newtonian
self-gravitational potential and is sourced by χ (®
r ):
∇2U (t, x) = 4πGm | χ (t, x)| 2 .

(1.11)

Recently other models have appeared that don’t allow spacetime to be in an arbitrary
superposition of states. This author is aware of two promising models: one is the
Correlated WordLines (CWL) theory, and the other is Tilloy’s et al. innovative use
of collapse models to source gravity [40, 39]. Similar to collapse models, Tilloy’s
et al.’s models are phenomonological but we can interpret the math in terms of
known linear quantum mechanics processes. The collapse model’s math maps to
an agent monitoring a certain degree of freedom (like the a smoothed out mass
density) everywhere in space. Gravity then emerges as feedback: the agent uses

the collected measurement results to apply a feedback force everywhere in space
and that is equal to the gravitational force. Compared to the Schroedinger-Newton
theory, Tilloy’s et al.’s proposals have one appealing feature: they have a clear
ontology: the wavefunction isn’t a probability density and a mass density at the
same time.
CWL modifies the path integral by having the different paths gravitate [36]. Its
structure is complicated because CWL takes quantum field theory and adds graviton
propagators between different Feynman-Keldysh diagrams . Moreover, Stamp et al.
are still working on understanding CWL’s structure, and making sure it is issue-free.
CWL’s exact formulation has been revised lately to ensure that CWL passes quantum
field theory consistency tests [2].
The author believes that understanding and contrasting the predictions of these
different models will greatly benefit the alternative quantum gravity community.
Since each prevents spacetime from being in arbitrary superposition, we believe
their predictions won’t be substantially different. Moreover, they would give us a
road map to design a class of experiments that could test the idea that gravity is
approximately classical. Although, each of the above models is likely to be refuted,
they direct us to what the signatures of an approximately classical spacetime might
look like. If such a concept is true, this class of experiments could find evidence for
it.
Another important research direction is to remedy the conceptual issues that these
models suffer from. As it did for us, such an endeavor could generate new ideas
and variations of the models above. Eventually, we also hope that addressing the
conceptual issues of these models would generate more widespread acceptance of
alternatives of quantum gravity to the broader physics research community.
1.3.2

Contributions to fundamental semi-classical gravity

This thesis focuses on the experimental signatures of fundamental semi-classical
gravity. Yang et al. determined the dynamics of an optomechanical system under
the Schroedinger Newton equation [44]. Specifically, they showed that if an object
has its center of mass’ displacement fluctuations much smaller than fluctuations of
the internal motions of its constituent atoms, then its center of mass, with quantum
state |ψi, observes
d |ψi
i~
= ĤNG + MωSN ( x̂ − hψ| x̂|ψi) |ψi
dt

(1.12)

where M is the mass of the object, ĤNG is the non-gravitational part of the Hamiltonian, x̂ is the center of mass position operator, and ωN S is a frequency scale that is
determined by the matter distribution of the object. For materials with single atoms
sitting close to lattice sites, we have
ωSN ≡

Gm
6 π∆xzp

(1.13)

where m is the mass of the atom, and ∆xzp is the standard deviation of the crystal’s
constituent atoms’ displacement from their equilibrium position along each spatial
direction due to quantum fluctuations. A pedagogical explanation of the derivation
of Eq. (1.12) can be found in Ref. [18].
Eq. (1.12) isn’t enough to predict the result of an experiment because it doesn’t
take into account thermal noise or measurements. Adding thermal noise is wellunderstood in linear optomechanics. We use the Langevin formalism to add a
fluctuating force to the Heisenberg equations of motion. We determine the strength
of the fluctuating
force
from the state of the bath, which is a mixed state of the
form exp −Ĥth /k BT where Ĥth and T are the Hamiltonian and temperature of the
thermal bath, respectively. However, Eq. (1.12) is non-linear, and in Non-Linear
Quantum Mechanics (NLQM), there is no Heisenberg picture, and we cannot use
density matrices. At first, we addressed these issues within the context of Eq.
(1.12). We showed that for the Hamiltonian in Eq. (1.12), we can define an effective
Heisenberg picture that depends on the initial or final state of the system. Moreover,
we argued (but did not prove) that the thermal bath’s state is composed of a mixture
of coherent states, whose quantum fluctuations is just vacuum and whose classical
fluctuations are related to the amplitude of the coherent states. Quantum fluctuations
introduce noise through expectation values over quantum states, whereas classical
fluctuations introduce noise through an ensemble average over a classical distribution
of the classical thermal force’s distribution.
In linear QM, Born’s rule tells us how to calculate the probability of an experiment
obtaining a particular measurement result. However, we discovered that in NLQM,
Born’s rule becomes ambiguous. We show this with a simple example. In linear
QM, the probability that an initial state |ii evolves under some unitary operator Û
and is then measured to be |mi is
pi→m = m Û i

(1.14)

We can write pi→m in many equivalent ways. For example,
pm→i = i Û † m

= pi→m .

(1.15)

The issue is that pi→m and pm→i become, in general, different in NLQM:
N LQM
pi→m
= |hm | Uii| 2 ,
N LQM
pm→i

i U†m

(1.16)

(1.17)

where |Uii is the evolved |ii and U † m is the backwards evolved |mi, under a nonlinear Hamiltonian. In Chapter 3, we didn’t explore the issue any further, and used
N LQM
N LQM
pi→m
and pm→i
as inspirations for two prescriptions that assign probabilities
N LQM
to measurements in NLQM. We termed the first, which is inspired by pi→m
as
N LQM
pre-selection, and we termed the second, which is inspired by pm→i as postselection. Pre-selection imposes the initial state of an experiment as the initial
condition for the non-linear evolution, whereas post-selection imposes the measured
states as conditions on the state of the system-environment at the final time of
the experiment. These prescriptions allowed us to explore the range of possible
signatures of fundamental semi-classical gravity, and the constraints on experiments
that would detect them. We concluded thatqboth prescriptions predict a deviation
2 + ω2 where ω
from quantum mechanics at the frequency ωcm
cm is the resonant
SN
frequency of the mechanical resonator. If ωcm
ωSN , pre-selection predicts a peak
that is easily detectable with current low-frequency torsion pendulum experiments.
Note that ωSN can be significantly boosted by constructing the mechanical resonantor
with the appropriate material. ωSN is at most about 0.49 for Osmium and is 0.36
for the more common Tungsten metal. On the other hand, post-selection predicts a
dip that is harder to detect. In addition to ωcm
ωSN , the input laser power has
to be fine tuned, the test mass’ quality factor has to be very high (around 107 ), and
the temperature has to be very low (around 1K). Even then, experimentalists have to
wait on the order of a month to rule out or verify the Schroedinger-Newton theory
with the post-selection prescription.

After this work, we endeavored to better understand measurements in NLQM.
In Chapter 4, we showed that the ambiguity of Born’s rule in NLQM can be
stated differently: different interpretations of quantum mechanics make different
predictions in NLQM. We also showed that this ambiguity can be thought of as
a degree of freedom to ensure that NLQM meets the no-signaling condition. We
showed that out of the well-known interpretations of quantum mechanics, the Everett

10
interpretation (which the pre-selection prescription is equivalent to) doesn’t violate
causality. However, for fundamental semi-classical gravity, it’s been ruled out
decades ago [25] and more recently by LISA pathfinder. We also showed that we can
craft a new prescription that doesn’t violate the no-signaling condition. Whenever
a measurement occurs at some spacetime location (x, t), only non-linearities in the
future light cone of (x, t) get updated to incorporate the result of this measurement.
We termed this prescription the causal-conditional prescription
In Chapter 5, we approach measurements in NLQM from a different perspective.
We show that once we fix the initial state of a system, evolution under a non-linear
Hamiltonian is equivalent to evolution under a linear Hamiltonian with feedback.
Specifically, we consider non-linear Hamiltonians of the form
i~∂t |ψ (t)i = ĤN LQM |ψ (t)i

(1.18)

where the non-linear Hamiltonian is
ĤN LQM = ĤL +

βi φ (xi, t, ψ (t)) V̂i .

(1.19)

ĤL is the linear part of Ĥ, and doesn’t depend on the wavefunction. The second
term represents a classical field that couples to our quantum system through V̂i at
positions xi . The classical field follows its own equation of motion:
Lφ (x, t) = S (x, t, ψ (t))

(1.20)

where L is a differential operator and S (x, t, ψ (t)) is a source term that, in general,
depends on ψ (t). The βi are constants. Note that the sum i could in general
contain an integral.
The equivalence between NLQM and quantum feedback is powerful for two reasons. First, it allows us to leverage the tools and concepts developed for quantum
feedback. Second, it allows us to determine how measurements can be added to
NLQM in a causal way. Feedback is causal only when the feedback force depends
on measurement results collected in the past light cone of where the feedback is
applied. Therefore, NLQM is causal only when the classical field φ (x, t) depends
on measurement results collected in the past light cone of (x, t).
With a causal prescription for adding measurements to NLQM, and a mapping that
allows us to leverage the tools of quantum feedback, we calculated the signature
of the Schroedinger-Newton theory with the causal-conditional prescription. The

11
last outstanding issue of the Schroedinger-Newton theory is ensuring that it doesn’t
violate the Bianchi identity. We hope it will be resolved some day.
1.4
1.4.1

Overview of contributions to the theory of optomechanics
Advances in optomechanics

Over the past decade, our understanding of quantum state preparation, verification and control in the field of optomechanics has improved substantially [11, 21,
23]. Concurrently, optomechanics technology has seen many breakthroughs. Researchers have cooled test masses to their quantum mechanical ground state cooling
[10]. They have also observed the asymmetry between the spectra of transmitted
red-detuned and blue-detuned light’s amplitude fluctuations when a test mass, or the
collective motion of atoms, are near their ground state [33, 37, 27, 8]. Researchers
now also have much better control over their experiment. In Refs. [29, 42, 17],
experimentalists substantially cool their test masses through feedback control.
There have been other breakthroughs in optomechanics. In Ref. [30], Purdy et
al. have detected signatures of the fluctuating radiation pressure force on the test
mass in the outgoing light’s fluctuations. Moreover, the center of mass motion of
test masses has been squeezed below the Heisenberg uncertainty level [43, 28, 20].
Optomechanics has also been used to generate squeezed light [31, 34, 9]. By driving
optomechanics setups in a special way, and by using photon counters instead of
homodyne detection, experimentalists have measured individual quanta of phonons
in mechanical resonators [19, 12].
1.4.2

Contributions to the theory of unmonitored linear optomechanics

In optomechanics, a system is driven by a continuum of light. As a result, the system
has interacted and is correlated with an infinity of modes. This can be conceptually
daunting, and could complicate calculations. Traditionally, researchers have worked
in the Fourier domain where, at steady state, degrees of freedom at different frequencies are independent. The Fourier basis is the preferred basis to work in if one is
interested in evaluating the sensitivity of an optomechanical system to a signal over
a certain bandwidth. The Fourier basis has also been successfully used to analyze
quantum state preparation and verification in optomechanics. However, the Fourier
basis has two main drawbacks. First, we are limited to steady state dynamics which
can be problematic in pulsed optomechanics, where the system’s initial state isn’t
necessarily forgotten. Second, an optomechanical system is correlated with, and
interacts with, a continuum of environment modes in the Fourier basis. This makes
it difficult to solve certain optimization problems, such as optimally picking the

12
quadratures of the outgoing light to measure in order to maximize the entanglement
between two degrees of freedom of the system (e.g. a cavity’s field and a test mass’
center of mass motion).
We developed two bases that separately simplify either the dynamics or the entanglement structure of a system with its environment. If a researcher is solely interested
in either the dynamics, or the entanglement structure, then using these new bases
mean that the researcher only has to consider m effective environment modes, where
m is the number of degrees of freedom of the optomechanical system. However,
there is a caveat. These bases have to be applied to the entirety of the environment,
which is composed of different fields (such as thermal and optical bosonic fields).
This means that the bases have limited practical applicability, because experimentalists can only measure optical environment modes. Nonetheless, the bases can
provide analytic insights to problems that seem intractable if we limit ourselves to
only measuring optical modes.
The first basis, which we discuss in detail in Chapter 6, simplifies the system’s dynamics. At any particular time τ, the optomechanical interaction can be considered
as a scattering process, during which Heisenberg operators of the incoming environmental modes, plus the system modes at the initial time of the experiment, are
transformed the Heisenberg operators of the outgoing environmental modes, plus
the system modes at time τ.
We’ll assume that the system has n degrees of freedom, whose ladder operators we
denote in vector form by
b=

b̂1 b̂2 ... b̂n

(1.21)

Denote the incoming environment modes’ ladder operators by
ain =

â1 â2 ... âN

(1.22)

where for ease of presentation we’ve assumed that environment is composed of a
single bosonic field, and we’ve discretized time from the initial time of the experiment till time τ into N time steps. Denote the time evolved counterparts of ain by
aout . In linear optimechanics, the scattering process can be written in matrix form:
w = Mv.

(1.23)

13
where
© ain ª
© aout ª
a† ®
a† ®
in ® .
(1.24)
w ≡ out ® , v ≡
b (0) ®
b (τ) ®
† (0)
† (τ)
By using that w satisfies the same commutation relations as v, we can show that
M must satisfy constraints that make it possible to define new effective incoming
environment modes Ain , and their time-evolved counterpart Aout , that simplify the
scattering process’ structure into two separable parts. b (0) and n modes in Ain
scatter into b (τ) and n modes in Aout . The remainder of the N − n modes in Ain
scatter into the remainder of the N − n modes in Aout , as is shown in Fig. 1.1.
The second basis, which we discuss in detail in Chapter 7, simplifies the system’s
entanglement structure with its environment at any particular time τ. Before we
present this basis, we will introduce some basic notation. The system’s modes
occupy a Gaussian Wigner function with a covariance matrix we denote by Vsys .
Similarly, the environment modes occupy a Gaussian Wigner function with a covariance matrix we denote by Venv . Moreover, denote the eigenoperators associated
the symplectic eigenvalues of Vsys by s, and the eigenoperators associated with the
symplectic eigenvalues of Venv by e. For example, to obtain s, we symplectically
diagonalize Vsys , which means we find a symplectic matrix S that satisfies
SVsys ST = Λ

(1.25)

where Λ is a block diagonal matrix of the form
⊕i=1

νi 0
0 νi

(1.26)

and we’ve assumed again that the system has n degrees of freedom. The existence
of S is guaranteed by the Williamson theorem. s is
Sx (τ)

(1.27)

where x (τ) is a vector of 2n operators in the Heisenberg picture. Each operator
corresponds to a quadrature of a degree of freedom of the system (for example, if
the system has one degree of freedom, the center of mass motion of a test mass, then
T
x (τ) = x̂ (τ) p̂ (τ) ).
The phase-space Schmidt decomposition theorem tells us that in the new basis e, all

General interaction

Figure 1.1: An optomechanical system with n degrees of freedom interacts with
(1)
(n)
n effective input modes, Âin
through Âin
, which evolve into n effective output
(1)
(n)
modes, Âout through Âout . The operators b̂1 through b̂n represent system degrees of
freedom. The remainder of the effective environment modes scatter into themselves
and nothing else.

modes, except n, are in vacuum [7, 1]. Each of these n modes is correlated with only
one mode of s and with nothing else. With this simplified entanglement structure,
we showed that we can understand the one-shot quantum Cramer-Rao bound in a
simple way, and seemingly intractable problems, like maximizing the entanglement
between a test mass and a cavity, become open to large-scale numerical analysis.
However, we have to make the assumption that we can measure any commuting
environment modes. Although this assumption is unrealistic, it allows to efficiently
sweep a large parameter regime, and so find promising optomechanics topologies
that we’d then analyze more carefully.
1.4.3

Contributions to conditional optomechanics

We cannot directly extract information from an optomechanical system. We need
a probe (usually a driving laser) to interact with the system. We then measure the
outgoing probe’s degrees of freedom. Processing the measurement results would
give us information about the state of the system, and on any external forces that act
on the system. Moreover, we can use the probe to steer the system towards quantum
states of interest. The system doesn’t just partially imprint its state on the probe.
The interaction between the system and the probe is reciprocal, so the probe, with
its own state and fluctuations, partially imprints itself on the system. Moreover,

15
the interaction entangles the system to the probe, so that when we measure the
probe, the system’s state changes through wavefunction collapse. In this section,
we are interested in the conditional dynamics of the system, which are its dynamics
conditioned on the measurement results of the probe.
The conditional dynamics are simplest when we linearly measure only commuting
observables of the probe (for light, this means a homodyne or heterodyne measurement scheme), when the probe’s state is Gaussian and when the probe drives the
system strongly enough that the interaction between them can be linearized. In such
a case, Helge et al. have shown that the optomechanical system can be mapped to a
classical dynamical system [23], opening up classical estimation and control theory
tools (which this author has enthusiastically used) to the quantum optomechanics
researcher. The mapping allows us to obtain analytic insights into the system’s
conditional dynamics, and poses an interesting question: is a linear Gaussian quantum optomechanical system effectively classical? A system is indisputably quantum
when its Wigner function is negative over some region, but the mapping tells us that
a linear Gaussian optomechanical system has a Gaussian and so positive Wigner
function. Nonetheless, such systems could be genuinely quantum because Gaussian
Wigner functions can exhibit entanglement, a distinctly quantum property, and more
importantly, a macroscopic test mass can have a Gaussian Wigner function that is
highly delocalized in space. Classical macroscopic objects are never at multiple
locations at the same time! On the other hand, in classical control theory, the
fundamental object whose dynamics we track is a probability distribution (i.e. our
uncertainty) over a system’s degrees of freedom. As a result, this mapping is a stark
example of a century old question in quantum mechanics: does the wavefunction
represent anything real or is it a subjective quantity containing our knowledge about
the world?
In Chapter 8, we show that studying the conditional dynamics of a cavity-optomechanical
system requires more care than studying the unconditional dynamics. When a cavity
is sufficiently lossy, it can be adiabatically eliminated from the dynamics. However,
if the system is measured fast enough, then such an approximation fails. We used the
phase-space Schmidt decomposition theorem to gain analytic insights on why this
occurs. As we drive an optomechanical system stronger and stronger, the system’s
fluctuations become increasingly dominated by the fluctuations of the optical bath
(i.e. the system becomes increasingly slaved by the optical bath). This is the ideal
case for quantum state preparation, as when we measure the outgoing light, we
eliminate our uncertainty about the optical bath’s state, and so we project the system
into a highly pure state. However, in a cavity-optomechanical system, there are

16
two degrees of freedom: the cavity field and the test mass’ center of mass motion.
Measuring the optical bath gives us information on their joint state. Entanglement
between them can limit us from extracting a lot information about either of them.
Adiabatic elimination fails when we become limited by the entanglement between
the cavity and the test mass. We’ve extracted so much information about the test
mass that we become limited by the information about the test mass that is locked
in the cavity.
In Chapter 9, we develop an analytic filter for calculating the state of an optomechanical system that is driven by Gaussian states and interacts linearly, conditioned on
the measurement results of a photon counter. This work was motivated by Galland
et al.’s proposal to use photon counters and appropriately detuned light to prepare
test masses in a Fock state [26]. The formalism they used to show that the test mass
is in a Fock state is only applicable to simple setups. We hope that our filter will help
researchers explore a range of optomechanics topologies that make use of photon
counters.

17
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21
Chapter 2

LISA PATHFINDER APPRECIABLY CONSTRAINS COLLAPSE
MODELS
B. Helou, B J. J. Slagmolen, D. E. McClelland, and Y. Chen, 2017, PRD 95, 084054
Abstract
Spontaneous collapse models are phenomological theories formulated to address
major difficulties in macroscopic quantum mechanics. We place significant bounds
on the parameters of the leading collapse models, the Continuous Spontaneous
Localization (CSL) model and the Diosi-Penrose (DP) model, by using LISA
Pathfinder’s measurement, at a record accuracy, of the relative acceleration noise
between two free-falling macroscopic test masses. In particular, we bound the CSL
collapse rate to be at most (2.96 ± 0.12)×10−8 s−1 . This competitive bound explores
a new frequency regime, 0.7 mHz to 20 mHz, and overlaps with the lower bound
10−8±2 s−1 proposed by Adler in order for the CSL collapse noise to be substantial
enough to explain the phenomenology of quantum measurement. Moreover, we
bound the regularization cut-off scale used in the DP model to prevent divergences
to be at least 40.1 ± 0.5 fm, which is larger than the size of any nucleus. Thus, we
rule out the DP model if the cut-off is the size of a fundamental particle.

2.1

Introduction

Spontaneous collapse models are modifications of quantum mechanics which have
been proposed to explain why macroscopic objects behave classically, and to address the measurement problem. The most widely studied collapse models are the
Continuous Spontaneous Localization (CSL) and the Diosi-Penrose (DP) models.
The CSL model is parametrized by two scales: λCSL , which sets the strength of
the collapse noise, and rCSL , which sets the correlation length of the noise. For a
nucleon in a spatial superposition of two locations separated by a distance much
greater than rCSL , λCSL is the average localization rate [1]. The quantity rCSL has
usually been phenomenologically taken to be 100 nm [6], and we will follow this
convention.
The DP model adds stochastic fluctuations to the gravitational field, and is mathematically equivalent to the gravitational field being continuously measured [11, 10, 6].
The latter statement leaves the DP model with no free parameters, but a regulariza-

22
tion parameter, σDP , is usually introduced to prevent divergences for point masses.
Nimmrichter et al., in [18], show that the effect of these models on an optomechanical
setup, where the center of mass position of a macroscopic object is probed, can be
summarized by an additional white noise force, F (t), acting on the system, and with
a correlation function of
hF (t) F (t 0)i = DC δ (t − t 0) .

(2.1)

For CSL, DC is given by
DCSL = λCSL

rCSL

(2.2)

with α a geometric factor [18]. LISA pathfinder has quasi-cubic test masses, which
we will approximate as perfect cubes. For a cube with length b
rCSL ,
α≈

4 b2
8πρ2rCSL

m02

(2.3)

where ρ is the material density, and m0 the mass of a nucleon. For the DP model,
DC is given by
3
G~
DDP ≈ √
Mρ
(2.4)
6 π σDP
with M the test mass’ mass, and a the lattice constant of the material composing the
test mass [18].
An optomechanics experiment would need to have very low force noise to significantly constrain collapse models. LISA pathfinder measures the relative acceleration noise between two free-falling test masses at a record accuracy of
Sa = 5.2 ± 0.1 fm s−2 / Hz for frequencies between 0.7 mHz and 20 mHz [4],
and so is a promising platform to test collapse models. We will use Sa , and relevant
details on the LISA pathfinder test mass which we present in table 2.1, to provide
an upper bound on λCSL and a lower bound on σDP .
We note that Sa has steadily decreased by about a factor of 1.5 since the start
of science operations in LISA pathfinder [4], and has continued to significantly
decrease since the results were published in June 2016 [22]. For the remainder of
this article, we will use the conservative value of 5.2 fm s−2 / Hz for Sa , but we

23
Table 2.1: LISA pathfinder test mass parameters (Ref. [3]). We estimated ρ and
a with weighted averages of the densities and lattice constants, respectively, of the
materials in the alloy that the test masses are made out of. The composition of this
alloy is 73% Au and 27% Pt.
Quantity
Description
Mass
Density
Lattice constant
Side length

Value
1.928 kg
19881 kg/m3
4.0 Å
46 mm

will also present bounds obtained from a postulated sensitivity level of

pos

= 3.5 fm s−2 / Hz,

which is about 1.5 times smaller than Sa .
2.2

Constraining the collapse models

We can bound the parameters of collapse models by measuring the force noise of
a test mass in an experiment, and attributing unknown noise to the stochastic force
F(t).
In LISA pathfinder, Brownian thermal noise provides the dominant contribution
to the differential acceleration noise at frequencies between 1 mHz and 20 mHz.
However, the value of this contribution is not precisely known. As a result, we
follow a simple and uncontroversial analysis which attributes all acceleration noise
to the collapse models’ stochastic forces:
Sa = 2SF /M 2,

(2.5)

where SF = 2DC is the single sided spectrum of the collapse force. The factor of 2
in Eq. (2.5) follows from the collapse noise on each test mass adding up, because
the spontaneous collapse force acts independently on each of the two test masses,
which are separated by about 38 cm, a distance much larger than rCSL and σDP .
Therefore, we can place an upper bound on DC of
DC ≤ DCmax = M 2 Sa /4.

(2.6)

24
Using Eq. (2.2), we can then bound λCSL to
max
λCSL ≤ λCSL

(2.7)

with
max
λCSL

m02

32π~2rCSL

Sa
b2

= 2.96 × 10−8 s−1,

(2.8)
(2.9)

where we have substituted in the values shown in table 2.1 for ρ, M and b. If we
pos
max to 1.34 × 10−8 s−1 .
use Sa instead of Sa , then we reduce λCSL
In addition, using Eq. (2.4), we can bound σDP to
min
σDP ≥ σDP

with
min
σDP

2~G ρ 1
3 π m Sa

1/3

a = 40.1 fm,

(2.10)

(2.11)

where we have substituted in the values shown in table 2.1 for ρ, M and a. If we
pos
min to 52.2 fm.
use Sa instead of Sa , then we increase σDP
2.3

Discussion

max is three orders of magLISA pathfinder provides a competitive bound on λCSL . λCSL
nitude lower than the bound 10−5 s−1 , which Feldmann and Tumulka [13] calculated
from Gerlich et al.’s matter wave inteferometry experiment of organic compounds
up to 430 atoms large [15]. Another matter wave interferometry experiment from
the same group [12] places a bound of 5 × 10−6 s−1 , as calculated in [20].
max is comparable to bounds on λ
Moreover, λCSL
CSL obtained from measuring spontaneous heating from the collapse noise. Bilardello et al. place a bound of 5×10−8 s−1
[7], by analyzing the heating rate of a cloud of Rb atoms cooled down to picokelvins
[17]. Note that Bilardello et al.’s bound depends on the temperature of the CSL
noise field, and on the reference frame with respect to which the CSL noise field is
at rest with [7]. The standard formulation of CSL has the collapse noise field at a
temperature of infinity, but the theory could be modified to include different temperatures. The incorporation of dissipation within CSL is based on the dissipative
CSL (dCSL) theory proposed by Smirne and Bassi [19].

Other competitive upper bounds have been obtained from cosmological data, the

25
lowest of which, 10−9 s−1 , is from the heating of the intergalactic medium [1].
However, this bound is also sensitive to the temperature of the collapse noise field
[19]. Moreover, our interest in this article is for controlled experiments.
In addition to providing an aggressive upper bound, LISA pathfinder explores the
max to
low frequency regime of 0.7 mHz to 20 mHz. In Fig. 2.1, we compare λCSL
pos
bounds obtained from experiments operating in different frequency regimes. If Sa
is used instead of Sa , then LISA pathfinder provides the smallest upper bound of all
experiments operating below a THz scale.
LIGO’s measurement of the differential displacement noise between two test masses
in the frequency regime 10 Hz to 10 kHz places upper bounds of at most about
10−5 s−1 . In [21], an upper bound of about 2 × 10−8 s−1 is obtained by analyzing the
excess heating of a nanocantilever’s fundamental mode at about 3.1 kHz. A record
upper bound of 10−11 s−1 is placed in [14, 9] by examining the spontaneous x-ray
emission rate from Ge. This bound could be greatly reduced if the collapse noise is
non-white at the very high frequency of 1018 s−1 [6].
max appreciably constrains the CSL model because it
Furthermore, the bound λCSL
overlaps with some of the proposed lower bounds on λCSL . Adler investigates the
measurement process of latent image formation in photography and places a lower
bound of λCSL ≃ 2.2 × 10−8±2 s−1 [1]. Moreover, Bassi et al. place a lower bound
of λCSL ≃ 10−10±2 s−1 by investigating the measurement-like process of human
vision of six photons in a superposition state [5]. Note that a lower bound of
about 10−17 s−1 , proposed by Ghirardi, Pearle and Rimini [16], is also sometimes
considered. Its justification comes from the requirement that an apparatus composed
of about 1015 nucleons settle to a definite outcome in about 10−7 s or less [2].

LISA pathfinder also provides a competitive bound on σDP . The nanocantilever
experiment [21] places a lower bound on σDP of about 1.5 fm, which is much lower
min . More importantly, the calculated value for σ min of 40.1 ± 0.5 fm is larger
than σDP
DP
min
than the size of any nucleus. Consequently, σDP rules out the DP model if the
regularization scale σDP is chosen to be the size of a fundamental particle such as a
nucleon.
2.4

Acknowledgments

We acknowledge support from the NSF grants PHY-1404569 and PHY-1506453,
from the Australian Research Council grants FT130100329 and DP160100760, and
from the Institute for Quantum Information and Matter, a Physics Frontier Center.

...
Figure 2.1: Upper and lower bounds on the CSL collapse rate λCSL obtained from
laboratory experiments operating at different frequencies. Blue, green, black and
gray regions: exclusion regions obtained from LISA pathfinder, LIGO, a millikelvincooled nanocantilever [21] and spontaneous emission from Ge [14, 9], respectively.
Our calculation of the bounds obtained from LIGO follow that of [8]. The dashed
blue line is the upper bound limit obtained from the LISA pathfinder results if
pos
Sa were used instead of Sa . The red and orange domains are regions in which
the collapse rate is too slow to explain the lack of macroscopic superpositions and
measurements, respectively. The red region is below the lower bound of 10−17 s−1
proposed by Ghirardi, Pearle and Rimini [16]. The orange region’s boundary
is the Adler lower bound 10−8±2 s−1 , below which latent image formation on a
photographic emulsion consisting of silver halide suspended in gelatine wouldn’t
occur fast enough [1]. The orange error bars reflect the uncertainty in this lower
bound.

27
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[3] F Antonucci et al. Lisa pathfinder: mission and status. Classical and Quant.
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[4] M. Armano et al. Sub-femto-g free fall for space-based gravitational wave
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[5] A. Bassi, D.-A. Deckert, and L. Ferialdi. Breaking quantum linearity: Constraints from human perception and cosmological implications. EPL (Europhysics Letters), 92(5):50006, 2010.
[6] Angelo Bassi, Kinjalk Lochan, Seema Satin, Tejinder P. Singh, and Hendrik
Ulbricht. Models of wave-function collapse, underlying theories, and experimental tests. Rev. Mod. Phys., 85:471–527, Apr 2013.
[7] M. Bilardello, S. Donadi, A. Vinante, and A. Bassi. Bounds on collapse models
from cold-atom experiments. ArXiv e-prints, May 2016.
[8] M. Carlesso, A. Bassi, P. Falferi, and A. Vinante. Experimental bounds on
collapse models from gravitational wave detectors. ArXiv e-prints, June 2016.
[9] C. Curceanu, B. C. Hiesmayr, and K. Piscicchia. X-rays help to unfuzzy the
concept of measurement. ArXiv e-prints, February 2015.
[10] L. Diósi. Models for universal reduction of macroscopic quantum fluctuations.
Phys. Rev. A, 40:1165–1174, Aug 1989.
[11] L. Diósi. A universal master equation for the gravitational violation of quantum
mechanics. Physics Letters A, 120(8):377 – 381, 1987.
[12] Sandra Eibenberger, Stefan Gerlich, Markus Arndt, Marcel Mayor, and Jens
Tuxen. Matter-wave interference of particles selected from a molecular library
with masses exceeding 10 000 amu. Phys. Chem. Chem. Phys., 15:14696–
14700, 2013.
[13] William Feldmann and Roderich Tumulka. Parameter diagrams of the grw and
csl theories of wavefunction collapse. J. Phys. A, 45(6):065304, 2012.
[14] Qijia Fu. Spontaneous radiation of free electrons in a nonrelativistic collapse
model. Phys. Rev. A, 56:1806–1811, Sep 1997.

28
[15] Stefan Gerlich, Sandra Eibenberger, Mathias Tomandl, Stefan Nimmrichter,
Klaus Hornberger, Paul J. Fagan, Jens Tüxen, Marcel Mayor, and Markus
Arndt. Quantum interference of large organic molecules. Nature Communications, 2:263, 4 2011.
[16] Gian Carlo Ghirardi, Philip Pearle, and Alberto Rimini. Markov processes in
hilbert space and continuous spontaneous localization of systems of identical
particles. Phys. Rev. A, 42:78–89, Jul 1990.
[17] Tim Kovachy, Jason M. Hogan, Alex Sugarbaker, Susannah M. Dickerson,
Christine A. Donnelly, Chris Overstreet, and Mark A. Kasevich. Matter wave
lensing to picokelvin temperatures. Phys. Rev. Lett., 114:143004, Apr 2015.
[18] Stefan Nimmrichter, Klaus Hornberger, and Klemens Hammerer. Optomechanical sensing of spontaneous wave-function collapse. Phys. Rev. Lett.,
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[19] Andrea Smirne and Angelo Bassi. Dissipative continuous spontaneous localization (CSL) model. Sci Reports, 5:12518, 2015.
[20] M. Toroš and A. Bassi. Bounds on Collapse Models from Matter-Wave Interferometry. ArXiv e-prints, January 2016.
[21] A. Vinante, M. Bahrami, A. Bassi, O. Usenko, G. Wijts, and T. H. Oosterkamp. Upper bounds on spontaneous wave-function collapse models using
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[22] W. J. Weber. Private Communication.

29
Chapter 3

MEASURABLE SIGNATURES OF QUANTUM MECHANICS IN
A CLASSICAL SPACETIME
B. Helou, J. Luo, H. Yeh, C. Shao, B. J. J. Slagmolen, D. E. McClelland, and Y.
Chen, 2017, PRD 96, 044008
Abstract
We propose an optomechanics experiment that can search for signatures of a fundamentally classical theory of gravity and in particular of the many-body SchroedingerNewton (SN) equation, which governs the evolution of a crystal under a selfgravitational field. The SN equation predicts that the dynamics of a macroscopic
mechanical oscillator’s center of mass wavefunction differ from the predictions
of standard quantum mechanics [29]. This difference is largest for low-frequency
oscillators, and for materials, such as Tungsten or Osmium, with small quantum fluctuations of the constituent atoms around their lattice equilibrium sites. Light probes
the motion of these oscillators and is eventually measured in order to extract valuable information on the pendulum’s dynamics. Due to the non-linearity contained
in the SN equation, we analyze the fluctuations of measurement results differently
than standard quantum mechanics. We revisit how to model a thermal bath, and
the wavefunction collapse postulate, resulting in two prescriptions for analyzing the
quantum measurement of the light. We demonstrate that both predict features, in
the outgoing light’s phase fluctuations’ spectrum, which are separate from classical
thermal fluctuations and quantum shot noise, and which can be clearly resolved with
state of the art technology.

3.1

Introduction

Advancements in quantum optomechanics has allowed the preparation, manipulation and characterization of the quantum states of macroscopic objects [1, 16, 6].
Experimentalists now have the technological capability to test whether gravity could
modify quantum mechanics. One option is to consider whether gravity can lead to
decoherence, as conjectured by Diosi and Penrose [21, 8, 7], where the gravitational
field around a quantum mechanical system can be modeled as being continuously
monitored. A related proposal is the Continuous Spontaneous Localization (CSL)
model, which postulates that a different mass-density sourced field is being continuously monitored [4]. In both cases, gravity could be considered as having a

30
“classical component”, in the sense that transferring quantum information through
gravity could be impeded, or even forbidden [14]. Another option, proposed by
P.C.E. Stamp, adds gravitational correlations between quantum trajectories [26].
In this paper, we consider a different, and more dramatic modification, where the
gravitational interaction is kept classical. Specifically, the space-time geometry is
sourced by the quantum expectation value of the stress energy tensor [24, 17, 5]:
G µν = 8π Φ|T̂µν |Φ ,

(3.1)

with G = c = 1, and where G µν is the Einstein tensor of a (3+1)-dimensional
classical spacetime. T̂µν is the operator representing the energy-stress tensor, and |Φi
is the wave function of all (quantum) matter and fields that evolve within this classical
spacetime. Such a theory arises either when researchers considered gravity to be
fundamentally classical, or when they ignored quantum fluctuations in the stress
energy tensor, Tµν , in order to approximately solve problems involving quantum
gravity. The latter case is referred to as semiclassical gravity [13], in anticipation that
this approximation will break down if the stress-energy tensor exhibits substantial
quantum fluctuations. In this article, we propose an optomechanics experiment that
would test Eq. (3.1). Other experiments have been proposed [12, 9], but they do not
address the difficulties discussed below.
Classical gravity, as described by Eq. (3.1), suffers from a dramatic conceptual
drawback rooted in the statistical interpretation of wavefunctions. In order for the
Bianchi identity to hold on the left-hand side of Eq. (3.1), the right-hand side must be
divergence free, but that would be violated if we reduced the quantum state. In light
of this argument, one can go back to an interpretation of quantum mechanics where
the wavefunction does not reduce. At this moment, the predominant interpretation
of quantum mechanics that does not have wave-function reduction is the relativestate, or “many-world” interpretation, in which all possible measurement outcomes,
including macroscopically distinguishable ones, exist in the wavefunction of the
universe. Taking an expectation over that wavefunction leads to a serious violation
of common sense, as was demonstrated by Page and Geilker [19].
Another major difficulty is superluminal communication, which follows from the
nonlinearities implied by Eq. (3.1) (refer to section §3.2 for explicit examples of nonlinear Schroedinger equations). Superluminal communication is a general symptom
of wavefunction collapse in nonlinear quantum mechanics 1. Entangled and iden1 We note that the issue of superluminal communication could be resolved by adding a stochastic

31
tically prepared states, distributed to two spatially separated parties A and B, and
then followed by projections at B and a period of nonlinear evolution at A, can be
used to transfer signals superluminally [22, 3, 10, 25].
In this paper, we do not solve the above conceptual obstacles. Instead, we highlight
an even more serious issue of nonlinear quantum mechanics: its dependence on the
formulation of quantum mechanics. Motivated by the time-symmetric formulation of
quantum mechanics [23], we show that there are multiple prescriptions of assigning
the probability of a measurement outcome, that are equivalent in standard quantum
mechanics, but become distinct in nonlinear quantum mechanics. It is our hope that
at least one such formulation will not lead to superluminal signaling. We defer the
search for such a formulation to future work, and in this paper, we simply choose two
prescriptions, and show that they give different experimental signatures in torsional
pendulum experiments. These signatures hopefully scope out the type of behavior
classical gravity would lead to if a non superluminal-signaling theory indeed exists.
This paper is organized as follows. In section 3.2, we review the non-relativistic
limit of Eq. (3.1), called the Schroedinger-Newton theory, as applied to optomechanical setups, and without including quantum measurements. We determine that the
signature of the Schroedinger-Newton theory in the free dynamics of the test mass
is largest for low frequency oscillators such as torsion pendulums, and for materials,
such as Tungsten and Osmium, with atoms tightly bound around their respective lattice sites. In section 3.3, we remind the reader that in nonlinear quantum mechanics
the density matrix formalism cannot be used to describe thermal fluctuations. As
a result, we propose a particular ensemble of pure states to describe the thermal
bath’s state. In section 3.4, we discuss two strategies, which we term pre-selection
and post-selection, for assigning a statistical interpretation to the wavefunction in
the Schroedinger-Newton theory. In section 3.5, we obtain the signatures of the preand post-selection prescriptions in torsional pendulum experiments. In section 3.6,
we show that it is feasible to measure these signatures in state of the art experiments.
Finally, we summarize our main conclusions in section 3.7.
3.2

Free dynamics of an optomechanical setup under the Schroedinger-Newton
theory

In this section, we discuss the Schroedinger-Newton theory applied to optomechanical setups without quantum measurement. We first review the signature of the theory
extension to the theory of classical gravity, as was proposed by Nimmrichter [18]. However,
although the theory removes the nonlinearity at the ensemble level, it also eliminates the signature
of the nonlinearity in the noise spectrum.

32
in the free dynamics of an oscillator, and discuss associated design considerations.
We then develop an effective Heisenberg picture, which we refer to as a state dependent Heisenberg picture, where only operators evolve in time. However, unlike the
Heisenberg picture, the equations of motion depend on the boundary quantum state
of the system that is being analyzed. Finally we present the equations of motion of
our proposed optomechanical setup.
3.2.1

The center-of-mass Schroedinger-Newton equation

The Schrödinger-Newton theory follows from taking the non-relativistic limit of
Eq. (3.1). The expectation value in this equation gives rise to a nonlinearity. In
r ), evolves as
particular, a single non-relativistic particle’s wavefunction, χ (®
~2 2
r ) + U (t, r®) χ (®
i~∂t χ (®
r, t) = − ∇ + V (®
r, t) ,
2m

(3.2)

where V (®
r ) is the non-gravitational potential energy at r® and U (t, r®) is the Newtonian
self-gravitational potential and is sourced by χ (®
r ):
∇2U (t, x) = 4πGm | χ (t, x)| 2 .

(3.3)

A many-body system’s center of mass Hamiltonian also admits a simple description,
which was analyzed in [29]. If an object has its center of mass’ displacement
fluctuations much smaller than fluctuations of the internal motions of its constituent
atoms, then its center of mass, with quantum state |ψi, observes
d |ψi
i~
= ĤNG + MωSN ( x̂ − hψ| x̂|ψi) |ψi
dt

(3.4)

where M is the mass of the object, ĤNG is the non-gravitational part of the Hamiltonian, x̂ is the center of mass position operator, and ωSN is a frequency scale that is
determined by the matter distribution of the object. For materials with single atoms
sitting close to lattice sites, we have
ωSN ≡

Gm
6 π∆xzp

(3.5)

33
Note that the presented formula for ωSN is larger than the expression for ωSN
presented in [29] by a factor of 2. As explained in [11], the many body non-linear
gravitational interaction term presented in Eq. (3) of [29] should not contain a factor
of 1/2, which is usually introduced to prevent overcounting. The SN interaction term
between one particle and another is not symmetric under exchange of both of them.
For example, consider two (1-dimensional) identical particles of mass m. The
interaction term describing the gravitational attraction of the first particle, with
position operator x̂1 , to the second is given by
−Gm

dx1 dx2

|ψ (x1, x2 )| 2
| x̂1 − x2 |

which is not symmetric under the exchange of the indices 1 and 2. Moreover, in
Appendix 3.8, we show that the expectation value of the total Hamiltonian is not
conserved. Instead,
E = ĤNG + V̂SN /2
(3.6)
is conserved, where V̂SN is the SN gravitational potential term. As a result, we take
E, which contains the factor of 1/2 present in expressions of the classical many-body
gravitational energy, to be the average energy.
If the test mass is in an external harmonic potential, Eq. (3.4) becomes
2
d |ψi
p̂
2 2
i~
+ Mωcm
x̂
dt
2M 2
+ MωSN ( x̂ − hψ| x̂|ψi) |ψi

(3.7)

where p̂ is the center of mass momentum operator, and ωcmistheresonant f requencyo f thecr ystal 0 smotion
Eq. (3.7) predicts distinct dynamics from linear quantum mechanics. Assuming a
Gaussian initial state, Yang et al. show that the signature of Eq. (3.7) appears in the
rotation frequency
ωq ≡

2 + ω2
ωcm
SN

(3.8)

of the mechanical oscillator’s quantum uncertainty ellipse in phase space. We
illustrate this behavior in Fig. 3.1.
As a consequence, the dynamics implied by the nonlinearity in Eq. (3.4) are most
distinct from the predictions of standard quantum mechanics when ωq − ωcm is
as large as possible. This is achieved by having a pendulum with as small of an
oscillation eigenfrequency as possible, and made with a material with as high of a

Figure 3.1: Left Panel: according to standard quantum mechanics, both the vector
(h x̂i, h p̂i) and the uncertainty ellipse of a Gaussian state for the center of mass
of a macroscopic object rotate clockwise in phase space, at the same frequency
ω = ωCM . Right panel: according q
to Eq.(3.7), (hxi, hpi) still rotates at ωcm , but the
uncertainty ellipse rotates at ωq ≡

2 > ω . (Figure taken from [29]).
ωcm 2 + ωSN
cm

ωSN as possible. The former condition leads us to propose the use of low-frequency
torsional pendulums. To meet the latter condition, we notice that ωSN depends
significantly on ∆xzp , which can be inferred from the Debye-Waller factor,
B = u2 /8π 2

(3.9)

where u is the rms displacement of an atom from its equilibrium q
position [20].

2 + ∆x 2
Specifically, thermal and intrinsic fluctuations contribute to u, i.e. u & ∆xzp
th
with ∆xth representing the uncertainty in the internal motion of atoms due to thermal
fluctuations.

In Table 3.1, we present experimental data on some materials’ Debye-Waller factor,
and conclude that the pendulum should ideally be made with Tungsten (W), with
ωW
SN = 2π × 4.04 mHz, or Osmium (the densest naturally occurring element) with a
theoretically predicted ωOs
SN of 2π × 5.49 mHz. Other materials such as Platinum or
N b = 2π × 1.56 mHz respectively, could
Pt
Niobium, with ωSN = 2π × 3.2 mHz and ωSN
be suitable candidates.
3.2.2

State-dependent Heisenberg picture for nonlinear quantum mechanics

In this section, we develop an effective Heisenberg picture for non-linear Hamiltonians similar to the Hamiltonian given by Eq.(3.7). We abandon the Schroedinger
picture because the dynamics of a Gaussian optomechanical system are usually

35
(10 kg/m3 )
2.33
7.87
5.32
8.57
21.45
19.25
22.59

Element
Silicon (Si)
Iron (Fe) (BCC)
Germanium (Ge)
Niobium (Nb)
Platinum (Pt)
Tungsten (W)
Osmium* (Os)

B2
(2 )
0.1915
0.12
0.1341
0.1082
0.0677
0.0478
0.0323

ωSN
(10−2 s−1 )
4.95
9.90
10.39
13.86
28.43
35.92
48.79

Table 3.1: Characteristic Schroedinger-Newton angular frequency ωSN for several
elemental crystals. Density is approximated by values at room temperature, and the
Debye-Waller factor B (at 1 K) is provided by Ref. [20]. *: Note that Osmium’s
Debye-Waller factor is solely obtained from theoretical calculations.

examined in the Heisenberg picture where the similarity to classical equations of
motion is most apparent.
We are interested in non-linear Schroedinger equations of the form
i~

d|ψi
= Ĥ(ζ(t))|ψi ,
dt
ζ(t) = hψ(t)| Ẑ |ψ(t)i,

(3.10)
(3.11)

where the Hamiltonian Ĥ is a linear operator that depends on a parameter ζ, which in
turn depends on the quantum state that is being evolved. Note that the Schroedinger
operator Ẑ can depend explicitly on time, ζ can have multiple components, and the
Hilbert space and canonical commutation relations are unaffected by the nonlinearities.
3.2.2.1

State-dependent Heisenberg Picture

We now present the effective Heisenberg Picture. Let us identify the Heisenberg
and Schroedinger pictures at the initial time t = t0 ,
|ψH i = |ψ(t0 )i,

x̂H (t0 ) = x̂S (t0 ),

p̂H (t0 ) = p̂S (t0 ),

(3.12)

where |ψH i is the quantum state |ψi in the Heisenberg picture, and we have used the
subscripts S and H to explicitly indicate whether an operator is in the Schroedinger
or Heisenberg picture, respectively. As we evolve in time in the Heisenberg Picture,

36
we fix |ψS (t0 )i, but evolve x̂H (t) according to
∂
i
ĤH (ζ(t)), x̂H (t) + x̂H (t) ,
x̂H (t) =
dt
∂t
ζ(t) = hψH | ẐH (t)|ψH i.

(3.13)
(3.14)

A similar equation holds for p̂H (t). We shall refer to such equations as statedependent Heisenberg equations of motion. Moreover, the Heisenberg picture of an
arbitrary operator in the Schroedinger picture
ÔS = f ( x̂S, p̂S, t) ,

(3.15)

including the Hamiltonian Ĥ(ζ(t)), can be obtained from x̂H (t) and p̂H (t) by:
ÔH (t) = f ( x̂H (t) , p̂H (t) , t) .
3.2.2.2

(3.16)

Proof of the State-Dependent Heisenberg Picture

The state-dependent Heisenberg picture is equivalent to the Schroedinger picture, if
at any given time
hψH |ÔH (t)|ψH i = hψS (t)|ÔS (t)|ψS (t)i.
(3.17)
Before we present the proof, we motivate the existence of a Heisenberg picture
with a simple argument. If we (momentarily) assume that the nonlinearity ζ (t) is
known and solved for, then the non-linear Hamiltonian Ĥ (ζ (t)) is mathematically
equivalent to a linear Hamiltonian,
Ĥ L (ζ (t)) = Ĥ (ζ (t)) ,

(3.18)

with a classical time-dependent drive ζ (t). Since there exists a Heisenberg picture
associated with Ĥ L (ζ (t)), there exists one for the nonlinear Hamiltonian Ĥ (ζ (t)).
We now remove the assumption that ζ (t) is known and consider linear Hamiltonians,
Ĥ L (ρ (t)), driven by general time-dependent classical drives λ (t). To each Ĥ L (λ (t))
is associated a different unitary operator Ûλ (t) and so a different Heisenberg picture
ÔH (λ, t) = Ûλ† (t) ÔS Ûλ (t) .

(3.19)

Next, we choose λ(t) in such a way that
hψH |ÔH (λ, t)|ψH i = hψS (t)|ÔS (t)|ψS (t)i.

(3.20)

37
is met. For the desired effective Heisenberg picture to be self-consistent, λ (t) must
be obtained by solving
λ(t) = ψH | ẐH (λ, t) |ψH ,
(3.21)
which, in general, is a non-linear equation in λ. We will explicitly prove that this
choice of λ(t) satisfies Eq. (3.20). Note that we will present the proof in the case
that the boundary wavefunction is forward time evolved. The proof for backwards
time evolution is similar.
We begin the proof by showing that λ and ζ are equal at t = t0 ,
λ (t0 ) = ψS (t0 ) | ẐS |ψS (t0 ) = ζ (t0 )
because the Schrodinger and state-dependent Heisenberg pictures are, as indicated
by Eq. (3.12), identified at the initial time t = t0 .
λ and ζ can deviate at later times if the increments ∂t λ and ∂t ζ are different. We
use the nonlinear Schroedinger equation to obtain the latter increment:
∂t ζ (t) = ∂t ψS (t) | ẐS |ψS (t)
ψS (r) | Ĥ (ζ (t)) , ẐS |ψS (t) .

(3.22)
(3.23)

Note that the equation of motion for ζ (t) is particularly simple to solve in the case of
the quadratic Hamiltonian given by Eq. (3.4), because the non-linear part of Ĥ (ζ (t))
commutes with x̂.
On the other hand, by Eq. (3.21),
∂t λ(t) =

ψH | ĤHL (λ (t)) , ẐH (λ, t) |ψH

Making use of Eq. (3.19), and of
Ĥ L (λ (t)) = Ûλ (t) ĤHL (λ (t)) Ûλ† (t) ,
we obtain
∂t λ(t) =

Ûλ (t) ψH Ĥ L (λ (t)) , ẐS Ûλ (t) ψH .

(3.24)

38
Furthermore, Ûλ (t) ψH evolves under
i~

d Ûλ (t) ψH
= Ĥ (λ (r)) Ûλ (t) ψH
dt

(3.25)

Notice the similarity with Eq. (3.10).
We have established that the differential equations governing the time evolution
of λ and Ûλ (t) ψH , are of the same form as those governing the time evolution
of ζ(t) and |ψS (t)i. In addition, these equations have the same initial conditions.
Therefore, λ(t) = ζ(t) for all times t. Eq. (3.20) then easily follows because we’ve
established that Ĥ (ζ (t)) and
Ĥ

ψH | ẐH (λ, t) |ψH

are mathematically equivalent for all times t.
3.2.3

Optomechanics without measurements

We propose to use laser light, enhanced by a Fabry-Perot cavity, to monitor the
motion of the test mass of a torsional pendulum, as shown in Fig. 3.2. We assume
the light to be resonant with the cavity, and that the cavity has a much larger linewidth
than ωq , the frequency of motion we are interested in.
We will add the non-linear Schroedinger-Newton term from Eq. (3.7) to the usual
optomechanics Hamiltonian, obtaining
( x̂ − hψ| x̂|ψi)2,
Ĥ = ĤOM + MωSN

(3.26)

where ĤOM is the standard optomechanics Hamiltonian for our system [6]. We have
ignored corrections due to light’s gravity because we are operating in the Newtonian
regime, where mass dominates the generation of the gravitational field. Ĥ generates
the following linearized state dependent Heisenberg equations (with the dynamics
of the cavity field adiabatically eliminated, and the "H" subscript omitted):
p̂
∂t p̂ = −Mωcm
x̂ − MωSN
( x̂ − hψ| x̂|ψi) + α â1

(3.28)

b̂1 = â1

(3.29)

∂t x̂ =

b̂2 = â2 + x̂,

(3.27)

(3.30)

Torsion
ﬁber

Bath at
temperature T0

Figure 3.2: The proposed low-frequency optomechanical experiment.

where â1,2 are the perturbed incoming quadrature fields around a large steady state,
and similarly b̂1,2 are the perturbed outgoing field quadratures (refer to section 2
of [6] for details). The quantity α characterizes the optomechanical coupling, and
depends on the pumping power Iin and the input-mirror power transmissivity T of
the Fabry-Perot cavity:
8Iin ~ωc 1
α2 =
(3.31)
T c2 T
Note that we have a linear system under nonlinear quantum mechanics because the
Heisenberg equations are linear in the center of mass displacement and momentum
operators, and in the optical field quadratures, including their expectation value on
the system’s quantum state.
3.3

Nonlinear quantum optomechanics with classical noise

To study realistic optomechanical systems, we must incorporate thermal fluctuations.
In linear quantum mechanics, we usually do so by describing the state of the bath
with a density operator. However, it is known that the density matrix formalism

40
cannot be used in non-linear quantum mechanics [3].
Our dynamical system is linear and is driven with light in a Gaussian state, so
all system states are eventually Gaussian. Moreover, our system admits a statedependent Heisenberg picture. Consequently, we can describe fluctuations with
distribution functions of linear observables which are completely characterized by
their first and second moments. In nonlinear quantum mechanics, the challenge will
be to distinguish between quantum uncertainty and the probability distribution of
classical forces. The conversion of quantum uncertainty to probability distributions
of measurement outcomes is a subtle issue in nonlinear quantum mechanics, and
will be postponed until the next section.
Once we have chosen a model for the bath, we will have to revisit the constraint,
required for Eq. (3.7) to hold, that the center of mass displacement fluctuations are
much smaller than ∆xzp . Thermal fluctuations increase the uncertainty in the center
of mass motion to the point that in realistic experiments, the total displacement of
the test mass will be much larger than ∆xzp . Nonetheless, after separating classical
and quantum uncertainties, we will show that Eq. (3.7) remains valid, as long as the
quantum (and not total) uncertainty of the test mass is much smaller than ∆xzp .
Finally, we ignore the gravitational interactions in the thermal bath, as they are
expected to be negligible.
3.3.1

Abandoning the density matrix formalism in nonlinear quantum mechanics

In standard quantum mechanics, we use the density matrix formalism when a system is entangled with another system and/or when we lack information about a
system’s state. The density matrix completely describes a system’s quantum state.
If two different ensembles of pure states, say {|ψi i} and {|φi i} with corresponding
probability distributions pψi and pφi , have the same density matrix

pψi |ψi i hψi | =

pφi |φi i hφi | ,

(3.32)

then they cannot be distinguished by measurements. Furthermore, when either
ensemble is time-evolved, they will keep having the same density matrix. However, this statement is no longer true in non-linear quantum mechanics because the
superposition principle is no longer valid.
Let us give an example of how our nonlinear Schroedinger equation, given by
Eq. (3.7), implies the breakdown of the density matrix formalism. Suppose Alice

41
and Bob share a collection of entangled states, |Φi, between Bob’s test mass’ center
of mass degree of freedom and Alice’s spin 1/2 particle, with |Φi given by
|Φi = √ (|↑i |ψ x i + |↓i |ψ−x i)
= √ (|→i |+i + |←i |−i)
where
|↑i + |↓i
|↑i − |↓i
|←i ≡
|→i ≡

(3.33)
(3.34)

and |ψ±x i are localized states around x and −x:
|ψ±x i = p √
σ π

(y ∓ x)2
|yi dy.
exp −
2σ 2

(3.35)

We choose σ
x so that hψ x |ψ−x i ≈ 0. Moreover,
|±i ≡ √ (|ψ x i ± |ψ−x i) .

(3.36)

Next, suppose that Alice measures her spins along the {|↑i , |↓i} basis, then Bob will
be left with the following mixture of states:
χ=

 |ψ x i

with probability 1/2

 |ψ−x i

with probability 1/2.

(3.37)

On the other hand, if Alice measured her spins along {|→i , |←i} basis, then Bob
will be left with the mixture
κ=

 |+i

with probability 1/2

 |−i

with probability 1/2.

(3.38)

In standard quantum mechanics, both mixtures would be described with the density

Figure 3.3: Two ways of forming the same Gaussian density matrix. In the
left panel, we have an ensemble of coherent states parameterized by a complex
amplitude α, which is Gaussian distributed. The red circle depicts the noise ellipse,
in phase space, of one such state. The green ellipse depicts the total noise ellipse
of the density matrix. In the right panel, we have an ensemble of squeezed states
with amplitudes ε, which achieves the same density matrix with a fixed squeeze
amplitude and a uniform distribution of squeeze angles.

matrix
|ψ x i hψ x | + |ψ−x i hψ−x |
|+i h+| + |−i h−| .

ρ =

(3.39)
(3.40)

However, under the Schroedinger-Newton theory, it is wrong to use ρ because under
time evolution both mixtures will evolve differently. Indeed, under time evolution
driven by Eq. (3.7) (which has a nonlinearity of h x̂i) over an infinitesimal period dt,
χ and κ no longer remain equivalent because h±| x̂|±i = 0, and so κ is unaffected
by the nonlinearity.
For this reason, we will have to fall back to providing probability distributions for
the bath’s quantum state. For a Gaussian state, there are many ways of doing so, as
is for example shown in Fig. 3.3. Since this distribution likely has a large classical
component (as we argue for in the next section), we will approach the issue of
thermal fluctuations by separating out contributions to thermal noise from classical
and quantum uncertainty.

43
3.3.2
3.3.2.1

Quantum versus classical uncertainty
Standard Quantum Statistical Mechanics

Let us consider a damped harmonic oscillator in standard quantum mechanics, which
satisfies an equation of motion of
M( xÜ̂ + γm xÛ̂ − ωcm
) = F̂th (t) ,

(3.41)

where γm is the oscillator’s damping rate and F̂th (t) a fluctuating thermal force.
We have assumed viscous damping. Other forms of damping, such as structural
damping, where the retarding friction force is proportional to displacement instead
of velocity [28], would reduce the classical thermal noise (which will be precisely
defined later in this section) at ωq , making the experiment easier to perform.
At a temperature T0
~ωcm /k B , which accurately describes our proposed setup
with a test resonant frequency under a Hz, the thermal force mainly consists of
classical fluctuations. We obtain F̂th (t)’s spectrum from the fluctuation-dissipation
theorem,
1 Im[G c (Ω)]
(3.42)
SF̂th,F̂th (Ω) = 2~
~Ω
e kB T0 − 1 2 |G c (Ω)|
where G c (Ω) is the response function of x̂ to the driving force F̂th (t),
G c (Ω) =

2 − Ω (Ω + iγ )
ωcm

(3.43)

and SF̂th,F̂th (Ω) is defined by
hF̂th (Ω) F̂th† (Ω0)isym = SF̂th,F̂th (Ω)2πδ(Ω − Ω0)

(3.44)

with

h ÂB̂ + B̂ Âi
(3.45)
Note that we have chosen a “double-sided convention” for calculating spectra.
h ÂB̂isym ≡

The fact that the motion of the test mass is damped due to its interaction with the
heat bath also requires that the thermal force has a (usually small but nevertheless
conceptually crucial) quantum component,

F̂th (t), F̂th (t 0) , 0 ,

(3.46)

which compensates for the decay of the oscillator’s canonical commutation relations

44
due to adding damping in its equations of motion (refer to section 5.5 of [2] for
details). Note that the second term in the bracket in Eq. (3.42) provides the zeropoint fluctuations of the oscillator as T → 0.
3.3.2.2

Quantum Uncertainty

Let the bath be in some quantum state |ΦB i over which we will take expectation
values. The thermal force operator acting on the system can then be conveniently
decomposed into
F̂th (t) = fcl (t) + fˆzp (t)
(3.47)
where we define
fcl (t) = hF̂th (t)i ,

fˆzp (t) = F̂th (t) − hF̂th (t)i.

(3.48)

We use the subscripts “cl” and “zp” because fcl (t) is a complex number, while
fˆzp (t) will be later chosen to drive the “zero-point” quantum fluctuation of the mass.
For any operator Â, we shall refer to h Âi as the quantum expectation value and
V[ Â] ≡ h Â2 i − h Âi 2

(3.49)

as its quantum uncertainty. We also define the quantum covariance by
Cov[ Â, B̂] = h ÂB̂isym − h Âih B̂i.

(3.50)

Suppose |ΦB i is a Gaussian quantum state, an assumption satisfied by harmonic
heat-baths under general conditions [27], then |ΦB i is completely quantified by the
following moments: the means
h fcl (t)i = fcl (t) ,

h fˆzp (t)i = 0,

(3.51)

the covariances that include fcl (t)
Cov [ fcl (t) , fcl (t 0)] = Cov fcl (t) , fˆzp (t 0) = 0,
and those that don’t
Cov F̂th (t) , F̂th (t 0) = Cov fˆzp (t) , fˆzp (t 0)
= h fˆzp (t) fˆzp (t 0)isym , 0.

(3.52)

45
3.3.2.3

Classical Uncertainty

The state |ΦB i is drawn from an ensemble with a probability distribution p(|ΦB i).
For each member of the ensemble, we will have a different quantum expectation
fcl (t), and a different two-time quantum covariance for fˆzp (t). We shall call the
variations in these quantities classical fluctuations, because they are due to our lack
of knowledge about a system’s wavefunction.
The total covariance of the thermal force, using our terminology, is given by:

F̂th (t)F̂th (t 0) + F̂th (t 0)F̂th (t)

=h fˆzp (t) fˆzp (t 0)isym + fcl (t) fcl (t 0) ,

(3.53)

where h i denotes taking an ensemble average over different realizations of the
thermal bath. Eq. (3.53) is the total thermal noise we obtain, and in standard
quantum mechanics there is no way to separately measure quantum and classical
uncertainties.
3.3.2.4

Proposed model

We shall assume that fˆzp ’s two-time quantum covariance, h fˆzp (t) fˆzp (t 0)isym , provides
the zero-point fluctuations in the position of the test mass, and that its ensemble
average is zero (i.e. the uncertainty in fˆzp (t) comes solely from quantum mechanics).
This results in fˆzp (t) having a total spectrum of:
qu

S fz p, fz p (Ω) = ~

Im[G c (Ω)]
= ~ΩMγm .
|G c (Ω)| 2

(3.54)

Moreover, we shall assume that fcl ’s two-time ensemble covariance, fcl (t) fcl (t 0),
provides the fluctuations predicted by classical statistical mechanics. This results in
fcl having a total spectrum of

3.3.3

Im[G c (Ω)]
≈ 2k BT Mγm .
− 1 |G c (Ω)|

S fcl (Ω) =

~Ω
k B T0

(3.55)

Validity of the quadratic SN equation

In general, the center of mass wavefunction |ψi follows the SN equation
i~

d|ψi
= ĤNG + V̂ |ψi,
dt

(3.56)

46
where the gravitational potential V̂ can be approximately calculated by taking an
expectation value of Eq. (8) in [29] with respect to the internal degrees of freedom’s
wavefunction:
V̂ =
E( x̂ − z) |hψ|zi| 2 dz
(3.57)
with E the “self energy” between a shifted version of the object and itself at the
original position. We calculate E to be

E(x) = GMm
− erf
∆xzp x
2∆xzp
x2
GMm √
x4
π−1+
= √
+ ... .
π∆xzp
12∆xzp
160∆xzp
As a result, V̂ is in general difficult to evaluate because it depends on an infinite
number of expectation values. When the center of mass spread
∆xcm ≡

( x̂ (t) − h x̂ (t)i)2

(3.58)

is much less than ∆xzp , E can be approximated to quadratic order in x, leading to
the simple quadratic Hamiltonian presented in Eq. (3.4) [29]. In this section, we
show that classical thermal noise does not affect the condition ∆xcm
∆xzp .
We include classical thermal noise in our analysis through the following interaction
term:
V̂cl (t) ≡ − fcl (t) x̂.
(3.59)
We will show that ∆xcm does not depend on fcl (t), even when we use the full
expression for V̂.
We first momentarily ignore V̂, and show that under the non-gravitational Hamiltonian, ĤNG , ∆xcm is unaffected by fcl (t). Since ĤNG is quadratic, then the timeevolved position operator under ĤNG , x̂ (0) , is of linear form
∫
Õ
(0)
x̂ (t) =
ci (t) q̂i + di (t) k̂i + G c (t − z) fcl (z) dz

r (t, z) â1 (z) dz +

s (t, z) â2 (z) dz,

(3.60)

where q̂i and k̂i are canonically conjugate operators of discrete degrees of freedom
such as the center of mass mode of the test mass, G c (t) is the inverse Fourier
transform of the response function defined by Eq. (3.43), and r(t) and s(t) are

47
c-number functions. As a result, the variance of x̂ (0) is unaffected by fcl (t).
The full time-evolved position operator (in the state-dependent Heisenberg picture
introduced in section II.B), can be expressed in terms of x̂ (0) in the following way:
x̂H (t) = ÛI† (t) x̂ (0) (t) ÛI (t) ,

(3.61)

where ÛI is the (state-dependent) interaction picture time-evolution operator associated with
(t)V̂(t)ÛNG (t).
(3.62)
V̂I (t) = ÛNG
Specifically, ÛI is defined by
Û = ÛNG ÛI ,

(3.63)

where ÛNG is the time-evolution operator associated with ĤNG . We will show that
V̂I and ÛI are independent of fcl (t).
We begin the proof, of V̂I independent of fcl (t), by conveniently rewriting |hψ|zi| 2
in Eq. (3.57) as the expectation value of an operator. We do so by writing the
projection |zi hz| as a delta function:
V̂ =
E ( x̂ − z) hδ ( x̂ − z)i dz,
(3.64)
which we then express in the Fourier domain
δ ( x̂ − z) =
dxδ (x − z) |xi hx|
−ik(x−z)
|xi hx| ∝
dx
dke
dke−ik( x̂−z) .
We then substitute this expression into V̂, transform E into the Fourier domain, and
obtain
V̂ ∝
F (l) e−il( x̂−z) e−ik( x̂−z) dk dl dz,
(3.65)
where F is the Fourier transform of E. Finally, we perform the integral over z,
obtaining
V̂ ∝

F (k) e−ik x̂ eik x̂ dk.

(3.66)

48
In the interaction picture,

F (k) e−ik x̂

(0) (t)

F (k) e−ik x̂

(0) (t)

V̂I (t) ∝

Ψ0 |eik x̂H (t) |Ψ0 dk

(0)
Ψ0 ÛI† (t)eik x̂ (t)ÛI (t) Ψ0

(3.67)

dk,

where |Ψ0 i is the initial wavefunction of the entire system. Notice that the linear
dependence of x̂ (0) on fcl (t) cancels out in Eq. (3.67). However, V̂I could still
depend on fcl (t) through ÛI . We will show that this is not the case.
The operator
V̂I (0) = V̂

(3.68)

ÛI (0) |Ψ0 i = |Ψ0 i

(3.69)

and the ket
do not depend on fcl (t) at the initial time t = 0. At later times, fcl (t) can only
appear through the increments dV̂I /dt or dÛI |Ψ0 i /dt. The latter is given by
i~

ÛI |Ψ0 i = V̂I ÛI |Ψ0 i ,
dt

(3.70)

while
dV̂I (t)
i~
dt

dkF (k)

(0)
e−ik x̂ , ĤNG ×

(0)
(0)
ÛI† eik x̂
ÛI + e−ik x̂ ×

D h
i E
ik x̂ (0)
ÛI e
, ĤNG + V̂I ÛI

(3.71)

where the expectation values h i 0 are taken over |Ψ0 i. In both terms in the
sum, the dependence of x̂ (0) on fcl (t) cancels out, and so fcl (t) does not explicitly
appear in the system of differential equations (3.70) and (3.71). fcl (t) does not also
appear in the initial conditions (3.69) and (3.71). Consequently, both V̂I and ÛI are
independent of fcl (t).
We then use Eq. (3.61) to establish that the center of mass position operator is
independent of fcl (t). As a result, the exact expression for ∆xcm is also independent
of fcl (t). If ∆xcm
∆xzp holds in the absence of classical thermal noise, it also
holds in the presence of it. We will have to check this assumption in order for the

49
linear Heisenberg equation to hold. Otherwise, if ∆xcm becomes larger than ∆xzp ,
the effect of V̂ becomes weaker, because V̂ becomes shallower than the quadratic
potential
( x̂ − h x̂i)2 .
MωSN
3.3.4

Heisenberg equations of motion with thermal noise included

The dynamics of our proposed model for an open optomechanical system are summarized by the following state-dependent Heisenberg equations:
d x̂ p̂
dt M
d p̂
x̂ − γm p̂ − MωSN
( x̂ − h x̂i)
= − Mωcm
dt
+α â1 + fcl + fˆzp

(3.73)

b̂1 =â1

(3.74)

b̂2 =â2 + x̂,

(3.72)

(3.75)

where the spectra of fˆzp (ω) and fcl (ω) are given by Eqs. (3.54) and (3.55),
respectively.
We solve Eqs. (3.72)–(3.75) by working in the frequency domain, and obtain at
each frequency ω,
b̂2 (ω) = Â (ω) +

αG c (ω)
fcl (ω) + B̂ (ω) .

(3.76)

We separately discuss the three terms. The operator Â (ω) is the linear quantum
contribution to b̂2 :
Â (ω) ≡ â2 (ω) +

αG q (ω)
α â1 + fˆzp (ω) ,

(3.77)

where â2 (ω) represents shot noise,
G q (ω) ≡

ωq2 − ω2 − iωγm

(3.78)

is the quantum response function of the damped torsional pendulum’s center of mass
position, x̂ (ω), to the thermal force, and α â1 and fˆzp are the quantum radiationpressure force and the quantum piece of the thermal force acting on the test mass,
respectively.

50
The second term in Eq. (3.76) represents classical thermal noise, with G c (ω) defined
in Eq. (3.43). Note that the classical and quantum resonant frequencies in G c (ω)
and G q (ω), respectively, differ from each other.
The third term in Eq. (3.76), h B̂(ω)i, represents the non-linear contribution to b̂2 (ω)
B̂ (ω) ≡

α∆G (ω)
α â1 (ω) + fˆzp (ω) ,

(3.79)

where we defined
∆G (ω) ≡ G c (ω) − G q (ω) .

(3.80)

In the next section, we discuss the subtle issue of how to convert the wavefunction
average h...i to the statistics of measurement outcomes.
3.4

Measurements in nonlinear quantum optomechanics

With the assumption of classical gravity, we will have to revisit the wavefunction
collapse postulate, because a sudden projective measurement of the outgoing optical field induces a change in the quantum state of any of its entangled partners,
including possibly the macroscopic pendulum’s state. As a result, we might obtain
an unphysical change in the Einstein tensor which violates the Bianchi identity.
Moreover, since the Schroedinger-Newton equation is nonlinear, we will show that
we have to address an additional conceptual challenge: there is no unique way of
extending Born’s rule to nonlinear quantum mechanics.
In this section, we propose two phenomenological prescriptions, which we term preselection and post-selection, for determining the statistics of an experiment within
the framework of classical gravity.
3.4.1

Revisiting Born’s rule in linear quantum mechanics

We will use the wavefunction collapse postulate as a guide. The postulate is
mathematically well defined, but can be interpreted in two equivalent ways, which
become inequivalent in nonlinear quantum mechanics.
The first interpretation is widely used, and describes a quantum measurement experiment in the following way: a preparation device initializes a system’s quantum
state to |ii, which evolves for some period of time under a unitary operator, Û, to
|ii → Û|ii .

(3.81)

The system then interacts with a measurement device, which collapses the system’s

Deterministic
evolution
Projected to

Deterministic (backwards)
evolution
Projected to
Figure 3.4: The two prescriptions, pre-selection (top) and post-selection (bottom),
that can be used to calculate measurement probabilities. Both prescriptions are
equivalent in linear quantum mechanics, but become different under non-linear
quantum mechanics.

state into an eigenstate, | f i, of the observable associated with that device. The
probability of the collapse onto | f i is
pi→ f ≡ |h f |Û|ii| 2 .

(3.82)

We will refer to this expression of Born’s rule as pre-selection.
Second, the unitarity of quantum mechanics allows us to rewrite Eq. (3.82) to
pi→ f = |hi|Û † | f i| 2 ≡ pi← f .

(3.83)

Interpreting this expression from right to left, as we did for Eq. (3.82), we can
form an alternate, although unfamiliar, narrative: | f i evolves backwards in time
to Û † | f i, and is then projected by the preparation device to the state |ii, as is
illustrated in Fig. 3.4. We will refer to the formulation of Born’s rule based on pi← f
as post-selection.
3.4.2

Pre-selection and post-selection in non-linear quantum mechanics

In non-linear quantum mechanics, the Hamiltonian, and so the time evolution operator, depends on the quantum state of the system. As a result, the pre-selection

52
version of Born’s rule, Eq. (3.82), has to be revised to
pi→ f = h f |Û|ii |ii

(3.84)

where Û|ii is the (non-linear) time evolution operator which evolves |ii forward in
time to Û|ii |ii.
Furthermore, the post-selection version of Born’s rule, Eq. (3.83), is modified to

pi← f ∝ hi|Û|†f i | f i ,

(3.85)

where Û|†f i is the (non-linear) time evolution operator which evolves | f i backwards
in time to Û|†f i | f i. The evolution can still be interpreted as running backwards in
time, because the non-linear Hamiltonians we are working with, such as in Eq. (3.7),
are Hermitian. Moreover, the proportionality sign follows from

hi|Û|†f i | f i

being not, in general, normalized to unity.
Notice that pi→ f and pi← f are in general different. Consequently, in non-linear
quantum mechanics, we can no longer equate the pre-selection and post-selection
prescriptions, and we will have to consider both separately.
3.4.3

Pre-selection and post-selection in non-linear quantum optomechanics

In our proposed optomechanical setup, the state |ii is a separable state consisting
of the initial state of the test object, and a coherent state of the incoming optical
field, which has been displaced to vacuum, |0iin by the transformation â1,2 →
δ â1,2 + â1,2 . In the pre-selection measurement prescription, as we reach steady
state, the test-mass’ initial state becomes irrelevant, and the system’s state is fully
determined by the incoming optical state.
The set of possible states | f i are eigenstates of the field quadrature b2 (t), which can
be labeled by a time series
|ξiout ≡ |{ξ(t) : −∞ < t < +∞}iout .

(3.86)

Similarly to what we discussed for pre-selection, as we reach steady state, the testmass’ initial state becomes irrelevant. This statement can easily be demonstrated if
pi← f is recast in a form, cf. Eq. (3.90), where the test mass’ state is forward-time

53
evolved and so is driven by light, and undergoes thermal dissipation.
Since |ξiout labels a collection of Gaussian quantum states, the distribution of the
measurement results ξ(t) will be that of a Gaussian random process, characterized
by the first and second moments. In standard quantum mechanics, they are given by
the mean hb̂2 (t)i and the correlation function
hb̂2 (t)b̂2 (t 0)isym − hb̂2 (t)ihb̂2 (t 0)i.
In nonlinear quantum mechanics, the situation is subtle because b̂2 (t) could
depend on the measurement results ξ(t).
To determine the expression for the second moment, we will explicitly calculate
pi→ f and pi← f . Since our proposed setup eventually reaches a steady state, we
can simplify our analysis by working in the Fourier domain, where fluctuations at
different frequencies are independent. Note that we first ignore the classical force
fcl (t). We will incorporate it back into our analysis at the end of this section.
The probability of measuring ξ in the pre-selection measurement prescription,
pi→ f = p0→ξ = | out hξ |Û|0iin |0iin | 2

(3.87)

is characterized by the spectrum of the Heisenberg Operator of b̂2 in the following
way:
dΩ |ξ(Ω) − hb̂2 (Ω)i0 | 2
p0→ξ ∝ exp −
(3.88)
2π
S A,A
where hb2 (Ω)i0 is the quantum expectation value of the Heisenberg operator b̂2 (ω),
calculated using the state-dependent Heisenberg equations associated with an initial
boundary condition of |0iin , and S A,A is the spectral density of the linear part of
b̂2 (Ω), Â, evaluated over vacuum:
D
0 E
2πS A,A(ω)δ ω − ω ≡ 0 Â (ω) Â† ω 0

sym

Note that the derivation of Eq. (3.88) is presented in Appendix 3.9. In the same
Appendix, we also show that in the limit of ωSN → 0, p0→ξ recovers the predictions
of standard quantum mechanics.
In post-section, the probability of obtaining a particular measurement record is given
by
pi← f = p0←ξ = h0|Û|ξi

out

|ξiout ,

(3.89)

54
which can be written as
p0←ξ = out hξ |Û|ξiout |0i

(3.90)

where Û|ξiout is the time-evolution operator specified by the end-state |ξiout . In
Appendix 3.9, we show that p0←ξ is given by

p0←ξ ∝ exp −

dΩ |ξ(Ω) − hb̂2 (Ω)iξ | 2
2π
S A,A

(3.91)

where hb̂2 (Ω)iξ is the quantum expectation value of b̂2 (Ω)’s Heisenberg operator,
obtained with the state-dependent Heisenberg equations associated with the final
state |ξi, but evaluated on the incoming vacuum state |0i for â1,2 .
Note that because hb2 (Ω)iξ depends on ξ, the probability density given by Eq. (3.91)
is modified. We extract the inverse of the new coefficient of |ξ 2 (Ω)| as the new
spectrum. We will follow this procedure in section §3.5 C. The normalization of
p0←ξ is taken care of by the Gaussian function.
Finally, we incorporate classical noise by taking an ensemble average over different
realizations of the classical thermal force, fcl (ω). For instance, the total probability
for measuring ξ in pre-selection is
p0←ξ =
D x p ( fcl (ω) = x (ω)) × p0←ξ(x(ω)),
(3.92)
where p ( fcl (ω) = x (ω)) is the probability that fcl at frequency ω is equal to x(ω),
and ξ(x(ω)) is the measured eigenvalue of the observable b̂2 given that the classical
thermal force is given by x. The above integral can be written as a convolution and
so is mathematically equivalent to the addition of Gaussian random variables. Thus,
assuming independent classical and quantum uncertainties, the total noise spectrum
is given by adding the thermal noise spectrum to the quantum uncertainty spectrum
calculated by ignoring thermal noise.
3.5

Signatures of classical gravity

With a model of the bath and the pre- and post-selection prescriptions at hand, we
proceed to determine how the predictions of the Schroedinger Newton theory for the
spectrum of phase fluctuations of the outgoing light differ from those of standard
quantum mechanics. We expect the signatures to be around ωq , the frequency where
the Schroedinger Newton dynamics appear at, as was discussed in section §3.2 and
in [29].

55
3.5.1

Baseline: standard quantum mechanics

We calculate the spectrum of phase fluctuations predicted by standard quantum
(ω), by setting ωSN to 0 in Eq. (3.76). Making use of
mechanics, Sb(QM)
2,b2
Sa1,a1 = Sa2,a2 = 1/2

Sa1,a2 = 0

(3.93)

for vacuum fluctuations of â1 and â2 , we obtain
(ω) =
Sb(QM)
2,b2

1 α4
α2 cl
(ω) ,
+ 2 |G c (ω)| 2 + 2 Sx,x
2 2~

(3.94)

where the first and second terms on the RHS represent shot noise and quantum
radiation pressure noise respectively, and
cl
(ω) = 2k BT0
Sx,x

Im (G c (ω))

(3.95)

is the noise spectrum of the center of mass position, x̂(ω), due to the classical
thermal force, fcl (ω).
We are interested in comparing standard quantum mechanics to the SN theory, which
(ω) around
has signatures around ωq . Therefore, we would need to evaluate Sb(QM)
2,b2
ωq . The first two terms in Eq. (3.94) can be easily evaluated at ω = ωq , and in the
limit of ωcm
ωSN ,
α2 cl
= βΓ2
x,x
~2

(3.96)

where we have defined two dimensionless quantities,
α2
β≡
M~γm ωq

γm2 ωq2
k BT0
≡ 2
~ωq γm2 ωq2 + ωSN

(3.97)

β characterizes the measurement strength (as α2 is proportional to the input power),
and Γ characterizes the strength of thermal fluctuations. If Q
1, we can simplify
Γ2 to
2k BT0 2
γm .
(3.98)
Γ2 ≈
~ωSN

56
3.5.2

Signature of preselection

In pre-selection, we evaluate the nonlinearity in Eq. (3.76), B̂ (ω) , over the incoming field’s vacuum state, |0iin :
in

0 B̂(ω) 0 in = 0.

Consequently, we can directly use Eq. (3.88) to establish that under the pre-selection
measurement prescription, the noise spectrum of b̂2 is S A,A. Taking an ensemble
average over the classical force fcl adds classical noise to the total spectrum:
(pre)

Sb2,b2 (ω) = S A,A (ω) +

α2 cl
S (ω) .
~2 x,x

(3.99)

Making use of Eq. (3.93), we obtain
+ SRQ (ω)
α4
SRQ (ω) ≡
G q (ω) +
2~
α2 G q (ω) qu
S fz p, fz p (ω) .
~2

S A,A (ω) =

(3.100)

(3.101)

The first term in S A,A, 1/2, is the shot noise background level, and SRQ (ω) is the noise
from quantum radiation pressure forces and quantum thermal forces. Moreover,
qu
S fz p, fz p (ω), given by Eq. (3.54), is the noise spectrum from vacuum fluctuations of
the quantum thermal force fˆzp (ω).
Around ωq , in the narrowband limit γm
ωq , the quantum back action noise
dominates and so
(pre)
Sb2,b2 (ω) ≈
+ βΓ ×
β(β + 2)
.
1 +
2 1/2 + βΓ
(ω − ωq ) 
1+
4γm2
As a result, the signature of classical gravity under the pre-selection prescription

57
can be summarized as a Lorentzian
h pre

S(ω) ∝ 1 +
1+4

(ω − ωq )2

(3.102)

∆2pre

with a height and a full width at half maximum (FWHM) given by
h pre =

β(β + 2)
,
2 1/2 + βΓ2

∆ pre = γm ,

(3.103)

respectively. We plot the pre-selection spectrum around ωq in Fig. 3.5.
Limits on the measurement strength
Our results are valid only if the Schroedinger Newton potential can be approximated
as a quadratic potential, which is necessary for linearizing the state-dependent
Heisenberg equations, as we described in Sec. 3.3.3.
Specifically, we must ensure that the spread of the center of mass wavefunction
excluding contributions from classical noise is significantly less than ∆xzp , which is
on the order of 10−11 − 10−12 m for most materials (as can be determined from the
discussion in section 3.2.1 and Ref. [20]). We calculate ∆xcm at steady state to be
h x̂ i − h x̂i = α

∫ +∞
−∞

G2q (ω)

1 S fz p (ω) dω
2π
α2

β+2 ~
2 2Mωq

(3.104)

where the expectation value is carried over vacuum of the input field, |0iin .
3.5.3

Signature of post-selection

In post-selection, we evaluate the nonlinearity in Eq. (3.76), B̂(ω) , over the
collection of eigenstates measured by the detector, |ξiout . To determine
B̂(ω) ξ ≡ out ξ B̂(ω) ξ out ,

(3.105)

we will make use of the fact that |ξiout is also an eigenstate of Â (ω) with an
eigenvalue we call
η (ω) = ξ(ω) − B̂(ω) ξ .
(3.106)

58
The equality follows from Eq. (3.76) with classical thermal noise ignored, which
we will incorporate at the end of the calculation. Notice that if we express B̂(ω) ξ
in terms of η (ω), we can also express it in terms of ξ (ω).
Our strategy will be to project B̂ (t) onto the space spanned by the operators Â (z)
for all times z:
B̂ (t) =

K (t − z) Â (z) dz + R̂ (t),

(3.107)

−∞

where R̂ (t) is the error operator in the projection. As a result,
B̂ (t) ξ =

∫ T
−∞

K (t − z) η (z) dz + R̂ (t) ξ ,

(3.108)

where we made use of the definition of η(t). In Appendix 3.10, we show that if we
choose K(t) in such a way that R̂(t) and Â(z) are uncorrelated for all times t and z,
in 0 R̂ (t) Â (z) 0 in + in

0 Â (z) R̂ (t) 0 in = 0

(3.109)

then R̂ (t) ξ = 0.
In the long measurement time limit, T
1, we make use of Eq. (3.107) to express
R̂(t) in terms of B̂(t) and Â(z) and then Fourier transform Eq. (3.109) to solve for
K(ω). We obtain
SB,A (ω)
(3.110)
K (ω) =
S A,A (ω)
Making use of Eq. (3.106), we express B̂ (ω) ξ in terms of ξ(ω),
b̂2 (ω) ξ = B̂ (ω) ξ =

ξ (ω)
1 + K (ω)

(3.111)

which we then substitute into Eq. (3.91) to establish that post-selection’s spectrum
(without classical thermal noise) is given by
|1 + K (ω)| 2 S A,A (ω) .
We finally add the contribution of classical thermal noise to b̂2 ’s spectrum, and
obtain
α2 cl
(post)
(ω) .
(3.112)
Sb2,b2 (ω) = |1 + K (ω)| 2 S A,A (ω) + 2 Sx,x

Classical
thermal
noise peak

Pre-selectionʹs
signature

Post-selectionʹs
signature

Figure 3.5: A depiction of the predicted signatures of semi-classical gravity. The
pre-selection measurement prescription’s signature is a narrow and tall Lorentzian
peak, while the post-selection measurement prescription’s signature is a shallow but
wide Lorentzian dip. Both prescriptions predict a Lorentzian peak of thermal noise
at ωcm . Note that the figure is not to scale and throughout this article, we follow the
convention of 2-sided spectra.

Around ωq , we apply a narrowband approximation on G q (ω) , and obtain
(post)
Sb2,b2

ω ≈ ωq

+ βΓ (1 + D (ω)) ,

where

(3.113)

β (β + 2) γm2

D (ω) ≡ −

1/2 + βΓ2

(β + 1)

γm2 + 4

ω − ωq

(post)

(ω), given by Eq. (3.94), we
is a Lorentzian. By comparing Sb2,b2 (ω) with Sb(QM)
2,b2
conclude that 1 + D (ω) is the signature of post-selection. We summarize it in the
following way:
dpost
(3.114)
1 + D(ω) = 1 −
(ω − ωq )2
1+4
∆2post
with the depth of the dip, and its FWHM given by
dpost =

β (β + 2)
2 1/2 + βΓ2 (β + 1)2

∆ post = (β + 1) γm ,

(3.115)

respectively. A summary of the post-selection spectrum around ωq is depicted in
Fig. 3.5.

60
3.6

Feasibility analysis

In this section, we determine the feasibility of testing the Schroedinger-Newton
theory with state of the art optomechanics setups. We will evaluate how long a
particular setup would need to run for before it can differentiate between the flat
noise background predicted by standard quantum mechanics around ωq :
ω ≈ ωq = 1/2 + βΓ2,
Sb(QM)
2,b2

(3.116)

and the signatures of the pre- and post- measurement prescriptions,

(pre)
Sb2,b2

ω ≈ ωq

Sb2,b2 ω ≈ ωq

(post)

with h pre and ∆ pre defined by Eq. (3.103), and dpost and ∆ post defined by Eq. (3.115).
Note that our analysis holds when the classical thermal noise peak is well resolved
from the SN signatures at ωq . Specifically, we require that ωq − ωcm be much larger
than γm . For torsion pendulums, this is not a difficult constraint, as ωSN is on the
order of 0.1 s−1 for many materials, as is shown in Table 3.1.
3.6.1

Likelihood ratio test

We will perform our statistical analysis with the likelihood ratio test. Specifically,
we will construct an estimator, Y , which expresses how likely the data collected
during an experiment for a period τ is explained by standard quantum mechanics or
the Schroedinger-Newton theory.
The estimator Y is given by the logarithm of the ratio of the likelihood functions
associated with each theory:
Y = ln

p (D|QM)
p (D|SN)

61
where p (D|QM) is the likelihood for measuring the data
D = {ξ(t) : 0 < t < τ}
conditioned on standard quantum mechanics being correct, and p (D|SN) is the
probability of measuring the data conditioned on the Schroedinger-Newton theory,
under the pre-selection or post-selection measurement prescription, being true. Note
that we will compare the predictions of standard quantum mechanics with the
Schroedinger Newton theory under each prescription separately. All likelihood
probabilities are normal distributions characterized by correlation functions which
are inverse Fourier transforms of the spectra presented at the beginning of this
section.
We can form a decision criterion based on Y . If Y exceeds a given threshold, yth ,
we conclude that gravity is not fundamentally classical. If Y is below the negative
of that threshold, we conclude that the data can be explained with the Schroedinger
Newton theory. Otherwise, no decision is made.
With this strategy, we can numerically estimate how long the experiment would
need to last for before a decision can be confidently made. We call this period τmin
and define it to be the shortest measurement time such that there exists a threshold
yth which produces probabilities of making an incorrect decision, and of not making
a decision that are both below a desired confidence level p.
3.6.2

Numerical simulations and results

We determined in the last section that the signatures of pre-selection and postselection are both Lorentzians. By appropriately processing the measurement data,
ξ(t), the task of ruling out or validating the Schroedinger Newton theory can be
reduced to determining whether fluctuations of data collected over a certain period
of time is consistent with a flat or a Lorentzian spectrum centered around 0 frequency:
Sh(d) (ω) = 1 +

h(−d)
1 + 4ω2 /γ 2

S (ω) = 1,

(3.117)

where γ is the full width at half maximum, Sh(d) corresponds to a Lorentzian peak
(dip) with height h (depth d) on top of white noise.
The data can be processed by filtering out irrelevant features except for the signatures

62
of post- and pre-selection around ωq , and then shifting the spectrum:
˜ ≡ e−iωq t
ξ(t)

∫ ωq +σ
ωq −σ

ξ (Ω) eiΩt dΩ,

(3.118)

where ξ(Ω) is the Fourier transform of ξ(t), and σ has to be larger than the signatures’
width but smaller than the separation between the classical thermal noise feature
at ωcm and the signatures at ωq . Two independent real quadratures can then be
constructed out of linear combinations of ξ(t):
˜ + ξ˜∗ (t)
ξ(t)
ξ˜c (t) ≡

˜ − ξ˜∗ (t)
ξ(t)
ξ˜s (t) ≡
2i

(3.119)

We will carry out an analysis of the measurement time with ξ˜c (t) in mind.
We numerically generated data whose fluctuations are described by white noise, or
lorentzians of different heights and depths. For example, in Fig. 3.6, we show the
distribution of Y for two sets of 105 simulations of ξ˜c (t) over a period of 200/γ (with
γ set to 1). In one set, ξ˜c (t) is chosen to have a spectrum of Sd with d = 0.62, and in
the second set, ξ˜c (t) has a spectrum of 1. The resultant distribution for both sets is a
generalized chi-squared distribution which seems approximately Gaussian. Fig. 3.6
is also an example of our likelihood ratio test: if the collected measurement data’s
estimator satisfies Y < −yth , for yth = 2, we decide that its noise power spectrum
is Sd , if Y > yth , white noise and if −yth ≤ Y ≤ yth , no decision is made. In
table 3.2, we show the associated probabilities of these different outcomes. Note
that the choice of yth is important, and would drastically vary the probabilities in
this table.
We then determined the shortest measurement time, τmin , needed to distinguish
between a lotentzian spectrum and white noise, such that the probability of making
a wrong decision and of not making a decision are both below a confidence level,
p, of 10%. Our analysis is shown in Fig. 3.7. Since ξ˜c (t) and ξ˜s (t) are independent,
we halved τmin , as an identical analysis to the one performed on ξ˜c (t) can also be
conducted on ξ˜s (t).
As shown in Fig. 3.7(a), numerical simulations of the minimum measurement time
needed to decide between white noise and a spectrum of the form Sh , are well fitted
by
27
τmin (h) ≈ 0.73 ×
(3.120)
γ/2
where 1/(γ/2) is the Lorentzian signature’s associated coherence time. The fit

63
Data has Sd spectrum
Data has S = 1
spectrum

P (correct)
78.7%

P (wrong)
1.1%

P (indecision)
20.2%

80.2%

2.1%

17.7%

Table 3.2: The probabilities of the different outcomes of the likelihood ratio test on
a particular measurement data stream with an estimator following either of the two
distributions shown in Fig. 3.6. The three possible outcomes are (1) deciding that
the data has a spectrum of Sd , (2) deciding that it has a white noise spectrum (S = 1)
or (3) making no decisions at all. P (correct) stands for the probability of deciding
(1) or (2) correctly, P (wrong) is the probability of making the wrong decision on
what spectrum explains the data, and P (indecision) is the probability of outcome 3.
Note that a different table would have been generated if a different threshold, yth ,
had been chosen in Fig. 3.6.

breaks down for heights less than about 10. However, as we show in the next
section, current experiments can easily access the regime of large peak heights.
In Fig. 3.7(b), we show that numerical simulations of the minimum measurement
time needed to decide between white noise and a spectrum of the form Sd , are well
fitted by
18.3 10.7
(3.121)
τmin (d) ≈
γ/2
This fit is accurate, except when d is close to 1. In the next section, we show that
this parameter regime is of no interest to us.
Moreover, we ran simulations for higher confidence levels p (in %). We show our
numerical results for pre-selection in Fig. 3.8. For h between 1000 and 4000, a
decrease in p from 10% to 1% results in a 4.5-5.5 fold increase in τmin . Our results
for post-selection are presented in Fig. 3.9. For d = 0.62 (which, as we show in
the next section, is the normalized depth level at which most low thermal noise
experiments will operate at), then τmin as a function of p is well summarized by
p 2
τmin (d = 0.62, p) ≈ 2.94 − 7.38 × erfc−1
100
γ/2
We can also fit τmin (d, p) at other values of d by a function of this form.
In the following sections, we present scaling laws for the minimum measurement
time, τmin , given a confidence level of 10%, in terms of the parameters of an
optomechanics experiment, and with the measurement strength β optimized over,
for both the pre-selection and post-selection measurement prescriptions.

Simulations of Y assuming a
spectrum of
Simulations of Y assuming a
white noise spectrum

Figure 3.6: A histogram showing the distribution of two sets of 105 realizations
of ξ˜c (t) over a period of 200/γ (with γ set to 1), and a time discretization of
dt = 0.14/γ. In one set, ξ˜c (t) is chosen to have a spectrum of Sd with d = 0.62,
and in the second set, ξ˜c (t) has a spectrum of 1. yth , which is chosen to be 2 in this
example, allows us to construct a decision criterion: if the collected measurement
data’s estimator satisfies Y < −yth , we decide that its noise power spectrum is Sd , if
Y > yth , white noise and if −yth ≤ Y ≤ yth , no decision is made.
3.6.3

Time required to resolve pre-selection’s signature

The normalized pre-selection signature’s height, h pre given by Eq. (3.103), is a
monotonically increasing function of β. Consequently, the larger β is, the easier
it would be to distinguish pre-selection from standard quantum mechanics. Using
Eq. (3.103) and the fit given in Fig. 3.7a of 13.5/h0.73 (in units of the Lorentzian
signature’s coherence time), τmin in the limit of large β scales as approximately
0.73
27 2Γ2
τmin ≈
γm β

(3.122)

It seems that arbitrarily increasing the measurement strength would yield arbitrarily
small measurement times. However, as explained in subsection 3.5.2, our results
hold for ∆xcm
∆xzp , which places a limit on β of

2∆xzp

~/ 2Mωq

where we made use of the expression for ∆xcm given by Eq. (3.104).

(a) Time required to distinguish a flat spectrum from a Lorentzian peak. The dashed line is
a fit of 13.5/h0.73 .

(b) Time required to distinguish a flat spectrum from a Lorentzian dip. The dashed line is
a fit of 18.3/d 2 − 10.7/d.

Figure 3.7: Simulation results showing the minimum measurement time, τmin ,
required to distinguish between a Lorentzian spectrum and a flat background in such
a way that the probabilities of indecision and of making an error are both below
10%. Plot (a) shows results for a Lorentzian peak, while plot (b) is for a Lorentzian
dip. The coherence time is given by the inverse of the half width at half maximum
of the Lorentzian. Note that both plots are log-log plots.

Figure 3.8: Simulation results showing the minimum measurement time, τmin , required to distinguish between the Schroedinger-Newton theory with the pre-selection
measurement prescription (which has the signature of a Lorentzian with depth h)
and standard quantum mechanics in such a way that the probabilities of indecision
and of making an error are both below p%. The coherence time is given by the
inverse of the half width at half maximum of the Lorentzian. Note that the y-axis is
on a log scale. Moreover, the dashed lines are only to guide the eye (and are fits of
the form a ln (p) + b).
Placing the limit on β at 1/10 the quoted value above, for h & 10, τmin scales with
the experimental parameters in the following way:
0.73
0.47
ωcm
T0
τmin ∼ 1.6 hours ×
300 K
2π × 10 mHz
0.49
0.73
184 amu
200 g
4 0.47
1.96
10
0.359 s−1
ωSN

(3.123)

where m is the mass of a constituent atom of the test mass, and we have assumed
that the test mass is made out of Tungsten.
Using the expressions for the measurement strength and for α2 , given by Eq. (3.97)
and Eq. (5.44), respectively, we determine that the input optical power needed to
reach the above quoted value of τmin is

Figure 3.9: Simulation results showing the minimum measurement time, τmin ,
required to distinguish between the Schroedinger-Newton theory with the postselection measurement prescription (which has the signature of a Lorentzian with
depth d) and standard quantum mechanics in such a way that the probabilities of
indecision and of making an error are both below p%. The coherence time is given
by the inverse of the half width at half maximum of the Lorentzian. Note that the
x-axis is scaled by the inverse of the complimentary error function, er f c−1 , and the
y-axis is on a log scale. Moreover, the dashed lines are to guide the eye and are fits
2
of the form a − b × er f c−1 (p/100) .

2/3 M 2
104
Iin ≈ 432 mW ×
184 amu
200 g

ωcm
ωSN 2/3
2π × 10 mHz
0.359 s−1

2
2π × 0.2 T Hz
ωc
10−2

(3.124)

We are allowed to make use of the fit presented in Fig. 3.7(a), of τmin = 27/h0.73 (in
units of the coherence time), which holds only for h & 10, because the pre-selection
signature’s normalized peak height can be easily made to satisfy this constraint.

68
Indeed, for the parameters given above
2
2/3 M
h ≈ 8235 ×
184 amu
200 g
104
2
ωSN 8/3
2π × 10 mHz
300 K
ωcm
T0
0.359 s−1

3.6.4

Time required to resolve post-selection’s signature

As indicated by Eq. (3.115), the depth and width of post-selection’s signature are
determined by 3 parameters: β, Γ2 and γm . For a given Γ2 , we can determine the
optimal measurement strength β that would minimize τmin . We numerically carried
out this analysis, and we show our results in Fig. 3.10. For Γ2 less than about 0.1, the
optimal choice of the measurement strength seems to follow a simple relationship:
βopt ≈

0.31
Γ2

with a corresponding measurement time, τmin , of about 200Γ2 /γm . Note that this is
a soft minimum, as large deviations from βopt still yield near optimal values of τmin .
Specifically, measurement strengths roughly between 0.1/Γ2 and 0.7/Γ2 achieve
measurement times below 225Γ2 /γm .
Moreover, in the parameter regime of Γ2 < 0.1, the normalized post-selection
dip depth at βopt is 0.62, which falls well in the region where the fit presented in
Fig. 3.7(b), of τmin = 18.3/d 2 − 10.7/d (in units of the coherence time), is accurate.
In the limit of ωSN
ωcm , the optimal measurement time scales as

107
T0
τmin ∼ 13 days ×
1K
0.488 s−1
ωcm
ωSN
2π × 4 mhz

(3.125)

where we assumed that the mechanical oscillator is made out of Osmium. Moreover,
the input optical power needed to reach the above quoted value of τmin is

2
2
1K
Iin ≈ 4.8 nW ×
T0
200 g
107

2π × 4 mHz
ωSN 4
ωcm
0.488 s−1

2
2π × 0.2 T Hz
ωc
10−2

(3.126)

69
Finally, we note that the experiment does not need to remain stable, or to operate, for
the entire duration of τmin . Since the coherence time of the post-selection signature,
(βopt + 1)γm
is much less than τmin (in the example above, the coherence time is 5 hours), the
experiment can be repeatedly run over a single coherence time. Alternatively,
numerous experiments can be run in parallel.
3.7

Conclusions

We proposed optomechanics experiments that would look for signatures of classical
gravity. This theory appreciably modifies the free unmonitored dynamics of the test
mass when the following two criteria are met. First, the choice of material for the
test mass is crucial. We recommend crystals with tightly bound heavy atoms around
their lattice sites. Tungsten and Osmium crystals meet this criterion. Second, we
recommend that the resonant frequency of the test mass be as small as possible.
Torsion pendulums meet this requirement.
When adding thermal noise and measurements to our analysis, we encountered two
conceptual difficulties. Both appear because the Schroedinger-Newton equation is
non-linear. The first difficulty is the breakdown of the density matrix formalism.
As a consequence, we had to propose a specific ensemble of pure states to describe
the quantum state of the thermal bath.
The second difficulty is generalizing Born’s rule to nonlinear quantum mechanics.
In section §3.4, we provided two prescriptions for calculating probabilities in the
Schroedinger-Newton theory. The first prescription, which we term pre-selection,
takes the probability of obtaining a particular measurement result to be the modulus
squared of the overlap between the forward-evolved initial state, which we choose
as a boundary state for the non-linear time evolution operator, and the eigenstate
corresponding to that measurement result. The second prescription, which we term
post-selection, takes the probability of obtaining a particular measurement result to
be the modulus squared of the overlap between the backwards-evolved measured
eigenstate, which we choose as a boundary state for the non-linear evolution operator,
and the initial state. Note that the predictions of both pre-selection and postselection are consistent with that of linear quantum mechanics in the limit that the
Schroedinger-Newton nonlinearity vanishes (i.e. ωSN → 0).
We then proceeded to obtain the signatures of classical gravity predicted by both

(a) τmin for different values of Γ2 , β and γm

(b) τmin for fixed values of Γ2

Figure 3.10: Minimum measurement time required to distinguish between the
Schroedinger-Newton theory with the post-selection measurement prescription and
standard quantum mechanics in such a way that the probabilities of indecision and
of making an error are both below 10%. Note that we interpolated the data given in
Fig. 3.7 to create this figure.

71
these prescriptions in the spectrum of phase fluctuations of the outgoing light. Both
signatures are Lorentzians centered around the frequency ωq . The pre-selection
prescription predicts a peak, while post-selection predicts a dip. We summarize
these features in figure 3.5, which is valid when the resonant frequency of the
mechanical oscillator, ωcm , is much smaller than ωSN .
Finally, in the limit of the classical thermal noise peak being well separated from the
SN signatures, we numerically simulated the experiment’s expected measurement
results and determined that pre-selection is easily testable with current optomechanics technology. However, testing post-selection will be much more challenging,
although is feasible with state-of-the-art experimental parameters. In particular,
we require cryogenic temperatures and a high Q low frequency torsion pendulum
made out of a material with a high ωSN . Eq. (3.125) contains the scaling of the
minimum measurement time required to confidently test post-selection with these
experimental parameters.
Acknowledgments
We thank K. Thorne, J. Preskill, P.C.E. Stamp, H. Miao, Y. Ma, C. Savage, and
H. Yang for discussions. Research of Y.C. and H.L. are supported by NSF grants
PHY-1404569 and PHY-1506453, as well as the Institute for Quantum Information
and Matter, a Physics Frontier Center.

72
3.8

Appendix: Conservation of energy in the SN theory

Consider the SN equation for a collection of N particles of mass m:

V̂SN = −Gm2

N ∫
i j=1

dx j

pj xj

x̂i − x j

(3.127)

where p j x j is the probability distribution for the jth particle to be at location x j :
pj xj =

dyi δ y j − x j |Ψ (y1, y2, ..., yN )| 2 .

(3.128)

i=1

Ψ is the many-body wavefunction for these N particles.
Let us investigate conservation of energy within the SN theory. In standard quantum mechanics, the energy operator is given by the Hamiltonian. Our non-linear
Hamiltonian is
P̂i2
+ V̂NG ( x̂1, ..., x̂N ) + V̂SN ,
(3.129)
Ĥ =
2m
i=1
where V̂NG ( x̂1, ..., x̂N ) encodes the non-gravitational potential energy. Under the
non-linear SN theory, Ĥ is not conserved because of V̂SN ’s dependence on the
wavefunction:
d Ĥ ∂ Ĥ
= ∂t V̂SN , 0
(3.130)
dt
∂t
Is there a quantity that is conserved? Consider
Ê =

P̂2

2m
i=1

+ V̂NG ( x̂1, ..., x̂N ) + βV̂SN ,

(3.131)

where β is to be determined such that d Ê /dt = 0. We will show that β = 1/2
meets this condition.
We begin the proof with the Heisenberg equation of motion for Ê. By expressing Ê
as Ĥ − (1 − β) V̂SN , we obtain
d Ê
dt

∂ Ê
i
Ĥ, Ê +
(3.132)
∂t
N ∫
pÛ j x j
i (1 − β) Gm2 Õ Õ
2 p k (x k )
− βGm
dx j
dx k P̂i ,
2~m
x̂
x̂
i j=1
jk

Taking the expectation value of both sides, and evaluating the commutator in the

73
first term, we obtain
d Ê
dt

− (1 − β) Gm Õ
p k (x k )
p k (x k )
dx k P̂i
P̂i
| x̂i − x k | 2 | x̂i − x k | 2
ik
N ∫
(x
dxi
dx j
−βGm2
xi − x j
i j=1

We then evaluate the expectation value in the first term. Defining the vector x ≡
(x1, ..., xN ), we have

p k (y k )

| x̂i − y k | 2

Next, we integrate by parts multiple times, and use that
p k (y k )
|xi − y k |

= −∂xi

p k (y k )
|xi − y k |

(3.133)

to obtain

p k (y k )
p k (y k )
P̂i
P̂i
| x̂i − y k | 2 | x̂i − y k | 2
p k (y k )
dx
∂xi Ψ (x)∗ ∂xi Ψ (x) − Ψ (x) ∂xi Ψ (x)∗ .
|xi − y k |
This result can be connected to the continuity equation (which is satisfied by the SN
theory):
∂t ρ + ∇. ®j = 0,
(3.134)
where
®j = ~ (Ψ∗ ∇Ψ − Ψ∇Ψ∗ ) .
(3.135)
2im
We integrate over all variables except xi (which we denote by x,i ), obtaining
dx,i ∂t ρ + ∇. j = 0
= ∂t pi (xi ) +
dx,i
∂x j Ψ∗ ∂x j Ψ − Ψ∂x j Ψ∗ .
2im
ρ = |Ψ| 2 ;

74
For j , i,

dx j ∂x j Ψ∗ ∂x j Ψ − Ψ∂x j Ψ∗ = 0

(3.136)

by integration by parts. Thus,
∂t pi (xi ) = −
2im

dx,i ∂xi Ψ∗ ∂xi Ψ − Ψ∂xi Ψ∗

(3.137)

= (1 − β) Gm2

Õ∫
ji

dxi

dx j

pi (xi ) pÛ j x j
x j − xi

− βGm2

N ∫

dxi

dx j

i j=1

which is equal to 0 when
1−β = β

(3.138)

or β = 1/2.
3.9

Appendix: Derivation of p0 → ξ and p0 ← ξ

In this Appendix, we derive equations (3.88) and (3.91) presented in subsection
3.4.3:
dΩ |ξ(Ω) − hb̂2 (Ω)i0 | 2
p0→ξ ∝ exp −
(3.139)
2π
S A,A
dΩ |ξ(Ω) − hb̂2 (Ω)iξ | 2
p0←ξ ∝ exp −
(3.140)
2π
S A,A
They represent the probabilities of obtaining a particular measurement record
{ξ (t) : 0 < t < τ}

(3.141)

over a period τ in the pre- and post-selection measurement prescriptions, respectively.

pi (xi ) pÛ j x j
xi − x j

75
The probability of measuring ξ(t) is

pξ = out ξ Û 0 in ,

(3.142)

where Û is a shorthand for the pre-selection time evolution operator Û|0iin or the
post selection evolution operator Û|ξi out , |0iin is a vacuum state for the incoming
light, and |ξiout is the state of the outgoing light corresponding to the measurement
results ξ(t). We then rewrite pξ to
pξ = 0|Û † |ξ

ξ |Û|0 ,

(3.143)

where we have used the shorthand |ξi for |ξi out . Û † |ξi hξ | Û is a projection operator
that can be written as a path integral (refer to p.2 of [15] for a derivation):
P̂ =

∫

D k (t) exp i dtk (t) b̂2 (t) − ξ (t) .

(3.144)

Notice that in the limit that the SN non-linearity vanishes, P̂ agrees with the standard
quantum mechanics projector onto the measurement results ξ(t). This is due to the
fact that when ωSN vanishes, b̂2 becomes a linear operator which matches the
prediction of standard quantum mechanics. Consequently, in the limit of ωSN → 0,
p0→ξ and p0←ξ recover the probabilities predicted by linear quantum mechanics.
Substituting P̂ back into equation (3.142), we obtain
pξ =

∫
∫

D k (t) 0 exp i dtk (t) b̂2 (t) 0 exp −i dtk (t) ξ (t) .

(3.145)

Let us explicitly separate the mean of b̂2 (t) by defining Â in the following way:
b̂2 (t) ≡ Â (t) + 0 b̂2 (t) 0 ≡ Â (t) + b̂2 (t) .
We can then rewrite pξ to
pξ =

∫

∫

D k (t) 0 exp i dtk (t) Â (t) 0 exp −i dtk (t) ξ (t) − b̂2 (t)

Next, we make use of the fact that |0i is a gaussian state to rewrite the above

76
expectation value as
* ∫
2 +!
∫

pξ =
D k (t) exp − 0
dtk (t) Â (t) 0 exp −i dtk (t) ξ (t) − b̂2 (t)
(3.146)
Expanding the first exponent, we obtain

pξ =

D k (t) exp −

dzk (t) k (z) Â (t) Â (z)

exp −i

dtk (t) ξ (t) − b̂2 (t)
(3.147)

pξ is a functional Gaussian integral over k (t), which we evaluate to

pξ ∝ exp −

dz ξ (t) − b̂2 (t)

Â (t) Â (z)

−1

ξ (z) − b̂2 (z)

(3.148)
where Â (t) Â (z)
is the inverse of the function Â (t) Â (z) . Assuming we have
a time-stationary process, Â (t) Â (z) can be simplified to Â (t − z) Â (0) which
allows us to take a Fourier transform and obtain
−1

pξ ∝ exp −

dω ξ (ω) − b̂2 (ω)
2π
S A,A (ω)

(3.149)

Finally, we note that for post-selection 0| b̂2 (t) |0 is calculated with b̂2 (t) obtained
from an effective Heisenberg picture with the boundary state fixed to be the recorded
eigenstates by the measurement device: |ξi. For pre-selection, we obtain b̂2 (t) from
an effective Heisenberg picture with the boundary state given to be the initial state
of the light, vacuum.
3.10

Appendix: More details on calculating B̂(ω) ξ

In subsection 3.5.3, we calculated the spectrum of the outgoing light phase operator
b̂2 (ω) = Â (ω) + B̂ (ω) ξ ,

(3.150)

where we have neglected the contribution from classical thermal noise, as it is not
important for this Appendix. Both Â and B̂ are linear operators of the form
∫ ∞

Â (t) = â2 (t) +
L A (t − z) â1 (z) dz
−∞
∫ ∞
B̂ (t) =
LB (t − z) â1 (z) dz.
−∞

(3.151)
(3.152)

77
We presented their exact expressions in Eqs. (3.77) and (3.79). Moreover, B̂ (ω) ξ
is the expectation value of B̂ over the outgoing light state |ξiout corresponding to
the measured eigenstates of the outgoing light’s phase. In the calculation of the
spectrum, and in particular of B̂ (ω) ξ , we stated without proof that if Eq. (3.109)
0 Â (z) R̂ (t) 0 in = 0

in 0 R̂ (t) Â (z) 0 in +in

(3.153)

is satisfied then
out

ξ R̂(t) ξ out ≡ R̂(t) ξ = 0

for all times t. R̂ is defined by Eq. (3.107). In this Appendix, we present the proof.
We first rewrite |ξi out to
|ξi out = P̂ |0i ,
where

P̂ ∝

D kei

(3.154)

dt k(t)( Â(t)−η(t))

(3.155)

projects the initial state of the light, vacuum |0i, onto |ξi out . This form of P̂ can be
derived by referring to p.2 of [15] and by making use of the fact that since B̂ ξ is a
c-number, a measured eigenstate of b̂2 (t), |ξ (t)i, is also an eigenstate of Â (t) with
a different eigenvalue which we choose to call η (t).
Substituting Eq. (3.154) into R̂ (t) ξ , we obtain
R̂ (t) ξ = 0| P̂ R̂ (t) P̂|0 = −i∂µ 0| P̂ei µR̂ P̂|0

µ=0

(3.156)

Let us combine P and ei µR̂ into one exponential by repeated use of the Baker–Campbell–Hausdorff
formula. We begin with P̂ei µR̂ ,
i µ R̂

P̂e

D ke

dt k(t)( Â(t)−η(t))+i µ R̂

exp −

dzk (z) Â (z) , R̂ (t)

To evaluate the commutator, we make use of Eq. (3.107)
R̂ (t) = B̂ (t) −

∫ T
K (t − z) Â (z) dz.
−∞

(3.157)

78
Furthermore, since A(t) and B(t) are linear operators

Â (z) , B̂ (t)

∫ ∞

LB (t − z) [â2 (t) , â1 (z)] dz

(3.158)

−∞

= −i

∫ ∞

LB (t − z) δ (t − z) dz = −iLB (0) .

(3.159)

−∞

Substituting this result back into P̂ei µR̂ , we obtain
i µ R̂

P̂e

D ke

dt k(t)( Â(t)−η(t))+i µ R̂

i µLB (0)
exp −

dzk (z) .(3.160)

Returning to P̂ei µR̂ P̂, we have
i µ R̂

P̂e

P̂ =

D ke

dt k(t)( Â(t)−η(t))+i µ R̂ i

dzl(z)( Â(z)−η(z))

i µLB (0)
exp −

dzk (z)

Dl
D kei dzk(z)( Â(z)−η(z))+i dzl(z)( Â(z)−η(z))+i µR̂
∫
i µLB (0)
dzk (z) exp
dzl (z) Â (z) , R̂ (t)
× exp −
i µLB (0)
i dzk + (z)( Â(z)−η(z))+i µ R̂
dzk− (z)
D k+ e
D k − exp −

where we applied the Baker-Campbell-Hausdorff formula in the second line, and in
the third line, we defined k + = k (z) + l (z), and k − = k (z) − l (z).
Now,

i µLB (0)
D k − exp −

dzk− (z)

lim δ

n→∞

Ö
µLB (0)
µLB (0)
(3.161)

so
i µ R̂

∂µ P̂e

Ö µLB (0)
P̂ = ∂µ D k + e
µLB (0) n−1 µLB (0)
i dzk + (z)( Â(z)−η(z))+i µ R̂
+ lim n D k + e
×δ
n→∞

dzk + (z)( Â(z)−η(z))+i µ R̂

When µ is set to 0, the second term will vanish because δ (µLB (0) /2) vanishes
at µ = 0 (as can be easily determined by writing the dirac-delta function as a zero
mean Gaussian with a vanishing variance). Consequently, we only need to study the
first term.

79
Let take the expectation of ∂µ P̂ei µR̂ P̂ over vacuum,

i µ R̂

∂µ 0 P̂e

P̂ 0

= ∂µ

D k+ 0 e

dzk+ (z) Â(z)+i µ R̂

0 e

−i

dzk + (z)η(z)

Ö µLB (0)

We now analyze the first term in the integrand. Since |0i is a Gaussian state, the
expectation over |0i can be simplified to

dzk + (z) Â(z)+i µ R̂

1 2 2
= exp − µ R

dzk+ (z) Â (z) × R̂
× exp − µ R̂ ×
dzk+ (z) Â (z) +

2
1
× exp −
dzk+ (z) Â (z)

The second exponential is equal to unity by the assumption given by Eq. (3.109).
Thus,

2
1 2 2
1
i dzk+ (z) Â(z)+i µ R̂
0e
0 = exp − µ R
dzk+ (z) Â (z)
exp −
Once we differentiate over µ and then set it to 0, this product vanishes, giving
0| R̂|0 = ∂µ 0 P̂ei µR̂ P̂ 0 = 0

as desired.

(3.162)

80
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82
Chapter 4

DIFFERENT INTERPRETATIONS OF QUANTUM MECHANICS
MAKE DIFFERENT PREDICTIONS IN NON-LINEAR
QUANTUM MECHANICS, AND SOME DO NOT VIOLATE THE
NO-SIGNALING CONDITION
B. Helou, and Y. Chen, 2017, arXiv:1709.06639
Abstract
Nonlinear modifications of quantum mechanics have a troubled history. They were
initially studied for many promising reasons: resolving the measurement problem,
testing the limits of standard quantum mechanics, and reconciling it with gravity.
Two results substantially undermined the credibility of non-linear theories. Some
have been experimentally refuted, and more importantly, all deterministic non-linear
theories can be used for superluminal communication. However, these results are
unconvincing because they overlook the fact that the distribution of measurement
results predicted by non-linear quantum mechanics depends on the interpretation of
quantum mechanics that one uses. For instance, although the Everett and Copenhagen interpretations agree on the expression of Born’s rule for the outcomes of
multiple measurements in linear quantum mechanics, they disagree in non-linear
quantum mechanics. We present the range of expressions of Born’s rule that can
be obtained by applying different formulations of quantum mechanics to a class of
non-linear quantum theories. We then determine that many do not allow for superluminal communication but only two seem to have a reasonable justification. The first
is the Everett interpretation, and the second, which we name causal-conditional,
states that a measurement broadcasts its outcome to degrees of freedom in its future
light-cone, who update the wavefunction that their non-linear Hamiltonian depends
on according to this new information.

4.1

Introduction

Non-linear quantum mechanics (NLQM) has long been considered as a possible
generalization of standard quantum mechanics (sQM) [1, 2, 3], for three main reasons. First, the measurement process is controversial. If we assume that linear
quantum mechanics explains all processes, then it is very difficult to explain wavefunction collapse [4]. Phenomenological non-linear, stochastic, and experimentally
falsifiable extensions of quantum mechanics (QM) have been proposed to explain
the measurement process [5], and upper bounds on the parameters of such theo-

83
ries have been obtained in [5, 6, 7]. Second, we would like to test the domain
of validity of sQM. One possible feature to test is linearity. Experimental tests
of certain non-linear theories have been performed in [8, 9, 10, 11, 12], and all
have returned negative results. Third, non-linear and deterministic theories of QM
have been proposed to combine quantum mechanics with gravity. For instance,
the Schroedinger-Newton theory describes a classical spacetime which is sourced
by quantum matter [13], and the correlated worldlines theory is a quantum theory
of gravity which postulates that gravity correlates quantum trajectories in the path
integral [14].
NLQM became a much less credible theory after 1990 because Gisin showed that
deterministic NLQM could allow for superluminal communication [15]. The nosignaling condition states that one cannot send information faster than the speed
of light, and is a cornerstone of the special theory of relativity. The community
regards the condition as being inviolable. Gisin’s work was quickly followed by
others with similar conclusions [16, 17]. Additional work then showed that under
general conditions NLQM allows for superluminal communication [18, 19].
In [6], we showed that NLQM suffers from another serious conceptual issue: Born’s
rule cannot be uniquely extended from sQM to NLQM. Born’s rule provides a
prescription for predicting the distribution of measurement results in a particular
experiment, and has, so far, passed all experimental tests. Any non-linear theory
must make predictions that become equivalent to Born’s rule in sQM when the
non-linearity vanishes.
As in sQM, measurements in NLQM pose significant conceptual difficulties. Fortunately, in sQM, these difficulties do not result in practical challenges. Whether the
experimentalist is an Everettian or a proponent of wavefunction collapse does not
matter, as in both cases they can safely use Born’s rule to predict the distribution of
measurement results. This is no longer true in NLQM. Different interpretations of
quantum mechanics result in different predictions for the outcome of an experiment,
and so result in different expressions for Born’s rule.
In this article, we search for interpretations of quantum mechanics that do not violate
the no-signaling condition when are applied to NLQM. Since there could be interpretations that haven’t been discovered, our approach won’t be to extend all known
interpretations to NLQM. Instead, we will extend the mathematical expression of
Born’s rule in a general way to NLQM, without regards to interpretation. After
finding causal prescriptions, we speculate about their interpretation.
Note that our analysis doesn’t cover all possible non-linear theories. We only wish

84
to show that non-linear quantum mechanics does not necessarily violate the nosignaling condition. More importantly, how to write down a general non-linear
modification of quantum mechanics is still an open question. For example, the class
of non-linear theories proposed by Weinberg in [3] doesn’t include P.C.E. Stamp’s
proposal in [14]. We also do not, a priori, place any physical constraints on the
class of non-linear theories we study. We only place one mathematical constraint:
a single Dirichlet boundary condition is enough to completely specify a solution.
This article is outlined as follows. By introducing the formalism for multiple measurements in sQM, we show that linearity prevents two parties from communicating
with each other faster than the speed of light. We then motivate the dependence of
NLQM on the formulation of quantum mechanics by providing a simple example
involving a single measurement. By extending the notion of a time-evolution operator to NLQM, we generate extensions of Born’s rule in the context of multiple
measurements. Afterwards, we discuss what well-known formulations of quantum
mechanics, such as the Everett interpretation, predict for the distribution of measurement results in NLQM. We then present all possible prescriptions that do not violate
the no-signaling condition. Finally, we propose and discuss a sensible prescription,
which we name causal-conditional, that doesn’t violate the no-signaling condition.
It states that a measurement broadcasts its outcome to degrees of freedom in its
future light-cone, who update the wavefunction that their non-linear Hamiltonian
depends on according to this new information.
4.2

Multiple measurements in sQM and the no-signaling condition

In Fig. 4.1, we show the setup that is typically used to show that NLQM violates
the no-signaling condition. Charlie prepares a collection of identical arbitrary 2particle states |i, and then sends them to Alice and Bob, such that they each hold
one part of each of the states |Ψini i. Alice performs measurements on her ensemble
of particles at time t1 , and then Bob on his at a later time t2 . We assume that Alice’s
measurements are space-like separated from Bob’s, and so their particles do not
interact from t1 till t2 .
4.2.1

Born’s rule for multiple measurements

Denote the probability that Alice measures α in some basis Aa , and Bob measures
β in some basis Bb by p (α, β| Aa, Bb ). For example, if the particles were spins, Aa
could be the σ̂z eigenstates, |↑i , |↓i, and Bb the σ̂x eigenstates |±i = (|↑i ± |↓i) / 2.
We will first determine p (α, β| Aa, Bb ) according to sQM, and then discuss the
different ways of generalizing it to NLQM in the next section.

Figure 4.1: A spacetime diagram showing multiple measurement events. Event
C describes the preparation of an ensemble of identical 2-particle states |Ψini i by
Charlie. Event A (B) describes Alice (Bob) measuring her (his) particles. The
dashed lines show the light cone centered around each event.
In sQM, p (α, β| Aa, Bb ) is given by
p (α, β| Aa, Bb ) = hα, β|α, βi ,

(4.1)

where |α, βi is the unnormalized joint quantum state of Alice and Bob at t2 , conditioned on the measurement results α and β:
Ψc|α,β = IˆA ⊗ P̂β Û (t2, t1 ) P̂α ⊗ IˆB Û (t1, t0 ) |Ψini i ,

(4.2)

where IˆA ( IˆB ) is the identity operator acting on Alice’s (Bob’s) particle, Û(t, z) is
the total time evolution operator for both Alice and Bob’s particles from times z till
t. The projectors are P̂α = |αi hα| and P̂β = | βi hβ|. To simplify the formalism,
we only work with pure states. Our analysis is general because |Ψini i can always be
enlarged to include the initial state of the environment.

86
Alice and Bob’s measurement events are spacelike separated, so from t1 till t2 , the
interaction Hamiltonian between Alice’s and Bob’s particle is zero, and Û(t2, t1 ) is
separable
Û(t2, t1 ) ≡ Â(t2, t1 ) B̂(t2, t1 ).
(4.3)
Â acts on Alice’s particle’s Hilbert space, and B̂ on Bob’s. We can then rewrite Eq.
(4.2) to
Ψc|α,β = Iˆ ⊗ P̂β Â(t2, t1 ) B̂(t2, t1 ) P̂α ⊗ Iˆ Û (t1, t0 ) |Ψini i ,

(4.4)

which we substitute into Eq. (4.1)
p (α, β| Aa, Bb ) = Û † (t1, t0 )P̂α B̂† (t2, t1 )P̂β B̂(t2, t1 )P̂α Û(t1, t0 ) ,

(4.5)

where we’ve used that Â(t2, t1 ) commutes with P̂β . Since Alice and Bob’s measurement events are spacelike separated, P̂α and P̂β commute,

P̂α, P̂β = 0,

(4.6)

which we use to simplify Eq. (4.5) to
p (α, β| Aa, Bb ) = Û † (t1, t0 ) B̂† (t2, t1 )P̂β B̂(t2, t1 )P̂α Û(t1, t0 ) .
4.2.2

(4.7)

The no-signaling condition

Superluminal communication from Alice to Bob is possible when
p (β| Aa, Bb ) =

p (α, β| Aa, Bb )

(4.8)

is influenced by Alice’s choice of a measurement basis in a deterministic way. Since
Bob can easily estimate p (β| Aa, Bb ), he can determine the basis Alice chose for her
measurement results, which can form the foundation of a communication strategy.
For instance, both Alice and Bob can agree that a particular choice of Alice’s
measurement basis could be associated with sending the bit 0, while another choice
could be associated with the bit 1.
In sQM, superluminal communication can never occur because, using Eq. (4.7),
p (β| Aa, Bb ) =

Û † (t1, t0 ) B̂† (t2, t1 )P̂β B̂(t2, t1 )

P̂α Û(t1, t0 )

(4.9)

Û (t1, t0 ) B̂ (t2, t1 )P̂β B̂(t2, t1 )Û(t1, t0 ) ,

(4.10)

87
is clearly independent of Aa . We’ve shown that p (β| Aa, Bb ) = p (β| Bb ), so sQM
doesn’t violate the no-signaling condition.
4.3

Ambiguity of Born’s rule in NLQM

In this section, we discuss the ambiguity of Born’s rule in NLQM. We first present an
example with one measurement, and then present a general formalism for generating
prescriptions for calculating the distribution of the outcomes of an arbitrary number
of measurements at arbitrary spacetime points.
4.3.1

A simple example

For experiments with a single measurement, sQM states that the probability of
measuring an outcome f is
pi→ f =

f Û i

(4.11)

where | f i is the pointer state associated with the outcome f . The expression (4.11) is
usually interpreted in the following way. A preparation device prepares the system in
|ii, which evolves for some period of time under the time-evolution operator Û. The
system then interacts with the measurement device. What happens next depends
on one’s interpretation of quantum-mechanics. An Everettian would state that
decoherence splits the wavefunction into numerous branches that are approximately
classical. On the other hand, a proponent of objective collapse would state that,
due to stochastic modifications of the Schroedinger equation that become important
when a microscopic system interacts with a macroscopic one, the wavefunction
collapses onto | f i with probability pi→ f . As in Ref. [6], we will refer to the
formulation of Born’s rule based on pi→ f as pre-selection.
Eq. (4.11) admits even more interpretations. For instance, it can be rewritten as
pi← f = i Û † f

(4.12)

which can be interpreted as | f i evolves backwards in time to Û † | f i and is then
projected by the preparation device to the state |ii. We will refer to the formulation
of Born’s rule based on pi← f as post-selection.
In NLQM, the time evolution operator depends on the state it acts on. As a result,
Eqs. (4.11) and (4.12) become
NL
pi→
f = |h f |Ui ii| ,

E2
NL
pi←
i|U

(4.13)

88
where, under some non-linear dynamics, |Ui ii is the time-evolved |ii and U †f f
is the backwards time-evolved | f i. The superscript NL explicitly indicates that
N L follows from
NLQM is being used. Moreover, the proportionality sign in pi←
N L are not
being not, in general, normalized to unity. pi→ f and pi←
f i|U f f
necessarily equal, and so Born’s rule cannot be uniquely extended to NLQM.
4.3.2

Ambiguity in the boundary condition driving the non-linear time evolution

By extending Eq. (4.4) to NLQM, we can extend Born’s rule, given by Eq. (4.1), to
NLQM. However, because NLQM is non-linear, a time-evolution operator doesn’t
exist. Nonetheless, inspired by the state-dependent Heisenberg picture introduced in
[6], we will show that we can define a boundary-dependent time-evolution operator,
and that the choice of a boundary condition is the essential degree of freedom for
extending Born’s rule to NLQM.
For some theories in NLQM1, running time-evolution requires solving the non-linear
Schroedinger equation which contains a linear term, ĤL , and a nonlinear term V̂N L :
i~∂t |ψi = ĤL + V̂N L (ψ(x, t)) |ψi .

(4.14)

V̂N L (ψ(x, t)) is a shorthand for a non-linear potential that depends on the wavefunction |ψ (t)i expressed in some (possibly multi-dimensional) basis x. For instance,
the Schroedinger-Newton equation for a single non-relativistic particle of mass m
interacting with its own gravitational field is given by

i~∂t |ψi = ĤL − m G

|ψ (x, t)| 2
|ψi ,
d x
| x̂ − x|

(4.15)

where ψ (x, t) is the state |ψi expressed in the position basis |xi.
We chose to write the nonlinear Schroedinger equation in the form of Eq. (4.14)
to illustrate its similarity to the standard Schroedinger equation. Once we specify
the boundary conditions, we can have Eq. (4.14) be formally equivalent to a linear
Schroedinger equation. We will assume that a single Dirichlet boundary condition
is sufficient to solve Eq. (4.14), and that its solution with the boundary condition
1 As mentioned in the introduction, we will not investigate all possible non-linear theories.

In
particular we only consider theories with the form of Eq. (4.14), and whose solution can be uniquely
specified with one Dirichlet boundary condition.

89
|ψ(T)i = |φi is |ϕ(t)i. Consequently, the linear Schroedinger equation
i~∂t |ψi = ĤL + V̂N L (ϕ(x, t)) |ψi

(4.16)

is formally identical to Eq. (4.14) with the boundary-condition |ψ(T)i = |φi.
Heuristically, in the context of Eq. (4.16), we can interpret ϕ(x, t) as a timedependent classical drive. Eq. (4.16) has a time-evolution operator associated with
it, which we denote by Ûφ(T) . The subscript is to emphasize that the time-evolution
operator is associated with the boundary condition |ψ(T)i = |φi. We can now write
the solution to Eq. (4.14) as
|ψ (t)i = Ûφ(T) (t, T) |φi .

(4.17)

We are ready to present the extension of Eqs. (4.1) and (4.4) to NLQM:
pN L (α, β| Aa, Bb ) =

hα, β|α, βi ,

(4.18)

where if Alice and Bob’s measurement events are spacelike separated
Ψc|α,β = P̂β Âφ A(T A) (t2, t1 ) B̂φB (T B ) (t2, t1 ) P̂α Ûφ1 (T1 ) (t1, t0 ) |Ψini i ,

(4.19)

and N = α,β hΨc | α, β | Ψc | α, βi ensures that α,β pN L (α, β| Aa, Bb ) is normalized to unity. Moreover, Ûφ1 (T1 ) (t1, t0 ) is the time-evolution operator from t0 till t1
and is associated with the boundary |ψ(T1 )i = |φ1 i. Âφ A(T2 ) (t2, t1 ) ( B̂φB (T2 ) (t2, t1 )) is
the time-evolution operator associated with the boundary condition |ψ(T2A)i = |φ2Ai
(|ψ(T2B )i = |φ2B i), and acts on Alice’s (Bob’s) particle from t1 till t2 . The timeevolution of Alice and Bob’s joint system from t1 till t2 is separable because Alice’s
particle’s future light-cone at t1 does not overlap with Bob’s particle’s past light-cone
at t2 . As a result, their total interaction Hamiltonian, which includes contributions
from the linear Hamiltonian ĤL and from the non-linear potential V̂N L , must be zero.
Note that, by construction, Eq. (4.19) recovers the predictions of sQM when the
non-linearity V̂N L vanishes.
If the events A and B were time-like separated, then there are numerous schemes
for extending the time evolution of Alice and Bob’s particles to NLQM. Call the
solutions to Eq. (4.14) with the boundary conditions |ψ(T2A)i = |φ2Ai, |ψ(T2B )i =
|φ2B i and |ψ(T2AB )i = |φ2AB i as |ϕ A(t)i, |ϕ B (t)i and |ϕ AB (t)i respectively. We can

90
then write the non-linear potential V̂N L in Eq. (4.16) in the following general way:
V̂N L = V̂A ϕ A (x, t) ⊗ IˆB + IˆA ⊗ V̂B ϕ B (x, t) + V̂int ϕ AB (x, t) ,

(4.20)

where V̂A (V̂B ) is the free non-linear Hamiltonian acting on Alice’s (Bob’s) particle,
and V̂int is the non-linear interaction potential. However, we find it difficult to justify
why each term in V̂N L would be generated by a different boundary condition when
Alice and Bob’s particles are allowed to directly communicate and interact. We will
impose φ2A = φ2B = φ2AB and T2A = T2B = T2AB when the measurement events A and
B are timelike separated. We summarize our chosen form of Ψc|α,β by
Ψc|α,β =

 P̂β Ûφ2 (T2 ) (t2, t1 ) P̂α Ûφ1 (T1 ) (t1, t0 ) |Ψini i

A & B timelike,

 P̂β Âφ A(T A) (t2, t1 ) B̂φB (T B ) (t2, t1 ) P̂α Ûφ1 (T1 ) (t1, t0 ) |Ψini i

A & B spacelike.

The introduction of arbitrary boundary conditions |φ1 i at T1 and |φ2 i at T2 might appear artificial, but isn’t. Each formulation of quantum mechanics predicts different
boundary conditions after a measurement. For instance, in Eq. (4.14), an interpretation of quantum mechanics with wavefunction collapse states that |φ1 i = |αi
and T1 = t1 , while the Everett interpretation states that |φ1 i is the initial state of the
universe and T1 is when the universe began. Refer to section 4.1 for more details.
In sQM, we do not have to worry if and how the wavefunction collapses because the
time-evolution operator is well-defined independently of the wavefunction it acts on.
However, in NLQM, each boundary condition generates a different time-evolution
operator, and so how we formulate quantum mechanics matters in NLQM.
4.3.3

Extending the formalism to relativistic quantum mechanics

We can rigorously study superluminal communication only in quantum field theory,
where the total Hamiltonian consists of free and interaction (between different fields)
energy densities
Ĥ =

d 3 xĤ0 (x) +

d 3 xĤint (x) .

(4.21)

Assigning spatial locations for quantum degrees of freedom is crucial for placing
constraints on Ĥ to ensure that it is causal. Let Ĥint be the interaction energy density
in an interaction picture with respect to d 3 xĤ0 (x), and then Ĥint commutes over
spacelike distances [20]

Ĥint (t x, x) , Ĥint t y, y

= 0,

c tx − ty

− |x − y| 2 < 0.

(4.22)

91
We generalize Ĥ to include a dependence on a wavefunction:
NL
Ĥ (t) =
d xĤ0 x, ΨΦ(T) (t) +
d 3 xĤint x, ΨΦ(T) (t) ,

(4.23)

where ΨΦ(T) (t) is the solution to the non-linear Schroedinger equation
i~∂t |Ψ (t)i =

∫

d xĤ0 (x, |Ψ (t)i) +

d xĤint (x, |Ψ (t)i) |Ψ (t)i

(4.24)

with the boundary condition |Ψ (t = T)i = |Φi. We further generalize Ĥ N L by
allowing for different boundary conditions at each location
NL
Ĥ (t) =
d xĤ0 x, ΨΦT (x) (t) +
d 3 xĤint x, ΨΦT (x) (t) ,
(4.25)
where ΨΦT (x) (t) is the solution to Eq. (4.24) with boundary condition |Ψ (t = T(x))i =
|Φ(x)i.
The relativistic non-linear generalization of Ψc|α,β in Eq. (4.2) is
Ψc|α,β = P̂β ÛΦ(1) (x) (t2, t1 ) P̂α ÛΦ(0) (x) (t1, t0 ) |Ψini i ,

(4.26)

where x ∈ R3 , and ÛΦ(0) (t,x) (ÛΦ(1) (t,x) ) is the time-evolution operator associated
T
E
E
with the boundary condition Ψ t = T (0) (x) = Φ(0) (x) ( Ψ t = T (1) (x) =
Φ(1) (x) ).
4.4

The no-signaling condition in NLQM

As explained in section 2.1, Alice cannot communicate with Bob superluminally if
pN L (β| Aa, Bb ) = α pN L (α, β| Aa, Bb ) is independent of Alice’s choice of measurement basis Aa . The normalization factor in pN L (α, β| Aa, Bb ) (which we’ve shown
explicitly in Eq. (4.18)) won’t affect our analysis and can be safely ignored for the
remainder of this article2.
Similarly to how we derived Eq. (4.7), pN L (α, β| Aa, Bb ) can be simplified to (ignor2 If the unnormalized p N L (β| A , B ) is independent of the basis A for all β, then its normala b
ization, β p N L (β| Aa, Bb ), will also be independent of Aa . Moreover, it is obvious when the
normalization could help: p N L (β| Aa, Bb ) is of the form ( α f (α)) g (β) where f depends only on

α and g only on β. If such a scenario occurs, we will mention that the normalization eliminates the
dependence of p N L (β| Aa, Bb ) on Aa .

92
ing the irrelevant normalization factor)

(α, β| Aa, Bb ) =

Ûφ† (T ) (t1, t0 ) B̂† B B (t2, t1 ) P̂β B̂φB (T B ) (t2, t1 ) P̂α Ûφ1 (T1 ) (t1, t0 )
1 1
φ (T )

(4.27)
where we have used Eqs. (4.6), (4.18) and (4.19). Before we perform a general
analysis for arbitrary boundary states φ1 , φ2A and φ2B , we provide some examples.
4.4.1

Some example formulations

An interpretation that states that the wavefunction collapses after a measurement
predicts
NL
(α, β| Aa, Bb ) = Û(t† ) (t1, t0 ) B̂φ† (t ) (t2, t1 ) P̂β B̂φα (t2 ) (t2, t1 ) P̂α Û(t0 ) (t1, t0 ) ,
pcollapse
α 2

Û(t† ) (t1, t0 )

B̂φ† (t ) (t2, t1 ) P̂β B̂φα (t2 ) (t2, t1 ) P̂α
α 2

Û(t0 ) (t1, t0 ) .

Consider another choice for Alice’s measurement basis: Ad , corresponding to an
observable with eigenstates |δi and projection operators D̂δ , then
NL
(β| Ad, Bb ) = Û(t† ) (t1, t0 )
pcollapse

B̂ϕ† (t ) (t2, t1 ) P̂β B̂ϕδ (t2 ) (t2, t1 ) D̂δ Û(t0 ) (t1, t0 ) ,
δ 2

NL
NL
(β| Aa, Bb ) and pcollapse
(β| Ad, Bb )
where |ϕδ i ≡ D̂δ Û(t0 ) (t1, t0 ) |i. In general, pcollapse
aren’t equal and so a formulation based on immediate wavefunction collapse violates
the no-signaling condition. It also violates another tenet of special relativity: the
statistics of measurement outcomes is not the same in all reference frames. Refer to
the Appendix for more details.

On the other hand, a formulation of quantum mechanics in which collapse doesn’t
occur, such as the many-worlds interpretation, states
(α,
pNM.W
β|
ÛΦ†
a b

(t
(t
P̂
B̂
P̂
Û
(t
2 1
β Φini (tini ) 2 1
α Φini (tini ) 1 0 ,
ini (tini
ini (tini )
(4.28)
where the expectation value is taken over |i, tini is when the universe began and
|Φini i is the initial state of the universe and so is independent of α and β. When
(t , t ) B̂†
) 1 0 Φ

93
L (β| A , B ), the sum over α can be directly applied on P̂ resulting
calculating pNM.W
a b
in the identity operator, and so many-worlds does not violate the no-signaling
condition. In the case of fundamental semi-classical gravity, Eq. (4.28) has already
been ruled out [22].

In section 3.2, we discussed the prescriptions pre-selection and post-selection in the
context of a single measurement. For the multiple measurements setup shown in Fig.
4.1, pre-selection takes φ1 and φ2 to be the initial state of the experiment |Ψini i and
T2 = T1 = t0 . Post-selection takes φ1 and φ2 to be the final state of the experiment
|α, βi and T1 = T2 = t2 . Post-selection violates the no-signaling condition because
both φ1 and φ2 depend on the measurement outcomes α and β. Pre-selection
doesn’t violate the no-signaling condition. However, although [6] treated it as a
phenomenological prescription, it is equivalent to the Everett interpretation3.
4.4.2

A general analysis

From Eq. (4.27), we calculate pN L (β| Aa, Bb ) to be

Õ
NL
Ûφ† (T ) (t1, t0 ) B̂† B
p (β| Aa, Bb ) =

(t , t )
(t , t )P̂ B̂ B B (t , t )P̂ Û
φ2 (T2B ) 2 1 β φ2 (T2 ) 2 1 α φ1 (T1 ) 1 0

and pN L (α| Aa, Bb ) to be

Õ
NL
Ûφ† (T ) (t1, t0 ) B̂† B
p (α| Aa, Bb ) =

(t , t )
(t , t )P̂ B̂ B B (t , t )P̂ Û
φ2 (T2B ) 2 1 β φ2 (T2 ) 2 1 α φ1 (T1 ) 1 0

where for both probabilities, the expectation value is taken over |i. The no-signaling
condition is violated if pN L (β| Aa, Bb ) depends on Aa or if pN L (α| Aa, Bb ) depends
on Bb .
Notice that if φ1 depends on α then pN L (β| Aa, Bb ) depends on Aa and so pN L (β| Aa, Bb ) ,
pN L (β|Bb ). Similarly, if φ1 depends on β then pN L (α| Aa, Bb ) depends on Bb .
Consequently, φ1 must be independent of α and β. Similarly, φ2B must also be
independent of α and β. On the other hand, φ2A is unconstrained, and so our analysis
doesn’t result in a unique prescription. Nonetheless, we find it difficult to justify
why φ1 and φ2B would be anything other than the initial state of the experiment or of
3 Choosing |φ i and |φ i to be the initial states of an experiment is not a well-defined procedure.

Consider again the setup shown in Fig. 4.1, where Charlie prepared |Ψini i. He must have manipulated
some state, which we call Ψini , to prepare |Ψini i. If we choose Ψini to be the initial state of the
experiment, then pre-selection predicts that |φ1 i = |φ2 i = Ψini . This argument could be repeated
back to the initial state of the universe. As a result, pre-selection seems to be equivalent to the Everett
interpretation.

94
the universe. If we choose all boundary states to be the initial state of the universe,
then we recover the Everett interpretation. In the next section, we discuss another
reasonable prescription for assigning boundary states.
4.5

Causal-conditional: A sensible prescription that doesn’t violate the nosignaling condition

In this section, we propose and discuss a prescription, which we name causalconditional, for assigning boundary states to time-evolution operators in a way
that doesn’t violate the no-signaling condition. The causal-conditional prescription
updates the boundary states of degrees of freedom lying in the future light cone
of a particular measurement. We will be conservative and not explicitly assign a
mechanism for this process, be it objective collapse or emergent behavior after the
wavefunction branches. We only specify that the predictions of causal-conditional
are mathematically equivalent to sQM with causal feedback following each measurement event.
To precisely explain the causal-conditional prescription, we will present, using the
language of quantum field theory introduced in section 3.3, the quantum state of a
general collection of degrees of freedom at an arbitrary time t f . We’ll assume that
their initial state at time t0 is |Ψini i and that N measurements have occurred up to
the final time t f . The (unnormalized) conditional state at t f is
|Ψc i = ÛN P̂ (tN , xN ) ÛN−1 ... P̂ (t1, x1 ) Û0 |Ψini i ,

(4.29)

where P̂ (t 0, y) is a projection operator associated with a measurement at the spacetime location (t 0, y) and Ûi , for 0 ≤ i ≤ N, is the time-evolution operator from ti−1
till ti . After some explanation, we provide Ûi ’s exact expression in Eq. (4.30).
According to the causal-conditional prescription, a degree of freedom modifies the
boundary condition that the non-linearity at its spacetime location depends on (as
in Eq. (4.25)) when it receives information about a measurement outcome. This
information propagates along the future light cone of a measurement event. Assume
that for some 1 ≤ i ≤ N, a degree of freedom at x receives information, at times
(i)
s1(i), ..., sm
between ti−1 and ti , about mi measurement outcomes, then
(i)
s (i), s (i) Û (i)
Ûi = ÛΦ(i) s(i) ,x ti, sm
...
Û
(i)
(i)
Φ (t
Φ s ,x

(i)
i−1
, x) 1

i−1

(4.30)

Note that we have extended the definition of the boundary state |Φ (x)i to include a
dependence on time: |Φ (t, x)i. Moreover, no measurements occurred before t1 so

95
Û0 = ÛΦ(0) (t ,x) (t1, t0 ).

The causal-conditional prescription chooses the boundary states as follows. For
t0 ≤ t < t1 , no measurements have occurred, so Φ(0) (t, x) = |Ψini i for all t and
the boundary time is T (0) (x) = t0 . For 1 ≤ i ≤ N , T (i) (x) = ti . The Φ(i) (t, x) are
defined sequentially from i = 1 till i = N:
(t, x)

Φ(2) (t, x)

(1)

≡ P̂(t,x) (t1, x1 ) ÛΦ(0) (t,x) (t1, t0 ) |Ψini i ,

(4.31)

Φ(N) (t, x)

≡ P̂(t,x) (t2, x2 ) ÛΦ(1) (t,x) (t2, t1 ) Φ(1) (t, x) ,
..

(4.32)

≡ P̂(t,x) (tN , xN ) ÛΦ(N −1) (t,x) (tN , tN−1 ) Φ(N−1) (t, x) , (4.33)

for all t ≥ t0 and where h. We illustrate the assignment of boundary states after the
first two measurements in Fig. 4.2.
Finally, note that our scheme is similar to Adrian Kent’s proposal in [21]. He argued
that if the non-linear time evolution depends only on local states, which are obtained
by conditioning only on measurements in the past light cone of a degree of freedom,
then superluminal communication is not possible.
4.5.1

An example

Consider the setup shown in Fig. 4.3, which is a more elaborate version of Fig. 4.1.
The thought experiment now includes two additional parties: Dylan who prepares
an ensemble of two particles in the state Ψini and then sends them to Charlie,
and Eve who performs a measurement outside the future light cone of Alice on
Bob’s particle at time t3 . We’ve added Dylan to demonstrate that we don’t need to
know Ψini to predict the distribution of outcomes for measurements lying in the
future light cone of Dylan’s measurement. We’ve added Eve to show that even for a
complicated configuration of measurement events, our prescription does not violate
the no-signaling condition.
We begin our analysis with the predictions of sQM for the final unnormalized state
of the experiment conditioned on Charlie, Alice, Bob and Eve measuring γ, α, β
and with corresponding measurement eigenstates |Ψini i, |αi, | βi and |i, and
corresponding measurement bases Cc , Aa , Bb and Ee , respectively:

|ψc i = P̂ Û(t3, t2 )P̂β Û(t2, t1 )P̂α Û(t1, t0 )P̂γ Û(t0, tD ) Ψini .

(4.34)

Figure 4.2: Assignment of boundary conditions after two measurements according
to the causal-conditional prescription. M1 and M2 are two measurement events at
spacetime locations (t1, x1 ) and (t2, x2 ), respectively, and the dashed lines show the
light cones centered around each of them. To keep the figure uncluttered, we work
with a one-dimensional quantum field, and we have discretized space and time into
10 points each. Each degree of freedom of the field is represented by a dot on the
figure. How we fill the dot depends on what boundary condition (B.C.), which is
indicated on the legend at the top of the figure, is assigned to the time-evolution of
the wavefunction that the non-linear Hamiltonian at the spatial location of the dot
depends on (see section 3.3 for more details). Note that the initial state of the field
is |Ψini i.

97
The projection operators are
P̂γ = |Ψini i hΨini | ,

P̂α = |αi hα| ,

P̂β = | βi hβ| ,

P̂ = |i h | .

(4.35)

The structure of |ψc i can be simplified by noticing that Alice’s measurement’s future
light cone doesn’t overlap with Bob and Eve’s measurement events’ past light cone.
We obtain
|ψc i = P̂ Â (t3, t2 ) Ê P̂β Â (t2, t1 ) B̂ P̂α Û P̂γ V̂ Ψini ,

(4.36)

where to keep the notation concise, we have made the following definitions
V̂ ≡ Û (t0, tD ) ;

Û ≡ Û (t1, t0 ) ,

(4.37)

and B̂ and Ê are the time-evolution operators for Bob and Eve’s measured degrees
of freedom from t1 till t2 and from t2 till t3 , respectively.
According to the causal-conditional prescription, |ψc i extends to NLQM in the
following way:
|ψc i = P̂ Âα(t1 ) (t3, t2 ) Êφ3 (t2 ) P̂β Âα(t1 ) (t2, t1 ) B̂Ψini (t0 ) P̂α ÛΨini (t0 ) P̂γ V̂Ψ 0 (tD ) Ψ(4.38)
ini
ini

∝ Âα(t1 ) (t3, t2 ) Âα(t1 ) (t2, t1 ) P̂ Êφ3 (t2 ) P̂β B̂Ψini (t0 ) P̂α ÛΨini (t0 ) |Ψini i ,

(4.39)

where
|φ3 i = P̂β ÂΨini (t0 ) (t2, t1 ) B̂Ψini (t0 )ÛΨini (t0 ) |Ψini i .

(4.40)

We have also used that Alice’s particle doesn’t interact with the second particle
after t1 and so Â commutes with B̂, Ê, P̂β and P̂ . Bob’s past light cone does not
include Alice’s measurement event, but includes Charlie’s, so |Ψini i is the boundary
state associated with Bob’s particle’s time-evolution operator B̂. Moreover, Eve’s
past light cone includes Bob’s measurement event, so the conditional state |φ3 i is
the boundary state associated with Ê. Notice that Â in |φ3 i is associated with the
boundary state Ψini . Refer to Fig. 4.4 for more details.
Eq. (4.39) doesn’t violate the no-signaling condition and contains genuine non-linear
time evolution, such as ÛΨini (t0 ) |Ψini i. Moreover, notice that measurements within
the past light cone of Charlie, like that of Dylan’s, do not affect our analysis. Indeed,
preparation events are always in the past light cone of the final measurements of an
experiment because the measured particles’ speed is upper bounded by the speed
of light. Consequently, experimentalists do not need to know about measurements

98
occurring outside their experimental setup to calculate the predictions of the causalconditional prescription.
We show that our proposed prescription does not violate the no-signaling condition by looking at the marginal probabilities, p (α|C, B), p (β|C, B) and p ( |C, B),
conditioned on Charlie measuring |Ψini i and on the measurement bases B ≡
{Cc, Aa, Bb, Ee }. The probability of obtaining the measurement results α, β, , and
that Charlie measures |Ψini i is given by the norm of |Ψc i:
pN L (α, β, , C|B) =

Ψini V̂Ψ 0 (tD ) Ψini

ini

Ψini ÛΨ†

B̂†
(t ) Ψ

ini 0

P̂
Ê
P̂
Ê
P̂
B̂
P̂
Û
(t ) β φ (t ) φ3 (t2 ) β Ψini (t0 ) α Ψini (t0 ) ini

ini 0

3 2

We are interested in pN L (α, β, |C, B), so we have to divide by
pN L (C|B) =

pN L (α, β, , C|B) =

α,β,

Ψini V̂Ψ 0 (tD ) Ψini

ini

(4.41)

We obtain
pN L (α, β, |C, B) = Ψini ÛΨ†

P̂
P̂
P̂
Ê
Ê
B̂
P̂
Û
β φ3 (t2 ) φ3 (t2 ) β Ψini (t0 ) α Ψini (t0 ) ini .
ini (t0 )
(4.42)

B̂†
(t ) Ψ

ini 0

Can Alice send signals to Bob or Eve, or vice versa? We first calculate pN L (β|C, B):

(β|C, B) =

Ψini ÛΨ† (t ) B̂Ψ
P̂β B̂Ψini (t0 )ÛΨini (t0 ) Ψini
ini 0
ini (t0 )

(4.43)

which doesn’t depend on Aa . Next, we calculate Eve’s distribution of measurement
results:
pN L ( |C, B) =

Ψini ÛΨ† (t ) B̂Ψ
ini 0
ini (t0 )

P̂β Êφ† (t ) P̂ Êφ3 (t2 ) P̂β
3 2

B̂Ψini (t0 )ÛΨini (t0 ) Ψini ,

(4.44)
so Alice cannot send superluminal signals to Eve. Bob and Eve’s measurement
events are time-like separated so it is acceptable that they can communicate amongst
each other. Finally, Bob and Eve cannot communicate to Alice superluminally
because
pN L (α|C, B) =
doesn’t depend on Bb and Ee .

Ψini ÛΨ†

P̂ Û
(t ) α Ψini (t0 ) ini

ini 0

(4.45)

Figure 4.3: A setup similar to that described
by Fig. 4.1, but more elaborate. Event D
is Dylan preparing the state Ψini , Event C
is Charlie measuring the eigenstate |Ψini i.
Event A (B) describes Alice (Bob) measuring her (his) particles. Bob then sends his
particle to be measured by Eve at event E.
The dashed lines show the light cone centered
around each event.
4.5.2

Figure 4.4: Partioning of spacetime into different regions according to which boundary
state is associated with time evolution. There
are 4 measurement events: C, A, B and E, that
we’ve arranged identically as in Fig. 4.3. We
didn’t include Event D to limit clutter. The 4
events result in 6 regions. The boundary state
associated with the non-linear time-evolution
operator of each region is the time-evolved
initial state of the experiment conditioned on
measurement events presented in the legend
at the top of the figure.

Proof that the causal-conditional prescription doesn’t violate the nosignaling condition

We prove that the prescription discussed in this section does not violate the nosignaling condition. We first present a heuristic argument. The causal-conditional
prescription is mathematically equivalent to linear quantum mechanics with causal
feedback, and so doesn’t violate the no-signaling condition. In particular, whenever
a measurement occurs, the wavefunction that the non-linear potential depends on
isn’t modified instantaneously. Instead, a measurement transmits its outcome along
its future light cone. Degrees of freedom that receive this information update their
boundary state accordingly.
We now present a rigorous argument. Consider a general measurement configuration

100
as viewed in some reference frame. The unnormalized conditional state after the
final measurement is
|ψc i = Û1 P̂1 (α1 ) ...Û f P̂ f α f |inii ,

(4.46)

where |inii is the initial state of all degrees of freedom before the first measurement,
P̂i (αi ) is the projection operator (with outcome αi ) associated with the ith measurement, and the Û1 , Û2 , ..., Û f are boundary-dependent time-evolution operators.
Assume that Bob performs, at time tB , one of these measurement. We will show
that Bob’s probability of measuring a particular outcome β,
p (β|Ω) =

α1,...,α f

ini Û1† P̂1 (α1 ) ... P̂ f −1

α f −1 Û †f P̂ f

(4.47)
where Ω is the set of all chosen measurement bases, is independent of measurements
after tB , and outside Bob’s measurement’s past light cone. Note that the sum is over
all measurement outcomes except Bob’s. All measurements occurring after tB do
not matter because we can directly sum over them. Let’s first sum over α f . We
obtain
p (β|Ω) =

α1,...,α f −1

α f Û f P̂ f −1 α f −1 ... P̂1 (α1 ) Û1 ini ,

ini Û1† P̂1 (α1 ) ...Û †f −1 P̂ f −1 α f −1 Û f −1 ... P̂1 (α1 ) Û1 ini

(4.48)
because the final measurement does not lie in the past light cone of any other
measurement, and so no time-evolution operator would depend on α f . We can
repeat this procedure for all other measurements events after tB .
For this part of the proof, we label the projection operator corresponding to Bob’s
measurement by P̂β and assume that n measurements precede Bob’s. Let Ω̃ ⊂ Ω be
the set of measurements bases chosen by Bob and all experimentalists performing
measurements before Bob. After summing over all the outcomes of all measurements performed after Bob’s, we then obtain that
Õ D
p β| Ω̃ =
ini Û1† P̂1 (α1 ) ... P̂n (αn ) Ûn+1
P̂β Ûn+1 P̂n (αn ) ... P̂1 (α1 ) Û1 ini .
α1,...,αn

(4.49)
Consider the measurement occurring closest to tB , and that is outside Bob’s measurement’s past light cone, as shown in Fig. 4.5. Assume it corresponds to the
ith measurement event, and so according to the causal-conditional prescription,
the time-evolution operators Ûi+1 , ..., Ûn+1 contain boundary terms dependent on

101
αi . Let’s explicitly separate each of them into two components: Û j ≡ V̂j Ŵ j for
i + 1 ≤ j ≤ n + 1 and where V̂j doesn’t depend on the boundary αi whereas Ŵ j
does. The Ŵ j also evolve degrees of freedom inside the ith measurement’s future
light cone. Consequently, the Ŵ j commute with the V̂j , allowing us to simplify the
expectation value in Eq. (4.49) to
Û1 P̂1 (α1 ) ...Ûi P̂i (αi ) V̂i+1 ... P̂n (αn ) V̂n+1Ŵ P̂β Ŵ V̂n+1 P̂n (αn ) ...V̂i+1 P̂i (αi ) Ûi ... P̂1 (α1 ) Û1 ,
where the expectation value is taken over |inii and Ŵ ≡ Ŵn+1 ...Ŵi+1 . Since Ŵ †
commutes with P̂β , it can be moved to the right of it where it will act on Ŵ and
result in the identity matrix. Similarly, P̂i (αi ) can be moved to the right of P̂β and
we obtain
Õ D
p β| Ω̃ =
Û1† P̂1 (α1 ) ...Ûi†V̂i+1
... P̂n (αn ) V̂n+1
P̂βV̂n+1 P̂n (αn ) ...V̂i+1 P̂i (αi ) Ûi ... P̂1 (α1 ) Û1
α1,...,αn

The sum over αi can then be directly applied on P̂i (αi ) allowing us to eliminate it.
As a result, p β| Ω̃ is independent of basis chosen during the ith measurement.
The above argument can be applied sequentially and in reverse chronological order to
eliminate p β| Ω̃ ’s dependence on all bases associated with measurements outside
Bob’s measurement’s past light cone. Although we conducted our analysis in a
particular reference frame, and so assuming a particular ordering of events, our
arguments could be applied to any other reference frame (modulo a relabeling of
spacetime points). We would always arrive to the same conclusion: the causalconditional prescription does not violate the no-signaling condition.
4.6

Conclusions

We have shown that modifying linear quantum mechanics is not as simple as adding
terms in the Hamiltonian that depend on the wavefunction. One must also make a
choice on how to interpret measurements and the evolution of the wavefunction. By
breaking linearity, different formulations of quantum mechanics, such as the Everett
and Copenhagen interpretations, no longer make equivalent predictions.
By introducing the notion of a time-evolution operator that depends on the specified
boundary conditions for the quantum state of the system that is being time-evolved,
we were able to explore the range of possible prescriptions for assigning probabilities
to measurement outcomes in NLQM. For a certain class of non-linear theories, we
showed that two reasonable prescriptions do not violate the no-signaling condition.
The first is the Everett interpretation, and the second, which we named causal-

102

Figure 4.5: A general configuration of measurements, labeled by M j where 1 ≤ j ≤
n, occurring before an event B, which describes Bob performing a measurement.
The dashed lines shows the past light cone of event B.

conditional, states that a measurement event at a particular spacetime point X
updates the boundary state associated with the time evolution operator of quantum
degrees of freedom lying in the future light cone of X. The predictions of causalconditional are mathematically equivalent to standard quantum mechanics with
causal feedback. A measurement applies a feedback force (the details of which are
determined by the non-linear theory of interest) on degrees of freedom lying in the
future light cone of that measurement event.
Ackowledgments
We thank Craig Savage, Sabina Scully, Tian Wang, Maneeli Derakhshani, Yiqiu
Ma, Haixing Miao, Sean Carroll and Ashmeet Singh for useful discussions. We
also thank Antoine Tilloy for pointing out important issues with how we explained
certain concepts. This research is supported by NSF grants PHY-1404569 and PHY1506453, as well as the Institute for Quantum Information and Matter, a Physics
Frontier Center.

103
Appendix: The statistics of measurement outcomes in different reference
frames
We first review why sQM predicts, in all reference frames, identical statistics for
measurement results. We then show that this is no longer true in NLQM if we
adopt an interpretation where the wavefunction instantaneously collapses across all
of space. Finally, we discuss why the causal-conditional prescription predicts, in
different reference frames, identical statistics for measurement outcomes.
Consider the multiple measurements configuration shown in Fig. (4.1) where Alice
and then Bob measure their respective particles, as viewed in some reference frame
that what we’ll refer to as the lab frame. In this frame, the probability of Alice
measuring α and Bob measuring β, p (α, β| Aa, Bb ) is given by Eqs. (4.1, 4.2). To
conveniently transform p (α, β| Aa, Bb ) from one reference frame to another, we will
express p (α, β| Aa, Bb ) in a Heisenberg picture. Define
P̂α (x A, t1 ) ≡ Û † (t1, t0 ) P̂α Û (t1, t0 ) ,

P̂β (xB, t2 ) ≡ Û † (t2, t0 ) P̂β Û (t2, t0 ) , (4.50)

where we’ve explicitly denoted the location of Alice’s measurement at x A and of
Bob’s at xB . Since P̂α (x A, t1 ) and P̂β (xB, t2 ) commute, we can rewrite p (α, β| Aa, Bb )
to
p (α, β| Aa, Bb ) = Ψini P̂β (xB, t2 ) P̂α (x A, t1 ) Ψini .
(4.51)
Consider now a Lorentz-transformation Λ from the lab frame to any other frame. On
the Hilbert space of Alice and Bob’s particles’, Λ is realized by an operator V̂ (Λ).
For instance, V̂ (Λ) transforms a momentum eigenstate of a spinless particles to
V̂ (Λ) |ki = |Λk0i , where |ki is covariantly normalized to hk|k0i = hΛk|Λk0i [20].
We re-express p (α, β| Aa, Bb ) in terms of wavefunctions and projection operators
viewed in a different frame
p (α, β| Aa, Bb ) =

ΛΨini V̂ (Λ) P̂β (xB, t2 ) V̂ † (Λ) V̂ (Λ) P̂α (x A, t1 ) V̂ † (Λ) ΛΨ
(4.52)
ini
µ ν
µ ν
ΛΨini P̂β Λ ν xB P̂α Λ ν x A ΛΨini ,
(4.53)

where |ΛΨi ≡ V̂ (Λ) |Ψi for any |Ψi, x Aν is the 4-vector (x A, t1 ) and xBν is (xB, t2 ). If
we assume that the measured results do not change under Lorentz transformations
(e.g. photodetector clicks4), then Eq. (4.53) is just the probability of measuring α
4 For a Klein-Gordon field φ̂, the measured observable would be

jˆν .dΣν,

jˆν (x µ ) = i ∂ν φ̂− (x µ ) φ̂+ (x µ ) + h.c,

(4.54)

104
and β in a different reference frame. Therefore, in sQM, the statistics of measurement
outcomes are the same in all reference frames.
The extension of p (α, β| Aa, Bb ) = Ψc|α,β |Ψc|α,β , as calculated by an observer in
the lab frame, to NLQM coupled with an interpretation of QM with wavefunction
collapse is
collapse
Ψc|α,β

= P̂β ÛΦα (t1 ) (t2, t1 ) P̂α ÛΨini (t0 ) (t1, t0 ) |Ψini i ,

(4.55)

(4.56)

The extension of the Heisenberg picture projection operators in Eq. (4.50) to NLQM
is
collapse

P̂α

(x A, t1 ) ≡ ÛΨ†

ini (t0 )

(t1, t0 ) P̂α ÛΨini (t0 ) (t1, t0 ) ,

collapse

P̂β

(xB, t2 ) ≡ Û2† P̂
(4.57)
β Û2,

Û2 ≡ ÛΦα (t1 ) (t2, t1 ) ÛΨini (t0 ) (t1, t0 ) .

(4.58)

Consequently, the extension of Eq. (4.53) to NLQM is

collapse

(α, β| Aa, Bb ) =

collapse
ΛΨini P̂β

collapse
Λ ν xBν P̂α

Λ ν x Aν ΛΨini

(4.59)

Let’s consider a Lorentz transformation Λ̃ that takes the lab frame to one where Bob
measures his particles before Alice measures hers. An observer in that frame would
calculate that the probability that Alice measures α and Bob measures β is
p̃

collapse

(α, β| Aa, Bb ) =
collapse

P̃β

collapse
collapse
( x̃ A) P̃β
( x̃B ) Λ̃Ψini
Λ̃Ψini P̃α

Ũ2
Φ̃ β

x̃B0 , t˜0 P̂β ÛΛ̃Ψini (t0 ) x̃B0 , t˜0 ,
Λ̃Ψini (t0 )
0 0
0 ˜
≡ ÛΦ̃β ( x̃ 0 ) x̃ A, x̃B ÛΛ̃Ψini (t0 ) x̃B, t0 ,
= P̂β ÛΛ̃Ψini (t0 ) x̃B0 , t˜0 Λ̃Ψini ,

( x̃B ) = Û †

collapse

P̃α

x̃ A,B ≡ Λ̃ ν x A,B

( x̃ A) = Ũ2† P̂α Ũ2,

where t˜0 is when the experiment began in this new frame. Although pcollapse (α, β| Aa, Bb )
where V is the spacetime volume occupied by the photodetector during a single measurement run,
and φ̂+ and φ̂− are the positive and negative frequency components of φ̂, respectively [23].

105
can be re-written to
collapse
collapse
( x̃ A) P̂β
( x̃B ) Λ̃Ψini ,
pcollapse (α, β| Aa, Bb ) = Λ̃Ψini P̂α

(4.60)

collapse

it isn’t in general equal to p̃collapse (α, β| Aa, Bb ) because P̂β

( x̃B ) depends on
collapse
collapse
( x̃B ) doesn’t. Similarly, P̃α
( x̃ A) depends on Φ̃ β while
Φα while P̃β
collapse
( x̃ A) doesn’t. In other words, V̂ Λ̃ doesn’t connect Φα, Φ β and Ψini to
P̂α
each other.
The fact that pcollapse (α, β| Aa, Bb ) isn’t the same in all reference frame isn’t surprising. and can be understood heuristically when we view the non-linearity as a
feedback force that changes acausally after the wavefunction collapses. Consider
the following non-linear interaction energy density
V̂N L (x) = Ψ (t) Ô Ψ (t) M̂ (x) ,

(4.61)

where Ô is Lorentz-invariant (V̂ † (Λ) ÔV̂ (Λ) = Ô for any transformation Λ), and
we assume that M̂ (x) transforms as M̂ (Λx) under Λ (so as to maintain the the
requirement that the total interaction Hamiltonian density Ĥint (x) transforms as
Ĥint (Λx) - see sec. 5.5 of [20]). We can then view F (t) ≡ Ψ (t) Ô Ψ (t) as a
classical feedback force on M̂.
When a measurement occurs, Ψ(t) instantaneously changes, and so F (t) acting on
M̂ (x) for all x ∈ R3 changes instantaneously too. The problem is that the spatial
surface of time simultaneity is the not the same in all reference frames. In the case
of multiple spacelike-separated measurement events, we’d get that F (t) changes
differently in different frames depending on the ordering of the measurement events
in that frame. On the other hand, for the causal-conditional prescription, F (t) would
change causally after any measurement, and so there are no issues.

106
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108
Chapter 5

MEASURABLE SIGNATURES OF A CAUSAL THEORY OF
QUANTUM MECHANICS IN A CLASSICAL SPACETIME
5.1

Introduction

With the recent advances in quantum optomechanics, it is now feasible for such a
platform to test alternative theories of quantum mechanics. There have been many
proposals, and some experiments, for optomechanics to test alternative theories of
quantum mechanics. In this article, we focus on fundamental semi-classical gravity,
where the space-time geometry is sourced by the quantum expectation value of the
stress energy tensor
G µν = 8π Φ T̂µν Φ
(5.1)
with G = c = 1, and where G µν is the Einstein tensor of a (3+1)-dimensional
classical spacetime. T̂µν is the operator representing the energy-stress tensor, and
|Φi is the wave function of all (quantum) matter and fields that evolve within this
classical spacetime.
There have been many proposals for optomechanics to test the predictions of Eq.
(5.1) [2, 3, 5]. In this article, we propose a new version of the SchroedingerNewton theory that does not violate causality. We also calculate its predictions in a
low-frequency optomechanics experiment.
In particular, we will argue that once we fix the initial state of a system, its monitored
evolution under a large class of Non-Linear Quantum Mechanics (NLQM) theories
(of which Eq. (5.1) is a part of) is equivalent to evolution under a particular
quantum feedback scheme. Since quantum feedback can be causal, we show that
adding measurements to Eq. (5.1) doesn’t necessarily mean the theory violates the
no-signaling condition. Eq. (5.1) can be thought of as changing the dynamics of
a quantum mechanical system by applying a causal feedback force everywhere in
spacetime. Since this force is causal, it means that the applied force at location (t, x)
can only depend on measurement results in the past light cone of (t, x), and so the
expectation value in Eq. (5.1) would be over a state |Φi that is conditioned only on
measurement results in the past light cone of (t, x). We also illustrate the mapping
between NLQM and feedback with concrete examples. Finally, we calculate the
signature of this causal theory of fundamental semi-classical gravity in a torsion
pendulum experiment.

109
5.2

NLQM is formally equivalent to quantum feedback

We will show that a large set of non-linear quantum mechanical theories are formally
equivalent to linear quantum feedback. This mapping has two main advantages.
First, it allows us to leverage the tools developed for linear quantum mechanics to
understand NLQM. Second, since we understand when quantum feedback is causal,
we can come up with strategies to incorporate measurements in NLQM without
violating the no-signaling condition.
Specifically, the class of non-linear quantum mechanical theories that we consider
has the following evolution equation:
i~∂t |ψ (t)i = ĤN LQM |ψ (t)i

(5.2)

where the non-linear Hamiltonian is
ĤN LQM = ĤL +

βi φ (xi, t, ψ (t)) V̂i .

(5.3)

(5.4)

where L is a differential operator and S (x, t, ψ (t)) is a source term that, in general,
depends on ψ (t). The βi are constants. We wrote the non-linear Hamiltonian in
the form of Eq. (5.3) because it clearly separates out the nonlinearity to a single
parameter, φ. This separation will make it easier to understand the mapping from
NLQM to quantum feedback. Once we solve for φ and substitute back into ĤN LQM ,
we will show that evolution under ĤN LQM can be thought of as evolution under a
linear Hamiltonian. Note that the sum i could in general contain an integral. An
example non-linear theory is fundamental semi-classical gravity in the Newtonian
limit, which is typically called the Schroedinger-Newton theory. The non-linear
Hamiltonian for a free particle of mass m is
ĤSN = −m dxφ (x, ψ (t)) |xi hx|
(5.5)
and the classical gravitational field follows the equation of motion
∇2 φ (x, t) = 4πGm |ψ (x, t)| 2 .

(5.6)

110
5.2.1

No measurements

We first show that for Eq. (5.2), the unmonitored dynamics of the wavefunction,
once we fix its initial state, are the same as the dynamics of a wavefunction evolving
under a linear time-dependent Hamiltonian. Eq. (5.2) is a special case of evolution
under a time-dependent Hamiltonian
i~∂t |ψ (t)i = Ĥ (t) |ψ (t)i

(5.7)

where
Ĥ (t) = Ĥ0 +

αi (t) Ôi

(5.8)

and Ĥ0 is time-independent. Once we fix an initial state for Eq. (5.2) then we can
solve Eqs. (5.2-5.4) and obtain φ. If we pick αi (t) to be equal to φ (xi, t, ψ (t)) and
Ôi equal to V̂i then the evolution of the initial state under Eq. (5.2) is identical to the
evolution of the same initial state under Eq. (5.7). Therefore, once we fix an initial
state, we can use the tools of time-dependent quantum mechanics to examine Eq.
(5.2). Moreover, we are assured that Eq. (5.2) has the same properties as Eq. (5.7),
such as satisfying the no-signaling condition.
5.2.1.1

An example

We will give a simple example illustrating how NgLQM without measurement can
produce dynamics that are equivalent to those of time-dependent linear quantum
mechanics. Let’s assume that Alice is at position x A and has a spin that evolves
under the linear Hamiltonian
ĤA = E σ̂z(A),
(5.9)
where σ̂z(A) is Alice’s spin’s Pauli z matrix. The spin also couples to a classical field:
(A)
V̂N(A)
L (t) = ~ω A φ (x A, t) σ̂x

(5.10)

where ω A ∈ R and σ̂x(A) is the Pauli x matrix. Let’s assume that φ (x A, t) is
φ (x A, t) = ψ (t) σ̂z(A) ψ (t) .

(5.11)

A , Eq. (5.10) is equivalent to a timeGiven the initial state of Alice’s spin ψini
dependent linear Hamiltonian. In particular, Alice can calculate φ (x A, t) by solving

(t)
|ψ (t)i
i~∂t |ψ (t)i = Ĥ A + V̂N(A)

(5.12)

111
with the boundary condition
|ψ (tini )i = ψini

(5.13)

With φ (x A, t) Alice can then construct an experiment with a Hamiltonian that is
equivalent to Ĥ A + V̂N(A)
L (t). She would just have to apply a magnetic field along
the z direction with time-dependence given by −~γφ (x A, t) /(2ω A), where γ is the
gyromagnetic ratio of Alice’s particle.
5.2.2

Adding measurements

If we monitor the quantum system we are investigating, then |ψ (t)i depends on
the measurement record because the unitary evolution of |ψ (t)i is interrupted by
projection operators. Specifically, if n observables, Ŷ1 through Ŷn , are measured at
times t1 through tn , then |ψ (t)i is (up to a normalization factor)
|ψ (t)i ∝ Ûϕn (t, tn ) P̂n ... P̂2Ûϕ1 (t2, t1 ) P̂1Ûϕ0 (t1, t0 ) |ψ (t0 )i

(5.14)

where for i = 1...n, P̂i projects Ŷi at time ti to its eigenstate with eigenvalue yi . Each
evolution operator Ûϕ (t˜1, t˜0 ) evolves a wavefunction from t˜0 till t˜1 under Eq. (5.2)
and the boundary condition
|ψ (t˜0 )i = |ϕi .
(5.15)
As explained in Ref. [4], the boundary state |ϕi could depend arbitrarily on the
measurements results {yi }. For example, according to the Everett intepretation,
all boundary states are the initial state of the universe. On the other hand, the
Copenhagen interpretation states that the boundary states should incorporate all
measurements up the current time.
Similarly to Eq. (5.14), the wavefunction under feedback, ψ f b (t) , is (up to a
normalization factor)
ψ f b (t) ∝ Û f b (t, tn ) P̂n ... P̂2Û f b (t2, t1 ) P̂1Û f b (t1, t0 ) |ψ (t0 )i ,

(5.16)

where we’ve denoted the measurement record by
y=

yn ... y2 y1

(5.17)

and Û f b is the time evolution operator associated with feedback and a time-

112
independent Hamiltonian Ĥ0 :
Û f b (z2, z1 ) = exp − Ĥ0 (z2 − z1 ) +

∫ z2
dzα j (z, y) Ôi

(5.18)

α is the time-dependent feedback force applied on the degree of freedom associated
with Ôi . Note that since we’ve allowed the αs to depend on the entire measurement
record y, we haven’t restricted the feedback scheme to be causal (we will do so in
Sec. 5.2.3).
Once the initial state |ψ (t0 )i is fixed, we will show that Eq. (5.16) can match any
evolution under the non-linear Hamiltonian (5.3). Let’s assume that till time t, a
feedback scheme successfully matches the non-linear evolution under Eq. (5.14).
We will show that this feedback scheme can continue to match the non-linear
evolution until time z, which we assume is when the next measurement occurs.
Under NLQM,
ψN LQM (z) ∝ P̂ (z) exp − ĤL (z − t) +

∫ z

!!
dlφ x j , l, ψ̃ (l) , y V̂i

|ψ (t)i

(5.19)
where P̂ (z) is the projection operator associated with the measurement at time z.
ψ̃ (l) for t ≤ l ≤ z is the solution of Eq. (5.2) with the boundary condition
ψ̃ (t) = |ϕi ,

(5.20)

where the boundary state|ϕi depends on the interpretation of quantum mechanics
that one uses1. Moreover, it might seem odd that we’ve associated a unitary evolution
operator to the non-linear Hamiltonian (5.3) but, as discussed in Sec. 5.2.1, once
we solve for ψ̃ (t), we can think of ĤN LQM as a time-dependent Hamiltonian.
Under linear quantum mechanics and feedback, the state at time z is
ψ f b (z) ∝ P̂ (z) Û f b (z2, z1 ) |ψ (t)i .

(5.21)

If we choose Ĥ0 to be ĤL , the Ôi to be the V̂i and the α j s to be equal to the φ j then
ψ f b (z) = ψN LQM (z) .

(5.22)

1 It might seem surprising that the boundary state is not |ψ (t)i but we remind the user that this

is a feature of NLQM: different interpretations of quantum mechanics lead to different predictions
in NLQM [4].

113
The first two conditions can be easily met. To meet to third condition, we have to
first solve for ψ̃ (l) for t ≤ l ≤ z, which will allow us to calculate the classical field
φ by using Eq. (5.4). The feedback force is then crafted to be the same as φ.
We’ve shown that once we fix the initial state, a monitored system evolving under
NLQM indistinguishable from a monitored system evolving under a linear Hamiltonian and feedback. Moreover, even if we don’t know the initial state, evolution under
quantum feedback could asympotactically approach the non-linear quantum evolution because a monitored system tends to eventually be driven by measurements,
and forgets its initial state.
5.2.3

Causal NLQM

It is widely believed that adding measurements to NLQM breaks causality. We will
exploit the equivalence between NLQM and quantum feedback to show that we can
incorporate measurements in NLQM in a causal way.
Feedback is causal if the applied force at time t and location x depends only on
the measurement results an experimentalist can collect. These are the results of
measurement events in the past light cone of (t, x). Therefore, for NLQM to be
causal, the classical field φ (x, t) that appears in Eq. (5.3) can only depend on the yi
collected in the past light cone of (t, x).
To allow only measurements in the past light cone of (x, t), we rewrite Eq. (5.4) to
Lφ (x, t) = S (x, t, λ (t, x))

(5.23)

where |λ (t, x)i is a modified |ψi, given by Eq. (5.14), that only incorporates
measurements in the past light cone. If the n measurements occur at locations z1
through zn then
|λ (t, x)i ∝ Û→ (t, tn ) P̂(t,x) (tn, zn ) ... P̂(t,x) (t2, z2 ) Û→ (t2, t1 ) P̂(t,x) (t1, z1 ) Û→ (t1, t0 ) |ψ (t0 )i
(5.24)
where P̂(t,x) (t , z) is P̂ (t , z) if (t, x) lies in (t , z)’s future light cone, and the identity
operator otherwise:
 P̂ (t 0, z) t − t 0 − |z−x|
c ≥ 0
P̂(t,x) (t , z) =
 Iˆ
otherwise

(5.25)

where P̂ (t 0, z) denotes a projection at time t 0 and location z. Moreover, Û→ (t2, t1 )
denotes evolution under Eq. (5.2) from time t1 till t2 . The → denotes that the

114
boundary state is chosen to be the state that Û→ (t2, t1 ) acts on. For example, in
Û→ (t2, t1 ) |ϕi, the boundary state is chosen to be ϕ. We will call this prescription
causal-conditional.
5.2.3.1

An example

We revisit the example setup in Sec. 5.2.1.1, but now extend it to two parties: Alice
and Bob, as is shown in Fig. 5.1. We’ll assume that Alice and Bob are localized at
positions x A and xB . We will illustrate what happens when Alice measures her spin
at time t1 . The associated measurement event is M A.
For both NLQM and feedback, let’s assume that they both have the same linear part
of the Hamiltonian
ĤL ≡ ĤA + ĤB .
(5.26)
According to the causal-conditional prescription, the non-linear Hamiltonian is
(B)
(t)
(t)
(λ
σ̂x(A),
σ̂
t)
~ω
V̂N(A)
(B)
(λ
(t)
(t)
t)
~ω
σ̂
σ̂x(B)
V̂N(B)

(5.27)
(5.28)

where λ A/B (t) incorporates all measurements in the past light cone of t, x A/B . We
can think of |λ A (t)i (|λB (t)i) as the joint Alice-Bob quantum state as perceived by
Alice (Bob) with all the information she (he) could have gathered at time t.
In our simple example, only one measurement occurred at (t1, x A). Therefore, only
(B)
two λs are sufficient to completely describe V̂N(A)
L and V̂N L . The first, which we call
|λ0 (t)i, is the solution of
(B)
(ψ
(t)
(ψ
(t)
|ψ (t)i
i~∂t |ψ (t)i = ĤL + V̂N(A)
t)
V̂
t)
NL

(5.29)

with the initial condition
|ψ (t0 )i = |ψini i .

(5.30)

|ψini i is the initial state of Alice and Bob’s spins. The second, which we call |λ1 (t)i,
is the solution of Eq. (5.29) under the initial condition
|ψ (t1 )i = D

P̂α(A) |λ0 (t1 )i
λ0 (t1 ) P̂α(A) λ0 (t1 )

(5.31)

where P̂α(A) is the projection operator associated with Alice’s measurement, which

115
we assume has a measurement result of α.
We can now write down the non-linear potential as a function of time only:
(A)
(A)
 ~ω A λ0 (t) σ̂z λ0 (t) σ̂x
(A)
V̂N L (t) =
 ~ω A λ1 (t) σ̂z(A) λ1 (t) σ̂x(A)
(B)
(t)
(t)
σ̂x(B)
~ω
σ̂
(B)
V̂N L (t) =
 ~ωB λ1 (t) σ̂z(B) λ1 (t) σ̂x(B)

t0 ≤ t ≤ t1
t > t1

A|
t0 ≤ t ≤ t1 + |xB −x
A|
t > t1 + |xB −x

(5.32)

. (5.33)

Stated in this way, we can now think of the total Hamiltonian
Õ

E σ̂z(i) + V̂N(i)L (t)

(5.34)

i=A,B

as a linear time-dependent Hamiltonian. Since it is separable, we can assign an
evolution operator for each of Alice and Bob’s spins:
Ûi (t, t 0)

i = A, B

(5.35)

and the unnormalized quantum state at time t > t1 + |xB − x A | /c is
|ψ (t)i ∝ ÛB (t, t1 ) Û A (t, t1 ) P̂α(A)ÛB (t1, t0 ) Û A (t1, t0 ) |ψini i .

(5.36)

Alice and Bob can forge their own feedback schemes that would produce a wavefunction evolution that is identical to |ψ (t)i. In particular, they’d base their
feedback strategy on two options: Alice
(Bob) can apply
a Dfeedback force on
(A)
(B)
(A)
σ̂x (σ̂x ) that is equal to either ~ω A λ0 (t) σ̂z λ0 (t) (~ωB λ0 (t) σ̂z(B) λ0 (t) )
(A)
(B)
or ~ω A λ1 (t) σ̂z λ1 (t) (~ωB λ1 (t) σ̂z λ1 (t) ). The two feedback strategies
would have the following two feedback potentials associated with them
(A/B)
(t)
(t)
V̂ f(A/B)
~ω
σ̂
σ̂x(A/B),
A/B
b,0
(A/B)
(t)
(t)
V̂ f(A/B)
~ω
σ̂
σ̂x(A/B) .
A/B
b,1

(5.37)
(5.38)

Alice and Bob would alternate between the two feedback strategies in such a way
(B)
that they’d reproduce V̂N(A)
L (t) and V̂N L (t).
To calculate λ0 (t) σ̂z(A/B) λ0 (t) and λ1 (t) σ̂z(A/B) λ1 (t) , Alice and Bob would
first solve Eq. (5.29) for |λ0 (t)i. Once they do so, they will apply V̂ f(A)
and V̂ f(B)
b,0
b,0

116

Figure 5.1: Showing how causal NLQM and causal feedback are equivalent in a
simple example. At time t1 , Alice performs a measurement. The corresponding
measurement event is denoted byM A. The result of the measurement is broadcast
along M A’s future light cone. It reaches Bob at time t2 = t1 + |xB − x A | /c. In the
NLQM picture, at time t2 , the classical field at xB suddenly changes to incorporate
information about Alice’s measurement result. In the quantum feedback picture,
Bob switches his feedback strategy at t2 to incorporate information about Alice’s
measurement result. |λ1 (t)i and |λ0 (t)i are obtained from solving a non-linear
Schroedinger equation with initial conditions given by (5.30) and (5.31), respectively.

till time t1 . Once Alice obtains her measurement result, she would solve Eq. (5.29)
to obtain |λ1 (t)i and switch her feedback potential from V̂ f(A)
to V̂ f(A)
. Alice would
b,0
b,1
also share her measurement result with Bob, who would receive the result at time
t1 + |xB − x A | /c, at which point he would change his feedback scheme from V̂ f(B)
b,0
to V̂ f(B)
b,1

117
5.3

An example of continuously monitored optomechanical systems

We will provide a concrete example of how the causal-conditional prescription is
equivalent to causal feedback.
5.3.1

Setup

We will consider two parties, Alice and Bob, that are separated by a distance
∆x AB and that respectively monitor an optomechanical setup that is non-linearly
coupled to a classical field. We will show that the evolution equation governing
their setups is exactly identical to a particular feedback scheme that Alice and Bob
could implement. Therefore, although their setups evolve non-linearly, Alice and
Bob cannot communicate with each other superluminally.
Each of Alice and Bob’s setups, shown in Fig. 5.2, evolves under the following
Hamiltonian
Ĥ (i) = ĤL(i) + V̂φ(i) (t) ,
(5.39)
where i = A, B and ĤL(i) is the linear part of the Hamiltonian
(i)
(i)
ĤL(i) = Ĥtm
+ Ĥprobe
+ V̂I(i) .

(5.40)

(i)
Ĥtm
is the test mass’ free Hamiltonian:
(i)
Ĥtm

P̂i2 1
+ mω2 x̂i2,
2m 2

(5.41)

(i)
and Ĥprobe
is the driving light’s free Hamiltonian:
(i)
Ĥprobe

∫ ∞

dω
~ω âi† (ω0 + ω) âi (ω0 + ω) .
−∞ 2π

(5.42)

Note that we are working with the 2-photon formalism and the driving optical fields’
frequency is ω0 . V̂I(i) is the linearized interaction Hamiltonian of light with the test
mass:
V̂I(i) = −~α x̂i âi,1 (xi ) ,

(5.43)

where x A (xB ) is the center of mass location of Alice’s (Bob’s) test mass (quantum
fluctuations negligibly perturb x A and xB ), and
α = 8Iin

~ω0
c2

(5.44)

118
where Iin is the driving laser’s intensity.
(B)
(A)
removed.
+ Ĥprobe
It will be convenient to enter into an interaction picture with Ĥprobe
(B)
(A)
just propagate the optical fields forward. V̂I becomes
+ Ĥprobe
Ĥprobe

V̂I (i) (t) = −~α x̂ âi,1 (xi − t/c)

i = A, B

(5.45)

where âi,1 (xi − t/c) is the incoming optical degree of freedom at location xi − t/c.
We’ll assume that the non-linear interaction is given by
V̂φ(i) = ~φ (xi, t) x̂i

(5.46)

where φ obeys the field equation
Lφ (x, t) = κ A hλ (t, x)| x̂ A |λ (t, x)i δ (x − x A) + κB hλ (t, x)| x̂B |λ (t, x)i δ (x − xB )
(5.47)
where |λ (t, x)i is |ψ (t)i with all projection operators corresponding to measurement
events at (t 0, z) replaced by P̂(t,x) (t 0, z) (see Eqs. (5.24-5.25)). For simplicity, we will
assume that the differential operator L’s Green’s function is the retarded Green’s
function:

|x − x 0 |
0 0
G L (x, t; x , t ) = δ t − t +
(5.48)
so

|x − xi |
|x − xi |
, xi x̂i λ t −
, xi .
(5.49)
φ (x, t) =
κi λ t −
i=A,B
Substituting back into V̂φ(i) for i = A, B, we obtain
V̂φ(A) /~ = κ A hλ (t, x A)| x̂ A |λ (t, x A)i x̂ A

∆x AB
∆x AB
+κB λ t −
, xB x̂B λ t −
, xB x̂ A

(5.50)

and
V̂φ(B) /~

∆x AB
∆x AB
= κA λ t −
, x A x̂ A λ t −
, x A x̂B
+κB hλ (t, x)| x̂B |λ (t, x)i x̂ A .

(5.51)

119

...

Figure 5.2: Alice and Bob’s optomechanical setups. Note that i = A, B.
5.3.2

Stochastic Schroedinger Equation

Denote Alice and Bob test masses’ center of mass joint quantum state at time t by
|ψ (t)i. We will obtain a differential equation for |ψ (t)i by following three steps.
We first unitarily evolve |ψ (t)i under the linear part of the Hamiltonian Ĥ (A) + Ĥ (B) .
Second, we evolve |ψ (t)i under the non-linear part of the Hamiltonian, V̂φ(A) + V̂φ(B) .
Third, we project the outgoing light into eingenstates corresponding to Alice and
Bob’s measurement results at time t. Note that to leading order we are allowed to
separate the first and second step.
We first unitarily evolve |ψ (t)i till time t + dt under Ĥ (A) + Ĥ (B) . Denote the
corresponding evolution operator by ÛL :
ÛL = e

(A)
(B)
−i Ĥtm +Ĥtm dt/~

exp i

α j x̂ j â j,1 x j − t/c dt + O dt 2 .

(5.52)

j=A,B

ÛL will act on |ψ (t)i ⊗ |0i ⊗ |0i where |0i ⊗ |0i is the joint quantum state of the
light driving Alice and Bob’s test masses at time t. We’ve assumed
the driving light

to be in vacuum. Since exp i j=A,B α j x̂ j â j,1 x j − t/c dt is a shift operator, it is
convenient expand |0i ⊗ |0i into eigenstates of â A,2 and âB,2 and then shift them by
−α A x̂ A dt and −αB x̂B dt, respectively. We obtain that
2
dq A dqB
exp −
qi − α j x̂ j dt /2 |ψ (t) , q A, qB i ,
ÛL |ψ (t) , 0, 0i = Ûtm
π 1/4 π 1/4
i=A,B
(5.53)
where |ψ, l, ki is a shorthand for |ψi ⊗ |li ⊗ |ki, with |li (|ki) an eigenstate of â A,2

120
(âB,2 ) , and
Ûtm ≡ e

(A)
(B)
−i Ĥtm +Ĥtm dt/~

(5.54)

We then evolve ÛL |ψ (t) , 0, 0i under
( j)

ÛN L

and obtain that the fully evolved wavefunction is
Û |ψ (t) , 0, 0i = ÛN L Ûtm

2
dq A dqB
exp −
qi − α j x̂ j dt /2 |ψ (t) , q A, qB i .
π 1/4 π 1/4
i=A,B
(5.57)

Finally, we apply the projection operators corresponding to Alice and Bob’s measurements of the phase quadrature of the outgoing light. Denote these operators
by
P̂ j t, x j ≡ δ â j,2 x j − t/c − y j (t)
(5.58)
where j = A, B and y A (t) (yB (t)) is Alice’s (Bob’s) measurement result at time t.
Applying P̂A (t, x A) P̂B (t, xB ) on the LHS of Eq. (5.57), we obtain that
|ψ (t + dt)i ∝ ÛN L Ûtm exp −

yi (t) − α j x̂ j dt

/2 |ψ (t)i .

(5.59)

i=A,B

y A (t) and yB (t) follow a stochastic evolution which we can quantify by noting that
in the Heisenberg picture,
b̂i,2 (t) = αi x̂i (t) + â2 (t) ,

i = A, B

(5.60)

so y A (t) and yB (t) are the Gaussian random processes
dWi (t)
yi (t) = αi h x̂i i (t) + √
2dt
dWi (t)
dyi (t) = αi h x̂i i (t) dt + √
where i = A, B and the dWi are Wiener increments.

(5.61)
(5.62)

121
To get a Stochastic Schroedinger equation, we will have to expand all the exponentials
in Eq. (5.59) to order dt. We obtain
d |ψ (t)i
Õ Ĥ ( j)
Õ α2j
2
tm
|ψ (t)i dt +
= −i
x̂ j − x̂ j (t) |ψ (t)i dt −
j=A,B
j=A,B
dW j (t)
α j x̂ j − x̂ j (t) |ψ (t)i √
−i×
j=A,B
!+
xk − x j
xk − x j
κj λ t −
, x j x̂ j λ t −
, x j x̂ j |ψ (t)i dt(5.63)
j,k=A,B
where we’ve substituted Eq. (5.61) into Eq. (5.59), Taylor-expanded the exponentials to leading order in dt, and we’ve used that dWi2 = dt for i = A, B. |λ (t, x)i
can be interpreted as the estimate of an observer at (t, x) of |ψ (t)i, given that the
observer has access to the entire measurement record in the past light cone of (t, x).
We can also obtain a stochastic Schroedinger equation for |λ (t, x A)i and |λ (t, xB )i.
Consider |λ (t, x A)i, it only incorporates measurements in the past light cone of
(t, x A). This means that after t − ∆x AB /c we no longer assume that Bob’s test mass
is monitored, it just follows its own free evolution. It also means as we change t,
|λ (t, x A)i for instance, follows a different stochastic Schroedinger equation, because
with each time increment dt, we receive information about Bob’s measurement
results in the past at time t + dt − ∆x AB /c. Let |λz (t, x)i represent a quantum
trajectory that at time z = t equals |λ (t, x)i. |λz (t, x A)i follows the stochastic
Schroedinger equation
d |λz (t, x A)i
Õ Ĥ ( j)
α2A
tm
|λz (t, x A)i dt +
( x̂ A − h x̂ Ai λ (z))2 |λz (t, x A)i dt
= −i
j=A,B

(5.64)

αB2
+ ( x̂B − h x̂B i λ (z))2 θ̃ (t,x A) (z, xB ) |λz (t, x A)i dt
dW A (z)
−α A ( x̂ A − h x̂ Ai λ (z)) |λz (t, x A)i √
dWB (z)
−αB ( x̂B − h x̂B i λ (z)) θ̃ (t,x A) (z, xB ) |λz (t, x A)i √
!+
xk − x j
xk − x j
−i
κj λ t −
, x j x̂ j λz t −
, x j x̂ j |λz (t, x A)i dt
j,k=A,B

122
where h i λ (z) is to remind the reader that an expectation value is taken over
λz (t, x A) at time z, and θ̃ (t,x) (t 0, y) is non-zero only if (t 0, y) is in the past light cone
of (t, x):
 1 t − t 0 − |y−x|
c ≥ 0
θ̃ (t,x) (t , y) =
(5.65)
 0 otherwise
A similar expression exists for |λz (t, xB )i.
A concise way to represent Eq. (5.64) is
d |λz (t, x A)i =

 (−iH (z) + L (z)) |λz (t, x A)i

z ≤ t − ∆x AB /c

 (−iH (z) + L A (z)) |λz (t, x A)i

z > t − ∆x AB /c

(5.66)

H is a unitary differential operator, L is a differential operator that incorporates both
Alice and Bob’s measurements and L A incorporates just Alice’s measurements. In
particular,
!+
( j)
xk − x j
xk − x j
© Õ Ĥtm
κ j λz t −
, x j x̂ j λz t −
, x j x̂ j ® dt
H (z) =
j,k=A,B
« j=A,B
L (z) = L A (z) + L B (z)
(5.67)
αi2
dWi (z)
( x̂i − h x̂i i (z))2 dt − αi ( x̂i − h x̂i i λ (z)) √
Li (z) =
i = A, B.
(5.68)
We can then calculate that for any Hermitian operator Ô that (for example)
d Ô λ = i

H, Ô

Ô, L̃

+ W̃ Ô W̃ λ

(5.69)

where the expectation value is with respect to for example |λz (t, x A)i, and
L̃ =

L

z ≤ t − ∆x AB /c

LA

z > t − ∆x AB /c

(5.70)

Moreover,
W̃ =

 WA + WB

z ≤ t − ∆x AB /c

 WA
z > t − ∆x AB /c
Wi = −αi ( x̂i − h x̂i i λ ) i = A, B.

(5.71)
(5.72)

123
5.3.2.1

Analogy to feedback

We will show that the Stochastic Schroedinger equation (5.63) is indistinguishable
from evolution under a linear Hamiltonian and feedback. Tracking the evolution of
|ψ (t)i doesn’t tell us if Alice and Bob’s test masses are evolving under a non-linear
quantum-mechanical theory, or if Alice and Bob are applying feedback forces on
their test masses. The last term in Eq. (5.63) can be interpreted as a feedback
force. Alice (Bob) can calculate |λ (t, x A)i and |λ (t − ∆x AB /c, xB )i (|λ (t, xB )i and
|λ (t − ∆x AB /c, x A)i) and then craft a feedback scheme that would mimic Eq. (5.50)
(Eq. (5.51)).
5.4

Signature of SN with the causal-conditional prescription

In this section, we calculate the signature of the Schroedinger-Newton theory in a
torsion pendulum experiment. As derived in Ref. [8], when an object has its center
of mass’ displacement fluctuations much smaller than fluctuations of the internal
motions of its constituent atoms, then its center of mass, has the following non-linear
Hamiltonian
p̂2
2 2
( x̂ − h x̂i)2
+ Mωcm
(5.73)
x̂ + MωSN
Ĥcm =
2M 2
where M is the test mass’ mass, ωcm is the center of mass’ resonant frequency and
ωSN is a frequency scale that is determined by the matter distribution of the object.
For materials with single atoms sitting close to lattice sites, we have
ωSN =

Gm 3
√ ∆xzp
6 π

(5.74)

where m is the mass of the atom, and ∆xzp is the standard deviation of the crystal’s
constituent atoms’ displacement from their equilibrium position along each spatial
direction due to quantum fluctuations [5]. We will show if the phase quadrature of
the outgoing light is measured, then the Schroedinger-Newton theory will predict a
deviation from the predictions of standard quantum mechanics in the fluctuations of
measured observable at the frequency
ωq ≡

2 + ω2 .
ωcm
SN

(5.75)

Hereafter, with the analogy of NLQM to quantum feedback established in subsequent
sections, we will treat the total system-environment Hamiltonian as a quantum
feedback (linear) Hamiltonian. Consequently, we can present the torsion pendulum’s

124
(unconditional) equations of motion in the Heisenberg picture:
p̂
( x̂ − h x̂i) + α â1 + fˆzp + fcl,
∂t p̂ = −Mωcm
x̂ − γm p̂ − MωSN

∂t x̂ =

(5.76)
(5.77)

b̂1 = â1,

(5.78)

b̂2

(5.79)

= â2 + x̂,

where α is defined in Eq. (5.44), â1,2 are the perturbed incoming quadrature light
fields around a large steady state, and similarly b̂1,2 are the perturbed outgoing light
field quadratures (refer to section 2 of [1] for details). fcl (t) is the classical thermal
bath fluctuation force. In a single realization of the experiment, it is a deterministic
force but its correlation function has an ensemble average of (in the high-temperature
limit)
(5.80)
fcl (t) fcl (t 0) = 2mγm k BT δ (t − t 0)
where T is the thermal bath’s equilibrium temperature, and γm = ωcm /Q is the
oscillator’s damping rate and Q is the test mass’ quality factor. fˆzp (t) is the quantum
thermal bath’s fluctuation force originating from the zero-point fluctuations of the
bath’s modes. are given by Since the torsion pendulum is strongly driven by the
driving light, the quantum fluctuations of α â1 dominate over fˆzp and fˆzp will be
ignored for the rest of this section.
From Eq. (7.43), it is clear that x̂ (t) has a non-zero expectation value. We remove
this expectation value by linearizing the system. We replace any operator ô (t) with
its expected value and a small perturbed part (note that we use the same ô to denote
its perturbed part):
ô (t) → hô (t)i + ô (t) .
(5.81)
By using Eqs. (7.43-5.77) and that hâ1 i = fˆzp = 0, we determine that
h x̂ (t)i = fcl (t)
M ∂t2 + γm + ωcm

(5.82)

which can be Fourier transformed to determine that
h x̂ (ω)i = G cm (ω) fcl (ω)

(5.83)

where G cm (ω) is the classical response function of x̂ to external forces
−1
G cm (ω) ≡ M −1 ωcm
− ω2 − iωγm

(5.84)

125
Therefore, in the time-domain,
∫ t

G cm (t − z) fcl (z) dz + x̂ (t) ,

x̂ (t) →

(5.85)

−∞

b̂2 (t) →

∫ t

G cm (t − z) fcl (z) dz + b̂2 (t) .

(5.86)

−∞

We then move to the Fourier domain for the linearized center of mass position
operator
x̂ (ω) = G q (ω) α â1 (ω) + fˆf b (ω)
(5.87)
where G q (ω) is the response function of x̂ to external forces
−1
G q (ω) ≡ M −1 ωq2 − ω2 − iωγm ,

(5.88)

and, in anticipation to the analogy to feedback, we’ve defined (in the Fourier domain)
fˆf b (ω) ≡ C (ω) b̂2 (ω)

(5.89)

where C (ω) is a complex function that needs to be chosen in such a way that
h x̂i .
fˆf b (ω) = MωSN

(5.90)

Furthermore, from Eq. (5.77), the non-linear force that we are mapping to the
2 h x̂i appears as a c-number, whereas we’ve introduced
formalism of feedback, MωSN
it as an operator in Eq. (5.87). This is because when our system is monitored, h x̂i
is a conditional mean that depends on the measurement record. As a result, a
feedback force that is proportional to h x̂ (t)i is promoted to a quantum operator in
the Heisenberg picture because the measurement record is stochastic and some its
randomness comes from the quantum fluctuations of the measured operator b̂2 given
by Eq. (5.79). More details about feedback forces in the Heisenberg picture can be
found in Chapter 5 of Ref. [7].
To obtain the conditional mean of x̂, we will show that all we need to calculate is the
projection of x̂ (t) with no feedback, which we call x̂0 (t), onto the subspace spanned
(z), for z ≤ t. x̂0 and b̂(0)
by the b̂2 (z) with no feedback, which we call b̂(0)
2 are
x̂0 (ω) = G q (ω) (α â1 (ω) + fcl (ω)) ,
x̂0
b̂(0)
â

(5.91)
(5.92)

126
so that
x̂ (ω) = x̂0 (ω) + G q (ω) fˆf b (ω) ,
(ω)
b̂2 (ω) = b̂(0)
G q (ω) fˆf b (ω) .

(5.93)
(5.94)

As is shown in Ref. [6], causally projecting x̂0 onto b̂(0)
2 entails expressing x̂0 as
(ω) + R̂0 (ω)
x̂0 (ω) = K (ω) b̂(0)

(5.95)

with K (t) chosen in such a way that
for all t 0 ≤ t. Eq.
(ω) is
K (ω) b̂(0)

(t
R̂0 (t) b̂(0)
=0

(5.96)

(5.96) guarantees that the inverse Fourier transform of
∫ t
−∞

(z) dz.
K (t − z) b̂(0)

(5.97)

Substituting Eq. (5.95) and Eq. (5.89) into Eq. (5.93), we obtain
(ω) + G q (ω) C (ω) b̂2 (ω) + R̂0 (ω) .
x̂ (ω) = K (ω) b̂(0)

(5.98)

Using Eq, (5.94) and Eq. (5.89), we can express b̂(0)
2 in terms of b̂2 :
(ω)
(ω)
(ω)
b̂2 (ω) .
b̂(0)

(5.99)

Substituting back into Eq. (5.98), we obtain
K (ω) 1 − G q (ω) C (ω) + G q (ω) C (ω) b̂2 (ω) + R̂0 (ω)
≡ X (ω) b̂2 (ω) + R̂0 (ω) .
(5.100)

x̂ (ω) =

Assuming that b̂2 and b̂(0)
2 are causally related, then Eq. (5.96) implies that

(t
R̂0 (t) b̂(0)
=0

(5.101)

for all t 0 ≤ t. As explained in Ref. [6], this entails that
h x̂ (t)i =

∫ t
X (t − z) b2 (z) dz.
−∞

(5.102)

127
Therefore, using Eq. (5.90), we must have that
C (ω)
= X (ω)
MωSN

(5.103)

which, using the definition of X (ω), implies
! −1
α
K (ω) − 1 G q (ω)
K (ω) .
C (ω) =
MωSN

(5.104)

Note that we need to ensure that C (t) contains no delta functions because causal
feedback can only depend on the measurement result up to the time when the
feedback is applied.
Substituting Eq. (5.104) into Eq. (5.94), and using Eq. (5.86), we determine that
the unlinearized b̂2 is
αK (ω) /~
(0)
(ω)
b̂2 (ω) = 1 + −1
b̂
G cm (ω) fcl (ω) .
(5.105)
−1
G q (ω) / MωSN
To calculate the spectrum of b̂2 , we use that, as shown in Ref. [6], K is
√ ~ q
ω − Ω3
K (ω) = 2 i Ω − ωq2
(ω − Ω1 ) (ω − Ω2 )

(5.106)

where

α4
ωq4 + 2 2 ,
q ~ m q
≡ ± Ω + ωq − i Ω − ωq / 2.

Ω ≡
Ω1,2

(5.107)
(5.108)

If ωSN
ωcm , Q
1 and a weakly driven test mass
ωSN

(5.109)

we can show that b̂2 (ω)’s one-sided spectrum is around ωq (after taking an ensemble
average)
2 ! −1
1 + βΓ2 − 1 + 4
(5.110)

128
which is a dip of width
∆≡

M~ωq

(5.111)

and βΓ2 is the thermal noise background around ωq :
β ≡

α2
M~γm ωq

γm2 ωq2
k BT0
≡ 2
~ωq γm2 ωq2 + ωSN

(5.112)
(5.113)

129
Bibliography
[1] Yanbei Chen. Macroscopic quantum mechanics: theory and experimental concepts of optomechanics. Journal of Physics B: Atomic, Molecular and Optical
Physics, 46(10):104001, 2013.
[2] C. C. Gan, C. M. Savage, and S. Z. Scully. Optomechanical tests of a schrödingernewton equation for gravitational quantum mechanics. Phys. Rev. D, 93:124049,
Jun 2016.
[3] André Großardt, James Bateman, Hendrik Ulbricht, and Angelo Bassi. Optomechanical test of the schrödinger-newton equation. Phys. Rev. D, 93:096003, May
2016.
[4] B. Helou and Y. Chen. Different interpretations of quantum mechanics make
different predictions in non-linear quantum mechanics, and some do not violate
the no-signaling condition. ArXiv e-prints, September 2017.
[5] Bassam Helou, Jun Luo, Hsien-Chi Yeh, Cheng-gang Shao, B. J. J. Slagmolen,
David E. McClelland, and Yanbei Chen. Measurable signatures of quantum
mechanics in a classical spacetime. Phys. Rev. D, 96:044008, Aug 2017.
[6] Helge Müller-Ebhardt, Henning Rehbein, Chao Li, Yasushi Mino, Kentaro
Somiya, Roman Schnabel, Karsten Danzmann, and Yanbei Chen. Quantumstate preparation and macroscopic entanglement in gravitational-wave detectors.
Phys. Rev. A, 80:043802, Oct 2009.
[7] Howard M. Wiseman and Gerard J. Milburn. Quantum Measurement and
Control. Cambridge University Press, 2009.
[8] Huan Yang, Haixing Miao, Da-Shin Lee, Bassam Helou, and Yanbei Chen.
Macroscopic quantum mechanics in a classical spacetime. Phys. Rev. Lett.,
110:170401, Apr 2013.

130
Chapter 6

EFFECTIVE MODES FOR LINEAR GAUSSIAN
OPTOMECHANICS. I. SIMPLIFYING THE DYNAMICS
Abstract
For a linear optomechanical system with a finite number of internal modes that interacts with an environment with an infinite number of internal modes, we show that
the interaction can be reduced to one with finite degrees of freedom. Specifically,
the optomechanical interaction can be considered as a scattering process, during
which Heisenberg operators of the incoming environment modes, plus the n initial
optomechanical modes (which correspond to 2n operators), are linearly transformed
into the Heisenberg operators of the outgoing environment modes, plus the n final optomechanical modes. In general, one can divide the incoming environment
modes into n interacting incoming modes and an infinity of non-interacting incoming modes (with operators from different modes commuting with each other), and
the outgoing environment modes into n interacting outgoing modes and an infinity
of non-interacting outgoing modes (with operators from different modes commuting with each other). The final optomechanical modes and the operators of the n
interacting outgoing modes depend only on the n interacting incoming modes and
the initial optomechanical modes. On the other hand, operators of the outgoing
non-interacting modes only depend on the incoming non-interacting modes. In this
way, the optomechanical interaction can be regarded as only taking place between
the interacting modes and the optomechanical system. Constructions of such interacting modes have been proposed in other works to analyze quantum engineering
protocols in simple optomechanical setups, but in this paper we demonstrate that
such interacting modes exist in general. We note that since the annililation operators
of the interacting modes can depend on both the annihilation and creation operators
of the original incoming modes, the interacting modes can be squeezed, and so
contain excitations, even when the initial state of the environment is at vacuum. As
a result, even though the interacting and non-interacting modes do not scatter into
each other, they may be statistically correlated, in a way that depends on the initial
state of the environment.

6.1

Introduction

Optomechanics studies the interaction of light with mechanical systems through
radiation pressure, and has many applications. Optomechanical setups can be a
high precision sensor of, for example, gravitational waves [2], or a transducer which
converts signals in the microwave regime to the optical regime [1]. They can also

131
be used to test alternative theories of quantum mechanics such as collapse models
[9].
All of these applications are mediated through light. Pulsed light has been of interest
lately because of proposals to prepare mechanical systems in non-Gaussian quantum
states. Hofer et al. proposed a scheme for entangling the center of mass mode of
a test mass with an outgoing light mode [5]. Vanner et al. developed a toolset
for transforming a test mass’ quantum state into any desired target state [10], and
Galland et al. proposed a post-selection scheme for preparing a test mass in a single
phonon state [3] which Hong et al. experimentally realized in Ref. [6].
Pulsed optomechanics has two appealing features. It can be resilient to thermal
loss, because a protocol’s duration can be designed to be much shorter than the
timescale of thermal dissipation. Moreover, because of the pulses’ finite duration,
protocols which are unstable in the continuous wave regime (such as those involving
blue-detuned light) are stable in pulsed optomechanics.
The pulses’ finite duration also entails that the initial state of the mechanics could
matter, and so the Fourier regime would be less useful because operators at different
frequencies would no longer be independent. We will have to construct a new set of
convenient effective modes.
Effective modes have been proposed in Refs. [5, 10, 3] to help analyze their proposed
quantum engineering protocols. In this article, we show that these effective modes
interact only with the system modes, and that no other environment degree of
freedom interacts with these modes or the system modes. We also present a general
method for constructing a set of effective modes which summarize the interaction
of a system of interest with the environment. If the joint system and environment
Hamiltonian is linear (equivalently, quadratic), then we can show that a system with
n degrees of freedom interacts only with n effective ingoing modes which evolve
into n effective outgoing modes, as is shown in Fig. 6.1.
This article is outlined as follows. In Sec. 6.2, we rigorously justify how the effective
modes were used in Ref. [5]. This example will serve as an introduction to the
general formalism presented in Sec. 6.3.
6.2

Effective modes for an optomechanical setup driven by pulsed bluedetuned light

In Ref. [5], Hofer et al. propose a protocol for entangling a mechanical oscillator
with an outgoing light pulse. We show their proposed setup in Fig. 6.2 where a
blue-detuned pulse impinges on a cavity with one movable mirror.

132

Figure 6.1: An optomechanical system with n degrees of freedom interacts with
(1)
(n)
n effective input modes, Âin
through Âin
, which evolve into n effective output
(1)
(n)
modes, Âout through Âout . The operators b̂1 through b̂n represent system degrees of
freedom.
In general, investigating the entanglement of an outgoing light pulse with the center
of mass mode of a test mass at a particular time is in general difficult because the
light modes form a continuum. For example, Miao et al. perform sophisticated
calculations in Ref. [8] to obtain the total entanglement between a mechanical mode
and the outgoing light modes, and to obtain the effective optical mode that the test
mass is most entangled with.
For the simple setup shown in Fig. 6.2, Hofer et al. postulate that the test mass
(H)
interacts with a single effective optical ingoing and outgoing mode, Âin
and Â(H)
out

133
respectively, where
(H)
Âin

Â(H)
out

∫ τ
2G
dt e−Gt âin (t)
1 − e−2Gτ 0
∫ τ
2G
dt eGt âout (t) .
2Gτ
−1 0

(6.1)
(6.2)

τ is the length of the pulse, âout (t) are the outgoing light modes, and
G≡

α2

(6.3)

α is the enhanced optomechanical coupling (and is defined by Eq. (2) of [5]), and
κ is the light amplitude decay rate from the cavity. Note that we’ve ignored thermal
noise, and we are in a rotating frame of +ωm for the mechanical mode and −ωm for
the cavity and incoming light modes (refer to section II.A of [5] for more details).
ωm is the mechanical oscillator’s resonant frequency. Moreover, we assume that we
are in the resolved sideband regime α
ωm
κ. This regime allows us to apply
the rotating wave approximation which approximates the interaction Hamiltonian
between the center of mass mode, âm , and the cavity mode, âc , to
~α

† †
âm âc + âm
âc

(6.4)

Since α
ωm
κ, the cavity is then adiabatically eliminated.
Hofer et al. claim that at time τ the mechanical oscillator is only entangled with
Âout . They show that Âin and Âout satisfy the following equations of motion
(0) ,
Âout = −eGτ Âin − i e2Gτ − 1âm
âm (τ) = eGτ âm (0) + i e2Gτ − 1 Âin

(6.5)
(6.6)

which represent pure two-mode squeezing between the mechanical oscillator and
the outgoing light mode Âout . However, they do not show that no other effective
output mode than Âout interacts with Âin or âm . In this section, we prove that âm
and Âin interact with each other and with nothing else .

134

...

Figure 6.2: The setup proposed by Hofer et al. in Ref. [5]. The cavity has a single
movable mirror with a center of mass mode denoted by âm . The incoming light
pulse, shown in dashed blue, is blue-detuned and of length τ. Note that the ingoing
and outgoing light modes, âin (t) and âout (t) respectively, form a continuum but we
show them as discrete modes for simplicity.

6.2.1

Structure of the equations of motion

We start the proof with the (approximate) equations of motion for the modes âm (t)
and âout (t):
α 2 †
α2
âm (t) + i √ âin (t),
∂t âm (t) =
√ !
α 2 †
âout (t) = 1 − √ âin (t) − i √ âm (t).

(6.7)
(6.8)

Since these differential equations are linear, they imply that both âm (t) and âout (t)
are linear combinations of the incoming light âin (t) and the initial state of the
mechanical oscillator âm (0). As a result, we can write the solution to Eqs. (6.7-6.8)
in matrix form:
w = Mv,
(6.9)

135
where w contains the outgoing light modes and the center of mass mode at time τ,
© ãout (0) ª
ãout (dt) ®
..
®≡
w =
ãout (τ) ®
†
(τ)
â

âout
(τ)
âm

(6.10)

v contains the input light and the mechanics at the initial time t = 0,
© ãin (0) ª
ãin (dt) ®
..
®≡
v =
ãin (τ) ®
†
(0)
â

âin
âm (0)

(6.11)

Note that to simplify the exposition of the proof, we’ve discretized the dynamics by
dividing the period τ into N time steps of length dt = τ/N. The limit N → ∞ can
be easily taken at the end of the proof. The ãin and ãout are normalized with 1/ dt
ãin ≡ âin / dt,

ãout ≡ âout / dt,

(6.12)

so that their commutation relation is normalized to unity (see Eq. (6.18)).
The matrix M describes the transformation of v into w, as is depicted in Fig. 6.3.
(τ) and âout (t). Because the
To obtain the entries of M, we need to solve for âm
equations of motion (6.7-6.8) are linear, âm (τ) and âout (t) are of the form

(0) +
β âm

∫ τ

dt g ∗ (t) âin (t) ,
∫ t
âout (t) = h (t) âm (0) +
dz T (t, z) âin (z) .
(τ)
âm

(6.13)
(6.14)

Consequently, M’s structure is
M≡

T h®
g®† β

(6.15)

where T shows how part of the outgoing fields are mixtures of the incoming fields,
h® and g® show the 2-mode squeezing between the light and the mechanics, and β is
the portion of âm (0) that appears in âm (τ). Note that β is a scalar, T ∈ MN×N is

136
...
...

...
...

Figure 6.3: The transformation of input degrees of freedom to output degrees of
freedom under the matrix M.
® g® are N-sized column vectors. We pick the convention
an N × N matrix, and h,
of writing matrices in capital letters, vectors in lower case form with an overhead
arrow, and scalars in greek letters.
6.2.2

The effective modes that simplify M’s structure

In general, M’s structure is complicated and it would seem like âm interacts with
uncountably many modes. We will show that a simple narrative, that of a mechanical
mode and an optical mode interacting with each other and with nothing else, is
possible.
Consider a new set of input and output modes, which we name ĉin and ĉout respectively, that are a linear combination of the ãin s and ãout s:

ĉin

(6.16)

ĉout

(6.17)

137
where U and V are unitary N × N matrices. They are unitary because we impose
that the effective modes ĉout and ĉin are separate degrees of freedom, just as âout and
âin are:

†i
†i
ĉout/in i , ĉout/in j =
âout/in i , âout/in j
= δi j

(6.18)

for 1 ≤ i, j ≤ N. By substituting ĉin and ĉout into Eq. (6.9) and using the expression
for M in Eq. (6.15), we find that
ĉout
(τ)
âm

UTV † U h®
g®†V †

ĉin
âm (0)

(6.19)

âm (0) and only a single optical mode interact with each other and with nothing else
if Eq. (6.19) is of the form
© Âin ª
© Âout ª
†
†
âm (τ) ®
θ φ 0 âm (0) ®
ª
® ©
ĉout (2) ® = γ β 0 ® ĉin (2) ® ,
®
®
..
..
® «
¬
(N)
(N)
ĉ
ĉ
« in
« out

(6.20)

where θ, φ, γ are scalars and W is a (N − 1) × (N − 1) matrix and
ĉin (1) ≡ Âin,

ĉout (1) ≡ Âout .

(6.21)

The second term on the RHS of Eq. (6.13) implies that âm (0) interacts with just one
effective input mode, so we postulate that:
Âin =

Nin−1
s∫

Nin ≡

∫ τ

dt g ∗ (t) âin (t) ,

(6.22)

dt |g (t)| 2,

(6.23)

where Nin ensures that Âin ’s commutation relation is normalized to unity. In
Appendix 6.5 (see Eqs. (6.113) and (6.117)), we show that Eq. (6.18) entails that
T g® = β∗ h,
h® †T ∝ g®†,

(6.24)
(6.25)

138
so from Eq. (6.14),
Âout =

∫ τ

−1
Nout

dt h∗ (t) âout (t) ,

(6.26)

s∫

Nout ≡

dt |h (t)| 2

(6.27)

interacts only with âm (0) and Âin .
If we show that no other effective output mode than Âout interacts with âm (0) and
Âin , then we would have proven that the test mass’ center of mass mode and one
optical mode interact with each other and nothing else. Consider a mode
∫ τ
Ô ≡

dt l ∗ (t) âout (t)

(6.28)

that commutes with Âout , so in the discrete limit
l®† h® = 0.

(6.29)

When combined with Eq. (6.14), this means that Ô cannot depend on âm (0).
Eq. (6.14) also tells us that Ô depends on the effective input mode l®†T âin , which
commutes with Âin ∝ g®† âin because of Eq. (6.24).
From Eqs. (6.19), (6.22) and (6.26), the effective modes follow the equations of
motion
© Âout ª
© Âin ª
†
†
0 ª âm (0) ®
âm (τ) ® © θ
® ĉ (2) ®®
ĉout (2) ® = k g® k β
(6.30)
® in
®
®,
®
..
..
0 W ¬
® « 0
(N)
(N)
ĉ
ĉ
out
in
where W is an (N − 1) × (N − 1) matrix,

continuous

∫ τ

limit

k®v k ≡ v® v −−−−−−−−→

dt |v (t)| 2

(6.31)

for any vector v®, and
θ=

h® †
h®

g®
k g® k

(6.32)

139
which, by using Eq. (6.24), we simplify to
g® k .
θ = β∗ k hk/k

(6.33)

(τ) satisfy the same commutation
By imposing that Âout and âm
relations as Âin and
® = k g® k, and that k g® k = | β| 2 − 1. The physics we are
(0), we conclude that k hk
âm

interested in is neatly summarized by
Âout = e

| β| Âin +

| β|
−iδ
iδ †
| β| âm (0) + | β| − 1e Âin ,
âm (τ) = e
−iδ

(0)
− 1eiδ âm

(6.34)
(6.35)

where δ ≡ arg β. Eqs. (6.34 - 6.35) describe two-mode squeezing with a squeezing
parameter re−i arg β . r can be formally solved for with the equations
cosh r = | β| ,

sinh r =

| β| 2 − 1.

(6.36)

Notice that to derive the effective modes, we never had to calculate T. Moreover, if
we are only interested in obtaining the equations of motion of the effective modes,
then we only need to know β.
Note that in Appendices 6.5 and 6.6, we follow two different approaches to show
why âm and a single effective optical mode interact with each other and nothing else.
Appendix 6.5 derives the effective modes from the constraints they have to satisfy.
Appendix 6.6 uses group theory to prove the existence of a single effective optical
mode.
We can use the above results to quickly obtain the effective modes mentioned in
Ref. [5]. We first extract h (t) and g (t) by solving for âm (t) and âout (t). Using Eqs.
(6.7) and (6.8),

∫ τ

(t) ,
dt e−G(t−τ) âin
∫ t
Gt †
âout (t) = −i 2Ge âm (0) − 2G
dz e−Gz âin (z)
√ !
+ 1 − √ âin,

âm (τ) = e âm (0) + i 2G
Gτ

and so
β = eGτ,

h (t) = −i 2GeGt,

g (t) = i 2Ge−G(t−τ),

(6.37)

140
with
® = k g® k =
k hk

| β| 2 − 1 =

e2Gτ − 1.

(6.38)

dt e−Gt âin (t)
Âin = √
1√− e−2Gτ 0
∫ τ
i 2G
Âout = √
dt eGt âout (t) ,
2Gτ
−1 0

(6.39)

Using Eqs. (6.22) and (6.26), we determine that
−i 2G

∫ τ

(6.40)

which agree with Eqs. (6.1) and (6.2) up to an irrelevant phase factor. Moreover,
the equations of motion for these effective modes are

(0) ,
Âin + e2Gτ − 1âm
Gτ
âm (τ) = e âm (0) + e2Gτ − 1 Âin

Âout = e

Gτ

(6.41)
(6.42)

which agree with the results of Ref. [5].
6.2.3

Applying our framework to heralded single-phonon preparation

Ref. [3] shows that a weak blue-detuned pulse can be used to prepare a mechanical
oscillator’s center of mass mode âm in a Fock state. As indicated by Eqs. (6.416.42), the pulse squeezes âm with Âout . Ref. [3] assumes that âm is initially almost
in vacuum and that the pulse is weak, so âm (τ) and Âout ’s joint quantum state is
approximately |0, 0i + |1, 1i. The outgoing light is measured by a photodetector. If
it clicks we infer that the center of mass mode must be in |1i. Such a measurement
scheme is usually difficult to analyze analytically because the detector measures
a nonlinear operator, the photon number operator. Fortunately, in the case of the
simplified setup discussed in [3], the analysis can be done analytically because the
effective modes given by Eqs. (6.39-6.40) have a simple form. In this section, we
rigorously derive the results of [3].
The (unnormalized) conditional state of an optomechanical setup under continuous
measurement is formally given by
|ψc i = P̂n Ûn ... P̂2 Û2 P̂1 Û1 |ψini i ,

(6.43)

where |ψini i is the initial state of the system and environment. We take the initial
state of the incoming light to be vacuum. Ûi is the unitary time evolution operator
from ti−1 to ti , and P̂i is the projection operator corresponding to a click or no click

141
at time ti . To simplify our analysis, we’ve taken the discrete limit and assumed that
ti and ti−1 are separated by a time interval dt. Moreover, we ignore the possibility
of multiple clicks at the same time because in the continuum limit the probability
of two clicks at the same time is 0.
We express the projection operators in the Heisenberg picture:
†
P̂iH = Ûi ...Û1 P̂i Ûi ...Û1 ,

(6.44)

which allows us to rewrite Eq. (6.43) to
|ψc i = Ûτ P̂nH ... P̂2H P̂1H |ψini i ,

(6.45)

where Ûτ is the total time evolution operator from t = 0 till t = τ:
Ûτ = Ûn ...Û2 Û1 .

(6.46)

The Heisenberg projection operator associated with a single click at time ti is
P̂ti = (|0i h0|) (τ) ⊗ .... ⊗ (|0i h0|) (ti+1 ) ⊗ (|1i h1|) (ti )
⊗ (|0i h0|) (ti−1 ) ⊗ .... ⊗ (|0i h0|) (0)
= âout (ti ) |0i h0| ⊗ Im âout (ti ) ,

where Iˆm is the identity
operator associated with the mechanical center of mass
mode. |0i h0| ⊗ Iˆm is a shorthand for the projection operator (in the Heisenberg
picture) associated with no clicks for the duration of the experiment. We then
express âout (ti ) in terms of the effective modes defined by Eq. (6.17):
âout (ti ) =

u jti ĉout ( j) ,

(6.47)

where u jti is a scalar and 1 ≤ j ≤ N. Substituting âout (ti ) into P̂ti , we obtain
P̂ti =

(ti ) |0i h0| ⊗ Iˆm ĉout (l) .
ulti âout

(6.48)

Using Eq. (6.45), the conditional state associated with a click at time ti is
ψti = Ûτ P̂ti |ψini i .

(6.49)

142
The optical part of |ψini i is vacuum so when ĉout (l) acts on |ψini i, only the l = 1
term, Âout , which is a linear combination of optical annihilation operators and a
mechanical creation operator, will result in a non-null state. As indicated by Eq.
(6.30), the remainder of the effective optical modes consist of only annihilation
operators, so they will not affect our analysis and will be ignored.
Let us assume that the center of mass mode is initially in vacuum, then when the
(0) component of Âout , given by Eq. (6.34), acts on |ψini i, it yields:
âm

P̂ti |ψini i = u1ti | β| 2 − 1
(ti ) |0i h0| ⊗ Iˆm âm
(0) |ψini i
×âout
= u1ti | β| 2 − 1
(ti ) |0i h0| ⊗ Iˆm |1, 0i .
×âout

(6.50)

We then expand Iˆm ,

|0i h0| ⊗ Iˆm

Õ1

(τ)
âm

(|0, 0i h0, 0|)H âm
(τ) .

(6.51)

l (τ) acts on |1, 0i, it can contain at
Substituting Eq. (6.51) in Eq. (6.50), when âm
l (τ) |1, 0i
most one mechanical annihilation operator. Therefore, using Eq. (6.35), âm
simplifies to
(τ) |1, 0i =
âm

| β| − 1

l/2 √

®.
|0,
× |1, li + r
1i
l | β| − 1

(6.52)

Eq. (6.35) indicates that âm (τ) and Âout are 2-mode squeezed with a squeezing

143
parameter re−i arg β . Therefore, |0, 0i H = Ûτ |0, 0i is
|0, 0i H =

ds |s, si

(6.53)

(tanh r)s
ds = (−1)s e−is arg β
cosh r
s/2
| β| − 1
= e−is arg β
| β| s+1

(6.54)

(6.55)

l (τ) |1, 0i, given by Eq. (6.52), only the l = 1 term remains:
When h0, 0| H acts on âm

(τ) 1, 0 = δl1 e−i arg β −e−2i arg β tanh2 r + 1 ,
0, 0 H âm

(6.56)

where we used Eq. (6.36) to express everything in terms of r. Substituting Eq.
(6.56) back into P̂ti |ψini i, we obtain
P̂ti |ψini i = u1ti e

−i arg β

−e

−2i arg β

tanh r + 1 sinh r

(τ) |0, 0i H .
(ti ) âm
×âout

(6.57)

Substituting P̂ti |ψini i into Eq. (6.49), we obtain
(τ) |0, 0i H
(ti ) âm
|ψc i ∝ Ûτ âout

(6.58)

(ti ) âm
(τ) Ûτ† |0, 0i
= Ûτ âout

(6.59)

= |1, 1i .

(6.60)

The mechanical state is in the first Fock state, as expected. Notice that |ψc i doesn’t
depend on ti . The probability of a click at time ti is
ψini P̂ti ψini = u1ti

−e−2i arg β tanh2 r + 1 sinh2 r.

(6.61)

For arg β = 0 (as in Ref. [3]), we have
ψini P̂ti ψini

2 sinh r
= u1ti

cosh r

(6.62)

To calculate the probability of obtaining a single click over the period τ, we integrate
over ti :
sinh2 r
(6.63)
Prob (1 click) =
cosh4 r

144
where we have used that (in the continuum limit)
∫ τ

dt |u1 (t)| 2 = 1

(6.64)

because (as explained in Sec. 6.2.2) the transformation matrix relating âout and ĉout
is unitary, so its rows and columns are normalized to 1.
Finally, we note that a similar analysis can be performed to show that the test mass’
quantum state conditioned on n clicks is |ni, and the probability of measuring n
clicks is |dn | 2 = tanh2n r/cosh2 r. In addition, the conditional scheme described in
Ref. [3] can be extended to setups driven by two tones (instead of just a blue-detuned
or red-detuned laser). For instance, consider the setup described in Ref. [7], where a
cavity is driven by two tones with frequencies centered around the cavity’s resonant
frequency. In the good cavity limit, where the cavity linewidth is much smaller
than the mechanical resonant frequency, the authors show that the optomechanical
Hamiltonian can be approximated to be squeezing between a squeezed mechanical
operator and the driving light. As a result, the analysis performed in Ref. [3] and
in this section also applies to the setup considered in Ref. [7]. The only difference
is that the mechanical ladder operators would have to be replaced with squeezed
ladder operators.
6.3

Effective modes for general setups

In this section, we show that a general system with n degrees of freedom interacts
with only n effective environment modes, as shown in Fig. 6.1. For instance, the
cavity optomechanical setup shown in Fig. 6.2 has n = 2 degrees of freedom: the
cavity field (assuming it isn’t adiabatically eliminated) and the test mass’ center of
mass mode. This system would interact with only n = 2 effective modes.
We first introduce the notation and present the general structure of equations of
motion in linear optomechanics. We then show that the system modes and n
effective environment modes interact with each other and nothing else. Finally,
although the constructed effective modes simplify the dynamics between the system
and its environment, we show that they do not, in general, simplify the structure of
the entanglement between the system and its environment.
6.3.1

Setup

Consider a general linear transformation of a collection of modes v to w via M:
w = Mv,

(6.65)

145
with
© ãout (t1 ) ª
..
ã (t = τ) ®
out N
ãout (t1 ) ®
..
†
ãout (tN = τ) ®
â
out
®≡
w =
(6.66)
(τ)
b̂
b̂
..
b̂n (τ)
b̂1 (τ)
..
(τ)
b̂
where τ is any time after the experiment began, ãout and ãin is defined in Eq. (6.12),
and b̂1 through b̂n are the ladder operators corresponding to the degrees of freedom
of the optomechanical system of interest. âout contains the output optical modes
and is an N-size vector (because we’ve discretized time into N intervals), and b̂τ is
an n-size vector. v contains the input optical modes and the system modes evaluated
at the initial time of the experiment t1 :
© ãin (t1 ) ª
..
ã (τ) ®
in
†
ãin (t1 ) ®
..
†
ã (τ) ®
®≡
v = in
(t
b̂
1 1 ®
..
b̂n (t1 ) ®
†
b̂ (t1 ) ®
1
..
(t
b̂
« n 1 ¬

âin
b̂0

(6.67)

M is not an arbitrary matrix because we require that the output light modes be
separate degrees of freedom in the same way that the input light modes are. We

146
must have
MΩM † = Ω,
where the commutation matrix Ω kl ≡ vk , vl† is given by

Ω=

JN 0
0 Jn

Jz ≡

Iz 0
0 −Iz

(6.68)

(6.69)

(6.70)

For any positive integer z, Jz is

where Iz is the identity matrix of size z.
In contrast to Eq. (6.10), we have included both annihilation and creation ladder
operators in Eq. (6.65). In general, we cannot, as in Eq. (6.4), apply a rotating
wave approximation that would reduce the optomechanical interaction to only the
beamsplitter or squeezing interaction. Moreover, we have assumed that the environment consists only of optical modes. Thermal noise can be incorporated by having
an explicit model of the thermal bath as a collection of harmonic oscillators. Such
models are usually used to derive the Langevin equations of motion [4]. The thermal
bath ladder operators can be included in âin , and their time-evolved counterparts in
âout . If we include them, then the effective modes we derive in this section would
be super-modes consisting of both light and bath ladder operators.
We can decompose M into four blocks:
M≡

T H
G B

(6.71)

where T ∈ M2N×2N relates the output optical fields to the input optical fields,
B ∈ M2n×2n relates the system modes at τ to their initial state b̂0 , H ∈ M2N×2n
specifies the dependence of âout on b̂0 , and G ∈ M2n×2N specifies the dependence
of b̂τ on the input modes âin . Because v and w contain ladder operators and their
adjoints, T must have the following structure
T≡

T1 T2
T2∗ T1∗

∈ M2N×2N ,

(6.72)

where T1 ∈ MN×N indicates how the input optical fields get mixed amongst each

147
other, and T2 carries information about their multi-mode squeezing. Similarly, H
and G must have the following structure:
H =

G† =

h®1 ... h®n h̃1 ... h̃n

(6.73)

g®1 ... g®n g̃1 ... g̃n

(6.74)

where s̃ denotes the dual of a vector s®
s̃ ≡ KN s®∗,
with
KN ≡

0 IN
IN 0

(6.75)

a ’switching’ matrix that switches the top and bottom halves of s®.
6.3.2

The effective modes that simplify M’s structure

From Eqs. (6.65) and (6.71), the system modes interact with the n effective input
modes Gâin . In general, Gâin does not form a conjugate set of modes, as their
commutation relation is not equal to Jn . Consequently, we will choose the n effective
input modes to be
Âin = Sv† Gâin
(6.76)
where Âin contains the annihilation and creation operators of n effective modes, and
Sv ensures that
Âin, Âin
= Jn,
(6.77)
so
Sv† GJN G† Sv = Jn .

(6.78)

Note that since bottom half of Âin has to be the hermitian conjugate of the top half,
Sv has to be of the form
Sv = Sv Kn Sv∗ ,
(6.79)
where Sv is an N × N matrix.
In practice, we can construct Sv† with a symplectic Gram-Schmidt procedure. We

148
pick the first effective input mode to be

Âin

−1 †
= Nin,1
g®1 âin,

−1
Nin,1

Âin

†

−1 †
= Nin,1
g̃1 âin,

g®1† JN g®1 ,

(6.80)
(6.81)

where we’ve assumed that g®1† JN g®1 > 0. If it isn’t, then g̃1† JN g̃1 must be positive,
−1 g̃ † â corresponds to an annihilation operator with its conjugate given by
and Nin,1
1 in
−1
N g® âin . To construct the second effective mode, we find a vector, l®2 that is a
in,1 1

linear combination of g®1 , g̃1 and g®2 , and that JN -orthogonal to g®1 and g̃1 :
l®† JN g®1 = 0,

l®† JN g̃1 = 0.

(6.82)

We then continue this process until we’ve exhausted all the rows of G.
In Appendix 6.7 (see Eqs. (6.153) and (6.156)), we show that Eq. (6.68) implies
that
H JN T =

H JN H − Jn B−1 G,

T JN G† = −H Jn B† .

(6.83)
(6.84)

The first of these equations tells us that the n effective output modes in
Âout = Su† H † JN âout

(6.85)

depend only on the system modes and Âin = Sv† Gâin . Similarly to Sv , Su is a
2n × 2n matrix that ensures that

Âout, Â†out = Jn .

(6.86)

Note that Su has to be of the form
Su =

Su −Kn Su∗

(6.87)

where Su is an N × N matrix, because the bottom half of Âout has to be the hermitian
conjugate of the top half, and JN introduces a minus sign to the bottom half of âin .
If we can show that no other effective output modes than Âout interact with the
system modes and Âin , then we’ve realized the desired narrative of n system modes
and n effective optical modes interacting with each other and nothing else. Consider

149
a mode
Ô ≡ l®† âout

(6.88)

l®† JN2 H = 0 = l®† H

(6.89)

that commutes with Âout , then

as Su is invertible. Using Eqs. (6.65) and (6.71), we deduce that Ô doesn’t depend
on the system modes. Furthermore, Ô depends on the effective input mode l®†T † âin ,
which commutes with Âin because of Eqs. (6.76) and (6.84).
The system modes and Âin and Âout interact in the following way:
Â
© out ª
b̂τ ® =
Ĉ
out

Mn
0 Mother

Â
© in ª
b̂0 ® ,
Ĉ
in

(6.90)

where Ĉin and Ĉout are effective modes of the environment that commute with Âin
and Âout , respectively. Mother tells us how the Ĉin evolve into the Ĉout , and doesn’t
affect the system modes’ dynamics. Mn is
Mn =

Su† H † JN T JN G† Sv Jn Su† H † JN H
GJN G† Sv Jn

(6.91)

where we’ve used Eqs. (6.65) and (6.71), and we’ve used that since Sv† G in Eq.
(6.76) is a symplectic matrix, we have that

Sv† G

−1

= JN G† Sv Jn .

(6.92)

By using Eq. (6.84),we can remove Mn ’s dependence on T:
Mn =

−Su† H † JN H Jn B† Sv Jn Su† H † JN H
GJN G† Sv Jn

(6.93)

Moreover, we can use Eq. (6.154) to eliminate Mn ’s dependence on G, and Eqs.
(6.83) and (6.84) to eliminate Mn ’s dependence on H. We obtain
Mn =

!
−Su† Jn − B† Jn B Jn B† Sv Jn Su† Jn − B† Jn B
Jn − BJn B† Sv Jn

(6.94)

150
Finally, we note that Appendix 6.7 offers an alternative derivation of the effective
modes. We derive the effective modes from the constraints they have to satisfy.
6.3.3

Discussion

The effective modes we have developed in this article simplify the structure of the
dynamics, as shown in Fig. 6.1. However, in general they do not simplify the structure of the entanglement between the optomechanical system and its environment.
We argue for this statement with a simple example.
Consider the hypothetical configuration of effective modes shown in Fig. 6.4,
(1)
where the system degrees of freedom interact only with the effective modes Âin
(n)
in a beam-splitter type interaction that swaps the states of the system
through Âin
and effective environment modes. Assume that the initial state of b̂1 through b̂n is
(n)
vacuum, so Â(1)
out through Âout will be in vacuum.
(1)
(n)
b̂1 (τ) through b̂n (τ) will inherit the state of Âin
through Âin
, which could be
(n+1)
(N)
entangled with Âin through Âin , because the effective modes are, in general, of
the form
∫ τ
∫ τ
(t) ,
(6.95)
dtK (t) âin
Âin =
dt L (t) âin (t) +

where L (t) and K (t) are arbitrary functions.
(n+1)
As a result, even though the system degrees of freedom do not interact with Âin
(N)
through Âin
, they could still be entangled with them. Our formalism does not,
in general, say anything about the entanglement of the system of interest with the
environment.

6.4

Conclusions

We showed that any linear optomechanical system interacts with a finite number of
effective environment modes. If the system has n degrees of freedom, we showed
how to construct n effective input modes and n effective output modes that interact
with the system. Modes of the environment orthogonal to these effective modes
interact with each other and with nothing else.
This article isn’t the first to propose such effective modes. For example, Hofer et al.
used them in Ref. [5] to analyze a protocol for entangling a mechanical center of
mass mode with an outgoing effective optical mode, and Galland et al. used them
in Ref. [3] to analyze a protocol for heralded single-phonon preparation. However,
they did not show that no other environment mode interacts with the system or the
proposed effective modes. We do so in this article. We also showed that such a

151

Beam-splitter
interaction

General interaction

Figure 6.4: A diagram showing a hypothetical beam-splitter interaction that swaps
the quantum states of the system degrees of freedom, b̂1 through b̂n , with that of the
(1)
(n)
effective modes it interacts with: Âin
through Âin
. The remainder of the effective
modes are assumed to have an arbitrary interaction amongst themselves.

formalism can also be used to study heralded phonon states in a cavity driven by
two tones.
Finally, we found that the usefulness of our proposed effective modes might be limited to simple setups because the modes are a linear combination of both annihilation and creation ladder operators. As a result, even when the original environment
modes are in vacuum, the effective modes’ ground state could be a complicated
multi-mode squeezed vacuum. For the same reason, the system modes could be
entangled with modes orthogonal to the constructed effective modes, even though
they do not interact with them. In part II, we will show that a finite number of
effective environment modes are entangled with a linear Gaussian optomechanical
system.
Acknowledgments
We thank Haixing Miao, Yiqiu Ma, Mikhail Korobko, Farid Khalili, Alessio Serafini,
and Klemens Hammerer for useful discussions. This research is supported by
NSF grants PHY-1404569 and PHY-1506453, as well as the Institute for Quantum
Information and Matter, a Physics Frontier Center.

152
6.5

Appendix: Constructing effective modes that simplify the dynamics of a
cavity optomechanical setup interacting with a single sideband of light

To realize the desired narrative of âm (0) and only a single optical mode interacting
with each other and with nothing else, we must satisfy the four conditions implied
by Eq. (6.20):
1. âm (τ) couples to a single input optical mode, which we call Âin ≡ ĉin (1):

g® V ∝

1 0 ... 0

(6.96)

2. Only one outgoing light mode, which we call Âout ≡ ĉout (1), couples to
âm (0):
T
(6.97)
U h® ∝ 1 0 ... 0
3. Âout couples to a single input mode, Âin :
First row of UTV † ∝

1 0 ... 0

(6.98)

4. Only Âout , and no other outgoing optical mode, couples to Âin :
First column of UTV † ∝
6.5.1
6.5.1.1

1 0 ... 0

(6.99)

Satisfying the constraints
Meeting constraints 1 and 2

If we define

© u®1 ª
© v®1 ª
.. ®
. ®
U ≡ . ® , V ≡ .. ® ,
« N ¬
« N ¬
we can rewrite constraint (6.97) to
†®
© u®1 h ª © 1 ª
u®† h® ® 0 ®
® ®
U h = 2. ® ∝ . ® .
.. ® .. ®
® ®
† ®
« N ¬ « 0 ¬

(6.100)

153
® Since U is unitary, this
As a result, we require u®2, ..., u®N to be orthogonal to h.
implies
® h® ,
u®1 = eiφu h/
(6.101)
® is the 2-norm of the
where φu is a phase factor that we are free to choose, and k hk
vector h®
(6.102)
h® ≡ h® † h.
Similarly, if
v®1 = eiφv g®/k g® k ,

(6.103)

then the constraint (6.96) is met. φv is another phase factor that we are free to
choose.
6.5.1.2

Meeting constraints 3 and 4

We’ve fixed Âin and Âout up to a phase factor but there are two more constraints to
meet. Meeting the third and fourth constraints, given by Eqs. (6.98) and (6.99),
require

UTV †

(6.104)

(6.105)

for some scalar θ. Consequently, v®2, ..., v®N must be orthogonal to T † u®1 , and u®2, ..., u®N
must be orthogonal to T v®1 . Since U and V are unitary matrices, this implies
v®1 ∝ T † u®1,

u®1 ∝ T v®1 .

(6.106)

Eqs. (6.101) and (6.103) constrain u®1 and v®1 to be proportional to h® and g® respectively, so
g® ∝ T † h,
h® ∝ T g®.

(6.107)
(6.108)

It would seem that Eqs. (6.107-6.108) cannot be met in general. Fortunately,
by investigating the structure of M, we will show that h® and g® are connected in

154
such a way that Eqs. (6.107-6.108) can be satisfied. M is not allowed to be an
arbitrary matrix because we require that the output light modes be separate degrees
of freedom. Specifically, the modes contained in w must have the same commutation
relation as those in v:

wi, w †j

Mik M jl∗

vk , vl†

≡ Mik Ω kl Ml†j

k,l=1

vk , vl† = Ωi j

(6.109)

where 1 ≤ i, j ≤ N, the commutation matrix Ω kl ≡

Ω≡

IN 0
0 −1

vk , vl†

(6.110)

and IN is the identity matrix of size N. Eq. (6.109) can be conveniently written in
matrix form
MΩM † = Ω.
(6.111)
Notice that M is a SU(N,1) transformation. In Appendix 6.6, we use this to prove
that âm and a single optical mode interact with each other and with nothing else.
By using Eq. (6.111) and the explicit form for M given by Eq. (6.15), we can easily
show that
TT † − h® h® † = IN ,
T g® − β∗ h® = 0,

(6.113)

g®† g® − | β| 2 = −1.

(6.114)

(6.112)

Eq. (6.113) automatically satisfies constraint (6.108). As for constraint (6.107),
using Eq. (6.112), we have
† ®
TT h = IN + h h h = h 1 + h
†®

(6.115)

Using Eq. (6.113), we establish that
TT † h® ∝ T g®.

(6.116)

T, which encodes the dependence of the outgoing optical fields on the ingoing
optical fields, is a causal matrix and so is a lower triangular matrix. Moreover, at

155
each instant of time, part of the input field at the current time is reflected from the
cavity, and so T’s diagonal entries are non-zero. Consequently, T is invertible and
Eq. (6.116) is equivalent to
T † h® ∝ g®,
(6.117)
which satisfies the constraint (6.107).
6.6

Appendix: An alternative proof based on group theory

We can prove that the optomechanical system discussed in Sec. 6.2 interacts with a
finite number of environment modes with a proof that makes use of group theory.
Notice that Ω, defined by Eq. (6.110), has the form of a generalized Minkowski
metric, SU(N,1), with N spatial dimensions. Using Eq. (6.111), we conclude that
M is a generalized Lorentz transformation. Consequently, M can be decomposed
into a pair of pure spatial rotations and a pure boost:
M = R1 B ( χ) R2,

(6.118)

where R1 and R2 are unitary matrices, and
cosh χ
0 ... 0 −eiφ sinh χ ª
1 0 ...
..
.. . .
..
..
B ( χ) =
. .
....
−iφ
cosh χ
« −e sinh χ 0 ... 0

(6.119)

is a boost matrix. Then,

R1−1 w = B ( χ) (R2 v) ,
(6.120)
and the first mode of R1−1 w is pure two-mode squeezed with the mechanics.
Moreover, the first mode of (R2 v) and the test mass’ center of mass mode interact
with each other and with nothing else.
6.7

Appendix: Constructing effective modes that simplify the dynamics of a
general optomechanical setup

We will construct a new set of input and output modes, which we call ĉin and ĉout
respectively, in such a way that only n input and n output modes interact with the

156
system. Let these effective modes be linear combinations of the âin s and âout s:

ĉin

ĉ (1)
© in .
..
ĉin (N) ®
® = V âin,
≡ †
ĉin (1) ®
..
(N)
ĉ
« in

(6.121)

ĉout

ĉ (1)
© out.
ĉout (N) ®
® = U âout,
≡ †
(1)
ĉ
out
..
« ĉout (N) ¬

(6.122)

where U and V are 2N × 2N matrices. To ensure that the ĉout and ĉin are commuting
degrees of freedom (just as âout and âin are), we require
U JN U † = V JN V † = JN ,

(6.123)

where we’ve defined JN in Eq. (6.70). In addition, since the bottom halves of ĉin
and ĉout are adjoints of the top halves, the structure of V must be

V1 V2
V2∗ V1∗

(6.124)

V1 ∈ MN×N contains the contribution of annihilation operators âin to ĉin , while V2
contains the contribution of creation operators âin
to ĉin . An analogous expression
holds for the structure of U.
By substituting ĉin and ĉout into Eq. (6.65), we find that they satisfy
ĉout
b̂τ

UTV −1 UH
GV −1

ĉin
b̂0

(6.125)

where
V −1 = JN V † JN .

(6.126)

157
6.7.1

The constraints

To achieve the desired narrative of the system modes and only n effective optical
modes interacting with each other, we must satisfy four requirements:
1. The modes contained in b̂τ couple to only n effective input modes which,
without loss of generality, we choose to be ĉin (1) through ĉin (n):
GV −1 =

Gn 0 ... 0 Kn G∗n 0 ... 0 ,

(6.127)

where Gn is of size 2n × n, and we’ve defined Kn in Eq. (6.75).
2. Only n effective output modes, ĉout (1) through ĉout (n) , couple to b̂0 :
UH =

HnT

0 ... 0

Hn∗ Kn

0 ... 0

(6.128)

where Hn is of size n × 2n.
3. ĉout (1) through ĉout (n) couple only to ĉin (1) through ĉin (n), and to no other
effective input mode, so the rows of UTV −1 satisfy:
TR 0 TS 0 ,
N + 1 to N + n rows =
TS 0 TR 0 ,
The first n rows =

where TR, TS are n × n matrices.
4. ĉout (1) through ĉout (n), and no other effective output modes, couple to ĉin (1)
through ĉin (n), so the columns of UTV −1 satisfy:
© TR ª
0 ®
The first n columns = ∗ ® ,
TS ®
© TS ª
0 ®
The N + 1 to N + n columns = ∗ ® .
TR ®

158
We summarize requirements 3 and 4 by
TS
© TR
0 anything 0 anything ®
UTV −1 = ∗
®.
TS
TR
anything
anything
6.7.2

(6.129)

Meeting the constraints

To show how to meet requirements (6.127), (6.128) and (6.129), it will be convenient
to make the following definitions:
u®†
© .1 ª
. ®
. ®
† ®
u®n ®
U ≡ † ®® ,
ũ1 ®
. ®
.. ®
ũ
« n ¬

v®†
© .1 ª
. ®
. ®
† ®
v®n ®
V ≡ † ®®
ṽ1 ®
. ®
.. ®
ṽ
« n ¬

(6.130)

with the dual vector ṽi defined by KN v®i∗ (and similarly for ũi ). U and V have this
form because the bottom halves of ĉin and ĉout are adjoints of the upper halves.
With these definitions, and using Eq. (6.123), we must have that with respect to the
operation
h x®, y®i ≡ x®† JN y®,
(6.131)
all the v® j s and ṽ j s are orthogonal to each other, the v®i are normalized to unity and
the ṽi are normalized to −1. If two vectors are orthogonal to each other with respect
to this operation, we will say that they are JN -orthogonal.
6.7.2.1

Meeting the first and second constraints

The entries of GV −1 are
g1† JN ṽ1 ... −®
g1† JN ṽN ª
© g®1 JN v®1 ... g®1 JN v®N −®
..
..
..
..
...
...
®.
g̃
ṽ
...
g̃
g̃
...
g̃
ṽ
n N 1
n N N
n N N ¬
« n N 1

159
Meeting constraint (6.127) means that GV −1 would be of the form
g1† JN ṽ1 ... −®
g1† JN ṽn 0 ... 0 ª
© g®1 JN v®1 ... g®1 JN v®n 0 ... 0 −®
..
..
.. . . ..
..
..
.. . . .. ®
..
..
. .
. . ®.
g̃
...
g̃
...
g̃
ṽ
...
g̃
ṽ
...

Thus, we require that the v®n+1, ..., v®N and their duals be JN -orthogonal to the g®s.
Since the v®s are JN -orthogonal to each other and to their duals, a solution is that v®1
through v®n are a linear combination of the g®s:

v®1 ... v®n

g®1 ... g®n g̃1 ... g̃n

Sv,

(6.132)

where SvT must be a full rank n × 2n matrix. The duals of v®1 ,...,®vn are then given by

ṽ1 ... ṽn

g®1 ... g®n g̃1 ... g̃n

Kn Sv∗ .

(6.133)

Consequently,

v®1 ... v®n ṽ1 ... ṽn

= G † Sv ,

(6.134)

where
Sv ≡

Kn Sv∗

(6.135)

is a 2n × 2n invertible matrix.
Similarly, we can show that constraint (6.128) requires

u®1 ... u®n ũ1 ... ũn

= JN HSu

(6.136)

(6.137)

where
Su ≡

Su −Kn Su∗

is a 2n × 2n invertible matrix.
6.7.2.2

Meeting the third and fourth constraints

We’ve fixed the effective input and output modes up to irrelevant phase factors and
rotations, but we still need to satisfy Eq. (6.129). Specifically, we would like UTV −1

160
to be of the form
UTV −1

(6.138)

... u®1†T JN v®n 0
...
..
..
..
...
... u®nT JN v®n 0
...
...
..
...
anything
...
... ũ1T JN v®n 0
...
..
..
..
...

0 −®
u1†T JN ṽ1
..
..
0 −®
unT JN ṽ1
..
..

...
... −®
u1†T JN ṽn 0
..
..
..
...
... −®
unT JN ṽn 0
...
...
..
...
anything
...
... −ũ1T JN ṽn 0
...
..
..
..
...

−ũ1T JN ṽ1
..

... ũnT JN v®n 0
...
0 −ũnT JN ṽ1
...
..
..
...
anything
...

... −ũnT JN ṽn
...
..
...
...

0 ª
.. ®
. ®
0 ®®
®.
0 ®®
.. ®®
. ®
...
0 ®
anything

Consequently,
1. We require the upper left blue block to be 0:
u®1†T JN v®n+1 = .... = u®1†T JN v®N = 0,
.. .. ..
. . .
u®n†T JN v®n+1 = .... = u®n†T JN v®N = 0,
and so we must have that v®n+1 ...®vN are JN -orthogonal to the vectors T u®1† ...T † u®n .
Since v®1 ...®vn and their duals are JN -orthogonal to v®n+1 ...®vN , we want T † u®i for
1 ≤ i ≤ n to be a linear combination of the v®i s and ṽi s:
T†

u®1 ... u®n

v®1 ... v®n ṽ1 ... ṽn

Rv,

where Rv ∈ M2n×n . Imposing that the lower blue block be 0 requires
ũ1†T JN v®n+1 = .... = ũ1†T JN v®N = 0,
.. .. ..
. . .
ũn†T JN v®n+1 = .... = ũn†T JN v®N = 0,

(6.139)

161
so we must have
T † ũ1 ... ũn
v®1 ... v®n ṽ1 ... ṽn R̃v,

(6.140)

where R̃v ∈ M2n×n must be equal to
R̃v = Kn Rv∗ .

(6.141)

Combining Eq. (6.140) with Eq. (6.139), we require
T†

Rv ≡

Rv Kn Rv∗

u®1 ... u®n ũ1 ... ũn

v®1 ... v®n ṽ1 ... ṽn

Rv, (6.142)

where

(6.143)

must be a 2n × 2n invertible matrix.
2. We require the upper right green block to be 0:
u®1†T JN ṽn+1 = .... = u®1†T JN ṽN = 0,
.. .. ..
. . .
u®n†T JN ṽn+1 = .... = u®n†T JN ṽN = 0.
For the lower right green block to be 0, we need
ũ1†T JN ṽn+1 = .... = ũ1†T JN ṽN = 0,
.. .. ..
. . .
ũnT JN ṽn+1 = .... = ũnT JN ṽN = 0.
All these constraints are satisfied by Eq. (6.142).
3. We require the upper red block to be 0:
u®n+1
T JN v®1 = .... = u®†N T JN v®1 = 0,
.. .. ..
. . .
u®n+1
T JN v®n = .... = u®†N T JN v®n = 0.

Thus, we require that the T JN v®i s be a linear combination of the JN u® j and JN ũ j

162
for 1 ≤ j ≤ n:
T JN v®1 ... v®n
= JN u®1 ... u®n ũ1 ... ũn Ru

(6.144)

where Ru is M2n×n . Imposing that the lower red block be 0 requires
ũn+1T JN v®1 = .... = ũ N T JN v®1 = 0,
.. .. ..
. . .
ũn+1T JN v®n = .... = ũ N T JN v®n = 0.
Eq. (6.144) already satisfies these requirements.
4. We require the upper magenta block to be 0:
u®n+1
T JN ṽ1 = .... = u®†N T JN ṽ1 = 0.
.. .. ..
. . .
u®n+1
T JN ṽn = .... = u®†N T JN ṽn = 0,

so the T JN ṽ j s must be a linear combination of the JN u® j and JN ũ j for 1 ≤ j ≤ n:
T JN ṽ1 ... ṽn
= JN u®1 ... u®n ũ1 ... ũn R̃u,

(6.145)

where R̃u must be equal to Kn Ru∗ . Combining this constraint with Eq. (6.144),
we have
T JN v®1 ... v®n ṽ1 ... ṽn
= JN u®1 ... u®n ũ1 ... ũn Ru,

(6.146)

where
Ru ≡

Ru Kn Ru∗

(6.147)

must be an invertible 2n × 2n matrix. Imposing that the lower magenta block

163
be 0 requires
ũn+1
T JN ṽ1 = .... = ũ†N T JN ṽ1 = 0
.. .. ..
. . .
ũn+1
T JN ṽ2n = .... = ũ†N T JN ṽn = 0

Eq. (6.144) already satisfies these requirements.
Combining constraint (6.142) with constraints (6.134) and (6.136) implies
T JN h1 ... hn h̃1 ... h̃n Su
g®1 ... g®n g̃1 ... g̃n Sv Rv .

(6.148)

Since Su is invertible, we rewrite Eq. (6.148) to
T † JN H ≡ G† Sv Rv Su−1 .

(6.149)

Furthermore, combining constraint (6.146) with constraints (6.134) and (6.136)
implies
T JN g®1 ... g®n g̃1 ... g̃n Sv
= JN h1 ... hn h̃1 ... h̃n Su Ru .

(6.150)

Since Sv is invertible, we rewrite Eq. (6.150) to
T JN G† = HSu Ru Sv−1 .
6.7.2.3

(6.151)

Meeting constraints 3 and 4

Since Ru and Rv are not arbitrary matrices and must be of the form given by Eqs.
(6.147) and (6.143) respectively, it would seem that Eq. (6.149) and Eq. (6.151)
cannot be met in general. By investigating the structure of M, we will show that
H and G are connected in such a way that we can satisfy both Eq. (6.149) and Eq.
(6.151).

164
By substituting Eq. (6.71) into Eq. (6.68), we obtain
T JN T † + H Jn H † = JN ,

(6.152)

T JN G† + H Jn B† = 0,

(6.153)

GJN G† + BJn B† = Jn .

(6.154)

Eq. (6.153) automatically satisfies constraint (6.151) if
Ru = −Su−1 Jn B† Sv .

(6.155)

This equality can be met because the RHS of Eq. (6.155) has the same form as Ru
in Eq. (6.147).
We can also meet constraint (6.149). We use Eqs. (6.152)-(6.154) to show that the
T † JN h®i s are a linear combination of the g®s :
T JN H = G

−1

H JN H − Jn ,

(6.156)

where we assumed that B is invertible. Thus, we can meet Eq. (6.149) if
−1
Rv = Sv−1 B†
H † JN H − Jn Su .

(6.157)

Note that the RHS of the above equation can be shown to be of the same form as
that of Rv in Eq. (6.143).
In summary, the n effective input modes that the system interacts with are
© ĉin (1) ª © v®1 ª
.. ® .. ®
. ® = . ® âin = Sv† Gâin,
®
(n)
ĉ
in
¬ «

(6.158)

(6.159)

where Su and Sv are 2n × n matrices that need to be picked in such a way that the u®s
and v®s are JN -orthogonal, respectively.

165
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166
Chapter 7

EFFECTIVE MODES FOR LINEAR GAUSSIAN
OPTOMECHANICS. II. SIMPLIFYING THE ENTANGLEMENT
STRUCTURE BETWEEN A SYSTEM AND ITS ENVIRONMENT
Abstract
We show that a general linear optomechanical system with n degrees of freedom,
and that is driven by arbitrarily many environment bosonic modes in Gaussian states,
is entangled with only n effective environment modes. We provide a cavity optomechanical system as an example, and quantify its entanglement with the environment.
We also discuss potential applications. A simple entanglement structure allows
us to derive and understand the one-shot quantum Cramer Rao bound in a simple
way, and allows to provide bounds on how well we can perform different statepreparation tasks. If we limit the effective modes to consist of only optical modes
(which experimentalists can probe), then we cannot make any general statements
about the entanglement structure of an optomechanical system with its optical bath.
Nonetheless, the correlation between a system and its optical bath has a simple
structure. A system with n degrees of freedom is correlated with only n effective
optical modes, which in turn are only correlated to another n effective optical modes.
This correlation chain continues ad infinitum.

7.1

Introduction

The present series of articles explores new bases in which to view linear optomechanical systems driven by environment modes in Gaussian states. The most widely
used bases are the time and Fourier bases, which have simple interpretations. For example, the incoming optical bath mode in the time basis, âin (t), represents the light
mode that interacts with an optomechanical system at time t. We can also assign
some meaning to Fourier operators (as defined by the two-photon formalism [3]).
Their spectrum represents the strength of fluctuations at a particular frequency. For
(ω) âout,2 (ω), where âout,2 (ω)
example, the symmetrized expectation value of âout,2
is the outgoing phase-quadrature light operator with frequency ω, characterizes the
degree of fluctuations that a homodyne detector would register at the frequency
ω/2π. Moreover, Fourier operators Dare useful because they
E are, at steady state,
independent at different frequencies: âout,2 (ω1 ) âout,2 (ω2 ) for |ω1 | , |ω2 |.
Hofer et al. and Galland et al. in Refs. [5, 4] constructed a special basis in their
analysis of quantum optomechanical engineering protocols. We were intrigued

167
because it seemed like the optomechanical system they considered, and the effective
environment modes they proposed, interact with each other and with nothing else. In
part I, we investigated these modes in detail, and showed that they can be generalized:
a linear optomechanical system’s modes and a finite number of effective environment
modes interact with each other and with nothing else. Nonetheless, we cannot in
general limit our analysis to these effective modes, because the system could be
entangled with effective modes it doesn’t interact with. In this article, we will show
that an optomechanical system with n degrees of freedom is entangled with only n
environment modes.
We will first set up a general optomechanical system, and introduce the notation we
will use throughout the article. In Sec. 7.2, we then present the phase-space Schmidt
decomposition theorem, which will allow us to directly show that a system with n
modes is entangled with n environment modes. We then provide a cavity optomechanical system as an example, and quantify its entanglement with the environment.
In Sec. 7.4, we discuss potential applications. If we assume we can measure any
environment mode, the simple entanglement structure of a general optomechanical
setup makes it easy to understand the one-shot quantum Cramer Rao bound, and
obtain bounds on well we can carry out different quantum state preparation tasks.
Finally, in Sec. 7.5, we limit our analysis to effective modes that consist of only
optical degrees of freedom, because experimentalists can only control and measure
such modes. Although we cannot make general statements about the entanglement
structure of an optomechanical system with its optical bath, we show that the correlation structure can be reduced to a ’chain’. A system with n degrees of freedom is
correlated with only n effective optical modes, which in turn are only correlated to
another n effective optical modes. The chain continues ad infinitum.
7.2

Setup and notation

Consider the general optomechanical system shown in Fig. 7.1. It consists of n
degrees of freedom with corresponding quadratures
( x̂1, p̂1 ) , ..., ( x̂n, p̂n ) .

(7.1)

This system interacts with an environment consisting of m bosonic fields through a
linear (i.e. quadratic) Hamiltonian. We label the ladder operators corresponding to
the degrees of freedom of each field by â(i)
j , where 1 ≤ i ≤ m labels the field and
j (generally a continuous variable) indexes a particular harmonic oscillator of that
field. To simplify the analysis in this article, we will take the discrete time limit of

168

...

...
...

...

Figure 7.1: General optomechanical setup with n degrees of freedom interacting
with m bosonic environment fields.
the dynamics by dividing the length of an experiment, τ, into N time steps of length
dt = τ/N. The limit N → ∞ can be taken at the end of our analysis.
It will be convenient to normalize all operators such that their commutator is equal
to 2i. Denote the vector of dimensionless system quadratures by
vsys =

x̃1 p̃1 ... x̃n p̃n

(7.2)

vsys ’s commutation matrix is

vsys, vsys

(7.3)

169
where σy is the y−Pauli matrix
0 i
−i 0

σy =

(7.4)

and Ωsys is
Ωsys =

0 1
−1 0

i=1

(7.5)

Similarly, we denote the vector of dimensionless environment quadratures by

venv

venv, j

(7.6)

(7.7)

(i)
for j = 1...m. ã(i)
j,1 and ã j,2 are two orthogonal quadratures of the jth degree of
freedom of the ith field:

†
ã(i)
ã(i)
(i)
ã j,1 ≡

†
ã(i)
ã(i)
(i)
ã j,2 ≡
i 2

(7.8)

where i = 1, ...m and j = 1...N. venv ’s commutation matrix is

venv, venv
= 2iΩenv

(7.9)

where similarly to Ωsys , Ωenv is the direct sum of m × N σy /i. We combine vsys and
venv into a vector v:
v≡

vsys
venv

v, vT ≡ 2iΩtot,

where
Ωtot ≡

Ωsys
0 Ωenv

(7.10)

(7.11)

For example, consider a cavity optomechanical setup with n = 2 modes, which
we show in Fig. 7.2. x̃1 and p̃1 are the normalized center of mass position and

170

...

Figure 7.2: A cavity optomechanical setup. A test mass’ center of mass position
with corresponding quadratures ( x̂1, p̂1 ) is driven by the thermal bath field operators
b̂in (t). The cavity field with corresponding quadratures ( x̂2, p̂2 ) is driven by the
optical field operators âin (t). âin (t) and b̂in (t) then evolve into âout (t) and b̂out (t),
respectively.
momentum operators of a test mass with resonant frequency ωm and mass m
2 x̂1
x̃1 =
∆xzp
where

2 p̂1
p̃1 =
∆pzp

∆xzp =

2mωm

∆pzp =

~mωm

(7.12)

(7.13)

x̃2 and p̃2 are proportional to the amplitude and phase quadratures of the cavity field
x̃2 = x̂2 2.

p̃2 = p̂2 2.

(7.14)

The environment consists of the input optical field âin (t) and its time-evolved counterpart âout (t), and of the thermal bath field b̂in (t) and its time-evolved counterpart
b̂out (t).
7.3

Entanglement structure

The phase-space Schmidt decomposition theorem [2, 1] allows to simplify the entanglement structure of a generic optomechanical system. We first state the theorem,
and then show how its direct application simplifies the state of an optomechanical
setup with n degrees of freedom to a collection of n independent two-mode squeezed

171
system-environment pairs (and the rest of the effective environment modes would
be in vacuum). Finally, we quantify the entanglement of a cavity optomechanical
setup with its environment.
7.3.1

Phase-space decomposition theorem

Because we assume that the dynamics are linear and that the environment is initially
in a Gaussian state (which includes thermal states), the joint system-environment’s
quantum state, |Ψ (t)i, is eventually pure and Gaussian for t
ti where ti is the
initial time of the experiment. As a result, |Ψ (t)i is fully characterized by its mean,
and by its covariance matrix V. We will also assume that initially hvi = 0 which
guarantees that for t
ti , hv (t)i ≈ 0. As a result, |Ψ (t)i’s Wigner function is of
the form
1 T
−1
exp − ν® V (t) ν® ,
(7.15)
W (®
ν, t) = p
det (2πV (t))
where v® is a real vector of the same dimension as v, defined in Eq. (7.10), and v®0s
ith entry, for i = 1...n, corresponds to the degree of freedom in the ith entry of v.
V (t) is of the covariance matrix for v
V (t) =

vvT s =

σsys (t) σcT (t)
σc (t) σenv (t)

(7.16)

where σsys is the system’s covariance matrix, σenv the environment’s covariance
matrix, and σc the cross-correlation between them:
σsys (t) ≡

vsys vTsys s ,

(7.17)

σc (t) ≡

venv vTsys s ,
venv venv

(7.18)

σenv (t) ≡

(7.19)

where
ôlˆ s ≡ ôlˆ + lˆô /2

(7.20)

denotes a symmetric expectation over |Ψ (t)i.
The phase-space Schmidt decomposition theorem [2, 1] states that we can substantially simplify the structure of V by choosing a different basis than vsys for the n
system modes and a different basis than venv for the environment modes. In this
new basis, all effective environment modes, except n of them, are in vacuum. The
n environment modes that are not in vacuum are each in a two-mode squeezed state
with an effective system mode.

172
Denote the new basis of system and environment modes by

wsys ≡
w ≡

wenv ≡

ŝ1,1 ŝ1,2 ... ŝn,1 ŝn,2
wsys
wopt

ê1,1 ê1,2 ê2,1 ê2,2 ...

(7.21)
(7.22)

(7.23)

We choose the modes in w in such a way that they are linear combinations of the
modes in v:
Msys
w = Mv, M ≡
(7.24)
Menv
Moreover, we constrain w to satisfy the same commutation relation as v:

w, wT = v, vT = 2iΩtot .

(7.25)

Using Eq. (8.44), Eq. (8.45) implies that MΩtot M T = Ωtot . Matrices that satisfy
such a relation are called symplectic matrices. Similarly, we can also show that
Msys and Menv are symplectic matrices.
The theorem states that there exists an M such that

Vw ≡

C ≡

S ≡

Sk ≡

C S 0
ww s = MV M ≡ S C 0 ®® ,
« 0 0 I ¬
© C1 0 ... 0 ª
0 C2 ... 0 ®
νk 0
.
® , Ck ≡
.. . .
..
...
n ¬
© S1 0 ... 0 ª
0 S2 ... 0 ®
. . .
®,
.. 0 ®
.. ..
...
n ¬
© νk − 1 q 0
®,
2−1

(7.26)

(7.27)

(7.28)

(7.29)

where I is the identity matrix, νk ≥ 1 and 1 ≤ k ≤ n. In addition, notice that Msys

173
is the symplectic diagonalizing matrix for σsys :
Msys σsys Msys
= C,

(7.30)

so the νi for 1 ≤ i ≤ n are the symplectic eigenvalues of σsys . The Williamson
theorem guarantees the existence of an Msys that diagonalizes σsys into C [7].
7.3.2

A collection of two-mode squeezed states

The structure of Vw in Eq. (8.47) indicates that the system is entangled with only n
effective environment modes. In particular, each of the effective modes described
by wsys forms a two-mode squeezed state with a mode in wenv . This can directly
be seen by forming the covariance matrix of one of the system effective degrees
of freedom, say ŝi,1, ŝi.2 for 1 ≤ i ≤ n, with its correlated effective environment
counterpart êi,1, êi,2 :
σi =

λi

ŝi,1
ŝi,2

êi,1 êi,2

(7.31)

(7.32)

(7.33)

σi indicates that the modes ŝi and êi are two-mode squeezed of degree ri where
cosh 2ri = νi,

(7.34)

as we show in Fig. 7.3.
Because the system-environment state is Gaussian, we can quantify the entanglement
between each effective system mode and its entangled effective environment partner.
Consider the joint density matrix of such a pair, ρ(i)
sys−env where i = 1 . . . n. Since
(i)
ρsys−env is Gaussian, it is separable if and only if its partial transpose with respect
to the system is positive semidefinite. This is called the Peres–Horodecki criterion.
Say σi , given by Eq. (7.31), is the covariance matrix associated with ρ(i)
sys−env , then

174

Figure 7.3: Two-mode squeezing between the modes ŝi and êi for 1 ≤ i ≤ n. The
bottom two graphs show the phase space distribution of both of these modes.

the Peres–Horodecki criterion is equivalent to
σ̃i +

σy 0
0 σy

≥ 0,

(7.35)

where
σ̃i = θσi θ,

(7.36)

is the covariance matrix of the partial transpose of ρ(i)
sys−env with respect to one of
its degrees of freedom.
The logarithmic negativity quantifies the violation of the Peres–Horodecki criterion.
For the two-mode Gaussian state characterized by the covariance matrix σ̃i , where
i = 1 . . . n, the logarithmic negativity is
Ni = max 0, − log η̃−(i)

= max 0, − log 2νi νi − νi − 1 − 1
where η̃−(i) is the smallest symplectic eigenvalue of σ̃i .

(7.37)
(7.38)

175
7.3.3

An example

Consider the cavity optomechanical setup shown in Fig. 7.2. The test mass’ free
Hamiltonian is
P̂2 1
Ĥtm =
+ mωm x̂ 2,
(7.39)
2m 2
where m and ωm are the test mass’ mass and resonant frequency, respectively. The
test mass interacts with the cavity field via the interaction Hamiltonian
Ĥint = ~G0 x̂1 â† â,

(7.40)

where x̂1 is the test mass’ center of mass position, â is the cavity field annihilation
operator (associated with the quadratures x̂2 and p̂2 ), and G0 is the bare optomechanical coupling
ωc
(7.41)
G0 =
ωc is a resonant frequency of the cavity, which we assume is equal to the driving
laser’s frequency ω0 , and L is the length of the cavity. Moreover, the cavity field is
lossy and is driven by a coherent laser:
p
Ĥdrive = i~ 2γ âin â − h.c.

(7.42)

where âin is the ingoing optical field as is shown in Fig. 7.2.
s.s. , which is the covariance matrix for the
At steady state, we can calculate σsys
normalized test mass modes, x̃1 and p̃1 as given by Eq. (7.12), and the normalized
cavity modes, x̃2 and p̃2 as given by Eq. (7.14). Their linearized equations of
motion, in an interaction picture with the cavity’s free Hamiltonian and the ~ω0 part
of the external continuum removed, and in terms of dimensionless parameters only

176
are
∂t˜ x̃1 (t˜) = p̃1 (t˜) ,

(7.43)

p̃1 (t˜) √ 3/2
∂t˜ p̃1 (t˜) = − x̃1 (t˜) −
− 2ΓΘ x̃2 (t˜)
2n
b̃th (t˜) ,
x̃2 (t˜)
ã1,in (t˜) ,
∂t˜ x̃2 (t˜) = −
Γm
Γm
p̃2 (t˜)
∂t˜ p̃2 (t˜) = − 2ΓΘ3/2 x̃1 (t˜) −
Γm
ã2,in (t˜) ,
Γm

(7.44)
(7.45)

(7.46)

where t˜ = t × ωm , ωm and Q are the resonant frequency and the quality factor of the
test mass, respectively, and
ωm
k BT
n=
~ωm
~g 2 /m
ΓΘ3 =
ωm

Γm =

(7.47)
(7.48)
g ≡ G0 ā.

(7.49)

Γm indicates the quality of the cavity, n is thermal occupation of the test mass and
ΓΘ is a measure of the measurement strength. ā is the amplitude of the light inside
the cavity
ā ≡ hâi = 2I0 /γ~ω0 .
(7.50)
We’ve assumed that the driving laser light has an intensity of I0 , and is in a coherent
state. Moreover,
â1,in (t)
â2,in (t)
, ã2,in (t) = √
(7.51)
ã1,in (t) = √
ωm
ωm
are the dimensionless amplitude and phase quadratures of the incoming light. b̃th (t)
is the dimensionless thermal fluctuation operator. Its correlation function is
b̃th (t˜) b̃th (t˜0) = Q × δ (t˜ − t˜0) .

(7.52)

Following a procedure similar to that of section I of the supplemental information of
[4], we can analytically calculate the system’s steady state covariance matrix from
Eqs. (7.43-7.46). The results are shown in Appendix 8.7.

177
By obtaining the symplectic eigenvalues of Eq. (7.102), we can use Eq. (7.38) to
quantify how entangled each of the correlated system-environment pairs are. For
n = 1/2 and Q = 106 , we show the results in Fig. 7.4. Notice that the larger Γm and
ΓΘ are, the more information the environment contains about the optomechanical
system. Moreover, most of the information about the optomechanical system leaks
to only one environment mode as N1
N2
7.4

Applications

If we assume that we can measure any environment mode then the phase-space
Schmidt decomposition theorem makes it easy to devise optimal strategies for sensing and quantum state preparation tasks. Although this assumption is unrealistic, it
provides us with tractable toy models that we can draw lessons from. Moreover, it
allows us to obtain bounds on how well we can perform a certain task, and allows
us to explore large parameter regimes with little computational cost.
We will first discuss how the phase-space Schmidt decomposition theorem provides
a fresh and illuminating perspective on the one-shot Quantum Cramer-Rao Bound
(QCRB). We then discuss two state preparation tasks: squeezing a system operator
as much as possible, and maximizing the entanglement between two modes of our
system.
7.4.1

Connection to one-shot QCRB

Simplifying the entanglement structure of a general optomechanical setup allows us
to understand, in a simple way, the QCRB for estimating a single parameter. The
QCRB sets fundamental limits on how well quantum systems can estimate classical
parameters. For a single parameter θ that is coupled linearly to a quantum system
through an operator Ô:
Ĥint = −Ôθδ (t) ,
(7.53)
the minimum estimation error on θ is (see Chapter 2 of [8])
QCRB
σθθ

~2
4 ψ Ô 2 ψ

(7.54)

where we’ve assumed that ψ Ô ψ = 0 and |ψi is the state of the system at time t.
We will show how a linear optomechanical system can always saturate the bound
QCRB
σθθ
if we assume that we can measure any commuting environment modes.
The proof will be both mathematically and conceptually simple because the phasespace decomposition theorem allows us to simplify the seemingly complicated joint

178

Figure 7.4: Quantifying the entanglement between the cavity optomechanical setup
shown in Fig. 7.2 with its environment. N1 (N2 ) is the logarithmic negativity of
the first (second) diagonalizing symplectic system mode of Eq. (7.102) with the
effective environment mode it is correlated with. We set the equilibrium thermal
occupation of the test mass to 1/2 and the quality factor to 106 .

179
quantum state of the system and environment to a collection of pure 2-mode squeezed
states as shown in Fig. 7.3.
First, we decompose Ô into the system’s symplectic modes:
Ô =

βi cos φi ŝi,1 + sin φi ŝi,2

(7.55)

i=1

where the βs and φs are real, and we’ve assumed that our system has n degrees of
freedom. Since each ŝi,1 and ŝi,2 pair, for i = 1...n, are independent and occupy a
symmetric thermal state (see Fig. 7.3), we can redefine each ŝi,1 and ŝi,2 to
ŝi,1 new = cos φi ŝi,1 + sin φi ŝi,2,

(7.56)

ŝi,2 new = sin φi ŝi,1 + cos φi ŝi,2,

(7.57)

which allows us to conveniently express Ô as
Ô =

βi ŝi,1 .

(7.58)

i=1

Substituting Eq. (7.58) into Eq. (7.53), we obtain
Ĥint =

ŝi,1 × (βi θ) δ (t) .

(7.59)

i=1

Each ŝi,1 , for i = 1...n, is independent from all the others, lives in a different Hilbert
space, and interacts with an environment mode, êi , that is only correlated with ŝi and
with nothing else. As a result, it is better to think of the signal θ not as appearing
in a single system, but as appearing in n distinct systems. Since each system is
composed of a single optomechanical degree of freedom that is coupled to a single
environment mode in a simple way, we can apply the same strategy for extracting
the optimal amount of information about θ to all n systems.
Consider for example the ith system. Because of the simple correlation structure
between ŝi and êi , given by Eq. (7.31), any system observable can be estimated
with the same accuracy: 1/νi . So the optimal estimation strategy is to maximize
the signal. Consider the ith term in Ĥint : βi ŝi,1 θδ (t). Such an interaction leaves ŝi,1
unchanged but shifts any operator lˆ that doesn’t commute with ŝi,1 by
i ˆ
βi l, ŝi,1 θ.

(7.60)

180
Since ŝi,2 is the conjugate operator of ŝi,1 , it maximizes the norm of l,ˆ ŝi,1 to 2.
Consequently, ŝi,2 will be the operator we will estimate as accurately as possible.

As indicated by Eq. (7.31), ŝi,2 is correlated with only one environment operator,
êi,2 . Therefore, we measure êi,2 to estimate ŝi,2 . The conditional mean of ŝi,2 given
that êi,2 is measured to be ei is

ŝi,2 c =

ŝi,2 êi,2 s
D E ei =
ŝi,2

− νi2 − 1
νi

(7.61)

νi

(7.62)

and the conditional variance of ŝi,2 is
ŝi,2
= ŝi,2

ŝi,2 êi,2 s
D E
ŝi,2

where we’ve used Eq. (7.31). As a result, for the ith system, our optimal unbiased
estimator of θ is

i
Zi =
βi ŝi,2, ŝi,1

−1

ŝi,2 c =

ŝi,2 c
2βi

(7.63)

with a squared error of
~2 1
4βi2 νi

∆Zi =

(7.64)

We now have a collection of n estimators, Z1 through Zn , for θ. We will optimally
combine them to obtain a single estimator for θ:
Z=

αi

! −1 n

i=1

αi Zi

(7.65)

i=1

and choose the αi such that ∆Z is minimized. We obtain, by using that Z1 through
Zn are independent, that the minimum error is
∆Zmin =

i=1

! −1
∆Zi−1

(7.66)

~2
4 Ô 2
i=1 βi νi

(7.67)

Ín

181

where we’ve used Eqs. (7.58) and (7.64), and that ŝi,1
= νi . Notice that ∆Zmin is
just the one-shot QCRB given by Eq. (7.54)

7.4.2

Optimal squeezing

Ref. [6] discusses a strategy for optimally choosing the optical environment modes
to measure in order to squeeze a system operator as much as possible. Nonetheless,
the phase-space Schmidt decomposition theorem makes it easy to obtain a bound on
how well a particular operator can be squeezed, and allows us to efficiently sweep
large parameter regimes.
Consider a system operator Ô. We will obtain a lower bound on how well we can
squeeze Ô. To do so, we first project Ô onto the system symplectic basis:
® .wsys
Ô = α
αi,1 ŝi,1 + αi,2 ŝi,2

(7.68)
(7.69)

i=1

where we’ve assumed that our optomechanical system has n degrees of freedom.
Each operator in the sum (8.52) is correlated with only a single environment mode.
Therefore, optimally squeezing Ô is equivalent to optimally squeezing n operators
with a simple correlation structure. This structure can be made trivial by normalizing
each term in Eq. (8.52):
Ô =

n q
i=1

2 + α 2 αi,1 ŝi,1 + αi,2 ŝi,2 .
αi,1
i,2 q
2 + α2
αi,1
i,2

(7.70)

We then simplify each term in the sum (8.53) by using the same argument that we
employed in Sec. 7.4.1. Since each ŝi,1 and ŝi,2 pair, for i = 1...n, are independent
and occupy a symmetric thermal state, we can redefine each
αi,1 ŝi,1 + αi,2 ŝi,2
2 + α2
αi,1
i,2
to be ŝi,1 . Therefore,
Ô =

n q

2 + α 2 ŝ .
αi,1
i,2 i,1

(7.71)

(7.72)

i=1

By measuring êi,1 , ŝi,1 is squeezed to

ŝi,1

= 1/νi so Ô can be optimally squeezing

182

Figure 7.5: Optimal squeezing of x̃1 for Q = 106 , n = 1/2 and different values of
Γm and ΓΘ for the cavity
√ optomechanical setup discussed in Sec. 7.3.3. We remind
the reader that x̃1 =p 2 x̂1 /∆xzp , where x̂1 is the center of mass position of the test
mass, and ∆xzp = ~/2mωm is its zero-point fluctuations. Moreover, Q is the test
mass’ quality factor, n is the test mass’ thermal occupation number, Γm = ωm /γ,
where γ is the cavity
decay rate, and ΓΘ is the dimensionless measurement strength
3 .
ΓΘ3 = ~g 2 / mωm
to
Ô

min

n α2 + α2
i,1
i,2
i=1

νi

(7.73)

We provide an example based on the cavity optomechanical setup that we discussed
in Sec. 7.3.3. In Fig. 7.5, we show how well the test mass’ center of mass position
can be squeezed for Q = 106 , n = 1/2 and different values of the measurement
strength and the cavity decay rate.
7.4.3

Maximizing the entanglement between the cavity and test mass

The phase-space Schmidt decomposition theorem makes it easy to determine how to
measure the environment in order to optimally enhance the entanglement between
two subsystems in an optomechanical setup. Without the theorem and because
quantum mechanics allows us to only measure commuting observables, for each of
the infinitely many environment modes that the system seems correlated with, we

183
have to pick the optimal quadrature to measure. As a result, the optimization problem
is not tractable unless we apply the theorem and reduce the number of environmental
modes to consider to just n (the number of system degrees of freedom). We illustrate
this with an example based on the cavity optomechanical setup discussed in Sec.
7.3.3. We will show how to measure the environment in order to optimally enhance
the entanglement between the cavity and the test mass’ center of mass degree of
freedom.
In the symplectic basis, the total system-environment covariance matrix has the
simple structure
σsys σc
σ=
(7.74)
σc σenv
where

σsys

σc

ν12 − 1

− ν12 − 1

®.
ν2 − 1
− ν22 − 1 ¬

The system is coupled to only two environment modes, but we are limited to
measuring commuting observables. Therefore, we are limited to measuring the two
observables
êθ1 ≡ cos θ 1 ê1,1 + sin θ 2 ê1,2,

(7.75)

êθ2 ≡ cos θ 2 ê2,1 + sin θ 2 ê2,2,

(7.76)

where θ 1 and θ 2 are real numbers.

184
If we measured êθ1 and êθ2 then the conditional covariance matrix of the system is
σ (θ 1, θ 2 ) = σsys
−σcT (θ 1, θ 2 )

! −1

ν1
ν2

σ1 (θ 1, θ 2 )
σ2 (θ 1, θ 2 )

σc (θ 1, θ 2 )

(7.77)

(7.78)

where and we’ve used that since ê1,1 and ê2,2 are independent
∆eθ1 = ν1

∆eθ2 = ν2,

(7.79)

and σc (θ 1, θ 2 ) is the cross-correlation matrix between the system symplectic modes,
wsys , and êθ1 and êθ2 :
!+
ê
σcT (θ 1, θ 2 ) = wsys
êθ2
cos
1 ν1 − 1
− sin θ 1 ν 2 − 1
=
®.
cos θ 2 ν22 − 1 ®®
2−1
sin

(7.80)

(7.81)

Moreover, we can calculate that
νi − cos2 θi ξi − cos θi sin θi ξi
σi (θ 1, θ 2 ) =
− cos θi sin θi ξi νi − sin2 θi ξi
−1
ξi ≡ νi νi − 1

(7.82)
(7.83)

for i = 1, 2.
Finally, to evaluate the entanglement between the cavity and the test mass, we first
revert back to the original basis, vsys (which consists of the test mass’ center of mass
position and momentum, and the cavity’s phase and amplitude quadratures):
vsys = M −1 wsys,

(7.84)

where M is the symplectic transformation that diagonalizes the system’s covariance
matrix in the vsys basis. We then evaluate the logarithmic negativity, given by Eq.

185
−1 T

(7.37), of the system covariant matrix in the vsys basis, M −1 σc (θ 1, θ 2 ) M
, for
different values of θ 1 and θ 2 . We show example results for different values of Γm and
ΓΘ in Fig. 7.6. Notice that choosing the right commuting observables to measure is
crucial and can result in a much stronger entanglement between the cavity and the
test mass.
7.5

Correlation structure of a system with its optical bath

In optomechanics experiments, we can only control and measure optical degrees of
freedom. We will show that even when we restrict effective environment modes
to consist of only optical degrees of freedom, a system with n degrees of freedom
is correlated with only n effective optical modes. However, these optical modes
are correlated with the remainder of the optical bath. As a result, the systemenvironment quantum state doesn’t reduce to a collection of n independent two-mode
squeezed states. Instead, we will show that a system with n degrees of freedom is
correlated with only n effective optical modes, which in turn are only correlated to
another n effective optical modes. This correlation chain continues ad infinitum, as
is illustrated in Fig. 7.7.
7.5.1

Bipartite and multipartite entanglement

We denote the outgoing optical modes’ quadratures by
(i)
(i)
(t) , âout,2
(t)
âout,1

(7.85)

where i ∈ Z+ indexes a particular optical field (there could be multiple optical fields
interacting with the system) and t ∈ R represents time. Let us re-write each of the
effective environment modes êi , which are the ladder operators corresponding to the
quadratures given by Eq. (7.23), in a way that separates the optical environmental
degrees of freedom:
êi = αi ôi + βi tˆi,

(7.86)

where ôi (tˆi ) is a ladder operator that lives in the Hilbert space spanned by the optical
(i)
(i)
(t) and âout,2
(t) (non-optical environment modes). The αi and βi are
modes âout,1
complex numbers that we’d choose in such a way that ôi, ôi = 1 and ti, ti = 1,
respectively.
We’ll first show that only ô1 through ôn are correlated with the system, where n is
the number of system degrees of freedom. As we showed in Section 7.3, the system
is only correlated with êi and êi† for i = 1 . . . n, and so any mode that is orthogonal

186

Figure 7.6: The logarithmic negativity between the cavity and the test mass for the
setup discussed in Sec. 7.3.3, if we were to measure the two environment modes êθ1
and êθ2 given by Eqs. (7.75-7.76).

187
to (i.e. commutes with) them is not correlated with the system. An arbitrary optical
mode lˆ that is orthogonal to ô1 through ôn will necessarily be orthogonal with ê1
through ên , because lˆ lives in a different Hilbert space than the tˆi .
We’ll now show that, in general, the system is not only entangled with ô1 . . . ôn . Let
oc contain the effective optical modes that are correlated with the system:
oc =

ô1,1 ô1,2 ... ôn,1 ôn,2

(7.87)

where ôi,1 and ôi,2 are the two quadratures associated with the mode ôi for i = 1 . . . n.
Moreover, let oi contain the modes that are independent from the system:
oi =

ôn+1,1 ôn+1,2 ... ôNopt ,1 ôNopt ,2

(7.88)

Nopt is the total number of outgoing optical modes that interact with the system.
The limit Nopt → ∞ can be taken at any time. The covariance matrix between the
system and the optical bath, as expressed in the effective modes basis { ôi }, is
CT 0
© sys
Vsys−opt = C B1 DT ®® ,
D B2 ¬
« 0

(7.89)

where Vsys = vsys vTsys s with vsys given by Eq. (7.2), B1 = oc oTc s , C = oc vTsys s
and B2 = oi oTi s . D = oi oTc s is in general non-zero, and so the system is not
only entangled with oc .
7.5.2

A correlation chain

We can further simplify the structure of Eq. (7.89) by choosing a new basis, õi , for
the modes in oi in such a way that they’d be independent from the modes in oc (i.e.
oi oTc s = 0).
Let
õi = Sopt−opt oi,

(7.90)

where

(7.91)

188

is a symplectic transformation ensuring that õi, õTi = oi, oTi , and

Ñ ≡ Nopt − n.

(7.92)

Moreover, we’ll write D, which is defined in Eq. (7.89), in the following way:
D≡
then

d®1,1 d®1,2 ... d®n,1 d®n,2

(7.93)

T d®
T d®
... s®1,1
s®1,2
n,1
n,2 ª
... s®1,2 dn,1 s®1,2 dn,2 ®®
..
..
..
®.
(7.94)
... s®Ñ,1 d®n,1 s®Ñ,2 d®n,2 ®
... s®TÑ,2 d®n,1 s®TÑ,2 d®n,2 ¬
Fir oc to be correlated with only n modes in õi , we would like this matrix to be of
the form
T d®
T d®
®1,2
s®T d®
s®T d®
... s®1,1
n,1 s
n,2
© 1,1. 1,1 1,1. 1,2 .
..
..
..
..
..
T
s® d®1,1 s® d®1,2 ... s® d®n,1 s® d®n,2 ®
n,,2
n,2
n,2
n,2
®.
(7.95)
..
..
..
..
..
T ®
T ®
© s®1,1 d1,1 s®1,1 d1,2
s®T d®
1,2 1,1 s®1,2 d®1,2
..
..
Sopt−opt D =
T ®
s®Ñ,1 d1,1 s®Ñ,1 d®1,2
T ®
T ®
« s®Ñ,2 d1,1 s®Ñ,2 d1,2

Eq. (7.95) indicates that s®n+1,1 , s®n+1,2 through s®Ñ,1 , s®Ñ,2 must be orthogonal to the
vector space spanned by the d®1,1 , d®1,2 through d®n,1 , d®n,2 . Since Sopt−opt is a symplectic
matrix then all the s®n+1,1 , s®n+1,2 through s®Ñ,1 , s®Ñ,2 are orthogonal to Ωopt s®1,1 , Ωopt s®1,2
through Ωopt s®n,1, Ωopt s®n,2 , where Ωopt is a direct sum of Ñ σy /i. Therefore if D is
full rank, then we can achieve Eq. (7.95) by constraining the s®1,1 , s®1,2 through s®n,1 ,
s®n,2 to span the entirety of the vector space spanned by Ωopt d®1,1 , Ωopt d®1,2 through
Ωopt d®n,1 , Ωopt d®n,2 . If we do so then, for example,
s®TÑ,1 d®1,1 = s®TÑ,1

i=1

is equal to 0.

αi Ωopt s®i,1 + βi Ωopt s®i,2

(7.96)

189
The covariance matrix between the system and the effective optical modes
oc
õi

(7.97)

C2T
B2
õ˜ i õTc s

®.
õc õi s ®®
õ˜ i õ˜ Ti s ¬

(7.98)

B2 is equal to õc õTc s and C2 is equal to õc oTc s .
The same arguments that we used to simplify the structure of D in Eq. (7.89) allow
us to simplify the structure of õ˜ i õTc s in Eq. (7.98). This process can be repeated
until we express the optical modes in a basis that transforms the covariance matrix
between the sytem and the optical into a tri-block-diagonal form:

Vsys−new opt =

Vsys CT 0
C B1 C2T

B2
..

...
..

CÑ/n

0 ®® ,
CÑ/n
BÑ/n ¬

(7.99)

where all the non-zero blocks are n × n matrices.
Eq. (7.98) tell us that, at its simplest, the correlation structure of an optomechanical
system with its optical bath reduces to a chain of n-partite correlated systems, as
is shown in Fig. 7.7. It also tells us that an optomechanical system is bipartite
entangled with at most n optical modes, but could be multipartite-entangled with an
infinite number of modes.
7.6

Conclusions

We’ve shown that a general linear optomechanical system with n degrees of freedom,
and that is driven by arbitrarily many environment modes in Gaussian states, is
entangled with only n effective environment modes. Simplifying the entanglement
structure to this extent allows us to better understand how information leaks from
the system to the environment. For a cavity optomechanical system at resonance,

190

...

Figure 7.7: The correlation structure, at its simplest, of a general optomechanical
system with its optical bath. The optomechanical system consists of n degrees of
freedom, and is correlated with only n effective optical modes. These optical modes
are also only correlated with n effective optical modes. This correlation ’chain’
extends for the rest of the optical bath modes.

191
we’ve quantified how entangled the system is with its environment, and for a certain
parameter regime, we’ve determined that information about the system mostly leaks
to just one mode.
We then discussed how a simple entanglement structure, and the assumption that we
can measure any commuting environment modes, allow us to easily devise optimal
strategies for sensing and quantum state preparation tasks. For example, we derived
the one-shot quantum Cramer-Rao bound in a conceptually simple way. Moreover,
we provided bounds on how well we can squeeze different observables, and how
well we can enhance the entanglement between the cavity and the test mass in a
cavity optomechanical setup.
Finally, if we limit the effective environment modes to consist of only optical modes
(which experimentalists can probe), then we cannot make any general statements
about the entanglement structure of an optomechanical system with its optical bath.
Nonetheless, the correlation between a system and its optical bath has a simple
structure. A system with n degrees of freedom is correlated with only n effective
optical modes, which in turn are only correlated to another n effective optical modes.
This correlation chain continues ad infinitum.
Acknowledgments
We thank Haixing Miao, Yiqiu Ma, Mikhail Korobko and Farid Khalili for useful
discussions. This research is supported by NSF grants PHY-1404569 and PHY1506453, as well as the Institute for Quantum Information and Matter, a Physics
Frontier Center.

192
Appendix: The covariance matrix for the example setup in Sec. 7.3.3
In this appendix, we show the steady state covariance matrix for the setup shown in
Fig. 7.2. The optomechanical system’s degrees of freedom are
x=

x̃1 p̃1 ã1 ã2

(7.100)

where x̃1 and p̃1 are the normalized test mass’ center of mass position and momentum
operators, respectively. The normalization factor is shown in Eq. (7.12). Moreover,
ã1 and ã2 are the normalized cavity’s amplitude and phase quadratures, respectively:
ã1,2 = 2â1,2

(7.101)

where â1 and â2 are the cavity’s amplitude and phase quadratures.
x’s covariance matrix is
3 + nλ −1
(Q
QΓ
2QΓΘ3/2 Γm2 σx̃s.s.
1 p̃2
3/2
s.s.
−1
2 Q Γm ΓΘ + nλ
− 2QΓΘ Γm σp̃1 p̃2 ®®
s.s.
σsys = λ
®,
3/2 2
3/2
−1
3 Γ3 ®
2QΓ
2QΓ
QΓ
Θ m ®
s.s.
s.s.
s.s.
σx̃1 p̃2
σp̃1 p̃2
QΓΘ Γm
σp̃2 p̃2 ¬
(7.102)
where
λ−1 ≡ Γm + Γm2 Q + Q,
√ 3/2
2n(Q + Γm )
s.s.
σx̃1 p̃2 =
2ΓΘ Γm −QΓm 2Q + Γm Γm + 4 Q + 2Γm ΓΘ −
λ,
3/2 2
(2Q
QΓ
2n
σp̃s.s.
2QΓ
1 p̃2

3 3 6
4 3
σp̃s.s.
4Q
4nΓ
2QΓ
2n
m Θ
m Θ
2 p̃2

+λ Q2 4n Γm2 + 1 ΓΘ3 + Γm2 2 Γm2 + 4 ΓΘ6 + 1 + 2 Γm2 + 1 .
Moreover, γ is the cavity decay rate and is defined in Eq. (7.42), and
ωm
k BT
, n=
~ωm
~g 2 /m
ΓΘ =
, g ≡ G0 ā.
ωm

Γm =

(7.103)
(7.104)

where n is thermal occupation of the test mass and ā is the amplitude of the light
inside the cavity. G0 is the coupling strength between the test mass and cavity (see

193
Eq. (7.40)).

194
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[2] Alonso Botero and Benni Reznik. Modewise entanglement of gaussian states.
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[3] Carlton M. Caves and Bonny L. Schumaker. New formalism for two-photon
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3092, May 1985.
[4] Christophe Galland, Nicolas Sangouard, Nicolas Piro, Nicolas Gisin, and Tobias J. Kippenberg. Heralded single-phonon preparation, storage, and readout
in cavity optomechanics. Phys. Rev. Lett., 112:143602, Apr 2014.
[5] Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, and Klemens Hammerer. Quantum entanglement and teleportation in pulsed cavity optomechanics.
Phys. Rev. A, 84:052327, Nov 2011.
[6] H. Mueller-Ebhardt, H. Miao, S. Danilishin, and Y. Chen. Quantum-state
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195
Chapter 8

ADIABATICALLY ELIMINATING A LOSSY CAVITY CAN
RESULT IN GROSS UNDERESTIMATIONS OF THE
CONDITIONAL VARIANCES OF AN OPTOMECHANICAL
SETUP
8.1

Introduction

A cavity is a crucial component in many optomechanical setups. It can amplify
the light that a test mass interacts with, it can drastically change how the test
mass responds to the driving light (opening up applications such as cooling a test
mass to its ground state), and it can improve the sensitivity of gravitational wave
interferometers at different frequencies [5].
The theory of unmonitored quantum cavity-optomechanical systems is well understood. We know how a cavity modifies the dynamics of the test mass. When the
cavity is sufficiently lossy, it doesn’t alter the behavior of the test mass in any substantial qualitative way. We can eliminate the cavity from the dynamics, and focus
only on the test mass. Such a procedure is called adiabatic elimination and it is common in the optomechanics literature. In connection to this, certain linear degrees of
freedom can be non-adiabatically eliminated, even when the optomechanical system
is monitored [10].
When we monitor an optomechanical system, its dynamics conditioned on the measurement results become more complicated. Nonetheless, over the past decade, our
understanding of quantum control, and quantum state preparation and verification in
continuously monitored setups has improved substantially[4, 6, 8]. These advances
have also been used to describe the experiment in Ref. [9].
In this article, we show that adiabatic elimination accurately describes the conditional dynamics of an optomechanical system in a limited parameter regime. For
the canonical cavity optomechanical setup, we numerically determined that adiabatic elimination is accurate when the measurement rate is slower than the cavity’s
bandwidth. We then analytically analyzed a simpler setup where we can measure
any environment mode. This toy model showed us that if we measure the cavity
optomechanical system too strongly, then the test mass is so strongly slaved to the
environment that we become limited by the information that is locked in the cavity
and which cannot be retrieved by measuring the environment.

196

laser

Figure 8.1: The cavity optomechanical setup we examine in this article. y (t) is the
measurement record collected by the experimentalist.

8.2
8.2.1

Unconditional dynamics of a cavity optomechanical setup
Setup

Consider the strawman optomechanical setup shown in Fig. 8.1, where a cavity with
one movable mirror is pumped by an external source of light. We denote the test
mass’ center of mass position and momentum by x̂ and p̂, respectively. The ladder
operator associated with the cavity field is â, and we denote the ladder operators
associated with the incoming (outgoing) light continuum by âin (âout ).
The Hamiltonian for the setup is
p
p̂2 1
2 2
(t) â ,
+ mωm
x̂ + ~ (ωc − ω0 ) â† â + ~G0 x̂ â† â + i~ 2γ âin (t) â† − âin
2m 2
(8.1)
where m and ωm are the test mass’ mass and resonant frequency, respectively. G0 =
ωc /L is the bare optomechanical interaction strength, ωc is a resonant frequency
of the cavity and L is the length of the cavity. We are an interaction picture with
the cavity’s free Hamiltonian, and the ~ω0 part of the external continuum, removed.
We denote by ∆ the detuning between the laser and the cavity resonant frequency:
ĤI =

∆ ≡ ω0 − ωc .

(8.2)

The test mass is also driven by a thermal fluctuation operator fˆth , with a correlation
function of
ωm
fˆth (t) fˆth (t 0) = 2m
k BT δ (t − t 0) ,
(8.3)
where Q is the test mass’ quality factor, and T the temperature of the thermal bath.

197
Since the laser emits light in a coherent state with a large amplitude, the fluctuations
of âin are small compared to its large classical amplitude. Since the cavity mode
â is driven by âin , it will also have a large classical amplitude which allows us to
linearize the interaction x̂ â† â to
x̂ |a| 2 + ā∗ δ â + āδ â†
where â = δ â + ā and

ā =

(8.4)

2γ
I0
γ 2 + ∆2 ~ω0

(8.5)

γ is the half-bandwidth of the cavity and I0 is the input laser intensity. Moreover, to
simplify our analysis, we’ve chosen the convention that ā is real and positive. For
the remainder of the article, we will denote δ â by â.
To simplify the analysis in this article, we will present the cavity’s and test mass’
equations of motion in matrix form, and we will work with dimensionless parameters
only:
t˜ ≡ ωm t,
x̂/∆xzp
p̂/
x̃ = √ , p̃ ≡ √ ,
~mωm
∆xzp ≡
, ≡
2mωm
ãin ≡ âin / ωm,

ãout ≡ âout / ωm,

(8.6)
(8.7)
(8.8)
f˜th ≡

fˆth
(ωm /Q) 2mk bT

(8.9)

Let x denote the system’s dimensionless degrees of freedom:
x=

x̃ p̃ â1 â2

(8.10)

and n denote the environment noise operators:
n=

f˜th ãin,1 ãin,2

(8.11)

x’s equation of motion in the Heisenberg picture is
∂t˜x = Mx + Nn,

(8.12)

(8.13)

(8.14)

and
k BT
ωm
, Γ∆ =
, n=
ωm
~ωm
~g 2 /m
, g ≡ G0 ā.
ΓΘ3 =
ωm
Γγ =

(8.15)
(8.16)

ΓΘ has the dimension of frequency and is a measure of the strength of the optomechanical interaction. n is the thermal occupation number of the test mass.
The outgoing light’s equations of motion are
aout = Ax + Bn

(8.17)

where
aout ≡
A =
8.2.2

T
ãout,1 (t˜) ãout,2 (t˜) ,
0 0
2/Γγ
0 0
2/Γγ

(8.18)
B=−

0 1 0
0 0 1

(8.19)

Adiabatically eliminating the cavity

When Γγ
1, the cavity bandwidth is much larger than the mechanical resonant
frequency and the cavity mode responds almost instantaneously to the motion of
the test mass, so most analyzes of cavity optomechanics adiabatically eliminate the

199
cavity. Specifically, they set ∂t˜â1,2 (t˜) = 0 and solve for â1 and â2 :
â1 (t˜) ≈
â2 (t˜) ≈

2Γγ

Γ∆2 Γγ2 + 1
2Γγ
Γ∆2 Γγ2 + 1

ãin,1 (t˜) − Γ∆ Γγ ãin,2 (t˜) + ΓΘ3/2 Γ∆ Γγ3/2 x̃ (t˜) ,

(8.20)

3/2
Γ∆ Γγ ãin,1 (t ) + ãin,2 (t ) − Γγ ΓΘ x̃ (t ) .

(8.21)

We substitute these back into the equations of motion for x̃ and p̃ given by Eq. (8.12)
and obtain
∂t˜ x̃ (t˜) = p̃ (t˜) ,

(8.22)

2Γ3/2 Γγ−3/2
2Γ3 Γ∆
p̃ (t˜)
˜) − ãin,1 (t˜)
ã
− 1 + 2 Θ −2 x̃ (t˜) + 2Θ
in,2
Γ∆ + Γγ
Γ∆ + Γγ−2
2n ˜
(8.23)
fth (t˜) .

∂t˜ p̃ (t˜) ≈ −

We can simplify Eqs. (8.22-8.23) by defining
Γq2 ≡ 4ΓΘ3

Γγ−1
Γγ−2 + Γ∆2

(8.24)

and redefining the optical external continuum’s amplitude and phase quadratures to
ãout/in,1 (t˜) new =

ãout/in,1 (t˜) − Γ∆ Γγ ãout/in,2 (t˜)
1 + Γ∆ Γγ

(8.25)

ãout/in,2 (t˜) new =

ãout/in,2 (t˜) + Γ∆ Γγ ãout/in,1 (t˜)
1 + Γ∆ Γγ

(8.26)

With these definitions, we simplify Eqs. (8.22-8.23) to
∂t˘x̆ = M̆ x̆ + N̆n,

(8.27)

200
where
x̆ =
M̆ ≡

N̆ ≡

x̃ p̃

(8.28)

− 1 + Γq2 Γ∆ Γγ /2 −1/Q
0 0
2n/Q −Γq 0

(8.29)

(8.30)

Notice that these equations resemble those of a test mass directly interacting with
a driving laser’s light. The test mass has a shifted eigenfrequency from ωm to
ωm + Γq2 ωm ∆/(2γ), and it interacts with the light via an interaction strength of Γq .
The outgoing light’s equations of motions are
ãout,1 (t˜) new ≈

ãout,2 (t˜) new ≈

8.2.3

1 − Γ∆2 Γγ2

2Γ∆ Γγ
˜) new −
ã
ãin,2 (t˜) new
in,1
1 + Γ∆2 Γγ2
1 + Γ∆2 Γγ2
+2Γ∆ Γγ Γq x̃ t˘ ,

(8.31)

1 − Γ∆2 Γγ2

2Γ∆ Γγ
˜) new +
ã
ãin,1 (t˜) new
in,2
1 + Γ∆2 Γγ2
1 + Γ∆2 Γγ2
−Γq 1 − Γ∆2 Γγ2 x̃ (t) .

(8.32)

Adiabatically eliminating a lossy cavity accurately describes the exact
unconditional steady state dynamics

In this section, we show that adiabatically eliminating the cavity accurately describes
the unconditional (i.e. unmonitored) dynamics of the test mass at steady state. We
will set the detuning ∆ to 0 (non no-zero detuning, adiabatic elimination is justified
when ∆
ωm [4]), which simplifies Eq. (8.31-8.32) to
ãout,1 (t˜) new ≈ ãin,1 (t˜) new ,
ãout,2 (t˜) new ≈ ãin,2 (t˜) new − Γq x̃ (t) .
Eq. (8.12) describes the exact unconditional dynamics, and Eq. (8.27) describes the
dynamics of the test mass when we adiabatically eliminate the cavity. Since these
equations are linear and we’ve assumed that the thermal and optical bath degrees of
freedom are in zero-mean Gaussian states, at steady state the state of the system is
Gaussian and so is completely characterized by its first and second moments. For
example, the unconditional Wigner function for the test mass and cavity at a time t

201
when the optomechanical setup’s initial state is forgotten is of the form
1 T −1
W (x, p, a1, a2 ; t) = p
exp − x® Vxx (t) x® ,
det (2πVxx (t))

(8.33)

where x®T ≡ x p a1 a2 and Vxx (t) is the symmetric expectation value
of x (t), which follows the equation of motion (8.12), over the initial systemenvironment state|Ψi i
x (t) .x (t)T s
1
Ψi x (t) .x (t)T Ψi + Ψi x (t) .x (t)T Ψi

Vxx (t) ≡

(8.34)

Note that hΨi |x (t)|Ψi i = 0 because the thermal bath has zero mean and we’ve
linearized the incoming light (around its large classical amplitude), and assumed
the laser is in a coherent state.
We calculate the unconditional equation of motion for Vxx to be
∂t Vxx (t) = MVxx (t) + Vxx (t) M T + NVnn N T ,

(8.35)

where Vnn δ (t − t 0) is the covariance matrix of n (t) and n (t 0):

Vnn ≡

QΓγ−1 0
1/2 0 ®® .
0 1/2 ¬

(8.36)

We will denote the variances of an operator ô calculated with adiabatic elimination
by Voo . Vx̆x̆ follows the equation of motion
∂t Vx̆x̆ (t) = M̆Vx̆x̆ (t) + Vx̆x̆ (t) M̆ T + N̆Vnn N̆ T , .

(8.37)

We can obtain the steady state unconditional covariance matrix by setting ∂t Vxx in
Eq. (8.12) and ∂t Vx̆x̆ (t) in Eq. (8.37) to 0. The resultant equations are called

202
continuous Lyapunov equations. For zero detuning, we obtain at steady state that
Vx̆x̆ =

QΓq2 /4 + n
QΓq /4 + n

Vx̆x̆ ≡

x̆ (t) .x̆ (t)T s

ΓΘ3 Γγ Q2
Γγ + Γγ2 Q + Q

n≡

k bT
~ωm

(8.38)
(8.39)

ΓΘ3 Γγ2 Q

(8.40)

In the bad cavity limit, Γγ
1, and for Q
1, we obtain Vx̆x̆ ≈ Vx̆x̆ , with the
leading error in Vx̃ x̃ − Vx̃ x̃ is nΓγ2 .
8.3

Numerics showing the breakdown of adiabatic elimination in describing
conditional dynamics

Adiabatically eliminating a lossy cavity accurately describes the unconditional dynamics of the test mass at all measurement strengths ΓΘ , but the approximation
fails to describe the conditional dynamics for a large range of ΓΘ . To arrive at this
conclusion, we numerically calculated the steady state conditional variance of x̃,
exactly and if the cavity were adiabatically eliminated, for a large range of Q, ΓΘ
and Γγ .
First, we’ll numerically show that even if we always measure the outgoing light’s
phase quadrature âout,2 and don’t optimize over which quadratures to measure,
adiabatically eliminating the cavity breaks down for measurement rates larger than
phase
phase
the cavity decay rate. In Fig. 8.2, we show Vx̃ x̃ /Vx̃ x̃
as a function of ΓΘ
phase
for different values of Γγ, Q and n. Vx̃ x̃
is the conditional variance of x̃ if
the phase quadrature is always measured (i.e. θ (t) is set to π/2 in Eq. (8.66)
phase
for all times t) and if the cavity is adiabatically eliminated. Vx̃ x̃
is the exact
conditional variance of x̃ if the phase quadrature is always measured. We observe
that when Γγ ΓΘ ≈ 1/2 (which corresponds to a measurement strength & the cavity
half-bandwidth γ), adiabatically eliminating the cavity results in a large error in the
conditional variance of x̃:
phase
phase
Vx̃ x̃
≈ Vx̃ x̃ /2.
(8.41)
When we optimally choose the homodyne angle θ (t) in order to squeeze x̃ as much
min
as possible (refer to Appendix 8.6 for how to do so), Vx̃min
x̃ = Vx̃ x̃ /2 (where the
superscript min is to highlight the fact it is the minimum achievable conditional
variance) when
ΓΘ ≈ 0.7Γγ−0.97 Q−0.09 .
(8.42)

203

Figure 8.2: Predictions of the squeezing of x̃ after measuring the phase quadrature
âout,2 of the outgoing light and at different measurement strengths ΓΘ , when the cavity
phase
is adiabatically eliminated and when it isn’t. When âout,2 is measured, Vx̆ x̆
is the
steady state conditional variance of x̃ if the cavity were adiabatically eliminated, and
Vx̃min
x̃ the steady state conditional variance of x̃ under the full dynamics. The inset
phase
shows Vx̆ x̆
at different measurement strengths. We chose the thermal occupation
number n to be 1.
We determined this relationship through a large scale numerical analysis which we
min
show in Fig. 8.3. Specifically, we’ve set n = 1, and we show Vx̃min
x̃ /Vx̃ x̃ as a function
min
of ΓΘ for different values of Γγ, Q and n. In Fig. 8.4, we also plot Vx̃min
x̃ /Vx̃ x̃ against
ΓΘ for different combinations Γγ , Q and n.
8.4

Insights from a simplified version of the problem

To analytically obtain insights on why adiabatic elimination fails, we will assume that
we can measure any environmental mode, including thermal fluctuation operators.
This assumption is useful because it will allow us to use the phase-space Schmidt
decomposition theorem, which makes it easy to obtain tractable expressions for the
conditional variances of our system.
Since in an actual experiment we only have access to optical degrees of freedom, we
will set n = 0 throughout this section, so that optical noise is the primary source of
uncertainty in the test mass’ center of mass position and momentum. Equivalently,
when n = 0, most of the information we can recover about the test mass will be

204

Figure 8.3: Simulation results of the critical measurement strength, ΓΘc , when the
optimal squeezing variance for x̃ according to the exact dynamics is half of that
as when the cavity is adiabatically eliminated. Each black dot represents ΓΘc for a
different choice of Γγ and Q in the range 10−1 ≤ Γγ ≤ 10−6 and 104 ≤ Q ≤ 109 .
We chose the thermal occupation number n to be 1. The gray surface is fit to guide
the eye, and is equal to −0.155 − 0.09 log10 Q − 0.97 log10 Γγ .
contained in the outgoing light rather the thermal bath degrees of freedom.
8.4.1

The phase-space Schmidt decomposition theorem

The joint state of the test mass, cavity and environment is in a pure Gaussian
state characterized by zero mean and a covariance matrix Vtot . The phase-space
Schmidt decomposition theorem [3, 1] states that we can substantially simplify the
structure of Vtot by choosing a different
basis than
x for the joint test mass and cavity
system, and a different basis than a fth , where fth denotes the collection of
thermal bath operators, for the environment modes. In this new basis, all effective
environment modes, except two, are in vacuum, and each of these two modes is in
a two-mode squeezed state with an effective system mode.

205

Figure 8.4: Predictions of the optimal squeezing of x̃ at different measurement
strengths when the cavity is adiabatically eliminated and when it isn’t. Vx̆min
x̆ is
the minimum achievable conditional variance of x̃ if the cavity were adiabatically
eliminated, and Vx̃min
x̃ the minimum achievable conditional variance of x̃ under the
full dynamics. We chose the thermal occupation number n to be 1.

Denote the new basis of system and environment modes by
wsys ≡

ŝ1,1 ŝ1,2 ŝ2,1 ŝ2,2

wenv ≡

ê1,1 ê1,2 ê2,1 ê2,2 ...

w≡

(8.43)
We choose
them in such
a way that they are linear combinations of the modes in
v ≡ x a fth :
w = Mv,

M≡

Msys
Menv

(8.44)

For example, ŝ1,1 is of the form m1 x̃ + m2 p̃ + m3 â1 + m4 â2 , where m1, m2 , m3 and
m4 are real numbers. Moreover, the w satisfy the same commutation relations as v:

w, wT = v, vT = iΩ,

(8.45)

where Ω is a block diagonal matrix
Ω ≡ ⊕i=1
ω,

ω≡

0 1
−1 0

(8.46)

wsys
wopt

206
with N the total number of degrees of freedom in the system and environment.
Using Eq. (8.44), Eq. (8.45) implies that MΩM T = Ω. Matrices that satisfy such a
relation are called symplectic matrices. Similarly, we can also show that Msys and
Menv are symplectic matrices.
The theorem states that there exists an M such that

Vw ≡

C ≡

S ≡

(8.47)

(8.48)

(8.49)

where k = 1, 2 and I is the identity matrix. In addition, notice that Msys is the
symplectic diagonalizing matrix for Vxx :
Msys Vxx Msys
= C,

(8.50)

so ν1 and ν2 are the symplectic eigenvalues of Vxx , and ν1 ≥ 1/2 and ν2 ≥ 1/2. The
Williamson theorem guarantees the existence of an Msys that diagonalizes Vxx into
C.
The structure of Vw in Eq. (8.47) indicates that the system is entangled with only
two effective environment modes, so studying optimal conditional state preparation
is simple when we can measure any environment observable.
8.4.2

Applying the theorem to optimal state preparation

Consider an operator of the system Ô, such as the test mass’ center of mass position
operator x̃, that we are interested in squeezing as much as possible by measuring
the environment. We first project Ô onto the system symplectic basis:
® .wsys
Ô = α
αi,1 ŝi,1 + αi,2 ŝi,2 .

(8.51)
(8.52)

i=1

Each operator in the sum (8.52) is correlated with only a single environment mode.
Therefore, optimally squeezing Ô is equivalent to optimally squeezing 2 operators

207
with a simple correlation structure. This structure can be made trivial by normalizing
each term in Eq. (8.52):
Ô =

n q
i=1

2 + α 2 αi,1 ŝi,1 + αi,2 ŝi,2 .
αi,1
i,2 q
2 + α2
αi,1
i,2

(8.53)

Since each ŝi,1 and ŝi,2 pair, for i = 1, 2, are independent and occupy a symmetric
thermal state, we can redefine each
αi,1 ŝi,1 + αi,2 ŝi,2
2 + α2
αi,1
i,2

(8.54)

to be ŝi,1 . Therefore,
Ô =

2 q

2 + α 2 ŝ .
αi,1
i,2 i,1

(8.55)

i=1

As can be inferred from Eq. (8.47), each ŝi,1 is correlated with a single effective
environment mode’s quadrature, êi,1 . Therefore, estimating ŝi,1 as well as possible
entails measuring êi,1 . The error in this estimation can be calculated by noticing
that ŝi,1 ’s and êi,1 ’s joint Wigner function is Gaussian and has the covariance matrix

νi

νi2 − 1 ª
®.
νi

(8.56)

The variance of ŝi,1 conditioned on measuring êi,1 is
q
νi −

νi2 − 1/4
νi

4νi

(8.57)

Using Eq. (8.55), we deduce thatÔ can be optimally squeezing to
Ô

min

2 α2 + α2
i,1
i,2
i=1

4νi

(8.58)

Eq. (8.58) gives us some insights on why approximations that accurately predict
unconditional dynamics could horribly fail in predicting the conditional dynamics.
When ν1
ν2 , the unconditional uncertainty of Ô is dominated by the fluctuations
in ŝ1 and so it would seem reasonable to ignore ŝ2 . However, if we measure the

208
environment
estimate Ô, then
ignore ŝ2 if
in such a way as to optimally
we can only
2 + α2
α1,1
1,2 /4ν1
α2,1 + α2,2 /4ν2 . Interestingly, if α1,1 + α1,2 is of the same
2 + α 2 , then we can ignore ŝ only if ν
ν !
order of magnitude α2,1
2,2
8.4.3

When does adiabatic elimination fail?

Using Eq. (8.35), and assuming no detuning and n = 0, we can analytically calculate
the system’s steady state covariance matrix, which we show in Appendix 8.7. We
then symplectically diagonalize this covariance matrix. In the parameter regime of
small measurement strengths (ΓΘ
1), high Q and small Γγ , we obtain that the
symplectic eigenvalues are
ν1 ≈ QΓΘ3 Γγ,
α12 ≈ 1,
As a result,
Vx̃min
x̃ ≈

ν2 ≈ 1/2,

(8.59)

α22 ≈ 2ΓΘ3 Γγ4 .

(8.60)

+ ΓΘ3 Γγ4 .
4QΓΘ3 Γγ

(8.61)

Since we are in the mindset of doubting the validity of approximations, we checked
the accuracy of Eq. (8.61) with simulations. We calculated Vx̃min
x̃ without making
any approximations for 10 randomly chosen parameters satisfying
104 ≤ Q ≤ 1011 ;

10−5 ≤ Γγ ≤ 10−1 ;

10−5 ≤ ΓΘ Γγ ≤ 10−1,

(8.62)

and compared them with the approximate expression of Vx̃min
x̃ given by Eq. (8.61).
As we show in Fig. 8.5, we obtained that the discrepancy between Eq. (8.61) and
the exact calculation of Vx̃min
x̃ is almost always below 1% and at most 3%.
The second term in Eq. (8.61) represents how much unrecoverable information
about x̃ is stored in the cavity. We determined this by looking at the optimal steady
state conditional covariance matrix for
x̆ =

x̃ p̃

(8.63)

if the cavity is adiabatically eliminated. We call this covariance matrix Vx̆min
x̆ . Since
Vx̆x̆ is diagonal (see Eq. 8.38), it is easy to calculate. We first find the symplectic
eigenvalue, ν, of Vx̆x̆ with n = 0. Since Vx̆x̆ is already diagonal ν = QΓq2 /4. We
then use the results of Sec. (8.4.2) which tell us that x̃ and p̃ can be squeezed to

209
(4ν)−1 , thus obtaining
−1
3Γ
4QΓ
Θ γ
Vx̆min
−1 ®
x̆
4QΓΘ Γγ

(8.64)

where we’ve used Eq. (8.24) with the normalized detuning Γ∆ set to 0. Consequently,
Eq. (8.64) combined with Eq. (8.61) tells us that ΓΘ3 Γγ4 is the amount of information
about x̃ that is locked in the cavity and that cannot be recovered by measuring the
environment in any which way.
Adiabatic elimination fails when the cavity contains most of the unrecoverable
information about the test mass. This occurs when the second term in Eq. (8.61) is
larger than the first term, which happens for measurement strengths larger than
ΓΘ >
4QΓγ5

! 1/6

(8.65)

Notice that this scaling of Γγ−0.83 Q−0.17 is similar to the scaling of Γγ−0.97 Q−0.09 that
we obtained numerically when we assumed we can only measure the optical bath
(see Eq. (8.42)). Moreover, as we show in Fig. 8.6, Eq. (8.65) is, as is expected,
a lower bound of the critical measurement strength, ΓΘc , when adiabatic elimination
predicts a minimum conditional variance for x̃ that is half as large as the exact
prediction.
8.5

Conclusion

We’ve shown that although adiabatic elimination accurately describes the dynamics
of a cavity-optomechanical system in the bad cavity limit, it fails to accurately
describe the conditional dynamics when the measurement rate is large enough.
We’ve also illustrated how the phase-space Schmidt decomposition theorem can give
us a bound when an approximation could break down in conditional calculations.
When we approach this bound, we should be very suspicious of our approximation.
The theorem clearly showed us that information about the optomechanical system is
stored in three places: the environment, the cavity, and the the test mass. In the bad
cavity limit, the test mass couples much more strongly to the environment than the
cavity, and so it seems like we can adiabatically eliminate it. However, when the test
mass is strongly driven by the environment, it becomes slaved by the environment.
It forgets its initial state, and its fluctuations are strongly correlated with different

210

Figure 8.5: Eq. (8.61) is an accurate approximation of Vx̃, x̃ in the parameter regime
of interest. We generated 105 different possible values of the triplet Q, ΓΘ Γγ, Γγ ,
3 × 104 of which are shown in the inset to demonstrate that they cover most of
the regime 104 ≤ Q ≤ 1011 , 10−5 ≤ Γγ ≤ 10−1 , and 10−5 ≤ ΓΘ Γγ ≤ 10−1 . We
evaluated Vc,x exactly and Eq. (8.61) over them, and obtained that for 98.8% of the
triplets δ = Vx̃c exact − Vx̃c approx /Vx̃c exact ≤ 1%, and for 1.2% of the triplets δ is
between 1 and 3%.
modes of the environment, and weakly correlated with the cavity. When we measure
the environment, we recover all the information that is lost to the environment. If we
drive the optomechanical system strongly enough, the uncertainty of the test mass’
state becomes limited by the information that is stored in the cavity.
By having the test mass slaved by the environment, we can substantially squeeze
one of its degrees of freedom. The stronger the measurement strength, the stronger
the entanglement. Some information is also in the cavity because the test mass
is entangled with it (here also the entanglement gets stronger as we measure the
cavity). For moderate measurement strengths, we are limited by the information
that’s stuck in the test mass and not the cavity.

211

Figure 8.6: Testing the accuracy of Eq. (8.65). The simulation results are the same
as those in Fig. 8.3. The plane shows the value of ΓΘ for which α̃12 /4ν1 = α̃22 /4ν2
for different values of Q and Γγ .

8.6

Appendix: Introduction to quantum state preparation in optomechanics

In this section, we provide the background information needed to understand quantum state preparation in optomechanics. We also review how to measure the outgoing
light in order to optimally squeeze an operator of interest.
8.6.1

The conditional state of an optomechanical system

If we measure the outgoing light, which is entangled with the system, we modify
the dynamics of the system through wavefunction collapse. In general, obtaining
the conditional state of a system based on an experimentalist’s measurement record
is difficult. However, in our case, it is relatively simple because the system is in a
Gaussian state. Indeed, the unconditional state is Gaussian (see Eq. (8.33)), and
measuring linear combinations of the outgoing light quadratures projects Gaussian
states into Gaussian states. Furthermore, the âout (t) commute at different times,
so are simultaneously measurable and equivalent to a classical Gaussian random
process (see appendix D.3 of [4]). Consequently, we can apply Bayes’ theorem to
determine the Wigner function of a cavity optomechanical system conditioned on a
collection of measurement results.
To simplify the presentation of how we obtain the conditional state of the sytem, we

212
will discretize time into N time steps. The N → ∞ limit can be taken at the end of
the calculation. Assume that the following N commuting outgoing light operators
are measured:
© cos θ (t1 ) ãout,1 (t1 ) + sin θ (t1 ) ãout,2 (t1 ) ª
cos θ (t2 ) ãout,1 (t2 ) + sin θ (t2 ) ãout,2 (t2 ) ®
yθ ≡
(t
(t
(t
(t
cos
ã
sin
ã
out,1
out,2

(8.66)

where ti ≡ i × dt for i = 1 . . . N. Moreover, denote the measurement results
associated with yθ by y®θ . If we apply Bayes’ rule, we can obtain the Wigner
function of the cavity optomechanical system at the end time of the experiment tN
and conditioned on y®θ :
W (x, p, a1, a2, y®θ )
(8.67)
W (®yθ )
c −1
= p
exp − ( x® − hxi c ) Vxx ( x® − hxi(8.68)
c) ,
c )
det (2πVxx

W c (x, p, a1, a2 | y®θ ) =

where W (®yθ ) is the Wigner function for the measured observables yθ . It is a
zero-mean Gaussian with a variance of
Vyθ yθ ≡ yθ yθT s .

(8.69)

W (x, p, a1, a2, y®θ ) is the total Wigner function for the system and yθ . It is also a
zero-mean Gaussian with a variance of
Vxx Vxyθ
(8.70)
Vxy
Vyθ yθ
where Vxx is the covariance matrix for x at time tN (see Eq. (8.34)), and
Vxyθ ≡ x.yθT s

(8.71)

c and are the
is the cross-correlation matrix between x and yθ . Finally, hxi c and Vxx
conditional mean and variance of the system, respectively, and are given by1

hxi c = Vxyθ Vy−1
y®
θ yθ

(8.72)

Vxx
= Vxx − Vxyθ Vy−1
VT .
θ yθ xyθ

(8.73)

1 see section 2.3.3 of [2] for how to obtain the first and second moments of conditional Gaussians

213
c , we don’t use Eq. (8.73) because V
For practical calculations of Vxx
yθ yθ is a very
large (in the continuum limit, infinite) matrix and so it is numerically slow, and
analytically impossible, to invert it. Instead, we can proceed in two different ways.
c , we write it as
To analytically calculate Vxx
(t) = h(x − Kyθ )i s
Vxx

(8.74)

K ≡ Vxyθ Vy−1
θ yθ

(8.75)

where

is a 4 × N matrix (in general 2m × N matrix where m is the system’s number of
degrees of freedom). In the continuum limit, Eq. (8.75) becomes
∫ tf

dz h ŷθ (t) ŷθ (z)i s K (z) = x t f ŷθ (t) s

(8.76)

where t f is the end time of the experiment. It can be solved with the Wiener-Hopf
method [8], which for large systems quickly becomes intractable. For a system
with n degrees of freedom, we would need to solve for the roots of a 2n-th order
polynomial. Consequently, exact analytic calculations are impossible for more than
2 modes, and messy for systems with two modes (such as our cavity-optomechanical
setup).
c (t) with the Kalman filter, which iteraIn this article, we numerically calculate Vxx
c (t) by applying Bayes rule one measurement record at a time (as
tively solves for Vxx
opposed to Eq. (8.68), where we used conditioned on all the measurement results in
c (t) (see Appendix
one step). This allows us to obtain a differential equation for Vxx
B of [4]):
(t) = MVxx
(t) + Vxx
(t) M T + NVnn N T − Y (t) BVnn BT Y (t)T ,(8.77)
∂t Vxx

−1
Y (t) ≡ Vxx (t) A + NVnn B
BVnn B
(8.78)

where M and N are defined in Eq. (8.12), A and B in Eq. (8.17), and Vnn in Eq.
(8.36). At steady state, Eq. (8.77) reduces to a continuous time algebraic Riccati
equation.
8.6.2

Introduction to optimal conditional one-mode squeezed state preparation

Since quantum mechanics only allows simultaneously measuring commuting observables, experimentalists have to make the difficult choice of what to measure.

214
Specifically, in our cavity optomechanical setup shown in Fig. 8.1, experimentalists have to choose the function θ (t) in Eq. (8.66). The optimal choice of θ (t)
depends on the application. Let’s assume that the experimentalists are interested in
squeezing a degree of freedom as much as possible. Following Ref. [7], we will
summarize how to optimally pick θ (t) for such an application. We will first derive
a lower bound on how much we can squeeze a particular operator, and then show
that the bound can be saturated by measuring a particular combination of phase and
amplitude quadratures at each instant of time.
We first obtain a lower bound on how well we can squeeze x by assuming that we
can measure all quadratures of the optical light. Let V be the total covariance matrix
for the system degrees of freedom and for the outgoing light:
V≡

Vxx Vxa
Vxa
Vaa

(8.79)

where
a=

ãout,1 (t1 ) ... ãout,1 (tN ) ãout,2 (t1 ) ... ãout,2 (tN )

(8.80)

The optomechanical system’s Wigner function conditioned on measuring a is Gaussian and has the following mean and covariance matrix
−1
µ (a®) = VxaVaa
a®,
−1
min
(t) Vxa
(t) = Vxx (t) − Vxa (t) Vaa
Vxx

(8.81)
(8.82)

where a® are the measurement results of a, and we have used the superscript ’min’
min contains the minimum achievable conditional variances. Deto highlight that Vxx
riving Eq. (8.82) follows the same reasoning as in Sec. 8.66.
If we are interested in squeezing a single operator as much as possible, we can
min . To be concrete,
achieve the lower bound given by the corresponding entry in Vxx
let’s say that we are interested in squeezing the normalized center of mass position
operator x̃, which is defined in Eq. (8.7). The argument can be easily generalized
to any system operator Ô.
Reaching Eq. (8.82) corresponds to estimating x with
−1
e ≡ VxaVaa
a ≡ Ka,

(8.83)

215
because
min
(x − Ka) (x − Ka)T s = Vxx

(8.84)

Consequently, the optimal estimator for x̃ is ê x̃ ≡ K x̃ a where
K x̃ ≡

1 0 0 0

(8.85)

(8.86)

and we’ve used that
x̃ =

1 0 0 0

In the continuum limit,
e x̃ (t) =

∫ t

dz K x̃(1) (z) ãout,1 (z) + K x̃(2) (z) ãout,2 (z) .

(8.87)

If we choose θ(t) in such a way that
K x̃ (z) cos θ (z) ≡ K x̃(1) (z)

(8.88)

K x̃(2) (z)

(8.89)

K x̃ (z) sin θ (z) ≡

for all z ≤ t, and where K x̃ (z) is a real function, then
e x̃ (t) =

∫ t

dzK x̃ (z) cos θ (z) ãout,1 (z) + sin θ (z) ãout,2 (z) .

(8.90)

(t), we have to measure the quadraConsequently, to achieve the lower bound Vx̃min
x̃
tures cos θ (z) ãout,1 (z) + sin θ (z) ãout,2 (z) for all z ≤ t.
8.7

Appendix: The unconditional covariance matrix for the setup in Sec. 8.2.1

In this appendix, we show the steady state covariance matrix for the setup shown in
Fig. 8.1. The optomechanical system’s degrees of freedom are
x=

x̃ p̃ â1 â2

(8.91)

where x̃ and p̃ are the normalized test mass’ center of mass position and momentum
operators, respectively. The normalization factor is shown in Eq. (8.7). Moreover,
â1 and â2 are the cavity field’s amplitude and phase quadratures, respectively.
Using Eq. (8.35), and assuming no detuning and n = 0, x’s covariance matrix is

216

3/2

Q2 ΓΘ3 Γγ
3/2

QΓΘ Γγ
9/2
Q2 ΓΘ Γγ3 (2Q+Γγ )λ

QΓγ2 ΓΘ
3/2
QΓΘ Γγ
− √
2λ
QΓΘ3 Γγ3 Γγ3

Vx̃s.s.
ã2
9/2 3
Q ΓΘ Γγ (2Q+Γγ )λ ®
®,
QΓΘ3 Γγ3
s.s.
Vã2 ã2
(8.92)

where
λ−1 ≡ Γγ + Γγ2 Q + Q,
(8.93)
QΓΘ9/2 Γγ2 2Q2 + Γγ Γγ2 + 4 Q + 2Γγ2 λ
s.s.
Vx̃ ã2 = −
(8.94)
2
4Q ΓΘ Γγ + Γγ + 2Q 2Γγ ΓΘ + Γγ + Γγ + Q 2Γγ Γγ + 4 ΓΘ + Γγ + 1
s.s.
(8.95)
λ.
Vã2 ã2 =
Q is the quality factor of the test mass, and the dimensionless parameters Γγ and ΓΘ
are defined by Eqs. (8.15-8.16).
Assuming no detuning and n = 0, we can solve for Vxx at steady state from Eq.
(8.35): It seems that Eq. (8.92) can be easily simplified. For instance, we expect
that we can neglect the second term in the numerator of Vx̃s.s.
ã2 , which is about Γγ /Q
smaller than the first term. We obtained for the following parameters: Γγ = 10−2 ,
Q = 106 and ΓΘ = 1/2, that this approximation underestimates the minimum
conditional variance of x̃ by three orders of magnitude. This demonstrates how frail
the process of calculating conditional variances are.

217
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[6] Haixing Miao, Stefan Danilishin, Helge Müller-Ebhardt, Henning Rehbein,
Kentaro Somiya, and Yanbei Chen. Probing macroscopic quantum states with
a sub-heisenberg accuracy. Phys. Rev. A, 81:012114, Jan 2010.
[7] H. Mueller-Ebhardt, H. Miao, S. Danilishin, and Y. Chen. Quantum-state
steering in optomechanical devices. ArXiv e-prints, November 2012.
[8] Helge Müller-Ebhardt, Henning Rehbein, Chao Li, Yasushi Mino, Kentaro
Somiya, Roman Schnabel, Karsten Danzmann, and Yanbei Chen. Quantumstate preparation and macroscopic entanglement in gravitational-wave detectors. Phys. Rev. A, 80:043802, Oct 2009.
[9] Witlef Wieczorek, Sebastian G. Hofer, Jason Hoelscher-Obermaier, Ralf
Riedinger, Klemens Hammerer, and Markus Aspelmeyer. Optimal state estimation for cavity optomechanical systems. Phys. Rev. Lett., 114:223601, Jun
2015.
[10] Huan Yang, Haixing Miao, and Yanbei Chen. Nonadiabatic elimination of auxiliary modes in continuous quantum measurements. Phys. Rev. A, 85:040101,
Apr 2012.

218
Chapter 9

THE CONDITIONAL STATE OF A LINEAR
OPTOMECHANICAL SYSTEM THAT IS BEING MONITORED
BY A NON-LINEAR, PHOTON-COUNTING, MEASUREMENT
Abstract
We present an analytic method to obtain the conditional state of a linear optomechanical system that is driven by Gaussian states and that is being monitored by
a non-linear, photon-counting, measurement. We hope that our work will help
researchers explore a range of optomechanics topologies that make use of photon
counters. The conditional Wigner function we obtain is a polynomial multiplied by
a Gaussian, and its parameters depend on quantities that can be efficiently obtained
with the Kalman filter. Normalizing this Wigner function (equivalently, calculating
the probability of obtaining a particular measurement record) entails integrating a
polynomial multiplied by a Gaussian over a possibly high-dimensional space.

9.1

Introduction

Quantum mechanics has been spectacularly successful at predicting the behavior of
microscopic systems, but the macroscopic world around us never seems to behave
non-classically. In the past decade, experimentalists have made significant advances
in preparing macroscopic objects in non-classical states. In particular, in optomechanics, researchers have cooled test masses to their quantum mechanical ground
state cooling [2]. Moreover, the center of mass motion of test masses has been
squeezed below the Heisenberg uncertainty level [15, 13, 8].
Preparing a test mass in its ground state or in a squeezed state is an amazing feat, but
such states have Wigner functions that are completely positive and so they can be
characterized with a classical probability distribution. A test mass is indisputably
behaving quantum mechanically when its Wigner function takes on negative values.
To prepare such Wigner functions, we need to go beyond linear optomechanical
setups that are driven by Gaussian states, and that monitor a linear observable (such
as the phase quadrature of the outgoing light). Khalili et al. proposed injecting a
non-Gaussian optical state (such as a single photon pulse) into an optomechanical
system [7]. O’Connell et al. prepared a mesoscopic mechanical resonator in a
Fock state by coupling it to a superconducting qubit [12]. Galland et al. showed
that using non-linear photon counting measurements, and an appropriately detuned

219
driving laser, can prepare a test mass in a non-Gaussian state [5]. Their proposal
was realized in Refs. [3, 6] where they measured individual quanta of phonons in
mechanical resonators.
The formalism used in [5] can only be applied to simple optomechanical setups,
and is only exact in the absence of losses and thermal noise. In this article, we
present an analytic filter for calculating the state of a generic linear optomechanical
system that is driven by Gaussian light and where the number of outgoing photons
is continuously monitored.
The filter we present is genuinely quantum mechanical. A linear Gaussian optomechanical system can be mapped a classical dynamical system [11, 9], and so we
can obtain the state of a test mass conditioned on the experiment’s measurement
results in a tractable way. However, in classical control theory, although analytic
non-linear filters exist, they do not exist for discrete measurement equations [4].
Our derivation will make use of the fact that Fock states’ Wigner function can be
expressed as derivatives of a Gaussian Wigner function.
9.2

Setup

Consider a linear optomechanical system that is continuously probed by light. The
number of photons in the outgoing light is then counted with a photodetector. We
show an example simple setup in Fig. 9.1, where a movable test mass is probed
by a laser. We will denote the outgoing light at time t by âout (t). Furthermore, to
simplify the analysis, we discretize time into N time steps.
The conditional state of our quantum system, given a measurement record n =
(n1, ..., nN ) where ni ≥ 0 are integers and represent the number of photons detected
at time i × dt, is
ρ̂c (n) = P̂n ρ̂ini P̂n /p (n) ;

p (n) = Tr P̂n ρ̂ini P̂n

(9.1)

where p (n) is the probability of obtaining the record n, and P̂n projects the
output field into a subspace where the outgoing light’s number of photons operator,
(t) âout (t), agrees with the measurement record n:
n̂ (t) = âout
P̂n = P̂n N Û (tN ) ... P̂n1 Û (t1 ) .

(9.2)

ti ≡ i × dt

(9.3)

We’ve defined

220

laser

Figure 9.1: A simple example optomechanical setup. A free test mass is driven by
an incoming light continuum, labeled by âin . âin then interacts with the test mass’
center of mass motion position and momentum operators, x̂ and p̂. The reflected
light forms an outgoing light light continuum, labeled by âout . âout is continuously
monitored by a photon counter. We denote the resultant measurement record by
n (t).
for i = 1 . . . N. Each Û (ti ) evolves the system and probe from t = ti−1 till t = ti , and
each P̂ni , for i = 1 . . . N, projects the outgoing probe light to a Fock state at time ti
P̂ni = |ni i hni | .

(9.4)

It will be convenient to work in the Heisenberg picture. We rewrite P̂n in the
following way
P̂n = Û P̂nH
(9.5)
where Û evolves the system and probe from t = 0 till the end time of the experiment
tN , and P̂nH is the collection of projection operators expressed in the Heisenberg
picture:
N
P̂n =
Û † (i × dt, 0) P̂ni Û (i × dt, 0) .
(9.6)
i=1

221
9.3

Switching to the Wigner function

It will be convenient to express everything in terms of the Wigner function. The
conditional Wigner function of our system at time tN is
W ( x®; n) =

d γ®

e−i x® γ® J (γ®; n) ,

(2π)

(9.7)

where m is the number of degrees of freedom of our system, and x® is a vector that
T
runs over the system’s degrees of freedom. For example, x® = x p
for the
example setup shown in Fig. 9.1. J is the generating function:

i γ®T x̂

J (γ®; n) = Trsys e

ρ̂sys,c (n)

(9.8)

where the trace is over the system degrees of freedom, and x̂ are the observables
T
associated with the system’s degrees of freedom (e.g. x̂ = x̂ p̂
for the setup
shown in Fig. 9.1). ρ̂sys,c (n) is the conditional density matrix of the system given
the measurement record n, and is obtained by tracing ρ̂c (n) over the environment
degrees of freedom:
ρ̂sys,c (n) = Trenv ( ρ̂c (n)) .
(9.9)
Using Eq. (9.1), we can write that
J (γ®; n) =

h T
Tr ei γ® x̂ P̂n ρ̂ini P̂n .
p (n)

(9.10)

Using Eq. (9.5), that P̂n† P̂n = P̂n , that
Û † x̂Û = x̂ (tN )

(9.11)

and that [x̂ (tN ) , n̂ (t)] = 0 for t < t f (the future can’t influence the past), we rewrite
J to
h T
i γ® x̂(t N ) H
Tr e
P̂n ρ̂ini .
(9.12)
J (γ®; n) =
p (n)
9.4
9.4.1

The Projection operator in terms of the Wigner function
Total projection operator in terms of the Wigner function

How do we represent, for example, |ni hn| (t) with ladder operators? Any density
operator can be represented with the Wigner function. Consider a bosonic mode
whose two quadratures we’ll label â1 and â2 . We can write its state in terms of the

222
Wigner function
∫ ∫
ρ̂ = π
W (α1, α2 ) exp [i (β2 (â1 − α1 ) − β1 (â2 − α2 ))] dα1 dα2 dβ1 dβ2 (9.13)
Reference: [14] section 3.4
For a Fock state (From [1] eq. 1.106) |ni
Wn (α1, α2 ) =

(−1)n e−(α1 +α2 ) Ln 2 α12 + α22

(9.14)

where the Ln is the Laguerre polynomials

m
n x
(−1)
Ln (x) =
m!
m=0

(9.15)

A few example Wigner functions are
2 −(α2 +α2 )
e 1 2,
2 −(α2 +α2 ) 2
2 α1 + α2 − 1 .
W1 (α1, α2 ) =

W0 (α1, α2 ) =

(9.16)

We will express P̂nH , the total projection operator onto n in the Heisenberg picture,
in terms of the Wigner function. Using Eq. (9.13)

P̂nH =

lim π N

N→∞

dα1 dα2 dβ1 dβ2

Wni (α1 (ti ) , α2 (ti )) ×

i=1

 Õ
 N


exp i
β2 t j â1 t j − α1 t j − β1 t j â2 t j − α2 t j 
 j=1
 ∫
= lim π N
dα1 dα2 dβ1 dβ2 ×
N→∞
h
i
Wni (α1 (ti ) , α2 (ti )) exp i β2T (â1 − α1 ) − β1T (â2 − α2 ) (9.17)
i=1

where â1 (t) and â1 (t) are the two quadratures associated with the outgoing light at

223
time t, and to simplify the notation, we’ve defined

9.4.2

α1/2 ≡

α1/2 (t1 ) α1/2 (t2 ) . . . α1/2 (tN )

(9.18)

β1/2 ≡

β1/2 (t1 ) β1/2 (t2 ) . . . β1/2 (tN )

(9.19)

â1/2 ≡

â1/2 (t1 ) â1/2 (t2 ) . . . â1/2 (tN )

(9.20)

Generating projection functional

Since Wn (α1, α2 ) is non-Gaussian for n > 0, it seems that we cannot analytically
perform out the integrals in Eq. (9.17). However, the Wn have a special structure:
they can be written as derivatives of W0 (which is Gaussian):
Wn (α1, α2 ) = (−1)n Ln 2∂ 2f1 + 2∂ 2f2 eα1 f1 +α2 f2 W0 (α1, α2 )

f1 = f2 =0

(9.21)

For example
W1 (α1, α2 ) = 2∂ 2f1 + 2∂ 2f2 − 1 eα1 f1 +α2 f2 W0 (α1, α2 )

f1 = f2 =0

(9.22)

We will use Eq. (9.21) to develop an analytic prescription for obtaining the conditional state of the system based on n.
Using Eq. (9.21), we rewrite P̂nH to
P̂nH ≡

lim

N→∞

Ĝ ≡ π

(−1)ni Lni 2∂ 2f1 (ti ) + 2∂ 2f2 (ti )

i=1

Ĝ

(9.23)
f =0

exp i β2T (â1 − α1 ) − β1T (â2 − α2 )
T
f1 ≡
f1 (t1 ) f1 (t2 ) . . . f1 (tN )
T
f2 ≡
f2 (t1 ) f2 (t2 ) . . . f2 (tN )
T
f ≡
f1 f2

exp αT1 f1 + αT2 f2 (9.24)
(9.25)
(9.26)
(9.27)

224
Substituting Eq. (9.16), we obtain
Ĝ ≡ 2
dα1 dα2 dβ1 dβ2 exp −αT1 α1 − αT2 α2 ×
h
i
exp i β2T (â1 − α1 ) − β1T (â2 − α2 ) exp αT1 f1 + αT2 f2 . (9.28)
Notice that Ĝ is composed of only Gaussian integrals, which we can analytically
evaluate.
We can rewrite Ĝn in terms of known operators. Doing so will help us interpret
Ĝ, and evaluate some of the integrals in Eq. (9.28). Specifically, we will express
Ĝn in terms of operators that project a time-dependent quadrature into a continuous
measurement stream z (t). In the Heisenberg picture, they are of the form
∫
tf
P̂ẑ(t)=z(t) =
Dξ exp i 0 dt ξ(t) ( ẑ(t) − z(t)) .
(9.29)
where ẑ (t) is the quadrature we are measuring at time t[7]. If we discretize P̂ẑ(t)=z(t) ,
then we obtain

P̂ẑ(t)=z(t)
Π j dξ j exp i
ξ j ẑ t j − z t j
(9.30)

We rewrite Ĝn , given by Eq. (9.28), in terms of such projection operators:
Ĝ = 2

∫
dα1 exp

−αT1 α1 + αT1 f1

P̂âH1 =α1

∫
dα2 exp

−αT2 α2 + αT2 f2

P̂âH2 =α2

(9.31)
where
P̂âH1 =α1

P̂âH2 =α2 ≡

dβ2 exp iβ2T (â1 − α1 ) ,

(9.32)

dβ1 exp iβ1T (â2 − α2 ) ,

(9.33)

and we’ve redefined β1 → −β1 so that exp −iβ1T (â2 − α2 ) → exp iβ1T (â2 − α2 ) .
We can now interpret Ĝ. It projects a state into a subspace where both quadratures
of the outgoing light, â1 (t) and â2 (t), are simultaneously measured to be α1 (t) and
α2 (t). We then average over all possible realizations of α1 (t) and α2 (t) as Gaussian random processes with variance 1/2 and a mean given by − f1 (t) and − f2 (t)
respectively.

225
9.5

Calculation of the conditional state

Substituting Eq. (9.31) into Eq. (9.23), which we then substitute into Eqs. (9.12)
and (9.7), we obtain that the system’s conditional Wigner function at the end time
of the experiment tN is

2N (−1) i ni
W ( x®; n) =
p (n)
∫
Lni 2∂ 2f1 (ti ) + 2∂ 2f2 (ti )
(9.34)
dα exp −αT α + f T α Wc ( x®; α)
f =0

i=1

Wc ( x®; α) ≡

d γ®
2m

(2π)

e−i x® γ® Tr ei γ® x̂(t N ) P̂âH1 =α1 P̂âH2 =α2 ρ̂ini ,

(9.35)

where m is the number of system degrees of freedom, and

α ≡

α1 α2

To evaluate Wc ( x®; α), we follow the procedure described in Ref. [10]. We first
apply Bayes’ rule:
W̃ ( x®, α)
Wc ( x®; α) =
(9.36)
W̃ (α)
where W̃ ( x®, α) is the unconditional Wigner function of the system and outgoing
light, and W̃ (α) is the unconditional Wigner function of the outgoing light. Assuming the incoming light is in a coherent state, and that tN is large enough so that
the optomechanical system’s initial (possibly non-Gaussian) state is forgotten, they
are both zero-mean Gaussians. W̃ (α)’s covariance matrix is
Vaa ≡ hâ (tN ) â (tN )i s ,
where
â ≡

â1 â2

(9.37)

(9.38)

â (tN ) is the unitary evolution of â from time 0 till time tN in the Heisenberg picture
under the system-environment Hamiltonian. For any operators ô1 (t) and ô2 (t 0),
hô1 (t) ô2 (t 0)i s is the symmetric expectation value over the system-environment’s
initial state:
hô1 (t) ô2 (t )i s ≡ Tr (ô1 (t) ô2 (t ) + ô2 (t ) ô1 (t)) ρ̂ini .
(9.39)

226
W̃ ( x®, α)’s covariance matrix is
Vxx Vxa
Vxa
Vaa

(9.40)

where
Vxa ≡ hx̂ (tN ) â (tN )i s ,
Vxx ≡ hx̂ (tN ) x̂ (tN )i s .
Since W̃ (α) and W̃ ( x®, α) are Gaussian, Wc ( x®; α) is also a Gaussian:

Wc ( x®; α) = q
(2π)2m det Vc

exp − (x − xc )T Vc−1 (x − xc )

(9.41)

where
−1
Vc = Vxx − Vxa
Vaa
Vxa,

(9.42)

xc = Kα,

(9.43)

K ≡

−1
Vxa
Vaa

(9.44)

With Wc ( x®; α) at hand, we evaluate the integral over α in W ( x®; n), which is given
by Eq. (9.34). The integral is

− 12 xT Vc−1 x

T

exp
α Lα T
T −1
dα exp −
+ f + x Vc K α
dα exp −α α + f α Wc ( x®; α) = q
2m
(2π) det Vc
(9.45)
where
L ≡ 2 + K T Vc−1 K.
(9.46)

Eq. (9.45) is a standard Gaussian integral. We evaluate to be

(2π)N
αT Lα T
1 T
T −1
T −1
−1
T −1
dα exp −
+ f + x Vc K α = √
exp
f + x Vc K L
f + K Vc x .
det L
(9.47)

L seems intimidating to invert, but the Woodbury identity can simplify L −1 . The

227
identity tells us that for any 3 matrices A, B, C

A + CBCT

−1

= A−1 − A−1C B−1 + CT A−1C CT A−1 .

(9.48)

Applying the identity to L −1 , we obtain that

−1

2
8 − 2L − K K

(9.49)

Furthermore, in

+ x Vc−1 K

9.5.1

−1

f +K

= f T L −1 f +xT Vc−1 K L −1 K T Vc−1 x+2xT Vc−1 K L −1 f
(9.50)
−1
we ignore the f L f term because f will be set to 0 (see Eq. (9.34)).

Vc−1 x

Final result

Combining everything, we have that

Wn (x)
p (n)
p (n) =
dxWn (x)

Wn (x) =

Wn (x) = (−1)

i ni

(9.51)
(9.52)
Lni

2∂ 2f1 (ti ) + 2∂ 2f2 (ti )

W̃n (x, f )

(2π)

(9.53)
f =0

i=1

1 T
T −1
−1
W̃ (x, f ) = p
exp − x V0 x exp x Vc K L f (9.54)
(2π)m det Vc det L

where
L = 2 + K T Vc−1 K
2
−1
8 − 2L − K K
V0 = Vc−1 − Vc−1 K L −1 K T Vc−1 .

(9.55)
(9.56)
(9.57)

Notice that Wn (x) is a polynomial times a gaussian.
Vc , K and so L can be efficiently obtained with the Kalman filter. Our concern is
that the integral in Eq. (9.52) could be difficult to estimate numerically.

228
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C h a p t e r 10

CONCLUSIONS
Optomechanics has undisputably entered the quantum regime. Setups, such as
LIGO, have become limited by quantum noise over a certain frequency bandwidth.
Moreover, experimentalists have prepared test masses in squeezed states, and have
measured individual quanta of phonons. These breakthroughs build upon a rich literature of theoretical optomechanics results. We’ve made advances to the theory of
optomechanics that will hopefully help fuel the next generation of quantum optomechanics technology. In particular, we’ve proposed two bases to view the environment
with. The first simplifies the interaction of a linear Gaussian optomechanical system
with its environment. We showed that the interaction can be reduced to one with
finite degrees of freedom. The second basis reduces the entanglement structure of
a linear Gaussian optomechanical system and its environment at a particular time to
a finite collection of two-mode squeezed states.
We were motivated to develop the first basis because researchers had used it for one
particular setup to prove that a certain protocol, that makes use of photon counters,
can prepare a test mass in a Fock state. We were hoping that we can use the first basis
to develop protocols for more complicated optomechanical setups, but this wasn’t
in general possible because the effective environment modes can be squeezed and
so could contain excitations and could be correlated with an infinite number of bath
modes, even when the initial state of the environment is at vacuum. Nonetheless,
we developed an analytic filter for obtaining the state of a system conditioned on the
clicks of a photon counter. We used the second basis to derive the one-shot quantum
Cramer-Rao bound in a simple way, and to understand why adiabatic eliminating a
lossy cavity could fail to accurately describe the conditional dynamics of a linear
Gaussian otpomechanical setup.
Outside of LIGO, some researchers have wondered what quantum optomechanics
can be used for. We showed that it can be used to test alternative theories of quantum
mechanics. In particular, it can test objective collapse models, which if found true
would resolve the measurement problem. We showed that LISA pathfinder places
aggressive bounds on the parameters of two of the most popular collapse models:
the CSL and DP models. In addition, optomechanics can test whether gravity
is fundamentally classical. We showed that such a theory can be made to be

231
compatible with causality, and that state-of-the-art torsion pendulum experiments
could test it. Such experiments would also have merit even if they do not detect any
new physics. They would be indirect evidence for quantum gravity, and a stepping
stone for developing experiments that test quantum gravity theories, and theories
where spacetime resists being in a superposition.