Quantum Electromechanics with Two Tone D

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Quantum Electromechanics with Two Tone Drive
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Weinstein, Aaron Jacob
(2016)
Quantum Electromechanics with Two Tone Drive.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/Z95M63MJ.
Abstract
In the field of mechanics, it is a long standing goal to measure quantum behavior in ever larger and more massive objects. It may now seem like an obvious conclusion, but until recently it was not clear whether a macroscopic mechanical resonator -- built up from nearly 10
13
atoms -- could be fully described as an ideal quantum harmonic oscillator. With recent advances in the fields of opto- and electro-mechanics, such systems offer a unique advantage in probing the quantum noise properties of macroscopic electrical and mechanical devices, properties that ultimately stem from Heisenberg's uncertainty relations. Given the rapid progress in device capabilities, landmark results of quantum optics are now being extended into the regime of macroscopic mechanics.
The purpose of this dissertation is to describe three experiments -- motional sideband asymmetry, back-action evasion (BAE) detection, and mechanical squeezing -- that are directly related to the topic of measuring quantum noise with mechanical detection. These measurements all share three pertinent features: they explore quantum noise properties in a macroscopic electromechanical device driven by a minimum of two microwave drive tones, hence the title of this work: "Quantum electromechanics with two tone drive".
In the following, we will first introduce a quantum input-output framework that we use to model the electromechanical interaction and capture subtleties related to interpreting different microwave noise detection techniques. Next, we will discuss the fabrication and measurement details that we use to cool and probe these devices with coherent and incoherent microwave drive signals. Having developed our tools for signal modeling and detection, we explore the three-wave mixing interaction between the microwave and mechanical modes, whereby mechanical motion generates motional sidebands corresponding to up-down frequency conversions of microwave photons. Because of quantum vacuum noise, the rates of these processes are expected to be unequal. We will discuss the measurement and interpretation of this asymmetric motional noise in a electromechanical device cooled near the ground state of motion.
Next, we consider an overlapped two tone pump configuration that produces a time-modulated electromechanical interaction. By careful control of this drive field, we report a quantum non-demolition (QND) measurement of a single motional quadrature. Incorporating a second pair of drive tones, we directly measure the measurement back-action associated with both classical and quantum noise of the microwave cavity. Lastly, we slightly modify our drive scheme to generate quantum squeezing in a macroscopic mechanical resonator. Here, we will focus on data analysis techniques that we use to estimate the quadrature occupations. We incorporate Bayesian spectrum fitting and parameter estimation that serve as powerful tools for incorporating many known sources of measurement and fit error that are unavoidable in such work.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
electromechanics, quantum noise, back-action evasion, squeezing, optomechanics, nanomechanics
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Schwab, Keith C.
Group:
Institute for Quantum Information and Matter
Thesis Committee:
Schwab, Keith C. (chair)
Painter, Oskar J.
Adhikari, Rana
Chen, Yanbei
Defense Date:
13 October 2015
Non-Caltech Author Email:
aaron.weinstein.j (AT) gmail.com
Record Number:
CaltechTHESIS:01072016-143812513
Persistent URL:
DOI:
10.7907/Z95M63MJ
ORCID:
Author
ORCID
Weinstein, Aaron Jacob
0000-0002-2354-0777
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No commercial reproduction, distribution, display or performance rights in this work are provided.
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9362
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CaltechTHESIS
Deposited By:
Aaron Weinstein
Deposited On:
12 Feb 2016 18:53
Last Modified:
02 Jun 2020 21:49
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Quantum Electromechanics with Two Tone Drive

Thesis by

Aaron Jacob Weinstein
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2016
(Defended October 13, 2015)

2016

Aaron Jacob Weinstein

To my parents and brother, for their unending support.

iii

Acknowledgements
I would first like to express my sincere gratitude to my advisor Prof. Keith Schwab for the
continuous support and guidance over the course of my study. His motivation, innovation,
and knowledge made all of this work possible. My sincere thanks also go to the Schwab
group postdocs: Dr. Junho Suh, Dr. KC Fong, and Dr. Matt Shaw. Thank you for serving
as my daily mentors, for picking up my frantic calls at all hours, and for making these years
stuck in windowless rooms a little bit brighter.
Thank you to the numerous staff and postdocs for sharing their valuable time, experience, and perspective. To the Eisenstein group, Prof. Erik Henriksen and Prof. Johannes
Pollanen, thanks for all of their optimism, gusto, and low-temperature wizardry. To Mark
Gonzalez, thanks for the wonderful training and guidance in the machine shop. To our
theory collaborators Prof. Aashish Clerk, Prof. Florian Marquardt, Dr. Anja Metelmann
and Dr. Andreas Kronwald, thank you for the clear and concise explanations, most of this
analysis would not be possible without your support.
I would also like to thank my fellow labmates Laura DeLorenzo, Chan U Lei, and Dr.
Emma Wollman for the stimulating exchanges, morning coffee chats, and shared sleepless
days in the lab. Thank you for all the fun that we have had over the past years. To my
Caltech friends – Vanessa, Brett, JD, Lincoln, Dvin, Simon, Jasper, Amir, Jeff, Richard,
Alex, Tim, – thank you for making these years far more enjoyable than I had ever expected.
Last but not least, I would like to thank my parents and my brother for never losing faith
in me and for letting me find my own way in life.

Abstract
In the field of mechanics, it is a long standing goal to measure quantum behavior in ever
larger and more massive objects. It may seem like an obvious conclusion now, but up until
recently it was not clear whether a macroscopic mechanical resonator – built up from nearly
1013 atoms – could be fully described as an ideal quantum harmonic oscillator. With recent
advances in the fields of opto- and electro-mechanics, such systems offer a unique advantage
in probing the quantum noise properties of macroscopic electrical and mechanical devices,
properties that ultimately stem from Heisenberg’s uncertainty relations. Given the rapid
progress in device capabilities, landmark results of quantum optics are now being extended
into the regime of macroscopic mechanics.
The purpose of this dissertation is to describe three experiments – motional sideband
asymmetry, back-action evasion (BAE) detection, and mechanical squeezing – that are directly related to the topic of measuring quantum noise with mechanical detection. These
measurements all share three pertinent features: they explore quantum noise properties in
a macroscopic electromechanical device driven by a minimum of two microwave drive tones,
hence the title of this work: “Quantum electromechanics with two tone drive”.
In the following, we will first introduce a quantum input-output framework that we use
to model the electromechanical interaction and capture subtleties related to interpreting
different microwave noise detection techniques. Next, we will discuss the fabrication and
measurement details that we use to cool and probe these devices with coherent and incoherent microwave drive signals. Having developed our tools for signal modeling and detection,
we explore the three-wave mixing interaction between the microwave and mechanical modes,
whereby mechanical motion generates motional sidebands corresponding to up-down frev

quency conversions of microwave photons. Because of quantum vacuum noise, the rates of
these processes are expected to be unequal. We will discuss the measurement and interpretation of this asymmetric motional noise in a electromechanical device cooled near the ground
state of motion.
Next, we consider an overlapped two tone pump configuration that produces a timemodulated electromechanical interaction. By careful control of this drive field, we report a
quantum non-demolition (QND) measurement of a single motional quadrature. Incorporating a second pair of drive tones, we directly measure the measurement back-action associated
with both classical and quantum noise of the microwave cavity. Lastly, we slightly modify
our drive scheme to generate quantum squeezing in a macroscopic mechanical resonator.
Here, we will focus on data analysis techniques that we use to estimate the quadrature occupations. We incorporate Bayesian spectrum fitting and parameter estimation that serve
as powerful tools for incorporating many known sources of measurement and fit error that
are unavoidable in such work.

Contents
Acknowledgements

Abstract

1 Theory

1.1

Classical microwave circuit analysis . . . . . . . . . . . . . . . . . . . . . . .

1.1.1

Ideal circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.2

Scattering parameters . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.3

Bypass channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.4

Optomechanical coupling and sideband transduction . . . . . . . . .

1.1.5

Mechanical forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1

Transmission line quantization . . . . . . . . . . . . . . . . . . . . . .

1.2.2

Input-output relations . . . . . . . . . . . . . . . . . . . . . . . . . .

Noise detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.1

Linear detection

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.2

Nonlinear detection . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Optomechanical interaction . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

1.3

1.4

1.4.1

Linearized optomechanical Hamiltonian and quantum Langevin equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.2

Optomechanical output spectrum and mechanical spectrum

. . . . .

1.4.3

Motional noise spectrum . . . . . . . . . . . . . . . . . . . . . . . . .

vii

1.4.4

Calculation of bad cavity effects . . . . . . . . . . . . . . . . . . . . .

2 Fabrication and measurement
2.1

2.2

2.3

Device design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.1

Device modeling

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.2

Suppressed parametric effects . . . . . . . . . . . . . . . . . . . . . .

Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1

Device recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.2

NbTiN and SiO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.3

Germanium and Thermal Evaporation . . . . . . . . . . . . . . . . .

2.2.4

PMGI and E-Beam Evaporation . . . . . . . . . . . . . . . . . . . . .

2.2.5

Polymer and Sputtered Aluminum . . . . . . . . . . . . . . . . . . .

Measurement techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1

Device packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.2

Fridge circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.3

Drive circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Sideband asymmetry
3.1

3.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1

Toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Optomechanical sideband asymmetry . . . . . . . . . . . . . . . . . . . . . .

3.2.1

Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.2

Zero-point bath designations . . . . . . . . . . . . . . . . . . . . . . .

3.2.3

Dressed mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.4

Microwave spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.5

Symmetric noise detection . . . . . . . . . . . . . . . . . . . . . . . .

3.2.6

Photon counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.7

Spectrum comparison . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.8

Cooling tone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

3.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.1

Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.2

Sideband ratio and imbalance . . . . . . . . . . . . . . . . . . . . . .

3.3.3

Output port occupation . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.4

Noise floor calibration . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Back-action evasion detection

4.1

4.2

4.3

4.0.1

Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . .

4.0.2

Quadrature definitions . . . . . . . . . . . . . . . . . . . . . . . . . .

4.0.3

Noise spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.0.4

Scattering matrix formalism . . . . . . . . . . . . . . . . . . . . . . .

Back-action and imprecision definitions . . . . . . . . . . . . . . . . . . . . .

4.1.1

BAE configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.2

DTT configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.3

Microwave drive phase dependence . . . . . . . . . . . . . . . . . . .

BAE results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1

Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.2

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Double BAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.1

Double BAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.2

BAE phase locking . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.3

Double BAE results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Mechanical squeezing

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Squeezing and detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1

Generating steady-state mechanical squeezing . . . . . . . . . . . . .

5.2.2

Bolgoliubov mode detection . . . . . . . . . . . . . . . . . . . . . . .

5.3

5.4

5.5

Squeezing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1

Ideal pumping with RWA . . . . . . . . . . . . . . . . . . . . . . . .

5.3.2

General spectrum model . . . . . . . . . . . . . . . . . . . . . . . . .

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4.1

Calibration measurements . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.2

Noise spectrum measurement . . . . . . . . . . . . . . . . . . . . . . 102

Bayesian parameter estimation and error analysis . . . . . . . . . . . . . . . 104
5.5.1

Comparison to Monte Carlo calibration simulation . . . . . . . . . . . 109

5.6

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Bibliography

114

List of Figures
1.1

Electromechanical device and effective circuit schematics. . . . . . . . . . . . .

1.2

Equivalent microwave circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

Microwave circuit with bypass channel. . . . . . . . . . . . . . . . . . . . . . .

1.4

Canonical examples of dispersive parametric coupling in opto- and electromechanical systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5

Thermomechanical noise spectra. . . . . . . . . . . . . . . . . . . . . . . . . .

1.6

General two-tone pump configuration. . . . . . . . . . . . . . . . . . . . . . .

2.1

Device images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Fabrication steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Aluminum stress control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

PMGI adhesion issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

Device mounting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

Fridge circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7

Switching circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8

Drive circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

Sideband asymmetry pump configuration. . . . . . . . . . . . . . . . . . . . .

3.2

Comparison between linear detection and photon counting. . . . . . . . . . . .

3.3

Device, calibration, and measurement scheme. . . . . . . . . . . . . . . . . . .

3.4

Sideband imbalance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Sideband asymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

nth
r noise spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

BAE pump configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

System calibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

BAE and DTT noise spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

BAE and DTT occupations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

Double BAE pump configuration. . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

Mechanical noise ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

Quadrature variance over the full noise ellipse . . . . . . . . . . . . . . . . . .

4.8

Phase locking circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.9

Frequency halving circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.10

Backaction heating versus cavity noise. . . . . . . . . . . . . . . . . . . . . . .

5.1

Mechanical squeezing pump configuration. . . . . . . . . . . . . . . . . . . . .

5.2

General pump configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3

Squeezing noise spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4

Markov chains generated via emcee. . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5

Triangle plot of single and pairwise parameter distributions. . . . . . . . . . . 107

5.6

Parameter estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.7

Squeezing results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

xii

Chapter 1
Theory
In this chapter, we will first present a classical model for the microwave cavity which we then
expand to include dispersive coupling to the motion of a mechanical oscillator. Later, we
develop a quantum theory of the electromagnetic and mechanical modes via the input-output
framework. We explore how this theory relates to the construction of the output microwave
noise spectrum as well as the electromechanical interaction between the microwave and
mechanical modes.

1.1

Classical microwave circuit analysis

1.1.1

Ideal circuit

Motivated by the geometry of our device, we model the microwave circuit as a lumped
element parallel RLC circuit. Given the resistance R, capacitance C and inductance L,
we consider a parallel array accessed via independent ports, which for the purpose of the
experiment we separately label the right port “R” as the output and the left port “L” as
the input port. Each port constitutes a coupling capacitor connected to a 50Ω transmission
lines that serves to couple microwave signal into and out of the device.
Though the parallel RLC circuit is a prototypical example of an electromagnetic resonator
and has been extensively analyzed [1], we will derive the circuit voltages and scattering
parameters here for a few reasons. First, this analysis helps clarify the connection between the

50 μm

Figure 1.1: Electromechanical device and effective circuit schematics. a. Optical micrograph of a typical
device from the top view. A parallel plate capacitor (center) is connected to a planar spiral inductor.
Input and output coupling capacitors connect the cavity to external microwave waveguides. b. Simplified
rendition of the microwave circuit emphasizing the electromechanical nature of the device. The microwave
resonator is composed of lumped-element inductor and capacitor. The free-standing top gate of the capacitor
is a mechanical oscillator, such that changes in position modify the capacitance and hence the microwave
resonance frequency. c. Equivalent microwave circuit, including the most relevant experimental details:
input and output capacitors serve to couple microwave signals into and out of the circuit while internal
resistance degrades the quality of the resonator.

classical and quantum circuit models and will serve as useful consistency checks throughout
the following calculations. Second, we ideally treat our circuit as an RLC circuit given in
Fig. 1.1(a), but in reality our device has numerous features that are not captured in this
circuit. We will first analyze the ideal circuit and then extend the model to incorporate
technical issues specific to our system (Fig. 1.3(a)). Lastly, this circuit has been analyzed
previously within the Schwab group with Norton equivalent circuit analysis [2], which leads
to direct calculations for circuit parameters at the cost of potentially obscuring their physical
origins. As an alternative method, we will derive the circuit voltages and effective scattering
parameters via Kirchoff’s circuit law.
To simplify the calculation, we can split the circuit as a T-network with three effective impedances: one for the unloaded resonator, Zc (i.e., the RLC circuit ignoring the input/output couplers), one for the left port ZL , denoted in this work as the left port “L”, and
one for the right port ZR , denoted as the right port “R”. For each grouping, the impedance
is calculated directly from the constituent passive components. In this work, we use the
engineering convention of j = −1, which differs from the physics convention by a negative
sign: j → −i). Assuming the high-Q limit where we only consider frequencies confined

RL
2Vo

VSMR

Vamp
RL

c)
ZR

ZL
2Vo

κL

κR

out

ωc
κint

Figure 1.2: Equivalent microwave circuit. a. Full microwave circuit schematic including the supply Vo
incident on the input capacitor, the load impedances RL of the input and output transmission lines, the
capacitances of the input CL and output CR couplers, the effective impedances of the RLC microwave
resonator and the voltage drops across the resonator capacitor, Vsmr , and amplifier load Vamp . b. Equivalent
impedances. To simplify the calculation and aid intuition, the cavity is split into three effective impedances:
ZL is the input impedance, ZR is the output impedance, and Zcav is the impedance of the unloaded microwave
resonator. c. Scattering picture. In the high-Q limit, the microwave resonator and environment is fully
represented by a coupled-mode diagram.
P The microwave cavity (cyan) is characterized by a resonance
frequency ωc and dissipation rate κ = σ κσ . Each port represents a coupling channel that supports both
relaxation and excitation of the cavity mode. The cavity radiates energy into each bath which in turn
radiates incoherent noise, given by the equilibrium temperature Tσ , back into the cavity.

within a narrow bandwidth about ωo (i.e., κ
ωo ), so that (ω 2 − ωo2 ) ≃ 2ωo (ω − ωo ), we find
Zσ (ω) = RL +
jωCσ
−1
Zcav (ω) =
+ jωC +
jωL
ωo Zo
2 j(ω − ωo ) + κint /2
where we define the characteristic impedance Zo and bare resonance ωo as Zo =

(1.1)
(1.2)
(1.3)

and

ωo = √LC
. By formulating the impedance in this manner, we can now easily identify the

cavity susceptibility as a Lorentzian factor with peak centered about ωo and internal loss κint .
The internal loss rate, also defined as the full width at half max (FWHM) of the unloaded
resonator, is defined via the circuit parameters as κint = 1/RC.
The total impedance of the network, Ztot , is defined in terms of the sub-unit circuit

impedances as
−1
+ ZR−1
Ztot = ZL + Zcav

1.1.2

−1

Zcav ZL ZR
−1
Zcav
+ ZL−1 + ZR−1 .
Zcav + ZR

(1.4)
(1.5)

Scattering parameters

We now derive the scattering parameters via the voltage drops over all relevant sections of
the circuit. The voltage in the superconducting microwave resonator, Vsmr , is given by the
voltage across the capacitor,
ZL−1
Vsmr = 2Vo
−1 + Z −1 + Z −1
Zcav
j κL
= Vo
RL C j(ω − ωc ) + κ/2

(1.6)
(1.7)

where we have omitted the explicit frequency dependence of the effective impedances. The
microwave resonance frequency ωc and total linewidth κ now include the effects of the input
and output coupling channels,

ωc = ωo +

δωσ ,

(1.8)

σ=R,L

κ=

κσ .

(1.9)

σ=R,L,int

Arranging the cavity voltage in a Lorentzian format allows us to identify the scattering
parameters associated with each port in relation to the circuit elements. Assuming that the
2 2
coupling capacitors are sufficiently small (ω 2 Cin
RL
1), the induced frequency shifts scale

as
ωo Zo
× Im(Zσ−1 ),

ωo Cσ

δωσ = −

(1.10)
(1.11)

The loss rate associated with each port follows
κint =

κσ = ωo Zo × Re(Zσ−1 ),
2
Cσ
≃ ωo RL

(1.12)
(1.13)
(1.14)

Regarding the voltage drop across the amplifier load, the output impedance behaves as
a voltage divider,
RL
Vsmr ,
ZR
= j κR RL CVsmr (ω).

Vamp =

(1.15)
(1.16)

Since the transmission lines are impedance matched to the signal generator, the power
incident on the microwave circuit is given as the average voltage drop across the load resistance Pin = hVo2 i/RL . The energy stored in the resonator is calculated via the maximum
voltage drop across the capacitor and is expressed as the number of coherent photons in the
cavity, np ,
Uc =

|Vsmr |2
2C

(1.17)

κL
Pin ,
(ω − ωc )2 + (κ/2)2

= ~ωc np .

(1.18)
(1.19)

The internally dissipated power is calculated from the time-averaged voltage drop across the
internal resistance,
h|Vsmr |2 i

(1.20)

= ~ωc np κint .

(1.21)

Pint =

Similarly, the output power follows from the time-averaged power dissipated across the amplifier load,
h|Vamp |2 i
RL
κL κR
Pin ,
(ω − ωc )2 + (κ/2)2

Pout =

= ~ωc np κR .

(1.22)
(1.23)
(1.24)

This relationship is useful for calibrating the internal loss, as the the peak height on resonance
is given by the ratio κLκκ2 R .
Based on the format of these equations, we see that the previous definitions of the loss
rates κσ , initially motivated by identifying the circuit transmission as a Lorentzian lineshape,
are properly associated with the dissipation rate of each respective port.

1.1.3

Bypass channel

From Eq. (1.7), the magnitude of the microwave circuit transmission exhibits a Lorentzian
lineshape. However, a typical measurement of transmission through the microwave circuit
noticeably deviates from a Lorentzian lineshape at frequencies far beyond the resonator
linewidth. One distinct feature is the presence of an anti-resonance in the spectrum, indicating interference of multiple current channels at the output of the microwave circuit,
in addition to a nearly flat transmission background that far exceeds the noise floor of our
measurement apparatus (Fig. 1.3(b)). As a first step to understand this behavior, we assume
the presence of a bypass channel that provides an alternate current path between the input
and output ports of the sample (Fig. 1.3(a)).
Applying Kirchoff’s Circuit Law over the equivalent circuit model, we recover the voltage
grop across the resonator, the voltage drop across the amplifier load, and, assuming waveguide impedance matching, the complex transmission S21 (ω) = Vamp (ω)/Vo . For simplicity,
we omit writing the explicit frequency dependence for the circuit impedances in the following
equations. Under the assumptions that 1/(ωCL ), 1/(ωCR ), |Xb |
RL , the voltages simplify

Xb
Vsmr

CR
Vamp

S21

2V0
−5

−50

( ω ωc ) 2π (MHz)

Figure 1.3: Microwave circuit with bypass channel. a. Equivalent microwave circuit model of a parallel
RLC circuit probed with input and output capacitive couplers CL , CR , and including an additional bypass
channel characterized by the reactance Xb . b. Driven response data (dark blue) fit with a bypass model
(red) and ideal model (|Xb | → ∞).

to
j κL
Vsmr (ω) = Vo
RL C j(ω − ωc ) + κ/2
j2RL (ω − ωc )
Vamp (ω) = j κR RL CVsmr (ω) 1 −
Xb
κL κR
− κL κR
S21 (ω) =
j(ω − ωc ) + κ/2

(1.25)
(1.26)
(1.27)

We further assume that the bypass channel is approximately flat over the explored frequency
range. We can now pull out a frquency-dependent correction factor for the transmission
through the resonator,
Pout (ω) = ∆(ω)~ωc np κR ,
where,

(1.28)

∆(ω) = 1 −

j2RL (ω − ωc )
Xb
κL κR

(1.29)

Though the source of this reactance is not entirely clear, we believe this bypass channel
is associated with a sample package or chip mode of the device. Typically, we observe a
low quality microwave resonance between 9-12 GHz depending on the dimensions of our
fabricated chip. At the frequencies relevant to our measurements, the bypass transmission

through this low-Q mode is sufficiently flat and, due to the large frequency detuning, can be
modeled as a positive, imaginary impedance that is consistent with this model.
We have also explored modeling this behavior from other known device features, such
as the shunt capacitance of the external couplers or the impedance mismatches between
sections of the on-chip waveguides; however, these attributes do not replicate the observed
Fano lineshape for cavity transmission.

1.1.4

Optomechanical coupling and sideband transduction
a)

b)
ωc

ωc
ωm

ωm

Figure 1.4: Canonical examples of dispersive parametric coupling in opto- and electromechanical systems.
a. Optomechanical system. One end mirror of a Fabry-Perot cavity is a mechanical resonator such that
changes in position will alter the length of the optical cavity which in turns shifts the optical resonance
frequency. b. Electromechanical system. A mechanical resonator forms one gate of a parallel plate capacitor
in a lumped-element microwave resonator. Motion modifies the capacitance gap size which induces a shift in
capacitance and resonance frequency. This system is closely replicated for the devices studied in this work.

Motivated by the design of our system, we now consider an electromechanical device
that is composed of a LC microwave resonator with a flexible capacitor gate. The motion
of the resonator modulates the gap size of the capacitor which in turn shifts the microwave
resonance frequency
ωc =

(1.30)

We express the strength of the electromechanical coupling, g, as
g=

∂ωc
ωc ∂C
=−
∂x
2C ∂x

(1.31)

If we consider only a single resonant mode of the gate with resonance frequency ωm and

amplitude xo , the microwave resonance frequency now has explicit time modulation,
1 ∂C
xo cos(ωm t + φm ) .
ωc (t) = ωc 1 −
2C ∂x

(1.32)

To help understand how this system behaves, we can calculate Vsmr by first assuming that
the cavity field instantaneously responds to the motion of the capacitor. The parametric
coupling induces a combination of phase and amplitude modulation for the cavity field. For
a small frequency shift δωc = ∂ω
x(t) = gx(t)
ωc , we capture the first order corrections
∂x

by Taylor expanding the cavity transmission prefactor (Eq. (1.7)):

∂S21
Vsmr = Vo S21 (ω) +
δωc (t)
∂ω
ω=ωp
jg
= Vo S21 (ωp ) 1 +
x(t) ,
j(ωp − ωc ) + κ/2
= Vp ei(ωp t+φp ) +
V± ei[(ωp ±ωm )t+φ± ] .

(1.33)
(1.34)
(1.35)

Parametrically modulating ωc at ωm generates sidebands detuned about the coherent pump
at integer multiples of the mechanical resonance frequency with amplitude and phase determined by the pump detuning from cavity resonance.
This calculation hinges on the assumption that the cavity responds to the position of
the mechanics much faster than the mechanical period. Since we are in the sideband resolved regime, ωm
κ, this assumption is explicitly violated – the cavity field responds to

parametric modulation with a time constant on the order of the loaded dissipation rate κ−1 .
As a next step, we can extend the circuit model to account for the finite response time of

the cavity. The following treatment follows closely along with Sec. 2.4 of [2]. Motivated by
Fig. 1.2, the total current for a driven microwave resonator can be expressed as a parametric
Mathieu equation,
Io cos(ωp t) = (CV ) + V +
∂t

dtV,

(1.36)

where the parametric modulation is implicitly included as the time-dependent capacitance
C and the circuit parameters R, L, C include loading effects of the drive and measurement

circuits. Here, the microwave pump signal has been expressed as an incident source current
oscillating at drive frequency ωp and with amplitude Io which can be expressed in terms of
the cavity voltage Vsmr = Io Zo ωp √

(ωp −ωc )2 +(κ/2)2

As it pertains to our experiments, we consider driving the system near cavity resonance
at detunings spanning ±ωm . As the cavity is sideband resolved, higher-order sideband terms
will be highly suppressed by the cavity response and so we only consider voltage contributions
from the first order sidebands. We consider a trial solution that includes the carrier drive
and nearest-neighbor sidebands:
V (t) = Vp e−i(℘t+φp ) + V− e−i(ω− t+φ− ) + V+ e−i(ω+ t+φ+ ) .

(1.37)

Differentiating Eq. (1.36) and substituting in the trial solution, the phase and amplitude
for these three components can be solved under the following assumptions: the modulation
is sufficiently weak so that the coherent pump is unaffected and follows Vp eiφp = Vsmr =
j κL
Vo RL1 C j(ω−ωc )+κ/2
, the pump frequency is sufficiently close to cavity resonance so ωp ≃ ωc
simplifies the cavity susceptibilities, and ωm
κ ensures the coupling to higher order sidebands is negligible. In terms of the pump detunings, ∆± = ω± − ωc , the voltage components
follow
V− = −gxo p
Vp ,
(κ/2)2 + ∆2−
φ− = arctan
− φp + φm ,
2∆−
Vp ,
V+ = −gxo p
(κ/2)2 + ∆2+
− φp − φm .
φ+ = arctan
2∆+

(1.38)
(1.39)
(1.40)
(1.41)

With an entirely classical model, we already see that electromechanical systems exhibit
the Raman-like processes of up and down-conversion of photons resulting from the parametric
coupling between the mechanical motion and the electromagnetic modes of a resonant cavity.
These up and down-converted sidebands form the subject of this work and will be discussed
10

240

aW/Hz

20mK

50mK

100mK

150mK

200mK

40
-100 ∆ω (Hz) 100
2π

Figure 1.5: Thermomechanical noise spectra. Typical microwave noise spectrum (blue) with Lorentzian fits
(red) of the up-converted motional sideband. As the cryostat temperature rises from 20mK to 200mK, we
observe the sideband power increase accordingly, as well as a small linear increase in the intrinsic mechanical
linewidth.

extensively in the following work. Foreshadowing for later discussion, note that there is a
sign difference for the transduced mechanical phase φm between the up and down converted
mechanical sidebands; this serves an important role in quantum noise correlations that arise
between the imprecision and back-action noise forces.
From this result, we can derive a useful relation for the integrated microwave noise power
under the transduced mechanical sidebands. Following Eqs. (1.38)-(1.41) and assuming
that the mechanics is uncorrelated with the drive voltage, the integrated microwave noise
power, normalized by the drive power, of the motional sidebands that are up- (+) and
down-converted (−) into the cavity center are proportional to the rms motional noise hx2 i,
hV±2 i
hVp2 i

hx2 i.

(1.42)

hV 2 i

Since the normalized ratio hV±2 i is insensitive to fluctuations in the output gain of the mip

crowave circuit, this is a useful relationship to extract the electromechanical coupling g .
In our experiments, we extract out the prefactor ( 2g
) by monitoring the ratio between the
transduced motional sideband power (Fig. 1.5) and the coherent drive power (measured at
the detector output) as we regulate the cryostat temperature from base temperature up to
200mK via calibrated resistance thermometry. As a consistency check, we find that the mechanical bath thermalizes all the way down to the base temperature of our fridge (Fig. 4.2).

1.1.5

Mechanical forces

We have now shown that a dispersive parametric coupling between the mechanical and
electromagnetic resonator gives rise to microwave transduction of motion, but that is not the
full picture. Through the electromechanical coupling, the electromagnetic resonator energy is
now position dependent and gives rise to microwave-induced mechanical forces. As a simple
illustration for a microwave resonator with a flexible capacitor gate, the resonator energy is
now a function of position, leading to electromagnetic-driven forces on the mechanics,
Fmech = −

dUc
dx

= |Vsmr (t)|2

(1.43)

dC
dx

(1.44)

From this simple result, we find that the mechanical forces are driven by microwave signal
mixing, reminiscent of the three-wave mixing that leads to phonon-photon transduction.
Since these mechanical forces are driven by microwave mixing, we must also pay careful
attention to microwave noise at ωp ± ωm for two reasons. First, the mechanical resonator is
in the high-Q limit and only responds to forces that are close to ωm in the Fourier domain.
Second, we typically rely on intense coherent drives at ωp to enhance the electromechanical
coupling, meaning that the dominant force terms at ωm are generated from mixing between
the coherent microwave drive at ωp and microwave noise in the narrow band about ωp ± ωm .
This immediately sets up a feedback mechanism that will generally introduce correlations between the microwave and mechanical signals. Furthermore, this feedback system is
dynamic in nature: the mechanical signals are transduced into microwave signals which in
turn mix back down into mechanical forces.
This feedback mechanism is a source of much of the major results in opto- and electromechanics, such as optomechanical induced transparency [3–5], mechanical cooling [6, 7],
mechanical amplification [8], microwave squeezing [9,10]. These results represent major milestones in the fields of mechanical sensing and mechanical state preparation at the quantum
level; the purpose of this section is to illustrate an analogous, and perhaps more intuitive,

classical model for much of this behavior.

1.2

Quantum analysis

1.2.1

Transmission line quantization

As a first step to building up a quantum model for electromechanical measurements, we
consider the quantization of a microwave transmission line. This is a good place to start
since it will guide our treatment of microwave baths and will also shape our treatment for
noise spectrum calculations in later sections. This section closely follows the arguments and
calculations of [11].
For a transmission line with capacitance per length c and inductance per length l, it is
convenient to define the flux variable ϕ̂ and momentum conjugate q̂ (local charge density).
The Hamiltonian is given by
Ĥ =

1 2
q̂ + (∂x ϕ̂) .
dx
2c
2l

(1.45)

The Hamiltonian is simplified with a normal mode expansion for the standing waves on an
infinite transmission line. With periodic boundaries, q = nπ/L for n ∈ Z,
b̂q ≡ p
~ωq L

dxe−iqx

√ q̂ − i
2c

q2
ϕ̂ .
2l

(1.46)

The Hamiltonian becomes
Ĥ =

~ωq b̂†q b̂q ,

(1.47)

where ωq = vp |q| is the mode frequency and vp = 1/(lc) is the wave velocity of the transmission line.
The voltage at one end of an infinite transmission line is defined as the amplitude quadra-

ture of the electromagnetic field.
V̂ (x, t) =

~ωq h i(qx−ωq t)
† i(qx+ωq t)
b̂q e
+ b̂q e
2Lc

(1.48)

where the explicit time-dependence of the bath operators are given by the Heisenberg equation of motion. Ignoring any coupling between bath and system, b̂q (t) = b̂q e−iωq t .
Based on the form of Eq. (1.48), we can organize the voltage operator into right moving
fields, proportional to (t − vxp ), or left moving fields proportional to (t + vxp ):
V̂ (x, t) = V̂ → (t −

) + V̂ ← (t + ).
vp
vp

(1.49)

Expanding Eq. (1.49) and noting that ωq ≥ 0 for all q,

V (t) =

Z ∞

dω
2π

~ωZo h →
b̂ [ω]e−iωt + b̂†→ [ω]eiωt ,

(1.50)

where the right-moving annihilation operator is defined as

b̂ [ω] ≡ 2π

vp X
b̂q δ(ω − ωq ).
L q>0

(1.51)

Similarly, the left-moving field is
dω ~ωZo h ←
b̂ [ω]e−iωt + b̂†← [ω]eiωt ,
V (t) =
2π
vp X
b̂→ [ω] ≡ 2π
b̂q δ(ω − ωq ).
L q<0

Z ∞

(1.52)
(1.53)

Note that the right and left movers are separated in the Fourier domain into positive and
negative frequencies. This frequency separation ensures that the field commutators are
nonzero only for identically moving fields,

i h
b̂→ [ω], b̂†→ [ω 0 ] = b̂← [ω], b̂†← [ω 0 ] = 2π δ(ω − ω 0 ),
14

(1.54)

and also introduces Heaviside step functions into the correlators for the right-moving field,
hb̂†→ [ω]b̂→ [ω 0 ]i = 2π δ(ω − ω 0 )nσ Θ(ω),

(1.55)

hb̂→ [ω]b̂†→ [ω 0 ]i = 2π δ(ω − ω 0 )(nσ + 1)Θ(ω),

(1.56)

and correlators for the left-moving field,
hb̂†← [ω]b̂← [ω 0 ]i = 2π δ(ω − ω 0 )nσ Θ(−ω),

(1.57)

hb̂← [ω]b̂†← [ω 0 ]i = 2π δ(ω − ω 0 )(nσ + 1)Θ(−ω),

(1.58)

where the fields are taken to be in equilibrium with a bath at occupation nσ .
This transmission line quantization applies to standing electromagnetic waves on an isolated transmission line. This is a rather simplified example of what we typically encounter
in experiments. For a finite length transmission line loaded on either side by measurement
equipment, the boundary conditions at each termination introduce explicit relationships between the right- and left-moving waves. These conditions potentially alter the manner in
which we treat the microwave fields. For example, a semi-infinite transmission line terminated at x = 0 by an impedance-matched spectrum analyzer will absorb all incident signals.
For the left-moving signals, the microwave field propagates in the other direction and, at
least in theory, never reflects back to the analyzer. In this situation, the analyzer is only
sensitive to the right moving signal V̂ → . In Sec. 1.3, we explore how this behavior manifests
for the types of noise measurements considered in this work.

1.2.2

Input-output relations

We will now consider the dynamics between a quantum system coupled to a bosonic bath
continuum and will arrive at Langevin-Heisenberg equations for the system operator that
includes the effects of bath-induced dissipation and noise fluctuations. The arguments and
calculations presented here closely follow along with [11, 12].

The Hamiltonian for a system coupled to a bath is
Ĥ = Ĥsys + Ĥbath + Ĥint .

(1.59)

The bath is a continuum of independent harmonic oscillators labeled by the quantum number
q,
Ĥbath =

~ωq b̂†q b̂q ,

(1.60)

with commutation relations,

b̂q , b̂†q0 = δq,q0 .

(1.61)

We make the rotating wave approximation and only consider the resonant contributions to
the system-bath interaction Hamiltonian,
Ĥint = −i~

fq â b̂q − fq∗ â† b̂q

(1.62)

where fq represents the coupling strength between system and bath operators. Terms like
b̂† â† and b̂â have been neglected since they are unphysical and are fast oscillating in the
interaction picture.
Next, we take the Markov approximation and assume the coupling is frequency independent,

|fq |2 e−i(ωq −ωc )(t−t ) = κδ(t − t0 ).

(1.63)

Substituting between the Heisenberg equations for the system and bath operators, we arrive at the Langevin-Heisenberg equations for the cavity mode that includes both dissipation
and coupling to bath fluctuations,
i κ
ih
â˙ =
Ĥsys , â − â − κb̂in (t),

(1.64)

where the input and output field operators are defined in terms of the bath modes,
b̂in (t) ≡ √1κ

b̂out (t) ≡ √1κ

fq eiωq (t−t0 ) b̂q (t0 ),

(1.65)

fq eiωq (t−t1 ) b̂q (t1 ).

(1.66)

Solving for the system operator dynamics, we arrive at the Langevin-Heisenberg equation
for the system operator,
i κ
ih
â˙ =
Ĥsys , â − â − κb̂out ,

(1.67)

with the input-output boundary condition,
b̂out (t) = b̂in (t) +

κâ(t).

(1.68)

This final condition implies that the output microwave field consists of the system signal
coupling out of the external port in addition to the reflected input bath noise. This relationship captures the behavior of a system coupled to the environment through a single port. In
our work, we deal with cavities that are coupled to the environment through multiple ports
that arise from both external and internal dissipation. We can generalize the input-output
relations for multiple ports σ, each associated with separate scattering rates κσ , and input
bath operators b̂σ,in . The only modification to the relations above is that we must now specify
an input-output boundary condition for each port,
b̂σ,out (t) = b̂σ,in (t) +

κσ â(t).

(1.69)

Since the input-output conditions are linear, the system responds identically to the HeisenbergLangevin equation Eq. (1.68), except now the bath operator and scattering rate is given by

the weighted average of the port contributions,
κ=

b̂in =

κσ

(1.70)

κσ
b̂σin .

(1.71)

As a final note, the input-output relations are a useful tool to model the linear propagation
of noise through our system. As it pertains to calculating experiment measurables like the
noise spectral density radiating out of the cavity or the complex transmission through the
microwave circuit, we only consider the “right-moving” fields of Eq. (1.48) which restricts the
summation in Eqs. (1.71) to only positive quantum numbers, q > 0. This restriction implies
that the input bath annihilation operator only has spectral weight at positive frequencies.
This frequency separation introduces Heaviside step functions that depend on the timeordering of the bath operators. For example, Fourier terms like

dteiωt hdˆ†σ,in (0)dˆσ,in (t)i = nσ Θ(ω)

(1.72)

have spectral weight only at positive frequencies. Alternatively, terms like

dteiωt hdˆ†σ,in (t)dˆσ,in (0)i = nσ Θ(−ω)

(1.73)

only have spectral weight at negative frequencies. Since the output microwave field b̂out will
be linearly proportional to the input field b̂in , we omit output-field correlation terms that,
following the input field correlators, have no spectral weight at positive frequencies.

1.3

Noise detection

In this work, we apply continuous wave signals and measure in the frequency domain. Therefore, the bulk of the analysis is performed in the frequency domains and utilizes two main
techniques, microwave noise measurement and scanning homodyne detection. Here we con18

sider the microwave noise spectrum, also referred to as the power spectral density of the
output voltage noise.
Consider output voltage noise normalized to units of quanta,
V̂out = dˆout + dˆ†out .

(1.74)

The power spectral density of the outgoing voltage noise is given by
SV V [ω] = h|V̂out [ω]|2 i.

(1.75)

Assuming the voltage fluctuations are stationary and ergodic [11, 13], the Wiener-Kintchin
theorem connects the spectral density to the Fourier transform of the autocorrelator GV V (t) =
hV̂out (t)V̂out (0)i,
SV V [ω] =

1.3.1

dteiωt hV̂out (t)V̂out (0)i,

(1.76)

dteiωt hdˆ†out (t)dˆout (0) + dˆout (t)dˆ†out (0)i.

(1.77)

Linear detection

A general approach to measure the power spectrum of the microwave field is to first measure
the time-dependent quadrature amplitudes of the output field and then use these elements
to calculate a power spectrum. We do this in our experiment by using a linear amplifier to
measure the voltage associated with the outgoing field.
This detection scheme is formally equivalent to a diode plus filter. The diode serves a
large bandwidth square-law power detector that is sensitive to the integrated voltage noise
over the full frequency domain. To isolate only a sharp peak of the noise spectrum, we
introduce a well-behave, normalized bandpass filter function f [ω] sharply peaked at the
designated ω. The voltage at the filter output is
V̂f [ω] = f [ω]V̂out [ω].
19

(1.78)

With filter in place, the diode output is

I ∝ |V̂f | =

Z ∞

dω|f [ω]|2 SV V (ω),

(1.79)

(SV V [ω] + SV V [−ω]) ,

(1.80)

−∞

= S̄V V [ω].

(1.81)

The second line follows from the fact that the filter function is real in the time domain,
f [ω] = f [−ω]∗ so that |f [ω]|2 = |f [−ω]|2 . Moving to notation for the noise spectrum
emanating from the right port “R” port of the our device, we find that linear detection is
sensitive to the symmetrized output noise spectrum,
dteiωt h{V̂out (t), V̂out (0)}i,
S̄R [ω] =
dteiωt hdˆ†R,out (0)dˆR,out (t) + dˆR,out (t)dˆ†R,out (0)i.

(1.82)
(1.83)

This last line arises from the frequency separation of the transmission line modes to right- and
left-moving fields so that the output voltage is defined only for the single branch of positive
frequencies and terms with no spectral weight at the specified frequency are omitted.
Note that symmetric detection is sensitive to the shot noise of the electromagnetic field
since it contains terms like hdˆout dˆ†out i. Associating the field correlator hdˆ†out dˆout i with photon destruction and hdˆout dˆ†out i with photon emission, the symmetric spectrum can be rein-

terpreted as the average rate at which the detector absorbs and radiates energy with the
environment.
For the case of voltage digitization and fourier transformation, an ADC will sample a
time series of voltages across an impedance-matched load. Once the voltage stream is stored
as classical real data, the classical voltage commutes with itself at all times, giving rise to
a symmetrized noise spectrum. Alternatively, the voltage time series is complex conjugate
symmetric in the Fourier domain (V [ω] = V [−ω]∗ ), and so again the spectral density from
digitization must be symmetric in frequency.
20

1.3.2

Nonlinear detection

What if one performs direct photon counting instead of linear noise detection? Since photodetection is sensitive only to the absorption of photons, the corresponding noise spectrum
contains only to normal-ordered terms consistent with the Glauber formalism [11, 14]. In
terms of the microwave noise radiating out of the output port “R”, the photon-counting
spectrum is
SR [ω] =

dteiωt h: V̂out (t)V̂out (0) :i,

(1.84)

dteiωt hdˆ†R,out (0)dˆR,out (t)i.

(1.85)

Since the normal-ordered detection only absorbs energy from the system, this measurement
scheme is not sensitive to the vacuum fluctuations of the electromagnetic field.
Note that the symmetrized and normal-ordered spectra are related through the commutation relations of the output microwave field ([dˆσ,out , dˆ†σ,out ] = 1) and are thus necessarily
the same minus a white shot noise floor,
S̄R [ω] = SR [ω] + 21 .

(1.86)

Since there is little question whether a microwave field behaves quantum mechanically, it
seems straightforward to relate the rates of emission to that of absorption. By formally applying the output field commutation relation, measurements can be framed as measuring only
photon absorption or both photon emission and absorption. Is this an important distinction?
If the question is whether or not the outgoing light field behaves quantum mechanically, then
perhaps not. But, as we show later, if the question extends to broader statements about
proving the quantum mechanical essence of macroscopic mechanical objects, then yes this is
an important distinction and raises ambiguity in measurement interpretation.

ωm + δ

G−

ω−

ωc

G+

ω+

Figure 1.6: General two-tone pump configuration. The drive field consists of two microwave tones (red and
blue peaks) detuned above and below the cavity resonance ωc (cavity DOS in black) at frequencies ω+ and ω−
and with amplitudes α± leading to enhanced optomechanical coupling strengths G± . To simplify calculations,
we move into the interaction pictures with the cavity field rotating about ωc +∆ and mechanical field rotating
about ωm + δ. The detunings are defined via the microwave drive frequencies: ∆ = [ 21 (ω+ + ω− ) − ωc ] and
δ = [ 21 (ω+ − ω− ) − ωm ].

1.4

Optomechanical interaction

We are now ready to tackle modeling our electromechanical system within the input-output
framework defined above.

1.4.1

Linearized optomechanical Hamiltonian and quantum Langevin
equations

We consider the canonical optomechanical Hamiltonian,
Ĥ = ~ωc â† â + ~ωm b̂† b̂ − ~g0 â† â b̂ + b̂† + Ĥdiss + Ĥdrive ,

(1.87)

where â â† is the annihilation (creation) operator of the intra-cavity microwave field, b̂ b̂†
is the mechanical phonon annihilation (creation) operator, and g0 = ∂ω
x is the bare
∂x zp

optomechanical coupling. The term Ĥdiss models the cavity and mechanical dissipation
channels to their respective baths, consistent with input-output relations of Eq. (1.67), and
the final term Ĥdrive captures the coherent cavity driving.
We will now begin to tailor this calculation to our experimental system. In our experiment, the otpomechanical system is a two-port electromechanical system driven from the
left input port, which we designate (L), and measured via the right output port, which we
designate (R). We initially consider driving the system with two microwave tones detuned
22

above and below the cavity resonance. The drive Hamiltonian reads
Ĥdrive = ~

αν âeiων t + â† e−iων t ,

(1.88)

ν=±

where ω± = ωc + ∆ ± (ωm + δ) and α± are the red and blue pump amplitudes defined at
the input port. The detunings δ and ∆ are shown in Fig.1.6. In the following, we apply
standard linearization, i.e., we separate the cavity and the mechanical operators, â and
ˆ In the
b̂, into a classical part, ā or b̄, plus quantum fluctuations, dˆ or ĉ. E.g., â → ā + d.
interaction picture with respect to Ĥ0 = ~ (ωc + ∆) â† â+~ (ωm + δ) b̂† b̂, we find the linearized

optomechanical Hamiltonian
Ĥlin = ĤRWA + ĤCR .

(1.89)

Here,
ĤRWA = −~∆dˆ† dˆ − ~δĉ† ĉ − ~

i
G+ ĉ† + G− ĉ dˆ† + G+ ĉ + G− ĉ† dˆ

(1.90)

describes the resonant part of the linearized optomechanical interaction, whereas
ĤCR = −~ G+ e−2i(ωm +δ)t ĉ + G− e2i(ωm +δ)t ĉ† dˆ† − ~ G+ e2i(ωm +δ)t ĉ† + G− e−2i(ωm +δ)t ĉ dˆ
(1.91)
describes off-resonant optomechanical interactions. Note that G± = g0 ā± describes the
driven-enhanced optomechanical coupling. Here, ā± is the intracavity microwave amplitude
due to the red and blue pumps, and we have assumed ā± ∈ R for simplicity and without loss
of generality.
Let us first consider the good cavity limit (ωm
κ) which allows us to work within the
rotating wave approximation, Ĥlin ≈ ĤRWA . In this case, the linearized quantum Langevin
equations (Eq.( 1.67)) read
κ
√
dˆ = −
− i∆ dˆ + i G− ĉ + G+ ĉ† + κdˆin ,
γ2
√
− iδ ĉ + i G− dˆ + G+ dˆ† + γm ĉin .
ĉ˙ = −

(1.92)
(1.93)

Here, dˆin =

σ=L,R,I

p κσ

dˆσ,in is the total input noise of the cavity, where dˆσ,in describes

the input fluctuations to the cavity from channel σ with damping rate κσ . σ = L and
R correspond to the left and right microwave cavity ports, while σ = I corresponds to
internal losses. The noise operator ĉin describes quantum and thermal noise of the mechanical
oscillator with intrinsic damping rate γm . The input field operators satisfy the following
commutation relations:
dˆσ,in (t), dˆ†σ0 ,in (t0 ) = δσ,σ0 δ(t − t0 ),
ĉin (t), ĉ†in (t0 ) = δ(t − t0 ),

(1.94)
(1.95)

hdˆ†σ0 ,in (t)dˆσ,in (t0 )i = nth
σ δσ,σ 0 δ(t − t ),

(1.96)

hĉ†in (t)ĉin (t0 )i = nth
m δ(t − t ),

(1.97)

−1
th
is the
where nth
σ is the photon occupation in port σ, and nm = [exp (~ωm /kB T ) − 1]

thermal occupation of the mechanical oscillator. The total occupation of the cavity is the
P κσ th
weighted sum of the contributions from different channels: nth
c =
σ κ nσ .
We include multiple bath temperatures (nth
σ ) to describe the various sources of heating
in microwave circuits. Compared to optical cavities which are passively cooled well into the
ground state (< 104 K), microwave cavities can have significant thermal occupation even at
temperatures reached in the dilution refrigerator. Filtering on the input and output transmission lines suprresses incident room tepmrerature noise; however, other issues may remain,
like internal dissipation in the cavity [15], or thermal noise from refrigerator components [16].
Additionally, there are other issues common to both microwave and optical systems, such
as source-phase noise [9] and cavity-frequency jitter [17]. Whatever the source, noise in the
system can be generalized into two categories based on how the noise contributes to the
th
measured signal, either by radiating directly into the cavity (nth
l and ni ) or by radiating

into both the cavity and detector (nth
r ).
For microwave noise spectrum calculations, we will also encounter correlations between
the cavity operator and the output port bath. The input-output relations and cavity field

correlations yield

κR th
n δ(t − t0 )
κ r
κR th
hdˆR,in (t) dˆin (t )i = hdˆin (t) dˆR,in (t )i =
(nr + 1) δ(t − t0 )
hdˆ†R,in (t) dˆin (t0 )i = hdˆ†in (t) dˆR,in (t0 )i =

(1.98)
(1.99)

hdˆ†R,in (t)dˆ†in (t0 )i = hdˆR,in (t)dˆin (t0 )i = 0.

1.4.2

(1.100)

Optomechanical output spectrum and mechanical spectrum

In this section, we derive the optomechanical output spectrum and the mechanical quadrature spectrum, first within the RWA and later in Sec. 1.3 including bad cavity effects.
For this, we solve the quantum Langevin equations (Eqs. 1.92, 1.93) in Fourier space.
T
T
, D in = din , din , ĉin , ĉin and
It is convenient to define the vectors D = d, d , ĉ, ĉ
√ √ √
√
L = diag κ, κ, γm , γm . We then find the following solution to the quantum Langevin
equations in frequency space:
D̂ [ω] = χ [ω] · L · D̂ in [ω] ,

(1.101)

where
χ [ω] ≡ 

− i (ω + ∆)

−iG−

−iG+

iG+

iG−

− i (ω − ∆)

−iG−

−iG+

γm
− i (ω + δ)

iG+

iG−

γm
− i (ω − δ)

−1

(1.102)

We measure the output microwave spectrum through the undriven (right) cavity port.
√ ˆ
One finds the input-output relation dˆσ,out [ω] = dˆσ,in [ω] + κσ d[ω],
dˆ = χ11 κdˆin + χ12 κdˆ†in + χ13 γm b̂in + χ14 γm b̂†in ,

(1.103)

where all explicit frequency dependence has been omitted. The complex transmission spectrum (driven response) is given by
S21 [ω] = − κL κR χ11 (ω).

(1.104)

The symmetric noise spectral density (Eq. (1.83)) is given by
S̄R [ω] =

dteiωt hdˆ†R,out (0)dˆR,out (t) + dˆR,out (t)dˆ†R,out (0)i,

= κR κ |χ11 |2 + |χ12 |2 (nth
c + 1/2)
+ κR γm |χ13 |2 + |χ14 |2 (nth
m + 1/2)
+ [1 − κR (χ11 + χ∗11 )] (nth
r + 1/2).

(1.105)
(1.106)
(1.107)
(1.108)

It is also useful to consider the normal-ordered spectrum, S̄R [ω] = SR [ω] + 21 :
SR [ω] =

dteiωt hdˆ†R,out (0)dˆR,out (t)i

= κR κ|χ11 |2 nth
+ κR κ|χ12 |2 (nth
c + 1)
+ κR γm |χ13 |2 nth
+ κR γm |χ14 |2 (nth
m + 1)
+ [1 − κR (χ11 + χ∗11 )] nth
r .

1.4.3

Motional noise spectrum

The mechanical spectrum is obtained in a similar fashion. The mechanical annihilation
operator is defined via the scattering terms,
b̂ = χ31 κdˆin + χ32 κdˆ†in + χ33 γm b̂in + χ34 γm b̂†in .

(1.109)

The position noise spectrum is calculated in the lab frame as
Sxx [ω] =

dteiωt hx̂(t)x̂(0)i,

= x2zp |χ31 (ω)|2 + |χ32 (−ω)|2 κ(nth
c + 1)
+ x2zp |χ32 (ω)|2 + |χ31 (−ω)|2 κnth
+ x2zp |χ33 (ω)|2 + |χ34 (−ω)|2 γm (nth
m + 1)
+ x2zp |χ34 (ω)|2 + |χ33 (−ω)|2 γm nth
m.

(1.110)
(1.111)
(1.112)
(1.113)
(1.114)

The symmetrized motional noise spectrum is defined in terms of symmetrized bath contributions,
S̄xx [ω] =

dteiωt h{x̂(t), x̂(0)}i,

(1.115)

= x2zp |χ31 (ω)|2 + |χ32 (ω)|2 κ(nth
c + α/2)
+ x2zp |χ33 (ω)|2 + |χ34 (ω)|2 γm (nth
m + β/2),
−
+ γop
γop
x2zp γtot
α
γm
th
th
nm +
nc +
(|ω| − ωm )2 + ( γtot
)2 γtot
γtot
x2zp γtot
β̃
nm +
γtot 2
(|ω| − ωm ) + ( 2 )

(1.116)
(1.117)
(1.118)
(1.119)

If necessary, bad cavity effects can be incorporated using the same truncation techniques as
outlined in the section below.

1.4.4

Calculation of bad cavity effects

In the frequency domain, the explicit time-dependence of ĤCR couples sytem operators at
different frequencies to each other,
D̂ CR [ω] = χCR [ω] · LCR · D̂ CR,in [ω] .

(1.120)

Here, D̂ CR [ω] contains infinitely many sidebands detuned by Ω = 2 (ωm + δ),
D̂ CR [ω] = ... D̂[ω − 2Ω], D̂[ω − Ω], D̂[ω], D̂[ω + Ω], D̂[ω + 2Ω] ... ,
while D̂ [ω] is defined in the same manner as in Sec. 1.4.2.
The updated scattering matrix χCR iteratively builds up the bad-cavity couplings,

−1

−1
 χ− χ (ω − Ω)
χ+
0 
−1
χCR (ω) =  0
χ−
χ (ω)
χ+
0 
−1
 0
χ−
χ (ω + Ω) χ+ 
..
with

χ− = 

and

−iG+ 0

iG−

iG+

−iG− 0

 0
χ+ = 
 −iG+ −iG−

iG+

(1.121)

0 
,
0 

(1.122)

0 
.
0 

(1.123)

In order to solve the equations of motion, we truncate the number of sidebands that we
take into account, i.e., we truncate the length of D̂ CR to the nth sideband at frequency
(ωm ± nΩ). As the analytic solutions are unwieldy even for first order corrections, we instead
numerically calculate the spectrum at frequencies specified by the data. In this fashion,
the likelihood function can still be evaluated and, with proper choice of numerical methods,
maximum likelihood estimation techniques can still be utilized for data fitting and parameter
estimation.
28

Chapter 2
Fabrication and measurement
This chapter will discuss the measurement details that are common to the experiments
discussed in this work. We first describe the design and fabrication of our electromechanical
device, then describe relevant details following how we package, probe and analyze the
system.

2.1

Device design

Previous electromechanical work in the Schwab group at Cornell and Maryland utilized
metalized nitride nanowires coupled to superconducting half-wave resonators. Using these
kinds of devices, mechanical cooling [18] and back-action evasion measurements [19] closely
approached the quantum regime but were hindered by technical nonidealities induced by
relatively weak electromechanical coupling and sufficiently large second order capacitance
nonlinearities. Motivated by electromechanical devices fabricated at JILA [4,20], the work in
the Schwab group at Caltech focused on a similar device design that incorporates a vacuumgap planar capacitor and lumped element spiral inductor. Compared to the nanowires, the
drastic increase in capacitance and mechanical stiffness of the planar capacitor mitigated the
previous difficulties with cooling and BAE measurements. Furthermore, we hoped to develop
alternative processing techniques that would ideally suppress known microwave issues such
as two-level system noise. Our design of these devices is discussed below.

50 μm
Figure 2.1: Device images. a. Top view optical micrograph of the device. The microwave circuit is fabricated
from aluminum (grey) on a silicon substrate (blue). A parallel plate capacitor, center, is surrounded by a
spiral planar inductor to form a microwave resonator. Input and output coupling capacitors transmit signals
into and out of the device. b. Simplified circuit model. The top gate of the capacitor is a compliant
membrane that supports drumhead acoustic modes. c. Electron micrograph side view of the capacitor. The
gap between the capacitor gates is roughly 100nm at cryogenic temperatures.

2.1.1

Device modeling

We simulate the microwave resonators in Sonnet, a commercial 2.5D electromagnetic field
solver that incorporates the Method of Moments applied to Maxwell’s equations. Shooting
for a microwave resonance between 5 and 7 GHz, we model a 40µm×40µm capacitor placed
at the center of 6-turn spiral inductor. The capacitor is placed in the center of the inductor
for technical reasons; we have no way of producing clean interconnects between different
metalization layers so the capacitor serves as the interconnect between the bottom spiral
and top airbridge layers of the inductor. From simulation, we extract out the dissipation
rates associated with intrinsic dissipation and radiation loss channels. For high-resistivity
silicon with a loss tangent of 2 × 10−4 [21], we expect a loss rate on the order of 2π × 100 kHz
stemming from eddy currents induced in the substrate. Radiation loss is a much weaker
effect, only contributing around 2π × 5 kHz of loss.
Ignoring other dissipation sources such as those associated with fabrication imperfections,

like dirt in the capacitor gap or trapped between the inductor metalization and subtrate,
the total internal loss is expected to be about κint = 2π × 165kHz. To mitigate the effects of
dielectric loss, we have attempted to switch from the device substrate from silicon to sapphire
but have encountered difficulties with modifying the fabrication recipes as the subtrate alters
the behavior of the sacrificial layer.
We can also extract the equivalent circuit parameters for the inductance and parasitic
capacitance. We extract a total inductance of approximately 15 nH, consistent with analytic calculations for planar spiral inductors [22]. For the parasitic capacitance, the series
capacitance of the air bridges and the capacitive coupling of the inductor to ground account
for roughly Cp = 30 fF. Given that we can achieve capacitor gaps of approximately 100 nm,
the parasitics roughly account for 20% of the total capacitance and hence the participation
ratio of the position-dependent capacitance is on the order of one: η = C+C
≃ 0.8.

As an aside, we have fabricated suspended top gates in both squares and circle patterns.
Though we do expect to see improved mechanical behavior of circular boundaries [23], in
practice we observe worse performance probably due to fabrication issues associated with
the altered design.

2.1.2

Suppressed parametric effects

We can perform a simple calculation to compare the onset of parametric amplification between nanowire [19] and planar capacitor devices. In both devices, the capacitance depends
on the position as x1 and hence contains terms to all orders in a Taylor expansion. Only

considering the second order term, ∂∂xC2 , the mechanics responds to the electromechanical
coupling with a power dependent spring shift, ∆ωm = 12 kEM
, where k is the bare mechanical
spring constant and the induced electromagnetic spring constant is
kEM =

~ωc np ∂ 2 C
2C ∂x2

(2.1)

With two tone drive detuned by ±ωm , the spring shift modulates the mechanical resonance at twice the mechanical frequency. Such modulation supports mechanical parametric
31

amplification [12] such that the amplification factor scales with the ratio of the frequency
. Calculating this factor for a set optical scattering rate
shift compared to the linewidth, ∆ω
γm

(ignoring differences in xzp scales), we find that it scales inversely with spring constant and
participation ratio:
∆ωm
Qm kEM
γm
2 k

~γop Qm
Qc

(2.2)
(2.3)

Assuming the consistent resonator Q’s between nanowire and planar devices, we find that
the amplification factor at a given cooperativity is highly suppressed in the planar devices
by nearly six orders of magnitude, mainly due to the enhanced participation ratio.

2.2

Fabrication

In this section, we will first describe the fabrication recipe for processing the current generation of electromechanical devices. Though this recipe may appear circuitous, it is the
result of multiple years of work spread between four graduate students and postdocs and
is informed by many iterations of alternative failed recipes. To elucidate these steps and
hopefully save other graduate students untold hours in the cleanroom, the remainder of
the section will discuss process development details of three older recipes that incorporated
alternative choices for deposition techniques, film materials and sacrificial layers. The different recipes are denoted by their most relevant fabrication details: “NbTin and SiO2 ”,
“Germanium and thermal”, “PMGI and e-beam” and “Polymer and sputter”.

2.2.1

Device recipe

We start with high-resistivity silicon wafer (> 10kΩ/cm2 ) prepared with a modified RCA
clean and BOE oxide strip. A 100 nm thick aluminum layer is sputter deposited at 6 Å/s in
a 5 mTorr Argon environment via a UHV-compatible DC magnetron sputter gun mounted

surface cleaning
& oxide strip

sputter deposition

spin & pattern

2-step wet etch

polymer strip

sacrificial spin & bake

pattern & thin

sputter deposition

spin & pattern

2-step wet etch

sacrificial etch

CPD & descum

Figure 2.2: Fabrciation steps. The device is processed on a high-resistivity silicon wafer (blue). 100nm
aluminum (grey) is sputter deposited and patterned via a wet etch. Photoresist (green) serves as both an
etch-stop for the device and a sacrificial layer, defining the capacitor’s gap size and protecting the bottom
layer from later processing steps. The sacrificial layer is removed in a solvent soak in Remover PG (yellow)
and device is dried via critical point drying to ensure the top gate does not collapse.

in a UHV chamber with a base pressure of < 10− 9 Torr. The bottom layer design is
patterned with contact photolithography using an S1800 series photoresist. A two-step
subtractive wet etch, consisting of Transene Aluminum etchant followed by CD-26, removes
the exposed bottom layer. Following polymer removal via Remover PG, the sacrificial layer
is patterned and thinned via double-exposure contact photolithography. After reflow and
surface preparation of the sacrificial layer with O2 plasma descum, the top layer is sputter
deposited and patterned under the same conditions as the bottom aluminum layer. The
sacrificial layer is removed via overnight soaking in PG Remover. To avoid collapse of the
top gate, the device is dried in a critical point drier followed by a final O2 plasma clean.
Next, we will discuss previous iterations of the recipes for older generations of devices.

2.2.2

NbTiN and SiO2

In the Schwab group, the electromechanics work at Caltech began with the work of a postdoc,
Matt Shaw, who fabricated a suspended, planar capacitor device entirely out of NbTiN. The
aim for using such a material was to achieve exceedingly high microwave Q (Q ∼ 106 seen
in λ/4 resonators made of NbTiN [24]) and potentially overcome nonlinearities associated
with two-level system defects that are typically encountered in aluminum-based microwave
cavities [25]. Coincidentally, the group had access to a cleanroom at the Jet Propulsion
Laboratory (JPL) with a high-quality NbTiN sputter chamber up and running.
The device was fabricated on high-resistivity silicon substrate and the NbTiN was reactive
ion sputtered in argon environment [15]. The top gate of the capacitor was suspended via
a silica sacrificial layer removed in a buffered-oxide etch (BOE) release soak. This recipe
proved difficult to control as the BOE soak altered the film stress of the NbTiN. Despite
attempts to diagnose and fix the issue, this device exhibited low microwave Q and significant
Ohmic heating at the pump powers required for sufficient optomechanical coupling.
Interestingly, this device showed a thermal-induced parametric instability in BAE pump
configuration (see Fig. 4.1). With pumps detuned by twice the mechanical resonance frequency, the power in the cavity modulates at twice the mechanical frequency. The mechanical
mode had a small enough specific heat and fast enough thermal time constant for the phonon
bath to track along with the modulating cavity power. Due to a thermal-induced mechanical
resonance frequency, this pump configuration modulated ωm at twice its frequency, eventually reaching a parametric instability at elevated pump powers [15].
We attempted to diagnose the different possible causes for the low microwave Q and
associated heating issues by repeating fab processing steps on quarter wave resonators, but
we failed to find a smoking gun. After months, we moved on to fabricating new devices out
of aluminum and relying only on Caltech facilities.

2.2.3

Germanium and Thermal Evaporation

At the time, our group had a custom built thermal evaporator mounted in a UHV chamber that had only been exposed to aluminum. For the next device, we chose to work with
thermal-evaporated aluminum because of the group’s past experience and because the campus cleanroom had few options for producing clean superconducting films from Nb or NbTiN.
Since the group was also familiar with metallized nitride films for mechanical resonators
[18, 19], the first device iteration was designed around a vacuum gap capacitor with a thick
(∼ µm) top gate and mechanical mode in the bottom gate formed by a nitride membrane
coated with 100nm of aluminum. We used LPCVD nitride deposited on high-resistivity
silicon (> 10 kΩ/cm2 ). For the sacrificial layer, we chose to work with e-beam evaporated
germanium so that we could take advantage of a dry XeF2 etch that aggressively attacks
Si/Ge with undercut etch rates in excess of 40µm/min [26]. This etch step fit nicely into the
processing steps: the nitride coating protected the silicon substrate, XeF2 does not attack
aluminum, and the aluminum is not exposed to any flourine-containing plasmas.
Using this technique, we achieved reasonable gaps and optomechanical coupling. However, the device had significant frequency jitter that worsened with aging and thermal cycling.
The device also showed significant TLS nonlinearities [17]. Thout we were unable to diagnose
the problem, we believe that these defects were likely associated with the choice of sacrificial
layer (germanium alloys with aluminum and absorbs water) and the sacrificial etch (XeF2
slowly attacks nitride).

2.2.4

PMGI and E-Beam Evaporation

In an effort to develop as gentle a sacrificial layer removal etch as possible, we next began
working on devices based around polymer sacrificial layers. We kept the same device design
as before – a doubly clamped beam at the center of a spiral inductor – and also kept the
mechanics as a metalized membrane on the bottom plane.
To achieve sacrificial layers on the order of 200-500nm, we used an underlayer resist,
PMGI. We also moved from thermal-evaporated aluminum to e-beam evaporation in hopes
35

d)
20 μm

10 μm

Figure 2.3: Aluminum stress control. a. E-beam deposited aluminum with PMGI sacrificial layer. The
top gate stress was difficult to control. With nearly identical processing conditions, the top gate would b.
collapse, c. appear nearly flat or d. pop up from compressive stress.

of getting better deposition control. The aluminum was patterned with a wet etch and
the sacrificial layer was removed with an overnight soak in Remover-PG. The new devices
showed consistent optomechanical coupling rates but with improved TLS behavior. Despite
much effort to get better control of the capacitor gap size, the stress in the top gate was
inconsistent between depositions – sometimes tensile stress, sometimes compressive stress –
and hence it was very difficult to control the behavior of the top gate after release (Fig. 2.3).
We attempted to shape the top gate to control the deformation with compressive stress,
but the stress varies enough between runs that there is no way to do this repeatedly despite
attempts at thermal annealing [27]. As another last ditch effort, we split the e-beam deposition of the top layer into seven steps and mounted the device to a thermal heat-sink. Alas,
there was no obvious improvement and we were unable to control the film stress.

2.2.5

Polymer and Sputtered Aluminum

After the last iteration of devices, we needed a method to control stress of the aluminum
film. To accomplish this, we removed our thermal evaporator and replaced it with a UHV
compatible magnetron sputter system. The move to a sputter deposition system allows for
control of the aluminum film stress via control of the backing argon pressure.
Ideally, replacing the e-beam deposition with sputter deposition would work with minimal
recipe modification; unsurprisingly, this was not the case. The sputtered films differed from
36

1μm

2.5μm

Figure 2.4: PMGI adhesion issues. a. The PMGI sacrificial layer (black) exhibits poor adhesion to the
sputtered aluminum (light grey) and poor resistivity to the wet aluminum etchant. As evidenced by the
thinning of the inductor air bridges, the device could not survive full aluminum patterning . b. Poor adhesion
between the PMGI and aluminum causes the etchant to rapidly penetrate the edges of the top layer.

e-beam deposition in multiplecurcial ways. First, sputter deposition is nearly isotropic, so
that there is no option of lift-off – all patterning must be done with a subtractive etch
process.
Another important issue is that our sputtered films do not etch cleanly with transene
etchant. After sufficient etch time, the substrate surface is left with a spackled residue. XDS
analysis of the residue confirmed that it was aluminum oxide, which is not surprising since
we have no in-situ surface prep and are unable to control our subtrate surface chemistry.
Since we were unable to strip reside with designated aluminum etchant, we attempted a
low dosage of TMAH common to photoresist developers (CD-26), which worked to remove
the residue. Thus, we use a two-step wet etch: Transene aluminum etchant A followed by
CD-26. The final dip in CD-26 cleans off surface residue over a short enough time that the
remaining film is insignificantly attacked.
However, the recipe still did not work as is. Sputtered films that are annealed above 130◦
display hillock formation [28,29]. PMGI requires baking at temperatures in excess of 170◦ and
so this problem is somewhat unavoidable. Hillocks on the scale of 50nm significantly perturb
the behavior of the gap; however, the polymer can conformally coat such topolographical
defects and this recipe could still work.

Finally, the adhesion between the PMGI and sputtered aluminum is so poor that we are
unable to wet etch the top aluminum layer without destroying the air bridges of the spiral
inductor. After attempts to reactive-ion etch (RIE) and descum to activate and roughen
up the PMGI surface, there is still no improvement. Attempts to move to dry etch the top
layer were futile as the etch would chlorinate the top polymer layer, rendering it difficult to
remove. As an aside, we also considered a lift-off technique for sputtered films but, due to
the isotropic nature of the deposition, the sidewall quality tapers off over microns, which is
unsuitable for bridges or inductors.
Serendipitously, we explored aluminum adhesion with different polymers and a fellow
graduate student, Chan U Lei, explored new recipes based around the Shipley S-series photoresists. Initial tests exhibited good adhesion between sputtered aluminum and S1800. To
optimize the adhesion, we found that thinning the photoresist with a weak flood-exposure
followed by surface activation with a short oxygen plasma descum created sufficient adhesion to survive the remaining fabrication steps. Since S1800 spins down to around 1.5µm,
we moved to a thinner resist in the same series, S500, in hopes of directly spin-coating a
500 nm sacrificial layer with higher precision than the double-exposure techniques required
for S1800, however the thinner resist had similar adhesion issues despite similar surface
preparation steps.
Based on our experimentation so far, the only recipe compatible with sputtered aluminum
and polymer sacrificial layers is based around S1800 series spun to around 1.5µm and thinned
to around 500nm via flood exposure. To control the thickness of this layer, we must calibrate
with spin speed and exposure time. However, the contact photolithography equipment we
use only controls exposure time steps within 0.10s steps. Given the lamp power drifts and
aging, this is not a very accurate method to control the polymer thickness. Since the polymer
behavior also differs with age and humidity, the trick was to calibrate and then fab as many
devices as possible.
Though we did not have a chance to attempt different sacrificial layer types, combining a
dielectric sacrificial layer with a wet etch removal might be possible with alternative silicon
etches [30, 31] that do not attack aluminum.
38

Figure 2.5: Device mounting. a. The device is mounted and wirebonded into a copper sample package. b.
Each sample package is mounted to the mixing plate and shielded with superconducting and mu-metal shields.
The microwave switches (blue cans) are visible. c. Our dilution refrigerator is an Oxford Kelvinox400.

2.3

Measurement techniques

2.3.1

Device packaging

We process our devices on 6mmx6mm and 6mmx3mm silicon chips. We found that moving
to 3mm chips pushed out chip modes, doubled our fab output, and was directly compatible
with existing 6x6mm packaging equipment. These chips are clip-mounted into copper sample
packages though we have also explored PMMA gluing. The chip waveguides are wirebonded
out to grounded CPW tranmission lines onto an arlon PCB that is impedance matched
to the silicon wafer. The arlon circuit boards are indium soldered into the copper sample
packages and the PCB transmission lines are soft soldered to coax ferrule adapters that
launch the microwave signals through the sample package. These coax adapters mate to field
replaceable SMA connectors which are commercially available from Southwest Microwave in
both standard and non-magnetic options. We have tested both kinds of connectors and have
observed no obvious differences in device behavior.
The sample package is mounted to a cold finger on the mixing plate of an Oxford Kelvinox
400 wet dilution refrigerator. The mixing plate has a cryoperm radiation shield, however we
also shield each device within superconducting and mu-metal shields. Alternating between
different shielding configurations, we see no obvious signs of flux noise [32] or quasiparticle
induced dissipation [33] though our devices are less susceptible to these issues compared to
superconducting qubit circuits.

2.3.2

Fridge circuit

Since it takes days to cycle our dilution refrigerator, we have added a set of passive microwave
switches that allow us to probe up to 6 different devices with the same fridge circuit. Our
switches are commercially available from Radiall. We learned about them from other research
groups that modify these switches for optimal operation at cryogenic temperatures [34],
though we initially installed our switches out of the box (unmodified). For a single switching
operation composed of two pairs of reset and set pulses, we typically heat our mixing plate
40

Fridge circuit, ver. 1
300K

Fridge circuit, ver. 2
out

out

300K

HEMT
+37dB

19dB

CuNi

20dB

50Ω

1K
10dB

100mK

100mK
Isolators
4-8 GHz

9dB

10mK

10mK
Switch &
device

50Ω

Switch &
device
20dB coupler

Figure 2.6: Fridge circuit. a. Version 1 as used in asymmetry and BAE measurements. An input waveguide composed of copper-nickel coax (blue) provides approximately 55dB of total attenuation to sufficiently
suppress room temperature Johnson noise. The output of the device runs to a cryogenic amplifier (HEMT)
at 4K through superconducting niobium coax (red). A pair of isolators cut out the 4K noise radiating back
from the amplifier. b. Version 2 as used in squeezing measurements. To reduce heating of the mixing plate
due to Ohmic heating of attenuators under intense microwave drive, the input line utilizes a directional
coupler that does not dissipate the incident power. Furthermore, the first stage of the attenuator is shifted
to the mixing plate to ensure the output port is in the ground state (nth
r = 0).

flex

out
S1

S2
device
package

Figure 2.7: Switching circuit. Multiplexing with a latching microwave switch allows up to five sample
packages and an impedance matched through connection to be accessed in a single cooldown.

to nearly 150mK, requiring between 30min - 90min to cool back down to base temperature
depending on the fridge pump configuration. This is a minor nuisance for the kinds of
experiments we do though modifying the switches is something that should be done in the
future.
To probe the device, the fridge is wired with separate input and output microwave coax
lines. The input lines is composed mainly of CuNi coax with an additional 40dB of isolation distributed over the dilution fridge stage temperatures. This isolation filters out room
temperature Johnson noise while minimizing ohmic heating and re-emission of thermal noise
from attenuators. On the output side, superconducting Niobium coax feeds the microwave
signals first through a pair of circulators and then to a cyrogenic amplifier at 4K. The circulators provide over 25dB of isolation from 4K noise radiating out of the input port of the
amplifier. The cryogenic amplifier is a cryogenic compatible low noise HEMT amplifier with
gain of 37dB and a noise temperature of 3.6K (manufactured at Caltech by S. Weinreb).
The signals are then fed out to room temperature analysis. Refer to Fig. 2.6 for details of
the fridge circuit.

2.3.3

Drive circuit

For our measurements we rely on intense multi-tone pumping as well as weak noise injection.
To achieve this, we construct the circuit outline in Fig. 2.8. Multiple drive tones are produced
either via independent single tone sources, coupled into the same line with passive summers
or direction couplers, or with a multi-tone vector source (Agilent E8267C). At the necessary
42

Sources

Filtering

2 in

Noise Injection

to fridge

Figure 2.8: Drive circuit.a. Multiple microwave sources, both single tone and vector sources, are summed
generate the desired pump configuration. b. A bank of room temperature filter cavities isolate source phase
noise to ensure the cavity does not get excited by classical noise on the input line. c. Noise injection. A
white noise source can selectively inject classical noise into the cavity on the order of one to ten microwave
photons.

powers, the phase noise of the sources at detunings near ωm is large enough to directly
excited our microwave cavity with sufficiently large classical noise.
To remedy this, we implement a bank of tunable copper-can filter cavities that isolate
injected phase noise by over 30dB (typically 50dB) over a narrow band ≈ 450 kHz. These
cavities are mounted at room temperature and typically have loaded quality factors of Q∼
5 × 103 . Depending on measurement requirements, the cavities can be set up in either
transmission or rejection mode and hence can serve either as bandpass filters or notch filters.
Despite all of this filtering, we then add a pseudo-white microwave noise source to inject
a weak amount of classical microwave noise into our circuit. The purpose of this noise is
to allow us control over classical occupation of the microwave resonator with occupations
on the order of one. This serves as an important measurement feature since our system
almost always exhibits classical noise effects. By controlling the amount of classical noise in
the cavity, we can calibrate out classical noise contributions and reveal the desired quantum
behavior.

Chapter 3
Sideband asymmetry
3.1

Introduction

A fascinating aspect of quantum measurement is that the outcome of experiments and the
apparent nature of the object under study depend critically on the properties of both the
system and the measurement scheme [35]. An excellent illustration is found when considering
measurements of the quantum harmonic oscillator. If measured with an ideal energy detector,
the observed signals will demonstrate energy level quantization [36, 37]; measured instead
with an ideal position detector, no evidence of quantized energy levels are found and the
measured signals appear to be that of a very cold, classical oscillator [6, 18]. The details of
the measurement are as essential to the apparent nature of the system under study as the
properties of the system itself – succinctly expressed by Roy Glauber: “A photon is what a
photodetector detects.” [38]
To describe the measured noise of quantum systems, it is often useful to make use of
so-called quantum noise spectral densities, which in general are not symmetric functions
of frequency: Sxx (−ω) 6= Sxx (+ω), where Sxx (ω) is the spectral density of the observable
x(t), defined as the Fourier transform of hx̂(t)x̂(0)i [11]. For a quantum harmonic oscillator,
the negative and positive frequency sides of this spectral density describe the ability of the

system to emit or absorb energy,
Sxx [ω] =

dωeiωt hx̂(t)x̂(0)i

th
= x2zp |χm (−ω)|2 γm nth
m + xzp |χm (ω)| γm nm + 1 ,

(3.1)
(3.2)

with the mechanical susceptibility defined as χm (ω)s = [−i(ω − ωm ) + γm /2]−1 . In the

ground state, nth
m = 0, there is no ability for the harmonic oscillator to emit energy so that
Sxx (−ωm ) = 0. It can, however, absorb energy and as a result, Sxx (+ωm ) = γ4m x2zp , where
xzp =
~/2mωm is the amplitude of zero point fluctuations for a mechanical oscillator
with mass m, resonance frequency ωm , and damping rate γm . This asymmetric-in-frequency
motional noise spectrum was first measured in atomic systems prepared in quantum ground
states of motion [39–41], where the motional sideband absorption and fluorescence spectra
were detected via photodetection.
Analogous quantum noise effects can also be studied in macroscopic mechanical systems,
using electro-mechanical and opto-mechanical devices prepared and probed at quantum limits [6, 7, 10, 42]. These systems exhibit the Raman-like processes of up and down conversion
of photons, resulting from the parametric coupling between mechanical motion and electromagnetic modes of a resonant cavity; the rates of these processes should naturally mirror
the asymmetry in the mechanical quantum noise spectral density Sxx (±ωm ). Recent experiments in optomechanics have demonstrated this expected imbalance between up and down
converted sidebands [43, 44]. Here, we demonstrate the analogous physics in a quantum circuit, where it is now microwave photons (not optical photons) which probe the mechanical
motion.

3.1.1

Toy model

As a first step to understand how the motional sideband asymmetry manifests in an electromechanical system probed through the cavity mode, we consider the following simplified
expressions for the microwave noise spectrum. For a single microwave drive at ideal red

(∆ = −ωm ) and blue (∆ = ωm ) detunings, the Langevin equation for the cavity operator
under RWA is
(†)
χ−1
c (ω)d[ω] = − κ din [ω] − iG∓ b̂ [ω],
where b̂ (b̂† ) and G∓ is associated with detuning ∆ = ±ωm .
The microwave noise spectrum is specified by the output field correlators in the frequency
domain. Relating the microwave correlators with the corresponding mechanical correlators
yields different results depending on the drive detunings. For the up-converted signal,
hdR,out [ω] dR,out [ω ]i ∝ hb̂in [ω] b̂in [ω ]i, 
∝ nth
m δ(ω + ω ),

(∆ = −ωm )

(3.3)

whereas the down-converted signal includes quantum fluctuations,
hdˆ†R,out [ω] dˆR,out [ω 0 ]i ∝ hb̂in [ω] b̂†in [ω 0 ]i,

∝ (nth
m + 1) δ(ω + ω ),

(∆ = ωm ),

(3.4)

so it does appear that the detection of microwave sideband asymmetry serves as an accurate
proxy for motional sideband asymmetry. Per the discussion in Sec. 1.3.1, however, linear
microwave detection senses the symmetrized noise spectrum that is calculated as the average
of both normal and antinormal ordered terms,
h{dˆ†R,out [ω] , dˆR,out [ω 0 ]}i ∝ h{b̂in [ω] , b̂†in [ω 0 ]}i,

∝ (2nth
m + 1) δ(ω + ω ).

(∆ = ±ωm ).

(3.5)

Now it appears that the mechanical asymmetry is averaged out by the process of microwave
measurement. What happened to the mechanical zero-point fluctuations? In the following,
we will model this system within an input-output framework and resolve this contradiction.

Gcool

ωc − δ

ω−

ωc

ωc + δ
ω+

Figure 3.1: Sideband asymmetry pump configuration. Two probe tones (red and blue bar) monitor the up
and down-converted motional sidebands while a third tone (green bar) cools the mechanics. The probe tones
are balanced in power with associated optomechanical coupling strengthcs G and are detuned symmetrically
about cavity resonance (black line) at frequencies ω± = ωc ±(ωm +δ). The up-converted (red are) and downconverted (blue area) motional sidebands are sufficiently detuned to avoid sideband overlap. Generally, we
assume the cavity is occupied with classical noise (beige area) that can mix down to produce real forces on
the mechanics.

3.2

Optomechanical sideband asymmetry

For our analysis, we consider a sideband-resolved system simultaneously probed with two
pumps at near ideal red- and blue-detunings. This model is useful because it closely resembles our measurement routine, models multitone effects that could arise between direct and
indirect coupling between the probe tones, and simplifies to a single drive model in the limit
of setting either probe amplitude to zero.
In contrast, the actual experiment consists of a two-port electro-mechanical system that
we simultaneously pump with three microwave tones, all detuned from cavity resonance.
Two balanced probe tones are detuned symmetrically about the cavity center and are used
to simultaneously monitor the motional sidebands that are up- and down-converted near
cavity resonance but with sufficient detuning to avoid sideband overlap. A third red-detuned
cooling tone dampens the mechanical motion via dynamical back-action and is used to cool
the mechanics to near the ground state. Refer to Fig. 3.1 for a schematic of the pump
configuration. In a frame rotating at the cavity frequency ωc , the drive Hamiltonian is
Ĥdrive =

aν (âeiν(ωm +δ)t + â† e−iν(ωm +δ)t ) + acool (âei(ωm +δc )t + â† e−i(ωm +δc )t ),

(3.6)

ν=±

where drive amplitudes are all assumed to be real. We will initially ignore the coupling to
the cooling beam and instead model the electromechanical dynamics in response to the two
47

balanced drive tones. As we later show, separating the transduced sidebands (compared to
the effective mechanical damping rate) allows one to treat the drives independently so that
we will later incorporate the additional cooling tone without altering the following analysis.

3.2.1

Equations of motion

Following the steps in Sec. 1.4, we make unitary transformations to a rotating, displaced
ˆ with a(t) = hâeiωc t i and b̂ = ĉe−i(ωm +δ)t ). In this frame, the
frame with â = e−iωc t [a(t) + d]
coherent drive amplitude follows
a(t) = a− ei(ωm +δ)t + a+ ei(ωm +δ)t + acool ei(ωm +δc )t ,

(3.7)

where the phases of all drive tones can be arbitrarily defined and assumed zero so that a±,cool
are real. This assumption holds for sufficiently large detunings δ, δc
γtot that suppress
any direct correlations between the separate drives. If this were not the case, the relative
phases between the drive tones could play an important role. Along the lines that the probe
tones act independently, the cooling tone will be temporarily ignored. Now, the interaction
Hamiltonian becomes Ĥint = Ĥlin + ĤCR , where
Ĥlin = −iG− (dˆb̂† + dˆ† b̂) − iG+ (dˆb̂ + dˆ† b̂† ),

(3.8)

ĤCR = −iG− ei2ωm t (dˆb̂ + dˆ† b̂† ) − iG+ e−2iωm t (dˆb̂† + dˆ† b̂).

(3.9)

The enhanced optomechanical couplings are defined via the microwave amplitudes, G± =
g0 a± , which lead to optical scattering rates γop
= 4Gκ± . Given that we typically operate in

the sideband-resolved regime, we make the rotating wave approximation and only treat the
linear contribution, though we can later incorporate counter-rotating effects if necessary.
With Eq. (3.8) and standard input-output relations, the Heisenberg-Langevin equations
in the frequency domain are formatted in matrix notation: D̂ [ω] = χ(ω) · L · D̂in [ω], with

ˆ dˆ† , b̂, b̂† )T , bath operators D̂in = (dˆin , dˆ† in, b̂in, b̂† in)T , and L =
mode operators D̂ = (d,
√ √ √
diag( κ, κ, γm , γm ). As before, the microwave input field is defined as the weighted
48

average over the bath contributions from the input (L), output (R), and internal (int) ports,
dˆin = σ=L,R,int κσ /κ dˆσ,in .
To reiterate, the inverse scattering matrix for this system is

− iω

−1
χ (ω) = 
 −iG−
iG+

−iG−

−iG+

iG+

iG−

−iG+

γm
− i(ω + δ)

iG−

− iω

γm
− i(ω − δ)

.

(3.10)

The relevant scattering parameters are discussed below.

3.2.2

Zero-point bath designations

The bath commutators and correlations for the microwave and mechanical bath operators
are given by Eqs. (1.94)-(1.97). In the following analysis, we will instead designate the bath
commutation relations with separate variables α and β:
[dˆσ,in (t), dˆ†σ,in (t0 )] = αδ(t − t0 ),

(3.11)

[b̂in (t), b̂†in (t0 )] = βδ(t − t0 ).

(3.12)

Here, α represents the quantum fluctuations of the microwave field while β represents those
of the mechanics. Similarly, the updated bath correlations read
hdˆσ,in (t)dˆ†σ,in (t0 )i = (nth
c + α)δ(t − t ),

(3.13)

hb̂in (t)b̂†in (t0 )i = (nth
m + β)δ(t − t ).

(3.14)

Formally, the field operators obey the canonical commutation relations such that α =
β = 1. The purpose of this labeling is not to adjust the commutation relations but rather to
provide a straightforward method to track how the microwave and mechanical fluctuations
propagate throughout the measurement. The ultimate goal of such a labeling scheme is to

help clarify the origins of the sideband asymmetry: does the observed asymmetry stem from
zero-point motion of the mechanics or the shot noise of the microwave field?

3.2.3

Dressed mechanics

As a first consistency check, we can explore how the motional noise power is modified by
electro-mechanical coupling with microwave drive tones at near ideal red- and blue-detunings.
The Langevin equation for the mechanical operator b̂ (defined in the frame rotating at ωm +δ)
is defined via the scattering parameters
b̂ = χ31 κdˆin + χ32 κdˆ†in + χ33 γm b̂in + χ34 γm b̂†in ,

(3.15)

where all explicit frequency dependence has been omitted. In this notation, the mechanical
susceptibility is given by the scattering factor χ33 . We can explicitly calculate the change
in the mechanical frequency and damping due to the interaction with the cavity by looking
at the corresponding“self energy” (i.e. difference in inverse mechanical susceptibilities evaluated on resonance with and without the cavity, see e.g. [45]). Making the rotating-wave
approximation (ω
κ) and assuming sufficient frequency separation between the sidebands
(δ
γtot ), the total mechanical damping is
γtot = 2Re χ−1
33 (ωm )

(3.16)

= γm + γop
− γop

(3.17)

δωm = Im χ−1
33 (ωm )

(3.18)

δ +
= − (γop
− γop
).

(3.19)

while the optical spring shift is

Note that in both the damping and frequency shift, the effects of the red- and blue-detuned
drives conspire to cancel.
50

We can now calculate the motional noise spectrum in the lab frame, where we assume
the mechanics is in the high-Q limit so that the sidebands are tightly confined at positive
and negative frequencies and the bath operators can be assumed to behave Markovian:
Sxx [ω] =

dteiωt hx̂(t)x̂(0)i,

= x2zp |χ31 (ω)|2 + |χ32 (−ω)|2 κ(nth
c + 1)
+ x2zp |χ32 (ω)|2 + |χ31 (−ω)|2 κnth
+ x2zp |χ33 (ω)|2 + |χ34 (−ω)|2 γm (nth
m + 1)
+ x2zp |χ34 (ω)|2 + |χ33 (−ω)|2 γm nth
m,
γtot
γtot
x2 n .
γtot 2 xzp nm + β̃ +
2 zp m
(ω − ωm ) + ( 2 )
(ω + ωm )2 + ( γtot
Here the mechanical occupation factor is given by the detailed balance rate equation
nm =

− th
th
γm nth
m + γop nc + γop (nc + α)
γtot

(3.20)

while β̃ denotes the zero point fluctuations of the dressed mechanical mode with coupling to
both the intrinsic mechanical bath and additional optical bath channels,
β̃ =

)α
− γop
γm β + (γop
= 1.
γtot

(3.21)

γ±

1) or carefully balanced drives (γop
In the limit of sufficiently small cooperativity ( γop

γop
), the mechanical fluctuations consist almost entirely of the zero-point fluctuations of

the intrinsic bath β̃ → β. For a large red-detuned drive (γtot
γm ), the optical damping
dominates over the intrinsic dissipation and the mechanical dynamics is entirely determined
by the microwave field fluctuations, β̃ → α.
This behavior is notable for two reasons. First, the optical contribution to the mechanical
fluctuations ensures that the backaction-imprecision product for position measurement obeys
a strict lower bound enforced by quantum mechanics [11]. It would appear from Eq. (3.20)

that a single red-detuned drive with high cooperativity (γtot ≃ γop
) pumping a cavity with

zero classical noise (nth
c = 0) generates no back-action heating of the mechanical mode and

the detector noise product would be zero. This argument is incorrect because it ignores
the contribution of the quantum fluctuations of the optical field. The pump-dependent
asymptotic heating of β̃ serves as the measurement back-action and ensures adherence to
quantum bounds.
Second, distinguishing between the source of “quantum motion” is a serious issue for
measurements that aim to detect mechanical zero-point fluctuations. If such a measurement
implies the detection of the intrinsic mechanical bath, then the optically-induced damping
must be small. If instead the purpose of such an asymmetry measurement is to measure
quantum-induced motion, i.e., the mechanical response to quantum fluctuations of either
the intrinsic microwave or mechanical noise, then the source of such fluctuations is not
relevant. In this view, there is no limit to the back-action induced damping, which is more
aligned with typical measurement schemes to date, i.e., most systems require intense reddetuned pumping to generate sideband cooling near the motional ground state. However,
in such a limit the mechanical signal can be treated as a classical transducer of microwave
noise and the sideband asymmetry is then traced entirely back to the quantum shot noise
of the microwave field, regardless of how the noise is measured. From this viewpoint, such a
measurement is closely related to experiments that detect the shot noise of electromagnetic
fields via back-action heating of macroscopic mechanical transducers [10, 42].
To date, sideband asymmetry has been measured in mesoscopic mechanical resonators
cooled near the ground state via either passive cooling [46] or active sideband-cooling [16,
43, 44, 47], and thus both high and low cooperativity regimes (γtot ≃ γm vs. γtot
γm ) have
been explored. To simplify notation in what follows, we will ignore the subtlety in defining
“quantum motion” in an electromechanical device and associate β̃ → β with intrinsic zeropoint fluctuations of the mechanics.

As a final note, the symmetrized mechanical noise spectrum scales as expected,
S̄xx [ω] =

dteiωt h{x̂(t), x̂(0)}i,

x2zp γtot
)2
(|ω| − ωm )2 + ( γtot

3.2.4

β̃
nm +

(3.22)

(3.23)

Microwave spectrum

Amplitude of the output field V̂out (t) = dˆR,out (t) + dˆ†R,out (t). The output operator defined via
√ ˆ
standard input-output relations dˆσ,out = dˆσ,in − κσ d.
In terms of the scattering parameters,
dˆR,out = dˆR,in − χ11 κR κdˆin − χ12 κR κdˆ†in − χ13 κR γm ĉin − χ14 κR γm ĉ†in .

(3.24)

For illustrative purposes, we present the scattering parameters for the microwave field
assuming weak coupling (κ
γtot ) and large sideband separation (δ
γtot ):
±γop
χ11 (ω) =
1+
2i(ω
δ)
tot
p + −
γop
± γop
χ12 (ω) =
κ ± γtot − 2i(ω ∓ δ)
p −
2 γop
χ13 (ω) =
γtot − 2i(ω + δ)
p +
2 γop
χ14 (ω) =
γtot − 2i(ω − δ)

(3.25)
(3.26)
(3.27)
(3.28)

Note that χ11 captures the various different ways cavity noise can propagate through
the system. As we discuss below, |χ11 |2 will contain non-vanishing mixing terms that can
be interpreted either as interference between scattering channels or as correlations between
back-action and imprecision measurement noise.
For calibration, we typically measure the complex transmission through the device, given
by
S21 (ω) = − κL κR χ11 .
53

(3.29)

a)
ωc − ωm − δ

ωc

ωc + ωm + δ

b) Linear detection

c) Photodetection

S̄xx

Sxx

ωc − δ

ωc + δ

SII

S̄II
Shot Noise

ωc − δ

Detector Noise
ωc + δ

ωc − δ

Detector Noise
ωc + δ

S̄IF
ωc − δ

ωc + δ

SII,tot

S̄II,tot

ωc − δ

ωc + δ

Figure 3.2: Comparison between linear detection and photon counting. a. Pump scheme. We consider
a single microwave cavity (dotted line) pumped at ωc ± (ωm + δ) (green bars). The up -converted (red
bar) and down-converted (blue bar) motional sidebands are placed tightly within the cavity linewidth. For
figure clarity, the occupations of the microwave and mechanical modes are assumed to be zero. b. Linear
detection. The quantum contribution from the symmetrized motional noise S̄xx is present in both sidebands.
Microwave shot noise (brown band) and amplifier noise (beige band) combine to form the imprecision noise
S̄II . This measurement is sensitive to noise correlations between the microwave and mechanical odes (S̄IF ),
which results in asymmetric squashing (red region) and antisquashing (blue region) of the noise floor. c.
Photodetection. Normal-ordered detection is sensitive to the asymmetric motional noise spectrum Sxx . The
detector is not sensitive to microwave shot noise, and the noise floor (SII ) is from detector nonidealities
(beige band), analogous to dark counts for a photodetector. Although the source is different, the sideband
imbalance is identical in both photodetection and linear detection.

3.2.5

Symmetric noise detection

Since we implement a linear detection scheme in this experiment, we construct the symmetrized microwave noise spectrum by substituting the above scattering parameters into the
expression for symmetric detection, except now the mechanical and microwave zero-point

bath fluctuations are separately specified:
S̄R [ω] =

dteiωt h{V̂out (t), V̂out (0)}i,

(3.30)

dteiωt hdˆ†R,out (0)dˆR,out (t) + dˆR,out (t)dˆ†R,out (0)i,

(3.31)

= κR κ |χ11 |2 + |χ12 |2 (nth
c + α/2)
+ κR γm |χ13 |2 + |χ14 |2 (nth
m + β/2)
+ [1 − κR (χ11 + χ∗11 )] (nth
r + α/2),

γtot γop
α
κR X
nm +
± neff +
= S̄o +
κ ± (ω ∓ δ)2 + (γtot /2)2

(3.32)
(3.33)
(3.34)
(3.35)

The noise floor is
S̄o [ω] =

4κR th
+ nth
(nc − nth
r +
r ),

(3.36)

th
and we have defined neff = 2nth
c − nr . The noise floor is shaped by the cavity response due

to interference from noise that reflects off the port correlated with noise that is re-radiated
from the cavity. The underlying components of this spectrum are outlined in Fig. 3.2(b).
One sees explicitly that the sideband imbalance is proportional to (2neff + α) and hence
is entirely due to fluctuations in the microwave fields driving the cavity. This interpretation
th
is true both when this noise is thermal and when it is purely quantum (i.e., nth
r = nl = 0).

These terms in the spectrum result from the interference between the two ways the incident
field noise can reach the output: either by directly being transmitted through the cavity
or by first driving the mechanical resonator whose position then modulates the amplitude
quadrature of the outgoing microwaves. This interference is the basic mechanism of noise
squashing, which in the case of thermal noise was previously observed in a microwavecavity-based electromechanical system [18]. This mechanism can also be fully described
using a general linear measurement formalism [48], where it is attributed to the presence of
correlations between the backaction and imprecision noise of the detector [9, 49].
The above calculation also shows that both the thermal and zero-point force noise emanating from the mechanical bath contribute symmetrically and hence play no role in de55

termining the asymmetry of the sidebands. This suggests that the sideband asymmetry
observed using linear detection of the scattered field is not directly probing the asymmetric
quantum noise spectrum of the mechanical mode.
To simplify the notation in the following, we consider the mechanical noise occupations
inferred from the output spectrum about the up- and down-converted sidebands
dω
1 κ
S̄R [ω ∓ δ] − S̄o
γop κR
2π

α
= nm +
± neff +

n±
m =

3.2.6

(3.37)
(3.38)

Photon counting

An alternate measurement strategy to amplitude detection is to first filter the output signal
to a narrow bandwidth around a frequency ω and then perform direct photodetection. One
is thus measuring power directly without first measuring field amplitudes, and in a manner
that is only sensitive to the absorption of photons. As a result, such a measurement is
described by the normal ordered spectrum
SR [ω] =

dteiωt hdˆ†R,out (0)dˆR,out (t)i

= κR κ|χ11 |2 nth
+ κR κ|χ12 |2 (nth
c + α)
+ κR γm |χ13 |2 nth
+ κR γm |χ14 |2 (nth
m + β)
+ [1 − κR (χ11 + χ∗11 )] nth
r .
Substituting in the above scattering parameters, we find
γtot γop
κR X
SR [ω] = So +
κ ± (ω ∓ δ)2 + (γtot /2)2

nm +
± neff +

(3.39)

where the symmetrized noise floor is defined in the vicinity about cavity center as So = S̄o − α2 .

3.2.7

Spectrum comparison

Comparing the symmetrized microwave noise spectrum with the normal-ordered spectrum,
it is clear that the spectra appear identical up to a flat background given by the microwave
shot noise,
S̄R [ω] = SR [ω] +

(3.40)

This is no coincidence. Both spectra are formally related via the canonical commutation
relation of the output microwave field
[dˆR,out (t), dˆ†R,out (t0 )] = αδ(t − t0 ).

(3.41)

Regardless of the physical details of the detection scheme, if one assumes this commutation relation, then one can legitimately interpret symmetric or normal-ordered detection
as measuring the same thing. On a formal level, this substitution requires that the input
fluctuations of the microwave and mechanical baths are identical. If α 6= β, the output
commutator would differ from the input commutator,

dˆR,out (ω), dˆ†R,out (ω 0 ) = α +

±γm γop
(β − α) δ(ω + ω 0 ).
(ω ∓ δ)2 + (γtot /2)2

(3.42)

This commutation relationship is associated with the relationship between detector absorbs and emits energy into the system. There is a fundamental difference between the two
detection schemes: linear detection is sensitive to both absorption and emission of photons
(and hence does sense the microwave quantum shot noise) whereas photon counting only
absorbs energy from the environment (and hence does not sense the shot noise fluctuations
of the electromagnetic field). If the purpose of the sideband asymmetry experiment is to
directly compare the rates of energy scattering into and out of the mechanical mode, then
one can interpret the commutator substitution as also asserting the mechanical commutator.

However, a deeper issue is that it is beyond the scope of input-output theory to tune the
mechanical commutator since such a model would violate the canonical commutator for the
output microwave field.
In the face of such subtleties, we adhere to the interpretation that most closely approaches
the physical techniques used in the measurement: sideband asymmetry with symmetric
detection is most naturally attributed to the shot noise of the microwave field. Similarly,
sideband asymmetry via photon counting is most naturally attributed to the zero-point
fluctuations of the mechanical bath.
As a final note, Eq. (3.42) is asymptotically true for γtot /γm → ∞ with α 6= β, which
again supports the idea that under large optical damping, the mechanics acts as a classical
transducer responding to the microwave fluctuations. In this limit, the asymmetry is attributed to the shot noise of the microwave field regardless of the choice of detection scheme
or choice of interpretation.

3.2.8

Cooling tone

So far, we have modeled the motional sidebands transduced via two probe tones while ignoring a third cooling tone. From the format of Eq. (3.35), we can now generalize the system
to additional drive tones.
Assuming large frequency separation of the sidebands, the probe tones indirectly interact
by modifying the mechanical susceptibility and mechanical occupation factor. Following
the same behavior as the red-detuned probe, the additional cooling tone mainly increases
the mechanical damping rate and subsequently cools the mechanical occupation factor c.f.
Eq. (3.20). To reflect this behavior, the mechanical parameters in Eq. (3.35) are updated to
capture the effects of the cooling tone:
cool
γM = γm + γop

(3.43)

th
cool th
γM nth
M = (γm nm + γop nc ).

(3.44)

We consider moderate cooling in the weak-coupling regime and hence ignore deviations that
arise from strong coupling or bad cavity corrections.

3.3

Results

Having explored the interpretation subtleties associated with sideband asymmetry, we now
turn to presenting the experimental observation of this imbalance in a microwave-cavity
based electromechanical system.
Our system is composed of a superconducting microwave resonator, also referred to as
“cavity”, where the resonance frequency is modulated by the motion of a compliant membrane. This frequency modulation leads to the desired parametric coupling between microwave field and mechanical motion (Fig. 3.3(a)). Measurements of the cavity response
below 100 mK yield the resonance frequency ωc = 2π × 5.4 GHz, total loss rate κ = 2π × 860
kHz, output coupling rate κR = 2π × 450 kHz, and input coupling rate κL = 2π × 150
kHz. The capacitor top gate is a flexible aluminum membrane (40µm×40µm×150nm) with
a fundamental drumhead mode with resonance frequency ωm = 2π × 4.0 MHz and intrinsic
loss rate γm = 2π × 10 Hz at 20mK. Motional displacement of the top gate modulates the
microwave resonance frequency with an estimated coupling rate of g0 = ∂ω
x = 2π × 16
∂x zp

Hz.
In Fig. 3.3(c), we present a schematic of the measurement configuration used in this
work. Tunable cavity filters at room temperature reduce the source phase noise to the
thermal noise level at 300K; cryogenic attenuators further reduce the noise down to the shot
noise level [18]. A pair of microwave switches at the device stage select between the device or
a bypass connection for high precision noise floor calibration of the cryogenic amplifier. The
output signal passes through two cryo-circulators at ∼100mK followed by a cryogenic lownoise amplifier at 4.2K, and finally to a room temperature circuit for analysis. The occupation
−3
factor of the microwave resonator, nth
at
c , which is expected to thermalize below 5 × 10

temperatures below 50mK, can be increased and controlled by the injection of microwave
frequency noise from amplified room temperature Johnson noise. From careful measurements
59

b)
−6

40
-50

Pm / Pthru
10
10 μm

c)
Cooling

Red Probe

ωc − ωm − δ

Blue Probe

∆ω
50
2π (Hz)

240

−7

Filter Cavity

4K20mK

200mK

40
-100 ∆ω
100
2π (Hz)

T (mK)

20mK

100

HEMT

100mK

Noise Injection

ωc − ωm − δ c

20mK

aW/Hz

Spectrum
Analyzer

ωc + ωm + δ

Figure 3.3: Device, calibration, and measurement scheme. a. Electron micrograph of the measured device.
A suspended aluminum (grey) membrane patterned on silicon (blue) forms the electro-mechanical capacitor.
It is connected to the surrounding spiral inductor to form a microwave resonator. Out of view, coupling
capacitors on either side of the inductor couple the device to input and output co-planar waveguides. b.
Motional sideband calibration. The cryostat temperature is regulated while the mechanical mode is weakly
probed with microwave tones set at ωc + ωm + δ (blue) and at ωc − ωm − δ (red) detunings, with δ = 2π × 500
Hz. The observed linear dependence provides the calibration between the normalized sideband power and
the mechanical occupation factor. Inset, up-converted motional sideband spectra collected at 20mK (top)
and 200mK (bottom), with ∆ω = ω − (ωc − δ). c. Schematic of the microwave measurement circuit.

of the noise power emanating from the cavity at zero pumping and comparing this to power
spectra with the bypass switched in place, we conclude that there is a small contribution to
nth
c due to thermal radiation from the isolated port of the cryogenic circulators, given by the
occupation factor nth
r = 0.34 ± 0.03.
When a single microwave tone is applied to the device at ωp , the parametric coupling
converts mechanical oscillations at ωm to up and down-converted sidebands at ωp ± ωm . In
this experiment, we apply microwave tones at frequencies near ωc ± ωm and at powers given
by the mean number of photons in the resonator, np . The microwave resonance suppresses
motional sidebands outside of the linewidth and we consider only the contributions of signals
converted to frequencies near ωc . These are the Lorentzian components of the noise power
spectra of Eq. (3.35), which for the remainder of the paper are denoted by “+” and “-”,
respectively.
Throughout the measurement, we simultaneously apply three microwave tones. We place
a cooling tone at ωc − ωm − δc to control the effective mechanical damping rate, γM , and
mode occupation, nm , via back-action cooling [45, 50]. Two additional probe tones, placed
at ωc ± (ωm + δ), produce up and down converted sidebands symmetrically detuned from
cavity center (Fig. 3.5(a)). The detunings are chosen to ensure no interference between the
sidebands (δc = 2π × 30 kHz, δ = 2π × 5 kHz) so that we may consider the probe sidebands
as independent measurements of the dressed mechanical mode.

3.3.1

Calibrations

To convert the motional sideband powers into equivalent mechanical occupation, we turn
off the cooling tone and measure the probe sidebands (δ = 2π × 500 Hz) with low optical
damping (n+
p = np ≃ 5 × 10 ) and high mechanical occupation set by the cryostat temper-

ature. Regulating the temperature to calibrated levels between 20 to 200mK, we calculate
the integrated noise power under the sideband Lorentzians, Pm± , normalized by the respec±
tive microwave probe power transmitted through the device, Pthru
. In the limit of high

thermal occupation, the normalized power is directly proportional to nm [51]. As we vary

the cryostat temperature, T , we compare the normalized power to the thermal occupation
factor [exp( k~ωBmT ) − 1]−1 (Fig. 3.3(b)). A linear fit yields the conversion factors for the up-

converted (n−
m ) and down-converted (nm ) sidebands: nm = (9.9 ± 0.2) × 10 · Pm /Pthru and

n+
m = (5.4 ± 0.1) × 10 · Pm /Pthru . The factor of c.a. two between calibration factors at the

two pump detunings is due to the presence of a parasitic bypass channel in the microwave
circuit that allows pump signals to weakly transmit across the input and output ports of the
device while completely bypassing the microwave resonator (see Sec. 1.1.3).

3.3.2

Sideband ratio and imbalance

Further detuning the probe tones (δ = 2π × 5 kHz) and turning on the cooling tone (δc =

2π × 30 kHz), we explore the sideband ratio, n+
m /nm , over various mechanical and microwave

occupations. To reduce nm to values approaching 1, we increase the cooling tone power
up to ncool
= 4 × 105 . For sideband characterization, the probe tone powers are set to

n−
p = np = 10 and the probe sideband spectra are analyzed using the conversion factors
described above. The imbalance between n+
m and nm is clearly evident in the noise spectra

(Fig.3.5(b)).
As further demonstration of the asymmetry with respect to neff , we plot n+
m /nm as a

function of n−
m in Fig. 3.5(c). Each curve corresponds to one setting of injected microwave
noise. The data shows excellent agreement to the expected ratio, n+
m /nm = 1+(2neff +1)/nm .

This relationship highlights the combined effect of quantum and classical noise in Eq. (3.35).
By fitting each curve to a two parameter model, a + b/n−
m , we find an average constant offset
a = 0.99 ± 0.02 for all curves, accurately matching the model and confirming our calibration
techniques. Fitting for b, the data indicates neff spanning 0.71 to 4.5 with uncertainty all
within ±0.09 quanta.
To quantify the contributions due to quantum fluctuations and classical cavity noise, we
fix the cooling tone power at ncool
= 4 × 105 (γM = 2π × 360 Hz) and measure the imbalance
n+
m − nm as we sweep neff . At each level, we measure the average noise power density, η, over

a 250 Hz window centered at ωc and away from any motional sideband. Over this range, η

neff = 0.6

neff = 1.1

neff = 1.6

neff = 2.1

neff = 2.5

SR- So
(aW/Hz)

-103

∆ω
2π (Hz)

103

Figure 3.4: Sideband imbalance. Microwave spectrum centered about the up-converted (red) and downconverted (blue) sidebands and with the noise floor of the measurement chain subtracted off. As the classical
noise in the cavity is increased from neff = 0.6 to 2.5, the noise floor increases, the average sideband
occupation increases, and the sideband imbalance grows.

contains two contributions: the noise radiating out of the microwave resonator, proportional
to neff , and the detector noise floor, set by the noise temperature of the cryogenic amplifier
(TN ≈ 3.6K). We directly measure the detector noise floor by switching from the device to
an impedance-matched bypass connection and measure the noise power density, η0 , over the
same window with matching detected tone powers.
In Fig. 3.5(d), we plot the sideband imbalance against the noise floor increase, ∆η =
η − η0 , which is expected to follow: n+
m − nm = 2neff + 1 = 4λ · ∆η + 1, where λ is the

conversion factor for ∆η in units of cavity quanta, nth
c . The detected noise spectrum at each

noise level measured relative to the floor is shown in Fig. 3.4. The data clearly follows a
linear trend with a slope of λ = (2.7 ± 0.1) × 10−1 (aW/Hz)−1 . More importantly, we observe
an offset of 1.2 ± 0.2, in excellent agreement with the expected quantum imbalance of “+1”
from the quantum fluctuations of the microwave field.
As an additional check, we also consider the sideband average, (n−
m +nm )/2, as a function

of ∆η. Averaging the up and down-converted Lorentzian components of Eq. (3.35), we see
that the resulting occupation, nm + β2 , does depend on neff due to the coupling between the
γ cool

γop
op
th
th
cool
mechanical and microwave modes, nm = γγtot
nth
m + γtot (2nc + α) + γtot nc , where γop (γop )

is the optical coupling rate for the individual probe (cooling) tones. Accounting for this socalled back-action heating of the mechanical mode [42,45], we recover λ = (2.5 ± 0.2) × 10−1
(aW/Hz)−1 , consistent with the imbalance results above.

Notably, the average sideband occupation does contain contributions from mechanical
zero-point fluctuations. Future experiments could infer the mechanical quantum contribution

S II,tot

d)
nth
eff

n−

ωc − δ

n−
m / nm

ωc + δ

aW/Hz

4.54(9)
3.01(7)
2.30(7)
1.68(7)
0.71(5)

-5.5

-4.5
4.5
(ω − ωc )/2π (kHz)

5.5

0.54

1.1

1.6

2.1

2.7

+ +
( n−
m nm )/2

− +
n−
m nm

nth
eff

8 10
n+

Microwave shot noise

∆η (aW/Hz)

Figure 3.5: Sideband asymmetry. a. Pump scheme. Three tones are placed about the microwave resonance.
Two probe tones generate up-converted (red) and down-converted (blue) sidebands. An additional tone
(purple) cools the mechanical mode. b. Sideband spectra. S̄R [ω] measured at neff = 0.60 (blue) and 2.5
(orange) with nm = 4.7 ± 0.1. c. Sideband asymmetry. The ratio n+
m /nm vs. nm is plotted for increasing
noise injection. d. Sideband imbalance (blue) and sideband average (purple) vs. the measured noise increase,
∆η. Sideband imbalance, n+
m −nm , and average, (nm +nm )/2, exhibit a linear trend with ∆η. The imbalance
at ∆η = 0 is the quantum imbalance due to the squashing of fluctuations of the microwave field.

of β2 with a method to independently calibrate nm to high accuracy, for example, with a
passively cooled high frequency mechanical mode thermalized to a primary low temperature
thermometer.

3.3.3

Output port occupation

We estimate the occupation factor of the output port, nth
r , by measuring the microwave noise
spectrum absent any microwave pumping. In this setup, we assume that nth
c is solely due to
noise radiating into the device from the the isolated port of a cryogenic circulator, so that
th
nth
c = nr κR /κ. This noise source generates a dip in the broadband noise floor,

κ
κ2
th
th
S̄R [ω] =
− 1 nr +
αR + 2nr
λ κ2 + 4(ω − ωc )2 κ
4κR

(3.45)

Taking κκR from independent calibration measurements and λ from the sideband imbal−1
ance measurements, we fit the observed Lorentzian to find nth
r = (3.4 ± 0.3) × 10 . A typical

noise floor spectrum is shown in Fig. 3.6.

aW/Hz (arb. offset)

0.2
−0.2
−0.4
−0.6
−0.8
−1

−2

−1
( ω ωc ) 2π (MHz)

Figure 3.6: nth
r noise spectrum. Example spectrum of microwave noise taken at zero pumping (light blue)
with Lorentzian fit (dark blue).

3.3.4

Noise floor calibration

The increase in the device noise floor at cavity resonance is measured relative to the noise
floor of an impedance matched through connection with matching amplifier conditions. With
the device switched in place, the broadband microwave noise floor is
S̄o [ω] =

th
th
+ nth
r + α/2.
ω 2 + (κ/2)2

(3.46)

Since this noise floor dip is not present when switching in the impedance-matched through,
the observed noise floor increase has a small correction that is proportional to neff and nth
r ,

2κR − κ
∆η =
neff −
nth
2λ
2κR

(3.47)

th
th
where neff = 2nth
c − nr as above, and where λ is the conversion factor for ∆η in units of nc .

To see how this behavior affects our measurements, we consider the sideband powers in
th
the presence of classical noise nth
c , nr . Integrating the noise power under the transduced

sidebands of Eq. (3.35), we find that
n+
m − nm = 4λ∆η +

2κR − κ
κR

nth
r + 1,

(3.48)

and
n+
m + nm

cool
2γop + γop
γM

!
λ∆η +

4κR − κ
4κR

nth

γm th γop 1
n +
+ ,
γM m γM

(3.49)

= γop .
= γop
where we follow the notation of Eq. (3.43) and have set α = β = 1, γop

The nth
r contribution does not affect the slope of either data set. For sideband imbalance
and average measurements, we expect linear dependence on ∆η with slope proportional to λ.
The nth
r factor does, however, add fixed offsets to both data sets. For the sideband difference,
the contribution is suppressed relative to the quantum offset of “+1”. With the experimental
parameters nth
r = 0.34 ± 0.03, κ = 2π × (860 ± 10) kHz, and κR = 2π × (450 ± 30) kHz, we
2κR −κ
−2
estimate an offset correction of
nth
r ≈ (3 ± 4) × 10 , well within the measurement
κR
uncertainty for sideband imbalance. This is not the case for the sideband average, where
we expect a correction to the offset that is significant when compared to the mechanical
quantum contribution of “+1/2”.

3.3.5

Conclusion

In summary, we report the quantum imbalance between the up and down-converted motional
sideband powers in a cavity electro-mechanical system measured with a symmetric, linear
detector. We show that for linear detection of the microwave field, the imbalance arises
from the correlations between the mechanical motion and the quantum fluctuations of the
microwave detection field. For normal-ordered detection of the microwave field, however,
the imbalance arises directly from the quantum fluctuations of the mechanics. By further
assuming that the output microwave field satisfies the canonical commutator, which also
determines the quantum fluctuations of the mechanical mode, the measurement can be
interpreted as performing either symmetric or normal-ordered detection regardless of the type
of detector utilized. In both scenarios, the imbalance in motional sidebands is a fundamental
quantity originating from the Heisenberg’s uncertainty relations and provides a quantum
calibrated thermometer for mesoscopic mechanical systems.

Recently, four other groups have measured the motional sideband asymmetry in micronscale opto- and electro-mechanical systems with both linear [52, 53] and nonlinear [46, 47]
detectors, akin to the photon-counting techniques described here. Whatever the interpretation choice of the reader, quantum motion in a macroscopic mechanical resonator has now
been demonstrated unambiguously.

Chapter 4
Back-action evasion detection
In the previous section, we explored the behavior for detuned two-tone probing of an electromechanical system. Detuning the sidebands was crucial to simultaneously probe the upand down-converted motional sidebands without introducing direct correlations between the
two drive tones. Though the probes indirectly interact via the dressed the mechanical mode,
either by modifying the occupation factor or mechanical susceptibility, there are no backaction imprecision correlations between separate sidebands. What happens when this is no
longer the case?
Let us first consider a pair of balanced red- and blue-detuned drive tones (G− = G+ = G)
that are arranged so that the up- and down-converted motional sidebands perfectly overlap
(Fig. 4.1). In this configuration, the cavity field becomes

α± e−i(ω± t+φ± ) ,

(4.1)

= α± cos(ωm t)e−iωc t ,

(4.2)

α(t) =

with balanced drive amplitudes, α± = α+ = α− . At this point, we ignore the drive phases
and set φ± = 0 – refer to Sec. 4.1.3 for further discussion of the role of φ± .
The cavity field now consists of a fast oscillating carrier tone at ωc modulated at the
mechanical resonance ωm . Since cavity field drastically boosts the coupling between the
cavity and mechanics, the interaction strength will be modulated in time at the mechanical

ωc
ωc − ωm

ωc

ωc + ωm

Figure 4.1: BAE pump configuration. In a back-action evading (BAE) scheme, two drive tones are detuned
about the microwave cavity (black line) at frequencies ωc ±ωm and have balanced amplitudes with associated
optomechanical couplings G. The up- and down-converted sidebands are converted into the center of the
cavity and are perfectly overlapped (purple area). Backaction forces generated via cavity noise (beige area)
mixing exhibit correlations resulting from the red-detuned drive (red bar) and blue-detuned drive (blue
bar) mixing with the same region of cavity noise near ωc . Though suppressed by the cavity DOS, sidebands
converted outside the cavity (small red and blue peaks located ωc ±2ωm ) do not exhibit noise floor correlations
and hence will generate quadrature-independent heating.

resonance frequency, and hence the cavity detector will be sensitive only to motion that
is in-phase with the modulation signal while out-of-phase motion will average to zero over
many mechanical periods. To show that this is indeed the case, we can analyze this drive
scheme within the input-output framework.

4.0.1

Interaction Hamiltonian

From these equations, we can now consider the linearized interaction Hamiltonian
ˆ b̂e−iωm t + b̂† eiωm t ).
Ĥ = ~g0 [α(t)dˆ† + α(t)∗ d](

(4.3)

Substituting in the cavity field α(t), we separate the Hamiltonian into constant DC terms
and counter-rotating terms, Ĥ = Ĥlin + ĤCR ,
Ĥlin = G(dˆ + dˆ† )(b̂ + b̂† ),

(4.4)

ĤCR = G(dˆ + dˆ† )[(b̂ + b̂† ) cos(2ωm t) − i(b̂ − b̂† ) sin(2ωm t)].

(4.5)

with G = g0 α± and dˆ (b̂) is the annihilation operator for the microwave (mechanical) mode
defined in the interaction picture. Assuming sufficient sideband resolution, the counterrotating effects are suppressed by the cavity susceptibility and thus ĤCR will be initially
69

ignored in the following analysis.
The operator Langevin equations directly follow from the transformed Hamiltonian as
χ−1
c (ω) d[ω] = − κdin [ω] − iG± (b̂[ω] + b̂ [ω]),
ˆ†
χ−1
m (ω) b̂[ω] = − κb̂in [ω] − iG± (d[ω] + d [ω]).

(4.6)
(4.7)

γm
−1
In this rotating frame, the susceptibilities χ−1
m (ω) = −iω + 2 and χc (ω) = −iω + 2 are

Hermitian invariant, i.e., χm (ω) = χ∗m (−ω) and χc (ω) = χ∗c (−ω), so that the above equations
transform directly into a closed system of equations between the microwave and mechanical
field quadratures X̂1 = (b̂ + b̂† ) and Û1 = (dˆ + dˆ† ),
χ−1
c (ω) Û1 [ω] = − κ Û1,in [ω],
χ−1
m (ω) X̂1 [ω] = − γm X̂1,in [ω],

(4.8)
(4.9)

with bath inputs X̂1,in = (b̂in + b̂†in ) and Û1,in = (dˆin + dˆ†in ). It appears that the probe tones have
no effect on this specific pair of oscillator quadratures. If we instead consider the orthogonal
quadratures X̂2 = −i(b̂ − b̂† ) and Û1 = −i(dˆ − dˆ† ) with bath inputs X̂1,in = (b̂in + b̂†in ) and

Û1,in = (dˆin + dˆ†in ),

χ−1
c (ω) Û2 [ω] = − κ Û2,in [ω] − 2iG± X̂1 [ω],
χ−1
m (ω) X̂2 [ω] = − γm X̂2,in [ω] − 2iG± Û1 [ω],

(4.10)
(4.11)

with bath inputs X̂2,in = −i(b̂in − b̂†in ) and Û2,in = −i(dˆin − dˆ†in ).
The correlations between the backaction forces conspire to direct the measurementinduced backaction noise onto a single mechanical quadrature, i.e., the motional heating
is driven entirely by fluctuations in Û1 . Similarly, the mechanical-induced heating of the
microwave field (i.e., the motional sideband) is transduced entirely into the conjugate microwave quadrature. That is, X̂1 is transduced entirely onto Û2 while X̂2 is not sensed. Thus,
there is no time-dependent feedback mechanism to drive dynamical backaction effects like

damping or amplification.

4.0.2

Quadrature definitions

In the previous section, we briefly introduced the harmonic oscillator quadrature fields as
quantum variables that naturally arise from this balanced and overlapped drive scheme. Now,
we will now discuss the motivation and significance of the harmonic oscillator quadrature
fields.
In general, a one-dimensional oscillator is defined via the dynamics of two conjugate
variables and can thus be instantaneously defined classically in a two-dimensional phase
space. Motivated by the form of the electromechanical interaction, we shall consider the
quadrature amplitudes that are defined in the basis of the cosine (in-phase) and sine (outof-phase) components of the oscillating signal. Thus, we can explicitly separate the position
in to the quadrature components 1 ,
x̂ = xzp (ĉ + ĉ† ) = X̂1 cos(ωm t) + X̂2 sin(ωm t),

(4.12)

where the mechanical quadrature operators are defined via the ladder operators in the interaction picture (ĉ = b̂e−iωm t ) as
X̂1 = xzp (b̂ + b̂† ),

(4.13)

X̂2 = −ixzp (b̂ − b̂† ),

(4.14)

with canonical commutation relations,
[X̂1 , X̂2 ] = 2ix2zp .

(4.15)

There are two commonly used quadrature definitions: X̂1 = xzp (dˆ + dˆ† ) or X̂1 = (dˆ + dˆ† )/ 2. Relevant
to this discussion, the definitions carry different commutation relations: [X̂1 , X̂2 ] = 2ix2zp versus [X̂1 , X̂2 ] = i
and energy prefactors (dˆ† dˆ + dˆdˆ† ) = (X̂12 + X̂22 )/2x2zp versus (X̂12 + X̂22 ). In this work, we use the former
definition.

These definitions can be generalized to an arbitrary axes rotation in quadrature space,
X̂φ = xzp (eiφ b̂ + e−iφ b̂† ),

(4.16)

= X̂1 cos φ − X̂2 sin φ.

(4.17)

For the purpose of the experiment, this phase angle is controlled by the drive tone phases in
the lab frame; refer to Sec. 4.1.3 for more information.
In terms of the quadrature definitions, the bare dynamics of the mechanical resonator is
prescribed by the system Hamiltonian,
~ωm †
(â â + ââ† ),
~ωm 2
= 2 (X̂1 + X̂22 ).
4xzp

Ĥo =

(4.18)
(4.19)

Moving into the Heisenberg picture, the bare quadrature time evolution follows from the
quadrature commutator and explicit quadrature time-dependence:
X̂φ = [Ĥo , X̂φ ] + X̂φ ,
dt
∂t
= 0.

(4.20)
(4.21)

There is no dynamic coupling between the conjugate quadrature variables X̂1 and X̂2 (or any
other pair of orthogonal quadratures) and ignoring environmental dissipation, the quadratures are constants of motion. Perturbing one quadrature will not affect the other and hence
the quadratures are candidate quantum non-demolition (QND) measurables [54–56].
We have assumed the existence of the microwave field quadratures in the above. Since
the microwave cavity is also a harmonic oscillator defined in a two-dimensional phase space
for electromagnetic field analogs to position and momentum (i.e., conjugate variables defined
between functions of voltage and current), the microwave quadratures are well-defined quantities with near identical operator definitions, commutation relations, and bare dynamical

behavior,
Û1 = dˆ + dˆ† ,

(4.22)

Û2 = −i(dˆ − dˆ† ),

(4.23)

[Û1 , Û2 ] = 2i,

(4.24)

Ûφ |Ĥo = 0.
dt

(4.25)

We can reinterpret the interaction Hamiltonian as a linear coupling between the quadratures,
Ĥlin = ~G± X̂1 Û1 .

(4.26)

which immediately motivates features of Eqs.(4.8)-(4.11): the mechanical transduction of
X̂1 onto Û2 , the optical back-action of Û1 onto X̂2 , and the associated stationary behavior
for both X̂1 and Û1 .
The microwave field acts as a X̂1 detector and will induce optical back-action forces
on the orthogonal X̂2 that are dynamically uncoupled from the measurement. This is the
purpose of setting up balanced, overlapping drive tones – by isolating the back-action forces
from the detected parameter, the drive configuration forms a QND measurement of a single
mechanical quadrature. In the following, this measurement scheme is referred to as backaction evasion (BAE) detection [35, 57, 58]. This type of measurement is fundamentally
different than continuous position measurement where the measurement will increasingly
perturb both position and momentum with increasing measurement strength, leading to the
so-called standard quantum limit (SQL) for position detection [11]. Over the past decade,
position measurements have reached imprecision levels below that at SQL [59,60]. Similarly,
the back-action heating driven via quantum fluctuations of the measurement field have been
observed in mechanical system via electrons [61] and via photons [10, 62].

4.0.3

Noise spectrum

To support the claims of QND detection, we now calculate the mechanical quadrature spectrum and the output microwave noise spectrum. Starting with the mechanics, the symmetrized quadrature noise spectrum is,
dteiωt h{X̂φ (t), X̂φ (0)}i
S̄X̂φ [ω] =
dω 0
h{X̂φ [ω 0 ], X̂φ [ω]}i.
2π

(4.27)
(4.28)

Substituting in Eqs. (4.8-4.11) along with the quadrature bath correlations
hX̂φ,in [ω]X̂φ,in [ω 0 ]i = x2zp (2nth
m + 1) 2πδ(ω + ω ),

(4.29)

hÛφ,in [ω]Ûφ,in [ω 0 ]i = (2nth
c + 1) 2πδ(ω + ω ),

(4.30)

hX̂φ,in [ω]Ûφ,in [ω 0 ]i = 0,

(4.31)

yields the quadrature spectrum,
S̄X1 [ω] /x2zp =

γm
th
bad
γm 2 1 + 2(nm + nm ) ,
ω +( 2 )

(4.32)

S̄X2 [ω] /x2zp =

γm
th
bad
γm 2 1 + 2(nm + nba + nm ) .
ω +( 2 )

(4.33)

The back-action heating of X̂2 manifests via the occupation factor,
nba =

2γop
(2nth
c + 1),
γm

(4.34)

where the optical scattering rate is defined as γop = 4G2 /κ.
In Eqs.(4.32, 4.33), we have included bad-cavity effects generated by the counter-rotating
terms of ĤCR . To first order in drive power, the counter-rotating terms introduce weak backaction forces that are included here in the form of quadrature-insensitive heating given by the
κ 2
occupation factor nbad
m = 32 ( ωm ) nba [58]. For the parameter regime typically encountered

in our experiments, this heating term is of order unity at the highest measurement strengths
and hence plays no significant role for interpreting results.
The output microwave field is connected to the cavity field via standard input-output
relation dˆσ,out = κσ dˆ + dˆσ,in , and is defined in terms of the field leaving the output port
(right) of the device, dˆR,out . Solving for the output spectrum
S̄R [ω] =
dteiωt h{dˆR,out (t), dˆ†R,out (0)}i,
κR γop
S̄ [ω] + S̄o [ω].
κ x2zp X̂1

(4.35)
(4.36)

where the noise floor identifies the classical occupation of the microwave baths and the shot
noise of the microwave field,
S̄o [ω] =

4.0.4

κR κ
th
+ nr + .
κ 2
ω + (2)

(4.37)

Scattering matrix formalism

As a consistency check, we can rederive the above relations within the scattering framework.
With the drive configuration for BAE, the scattering matrix is
χ(ω) = 

−iω + κ2

−iG

−iω + κ2

−iG

−iω + γ2m

−1

(4.38)

Solving for the mechanical scattering terms,
4iG
4iG
χ31 (ω) = (γm −2iω)(κ−2iω)
, χ32 (ω) = (γm −2iω)(κ−2iω)

χ33 (ω) = γm −2iω

χ34 (ω) = 0.

(4.39)

The symmetrized quadrature spectra are
S̄X̂1 [ω] = S̄xx [ω] + ∆[ω],

(4.40)

S̄X̂2 [ω] = S̄xx [ω] − ∆[ω],

(4.41)

where ∆[ω] incorporates the cross terms associated with hb̂[ω]b̂[ω 0 ]i and hb̂† [ω]b̂† [ω 0 ]i,
∆[ω] = 2 Re(χ31 [ω]χ32 [−ω]) κ(2nth
c + 1)
+ 2 Re(χ33 [ω]χ34 [−ω]) γm (2nth
m + 1)
γop
th
= −2 2
γm 2 κ(2nc + 1).
ω +( 2 )

(4.42)
(4.43)
(4.44)

Solving for the quadrature spectra yields identical results as Eqs. (4.32), (4.33).
For the microwave spectrum, the relevant scattering terms are
χ11 (ω) = κ−2iω

χ12 (ω) = 0,

4iG
χ13 (ω) = (γm −2iω)(κ−2iω)

4iG
χ14 (ω) = (γm −2iω)(κ−2iω)

(4.45)

which reproduce the output BAE spectrum of Eq. (4.36).

4.1

Back-action and imprecision definitions

4.1.1

BAE configuration

In terms of spectral density, that we directly accesses via FFT analysis of the full measurement chain, SI , we must include the power gain, G 2 [ω], and uncorrelated noise floor, S̄add [ω],
of the amplifier chain,
SI [ω] = G 2 S̄R [ω] + S̄add [ω] .

(4.46)

The final expression references the measured spectrum in terms of the components of

the mechanical quadrature spectrum. By splitting up the spectrum into three main components, it is now easy to identify the contributions from the intrinsic thermal motion, nth
m,
measurement back-action, nba , and detector imprecision, nimp .
Regarding the amplifier noise floor, we initially assume the amplifier gain and noise floor
are sufficiently flat about the cavity resonance frequency over a bandwidth on the order of
κ, G 2 [ω] → G 2 and S̄add [ω] = ~ωc nadd [ω].
Per the typical definition of measurement imprecision [6, 11], nimp is defined as the
measurement-induced mechanical occupation evaluated at mechanical resonance. Here, however, the measurement system is a single quadrature detector opposed to a position detector
and hence we only consider the imprecision noise for the sensed parameter X̂1 :
± κR
SI [0] = γop

κ γm

th
nm +
+ nimp ,

(4.47)

where,
nimp =

κ γm
S̄
[0]
S̄
[0]
add
κR 8γop

(4.48)

For ideal BAE, there is no back-action heating of X̂1 and the single quadrature detector
noise product is zero. This does not break the quantum bounds on detector noise products
since the BAE measurement manifests through back-action imprecision correlations.
In practice, there is drive power dependent heating of the mechanics induced either
through resistive ohmic heating or other mechanical nonlinearities [42,63]. To conservatively
model any drive-dependent heating, we define the amount of motional backaction heating
as the net increase in mechanical occupation induced from the measurement drive tones
regardless of the microscopic origin. As such, the measured backaction, nba , is calculated in
this work as the net increase between the dressed mechanics in the presence of BAE driving
and the bare mechanics, which here refers to the system without the BAE tones (but still
including the cooling tones).
For a quantum limited phase-insensitive amplifier, S̄add = 12 and the minimum imprecision level for a zero-temperature cavity is equal to one quanta. Referenced to the mechanical

spectrum, however, the imprecision can be made arbitrarily small by increasing the drive
powers (γop ). Though this would seem to ensure that sufficiently high powers can, in principle, be applied to suppress imprecision below the xzp level, there are various cavity and
mechanical nonlinearities that pose limits to the applied power. Historically, such a measurement has proven difficult due to the effect of relatively small optomechanical couplings
in combination with mechanical Kerr nonlinearities that have combined in the past to limit
the imprecision above zero-point level [15, 17, 19].

4.1.2

DTT configuration

We have now shown that ideal BAE detection generates back-action forces on the mechanics
that are entirely routed to the un-detected mechanical quadrature X̂2 . To confirm that this
is indeed the case, we can directly observe the back-action heating generated by the BAE
tones in numerous different ways. As a first step, we compare the BAE detection scheme to
a hx̂2 i measurement with identical drive strengths. We designate this position measurement
scheme as a “detuned two-tone” (DTT) identical to the pump configuration summarized for
sideband asymmetry measurements in Fig. 3.1.
Detuned two-tone pump configuration consists of two balanced drive tones (γop
= γop

placed symmetrically about the cavity at frequencies ωc ± (ωm + δ). Compared to the total
mechanical linewidth, the transduced sidebands are sufficiently separated in frequency space
(δ
γm ) to prohibit direct drive interference.
Using the analysis developed for sideband asymmetry measurements of the previous section, we now further restrict the drive configuration to balanced red and blue drives, such

that G− = G+ = G and by extension γop
= γop
= γop = 4Gκ . The DTT output microwave

spectrum exhibits electromechanical noise contributions from the intrinsic mechanical motion, the measurement imprecision, and back-action imprecision correlations in the form of
noise squashing and anti-squashing.
We can reinterpret the detuned two tone (DTT) configuration as a measurement of
position (∝ hx̂2 i) by averaging the up and down-converted sidebands in Eq. (3.35) to remove
78

the effects of cavity noise correlations in the form of squashing and anti-squashing,
1X
κR γm γop
S̄R [ω ± δ] = S̄o [ω] +
(nm + 1/2),
2 ±
κ ω 2 + ( γ2m )2

(4.49)

where we have assumed the noise floor S̄o is flat over the designated frequency range. The
mechanical occupation now exhibits optically induced quadrature-independent back-action
γ±

op
th
heating, nm = nth
m + γm (2nc + 1).

Following the imprecision formalism from above, the detected output noise spectrum is
SIDTT [ω] = G 2

1X
S̄R [ω ± δ] + S̄add [ω] ,
2 ±

with DTT back-action factor as
nDTT
ba

γop
(2nth
c + 1),
γm

= nBAE
ba /2,

(4.50)
(4.51)

and imprecision factor,
nDTT
imp =

κ γm
S̄o [0] + S̄add [0]
κR 4γop

= 2nBAE
imp .

(4.52)
(4.53)

Note that the back-action and imprecision noise in DTT differs from the BAE configuration by a factor of two. The scaling between BAE and DTT back-action heating factors
implies that the total back-action heating for position is equal in both BAE and DTT pump
configurations. The different scaling arises because the heating is spread equally between
the quadratures in DTT configuration but is entirely concentrated to a single quadrature in
BAE configuration. The imprecision also differs by a factor of two due to the distribution
of the sidebands in frequency space. Spacing out the sidebands effectively adds twice the
amount of imprecision noise compared to the overlapped sidebands in BAE configuration.
79

4.1.3

Microwave drive phase dependence

So far, we have completely ignored the phases of the microwave drive tones. Now we can
explicitly include the phase information and explore how it may affect the BAE measurement. Modeling the cavity field generated via two balanced drives (α− = α+ ) symmetrically
detuned from ωc by ±ωm ,
α(t) =

α± cos(ω± t + φ± ),

(4.54)

= α± e−i(ωm t+∆) + ei(ωm t+∆) e−i(ωc t+φ̄) + c.c.,

(4.55)

where the above frequencies and phases are defined via the drive parameters,
(ω+ + ω− ) = ωc ,

(ω+ − ω− ) = ωm ,

(φ+ + φ− ) = φ̄,

(φ+ − φ− ) = ∆.

(4.56)

In terms of the incident microwave power and phases that are available for experimental
control,
α± =

κL
P ±,
ωm + (κ/2)2 in

φ̄ = φ̄in ,

(4.58)

∆ = arctan

(4.57)

2ωm

+ ∆in .

(4.59)

Plugging this drive field into the Langevin equations for the mechanical and microwave
ˆ −iωc t and ĉ → b̂e−iωm t ) with enhanced
fields in a rotating, displaced frame (â → [α(t) + d]e
optomechanical couplings G± = g0 α± and rotating wave approximation, the field quadratures
follow
−i(φ−∆)
χ−1
− e−i(φ−∆) xzp ,
m (ω) X̂φ [ω] = − γm X̂φ,in [ω] − iG± Ûφ [ω] e
−i(φ−φ̄)
−i(φ−φ̄)
χ−1
(ω)
Û
[ω]
Û
[ω]
iG
X̂
[ω]
/xzp .
φ,in
80

As one would expect for power detection, the output spectrum derived with drive phase
information has no absolute phase dependence,
κR γop
S [ω].
S̄R [ω] = S̄o +
κ x2zp X̂∆

The phase difference ∆ determines the mechanical quadrature axes in the lab frame. The
absolute phase φ̄ determines the lab frame axis of the microwave quadrature that carries
the mechanical quadrature signal. Our measurement techniques are unable to differentiate
between the microwave quadratures (this could be addressed with an IQ-mixer for quadrature
detection) so we only concern ourselves with the relative phase ∆.

4.2

BAE results

Having analyzed the BAE measurement within an input-output framework and confirmed
its QND behavior, we will now discuss our experimental results in an electromechanical device. For this experiment, we implement BAE with the same device and measurement circuit
utilized in sideband asymmetry measurements. To reiterate, we study a lumped-element microwave LC resonator (denoted in this work as “cavity”) in the high-Q regime with resonance
frequency ωc = 2π × 5.4 GHz and total linewidth κ = 2π × 860 kHz. From independent tests
at 300mK, the output port scattering rate is κR = 2π × 450 kHz. The capacitor top gate
that supports out-of-plane acoustic modes, of which we study the fundamental mode with
resonance ωm = 2π × 4.0 MHz and intrinsic dissipation γm = 2π × 10 Hz at 20mK. Motion
of the top gate modulates the capacitance and shifts ωc by 2π × 16 Hz (= g0 ) per xzp , where
xzp ≃ 1.8 fm.

4.2.1

Calibrations

To calibrate the measurement circuit, we perform linewidth-broadening and thermo-mechanical
noise calibrations for a single red-detuned drive. A microwave pump is placed at ωc −ωm and
and the complex cavity transmission is measured via sweeping heterodyne detection. As the
81

-6

γm /2π (Hz)

Pm/Pthru

-8

-2

T (K)

-1

Figure 4.2: System calibrations. a. Calibration of the up-converted motional sideband against calibrated
thermal motion (blue circles). At the base temperature of the fridge (and outside the range of resistance
thermometry) the mechanics thermalizes to an extracted temperature of 7.2 ± 0.2mK (light blue square)
consistent with the expected base temperature of the cryostat. b. Backaction damping and pump photon
calibration. As the system is pumped with elevated drive power, the total linewidth of the up-converted
sideband (blue circles) is monitored via scanning homodyne detection. Fit to back-action damping theory
(red line) result in a calibration for pump photons np versus detected output power.

pump power is increased, we monitor the detected output pump power, P− , as well as the
back-action damping γop from linewidth broadening of the transduced mechanical sideband.
Next, we decrease the pump power to a sufficiently low level (γop ≃ γm /100) and then plot
the integrated noise power in the up-converted sideband, Pm , as we sweep the calibrated
temperature of the cryostat between 20mK to 200mK. The two measurements are cast in
linear form and yield the calibration factors a− and b− ,
4G2
γop =
= a−
1 kB T
Pm
P−
b− ~ωm

P− ,

(4.60)
(4.61)

where κ is observed to be constant over the relevant pump configurations so that powerdependent linewidth shifts are ignored. Converting the calibration factors to readily accessible formats for measurement, we find

np
a−
= (2.25 ± 0.7) × 1011 W−1 ,
P−
g02
nm
= b− = (9.92 ± 0.16) × 108 .
(Pm /P− )
82

(4.62)
(4.63)

SX1 /x2zp Sx /x2zp
(Hz-1)

1.2
0.8

0.4

-5.2 -5 -4.8

-0.2 0 0.2

4.8 5 5.2

(ω − ωc )/2π (kHz)
Figure 4.3: BAE and DTT noise spectrum. Measured noise spectrum converted to units of motional zeropoint fluctuations. The BAE spectrum (red area) clearly shows reduced noise area compared to the sidebands
generated in DTT pumping (blue area).

Though BAE and DTT experiments incorporate both red and blue-detuned drives, accurately balancing the drive powers provide full system analysis entirely via the red-detuned
drive calibrations. Refer to Fig. 4.2 for calibration measurement results.

4.2.2

Measurement

For the actual experiment, a third cooling tone is added to suppress mechanical frequency
jitter on the scale of the intrinsic linewidth. To do this, we apply a cooling tone at ncool
∼ 105
detuned from cavity center by δc (refer to Fig. 4.5). Sufficiently large detuning (δc
γtot )
ensures that the cooling tone dresses the mechanics – it broadens the mechanical damping
and cools the mechanical occupation from the intrinsic bath the lower occupation – without
otherwise affecting behavior of the BAE detection scheme.
For the measurement, we aim to perform three tasks: measure the intrinsic mechanical
motion dressed via the cooling tone, balance and measure DTT noise spectrum, and measure
the BAE spectrum. We accomplish this with the following protocol:
1. First, measure the cavity transmission with all tones set to estimated powers and frequencies. Fit the transmission spectrum to extract system frequencies and detunings.
83

Reconfigure the pumps to ensure ωcool = ωc − (ωm + δc ) with δc = 2π × 35 kHz and
ω± = ωc ± (ωm + δ) with δ = 2π × 5 kHz.
2. Turn off BAE tones, check cooling tone fedthru power and take cooling spectra. Extract
γm and ncool
m . Treat these as the initial, unperturbed system parameters γm and nm ,

ncool = b−

Acool
Pcool

= nom .

(4.64)
(4.65)

3. Turn on DTT tones, take noise spectrum, and balance to γm . Power balancing is
achieved by matching the linewidth of the dressed mechanical system to the intrinsic
linewidth γm . We assume the intrinsic linewidth does not appreciably change with
applied BAE tone which is consistent with mechanical occupation extracted from BAE
measurement at lower powers.
4. Measure DTT noise spectra. Fit each sideband to a Lorentzian lineshape then calculate
the average integrated sideband power as ADTT = 21 (Ared + Ablue ). We then convert
the sideband powers to equivalent occupation,
nDTT = b−

ADTT
P−

= nom + nDTT
+ .
ba

(4.66)
(4.67)

A typical DTT noise spectrum is presented in Fig. 4.3.
5. Measure the BAE noise spectra. Reconfigure the pumps to overlap the mechanical
sidebands. Measure the fed through power of each drive, then measure the BAE
spectrum. Refer to Fig. 4.3 for a typical BAE spectrum. In comparison to the DTT
spectrum, it is clear that the back-action is significantly reduced in BAE configuration.
Extract the integrated area under the BAE Lorentzian, ABAE . With calibration factors,

convert the integrated power to equivalent quanta,
nBAE = b−

ABAE
2P−

= nom + nBAE
+ .
ba

(4.68)
(4.69)

Note that the conversion factor between integrated sideband power and mechanical
occupation differs between BAE and DTT configurations by a factor of two. Furthermore, the backaction definition also differs by two between BAE and DTT. To properly
account for both these effects, we define the number of pump photons in BAE as twice
the number of pump photons in DTT.
6. Extract the back-action and imprecision components of the BAE and DTT occupation
factors. Associate any mechanical heating deviation from step (1) as measurement
backaction nBAE
ba . For the two pump configurations (σ = BAE, DTT), the backaction
can be directly isolated in the above calculations, whereas the imprecision can defined
in terms of the detected noise floor, sideband peak height, and equivalent sideband
occupation:
nσba = nσ − nom ,

background
nimp = nσ +
peak

(4.70)
(4.71)

DTT
For probe powers spanning np = 104 − 107 , the detected nBAE
for are plotted in
ba , nba

Fig. 4.4.
At the highest powers reached in the experiment, we demonstrate BAE detection that
avoids total backaction heating by approximately 10dB. More importantly, we observe quadrature heating that is 8.5±0.4 dB below the noise associated with quantum limited back-action
heating from the microwave shot noise. Simultaneously, the quadrature imprecision is below the zero-point level, corresponding to hX̂12 iimp = 0.6x2zp . Reframing these quantities in
terms of the detector spectral densities associated with backaction (SX̂1 ,imp ) and imprecision
85

Total

Back-action

Imprecision

60
40
20
10

x2 imp /x2zp

X̂21 imp /x2zp

x2 ba /x2zp

100
80

X̂21 ba /x2zp

x2 /x2zp

200

Figure 4.4: BAE and DTT occupations. a. Integrated sideband power for DTT (dark blue), BAE (red), and
no-pump (light blue) configurations. There is a clear reduction in the total observed noise power between
DTT and BAE configurations. In no-pump, the BAE tones are turned off and the cooling tone sideband is
monitored to check system drift. b. Back-action noise for DTT (blue) and BAE (red) pumping defined as
the increase in motional noise relative to the undriven mechanical occupation, i.e., the occupation extracted
from no-pump measurements. Even with zero classical noise, the BAE quadrature detection reaches below
the level of quantum-limited backaction for position detection (green line). Example spectra at the highest
powers (dotted grey box) are featured in Fig. 4.3. c. Imprecision noise as a function of pump power. At the
highest powers, the BAE imprecision reaches 0.6 x2zp . A quantum limited amplifier would drastically reduce
this level (green line).

(SF,ba ), we reach a detector noise product of

SX̂1 ,imp SF,ba ≃ 2.5~ [42] that is lower than

other comparable micro- and nano-mechanical devices [11].

4.3

Double BAE

So far, we have used the DTT position measurement as a way to highlight the QND nature
of the BAE measurement. Comparing the two measurements, the effective sideband area in
BAE is highly suppressed, consistent with backaction avoidance. However, this measurement
does not entirely verify or validate that the BAE measurement is acting as intended. To
exclude any loopholes, such as miscalibrated pump power or electromechanical couplings
(which would effect both BAE and DTT measurements and therefore are not realistic issues)
or a mechanical parametric effect, we now consider directly measuring the backaction of one
set of BAE tones (denoted below as “pump”) with an additional, weaker BAE set (“probe”).
Refer to Fig. 4.5 for a schematic of the full pump configuration. In the following, we will
first confirm that double BAE works as we suspect and will then present results of the direct
measurement of BAE measurement backaction.
86

Gcool

ωc

ωc + δ

Gprobe
ωc

Figure 4.5: Double BAE pump configuration. Five drive tones are placed about the cavity (black line). A
pair of BAE pump tones (red bars), each inducing the enhanced optomechanical strength G, are placed at
ωc ± ωm . A weaker set of probe BAE tones (blue bars) are placed at ωc + δ ± ωm with coupling strength
Gprobe . A fifth cooling tone (green bar) with coupling strength Gcool is detuned outside of the BAE tones and
serves to optically damp the mechanics. Inset: the BAE pump (red area) and probe (blue area) sidebands
are sufficiently detuned to avoid overlap. During measurement, both sets of sidebands are monitored as we
vary the relative phases between the probe drives. Additionally, we regulate the cavity noise occupation
(beige) which resembles a flat noise floor over the relevant measurement bandwidths.

4.3.1

Double BAE

Assuming the pump and probe BAE are sufficiently isolated from each other, each set of
BAE tones are sensitive only to the quadrature phase set by their respective drive phases:
the pump signal at ω ≃ 0 senses φ = 0, the probe signal at ω ≃ δ measures φ = θ,
χ−1
c (ω)d[ω] = − κdin [ω] − iGX̂1 [ω]/xzp − iGprobe X̂θ [ω − δ]/xzp .

(4.72)

We immediately find that the microwave field will carry information about the desired
quadratures in spectrally-distinct regions of frequency space.
For δ
γtot , the mechanics now incorporates two distinct phase-dependent heating tones
with measurement axes rotated by φ2 .
−1
χm
(ω)X̂φ [ω] = −

γm X̂φ,in [ω]

dˆin [ω] + dˆ†in [ω]

− ixzp Gprobe ei(φ−θ) − e−i(φ−θ) dˆin [ω + δ] + dˆ†in [ω − δ] ,

− ixzp G eiφ − e−iφ

where we have ignored cross-terms that scale as ( γκm κδ nba ). Unlike the case of single BAE
drive, the mechanical quadratures now exhibit cross-correlations, i.e., SX̂1 X̂2 6= 0. However,
we only concern ourselves with the quadrature spectrum for the experimentally accessible
87

X̂1

X̂φ

X̂2

Figure 4.6: Mechanical noise ellipse. a. For a single BAE drive, the mechanical noise ellipse (blue) resembles
a thermal squeezed state whereby all the back-action noise is added to the undetected quadrature X̂2 of
the thermal noise ellipse (beige). With single BAE alone, we only access the X̂1 quadrature fluctuations
and there is no direct way to assess the full noise ellipse. b. Double BAE noise ellipse. Considered as
separate measurements, the pump BAE elongates the mechanical noise ellipse along the X̂2 quadrature axes.
Set to measure X̂φ , the probe BAE acts in kind and funnels noise into its associated conjugate quadrature
X̂φ+π/2 . Operating simultaneously, each BAE set is sensitive to the back-action of the other (green and
orange crosses).

phases φ = θ, 0:
SX̂θ [ω] =

th
th
(2n
1)
4γ
(2n
1)
sin
op
ω 2 + ( γ2m )2

= S̄X̂
[ω] cos2 θ + S̄X̂
[ω] sin2 θ.

(4.73)
(4.74)

where the spectra S̄X̂
denote the quadrature spectra derived in Eqs. (4.32)-(4.33) for
,X̂

single BAE drive. Additionally, the pump BAE signal will be perturbed by the probe tones
in a symmetric fashion,
SX̂1 [ω] =

ω 2 + ( γ2m )2

probe
γm (2nth
(2nth
m + 1) + 4γop
c + 1) sin θ .

(4.75)

Though the full system Hamiltonian can no longer be considered QND, the sets of pump
and probe BAE drive act as independent QND measurements of their respective quadrature
axes; each respective signal is only sensitive to the back-action heating of the other (Fig. 4.6).
By sweeping the phase θ of the probe BAE, we perform a QND measurement of the rotated
quadrature X̂θ and directly measure the back-action forces of the pump BAE measurement
in-situ.

X̂2φ /x2zp

X̂1

X̂21 /x2zp

X̂2φ

X̂2

50
40
30

−π/2

π/2

Figure 4.7: Quadrature variance over the full noise ellipse. a. The probe BAE senses the noise ellipse
generated by the pump BAE tones. The quadrature ellipse follows the expected trend whereby all the
measurement back-action is added to only a single quadrature. The pump BAE senses the back-action of
the probe BAE which here are too weak to be detected above measurement noise. b. Quadrature noise
variance in polar axes. Since we directly measure the noise variances, the noise ellipse resembles a peanut
instead of an oval.

4.3.2

BAE phase locking

According to Eq. (4.74), introducing a second pair of probe BAE tones measures the mechanical noise ellipse dressed by the pump BAE drive. However, implementing this routine in
our experiment requires one new technique that has not yet been discussed: phase control of
the drive tones. In the following, we will discuss technical details for phase locking between
the pair of BAE drive tones.
For simplicity, we align our lab frame clock such that the phase difference between the
pair of pump BAE drives is zero. In this frame, the probe BAE measurement quadratures
are rotated by θ = (φ+ − φ− )/2, where φ+ and φ− are given by the incident phases of the
probe drive tones. Due to the phase reduction by a factor of two, the incident drive phases
must be controlled within [0, 4π] to achieve quadrature detection over a full 2π rotation.
We can imagine performing this with open loop control by manually setting the relative
phase of the microwave signal generators with internal or external phase control. Open loop
control is accurate up to the quality of sources’ phase drift. For phase stability, all signal
generators are locked via a 10 MHz rubidium standard. However, in practice, we observe
rms phase drifts on the scale of one degree per 10 minutes. We were unable to reduce the
89

Pump

Drive Division

Drive Conditioning
fridge
A1

10dB

ωc ± ωm

ωc

φo

10dB

ωc + ωm + δ

φo

2.9-8.7GHz

Noise
Generator

Drive Conditioning

20dB

Sig

Lock-in

Ref

Phase Readout

ωc − ω m + δ

Figure 4.8: Phase locking circuit.

phase drifts beyond this nominal level despite optimizing the BNC timing circuit to increase
mechanical stability, minimize delays between different sources and reduce pickup noise.
To remedy this phase drift issue, we implement closed loop control with manual feedback.
Though we did implement an active feedback circuit for real-time phase control, we found
that the manual phase control was more robust and more than adequate to deal with the
timescale of our phase drifts. The feedback circuit is designed around phase readout for the
pump BAE detection quadrature θ.
We will now discuss how we generate an error signal that is proportional to this phase.
First, we peel off a small amount of power from a set BAE tones and feed it into a power diode
(square law detector) followed by a RF low-pass filter with 11MHz cut-off frequency. This
combination generates a low-frequency (LF) signal at the modulation frequency. Focusing
on the component oscillating at twice the mechanical frequency,
LF(t) = (α(t))2 2ωm
= α± cos(2ωm t + 2θ),

(4.76)
(4.77)

Note that this error signal doubles both the frequency and phase of the modulation signal
which causes complications for the detection. Suppose we generate the LF modulation signals
from both pump and probe BAE signals and then read out the phase difference with a lockin amplifier. The lock-in is sensitive to a single branch of phase spanning 2π and so, since
mixing down to produce the LF signal doubles the phase shift, the lock-in can only supply

Frequency Divider

VCO
err

out

ωm

10dB
28dB

2ω
28dB

11MHz

1.9MHz

Figure 4.9: Frequency halving circuit.

half of the full sweep, θ ∈ [0, π].
To get closed loop control over the entire 2π range, we use an additional RF signal generator and create two signals on separate channels, oscillating at the mechanical frequency.
Each channel is further divided in two, feeding into a frequency doubling circuit and a lockin amplifier. The frequency-doubled signal from both channels are fed into separate PLL
circuits that phase lock each channel to the pump and probe BAE LF signals, respectively.
Refer to Fig. 4.8 and 4.9 for a schematic of the circuit. The phase difference between the
un-doubled signals is monitored via a lock-in amplifier, which now supplies a one-to-one correspondence between error signal and relative BAE phase. We now have close-loop control
over the entire quadrature noise ellipse.

4.3.3

Double BAE results

Using these phase-locking techniques, we are now able to directly measure mechanical backaction forces generated by a single BAE drive at np = 1.1 × 106 . We place a second set of
probe BAE tones, 20dB weaker and detuned than the pump and detuned by 2π × 30 kHz.
As we rotate the probe phase, it is clear that the back-action onto the mechanics is highly
quadrature-dependent and follows closely along with Eq. 4.74 (Fig. 4.10(a)). Similarly, we
see no apparent change in the X̂1 quadrature variance as measured by the pump tones which
is consistent with the reduced back-action generated from the much weaker probe tones.
Fitting the probe noise ellipse to this model, we can now directly extract out the backaction onto the X̂2 quadrature. As we inject more and more classical noise into the cavity
(following the same noise injection and noise floor calibration techniques of Sec. 3.3.2), we
91

b)
X̂22 ba / X̂22 qba

X̂2φ /x2zp

300
200
100

−π/2

π/2

2.2

3.3

4.4

1.1

Δη (aW/Hz)

Figure 4.10: Backaction heating versus cavity noise. a. The mechanical quadrature variances are measured
for increasing levels of cavity occupation. At elevated noise levels, the back-action heating of X̂2 increases
while that of X̂1 stays fixed. b. As the cavity occupation increases, the noise floor shift ∆η, representing
the peak height of the cavity noise Lorentzian, is plotted against the X̂2 quadrature heating. Extracting to
zero classical noise reveals the contribution of the quantum backaction induced by the quantum fluctuations
of the microwave field.

repeat this measurement at each setting and extract out the back-action heating onto the
X̂2 quadrature (hX̂22 iba ) as a function of the noise floor increase (Fig. 4.10(a)). According
to Eq. (4.34), the measurement back-action is proportional to both classical and quantum
th
noise, hX̂22 iba = (2γop /γm )(2nth
c + 1), such that the intercept at nc = 0 reveals the back-

action heating entirely due to the quantum fluctuations of the microwave field. Fitting our
data (Fig. 4.10(b)) reveals an intercept of 1.1 ± 0.1, in excellent agreement with the expected
quantum contribution of “+1”. This is the first time that the quantum fluctuations of
the microwave field [64, 65] is demonstrated via mechanical detection. Notably, this point
corresponds to real mechanical motion that is driven by a completely empty cavity.

Chapter 5
Mechanical squeezing
5.1

Introduction

We now focus on generating a quantum squeezed mechanical state where the variance of a
single motional quadrature is suppressed below the zero-point level [66], similar to squeezed
states that have been produced in various other systems [67–73], . Given that ideal BAE
detects the X̂1 quadrature while avoiding backaction heating, it would seem that BAE itself could generate a squeezed mechanical state. However, this is not the case. Due to the
QND nature of this kind of measurement, the X̂1 quadrature is completely unaffected from
measurement. For a single realization of the measurement on the timescale governed by the
total mechanical dissipation, the mechanics is indeed squeezed. After repeated continuous
measurement, however, the state is averaged out over the thermal distribution of the mechanics and we instead measure a conditional squeezed state. In this fashion, we observe
no reduction in the quadrature noise from the initial thermal level as expected for a QND
measurement.
There are numerous other proposals to generate mechanical squeezing via continuous
measurement plus feedback, such as BAE quadrature detection with feedback [56,58,74,75],
position measurement with detuned parametric drive [76]. Implementing BAE with feedback
is technically challenging as it requires an active feedback circuit for signal processing and
sufficiently small measurement imprecision noise. Alternatively, we could implement a simpler parametric squeezing scheme [77], though single quadrature cooling is limited to 3 dB
93

before the mechanics reaches a parametric instablility [78] and hence quantum squeezing via
parametric modulation is beyond our current capabilities. However, there are many other
theoretical proposals to surpass this 3 dB limit [79–82].
Instead, we consider a dissipative bath engineering scheme that can produce arbitrarily
large steady-state mechanical squeezing [83]. By simply increasing the strength of the reddetuned drive in the two-tone BAE drive configuration (Fig. 5.2), we no longer perform a
QND measurement on X̂1 . Instead, the backaction fores are correlated with the mechanical
motion in such a way that, instead of quadrature-insensitive cooling, the electromechanical
interaction forms a coherent feedback circuit that generates both steady-state mechanical
squeezing and net cooling of both quadratures from their initial thermal levels. This scheme
can be interpreted as a manifestation of reservoir engineering [84] realized in other bosonic
systems [85–88].
Recently, we performed such a measurement and observed quantum squeezing [63]. Since
then, two other groups have reported similar results in the microwave domain [89, 90]. In
this chapter, we will briefly discuss the electromechanical squeezing interaction as it directly
pertains to our system, followed by discussion of measurement and analysis techniques. The
bulk of this chapter will address the issue of error analysis in such a routine: how can we
systematically address correlated error between calibration uncertainty and fit uncertainty
for error propagation to the quadrature occupations? Though there are multiple ways to
tackle this problem, we will explore a Bayesian analysis routine that provides a relatively
direct way to resolve this issue.

5.2

Squeezing and detection

We will first outline the squeezing interaction and, making a series of approximations, show
that arbitrarily large steady-state mechanical squeezing is achievable with two-tone driving
of an electromechanical system such as ours. However, as many of these approximations
turn out to be violated due to nonidealities like power-dependent cavity [17] and mechanical
heating [63], we will later relax these assumptions and show that mechanical squeezing is
94

still within reach.

5.2.1

Generating steady-state mechanical squeezing
G−

G+

ωc − ωm

ωc

ωc + ωm

Figure 5.1: Mechanical squeezing pump configuration. A red- and blue-detuned pump (red and blue bars)
are placed at ideal detunings ωc ± ωm about the cavity Lorentzian (beige). The pumps are power imbalanced
such that there is excess red power, G− > G+ .

As a first step, we model the interaction in the good cavity limit ( ωκm
1) with perfectly
aligned and overlapped sidebands (δ, ∆ = 0) for a cavity mode deep in the ground state
(nth
c = 0). Moving to the interaction picture rotating at ωc , ωm and displaced by a large
coherent field |ā± |
1, the Hamiltonian simplifies to
Ĥint = −~dˆ† (G+ b̂† + G− b̂) + H.c. ,
= −~G(dˆ† β̂ + dˆβ̂ † ).

(5.1)
(5.2)

with enhanced optomechanical couplings G± = g0 ā± . In the second line, we have expressed
the electromechanical interaction in terms of a mechanical Bolgoliubov mode, β̂, where
β̂ = b̂ cosh r + b̂† sinh r,

(5.3)

with squeezing factor
G+
G−

(5.4)

G2− − G2+ .

(5.5)

tanh r =
and effective coupling strength
G=

Analogous to sideband cooling (G+ = 0) where the mechanical mode is cooled via cou95

pling to a cavity mode that is near the ground state, two tone drive (G− > G+ > 0)
instead couples a mechanical Bolgoliubov mode to the cavity mode. For the imbalanced and
overlapped two-tone drive configuration of Fig. 5.1, the mechanical Bolgoliubov mode, β̂,
represents the annihilation operator of a squeezed mechanical state. That is, β̂ Ŝ(r) |0i = 0

where Ŝ(r) = exp[r(b̂ b̂ + b̂† b̂† )/2] is the squeeze operator. From the form of this interaction,
cooling via cavity dissipation is capable of generating steady-state mechanical squeezing. In
Sec. 5.3 we relax some of the above assumptions but show, via an input-output framework
that we can indeed generate mechanical squeezing for experimentally accessible regimes.

5.2.2

Bolgoliubov mode detection

Following the assumptions outlined above – perfectly overlapped and centered drive tones, a
cavity deep in the ground state (nth
c = 0) and highly suppressed bad-cavity effects – we can
analyze and detect mechanical squeezing by following the measurement protocol outlined
in [83]. Here, the expressions for the quadrature occupation and squeeze factor simplify
to compact equations in terms of the red drive cooperativity, C = 4G2− /(κγm ), and the

mechanical bath occupation, nth
m . The squeeze factor r follows
e−2r ≈

1 + 2nth

(5.6)

while the squeezed quadrature occupation is
γm
(1 + 2nth
2hX̂12 i ≈
m) +

1 + 2nth

(5.7)

In terms of experimental measurables, the squeezing can be estimated directly from the
output microwave noise spectrum, SR [ω] = dteiωt hdˆ†R,out (0)dˆR,out (t)i. Specifically, the integrated noise power is proportional to the Bolgoliubov mode occupation,

4κG2
dω
SR [ω] ≃
hβ̂ † β̂i.
2π
4G2 + κ(κ + γm )

(5.8)

For the limit of high cooperativity such that nth
m /C
1, this relationship is sufficient for

quadrature detection. In terms of β̂ and β̂ † , the quadrature occupation is
e−2r †
† †
hβ̂ β̂i + hβ̂ β̂ i + hβ̂ β̂ i + hβ̂ β̂i .
hX̂1 i =

(5.9)

Asserting the Cauchy-Schwarz inequality, the squeezing occupation obeys a rigorous upper
bound
hX̂12 i ≤ e−2r [1 + 2hβ̂ † β̂i].

(5.10)

This bound approaches an equality up to corrections on the order of 1/ C.
The above Eqs. (5.8)-(5.10) show that the integrated noise output noise spectrum serves
as a proxy for mechanical squeeze detection. In practice, however, we implement a different
measurement routine for the following reasons. First, we must relax various assumptions in
the above analysis that are not applicable to our measurement system. Most importantly, we
must account for classical cavity noise (nth
c > 0) that is pump power dependent and internally
generated within the microwave cavity. The cavity occupation changes the dependence of
Eqs. (5.6)-(5.7) on both optimal cooperativity and optimal pump ratios. Furthermore, the
sideband resolution of our device ( ωκm ≈ 8) introduces corrections to the microwave spectrum
lineshape and quadrature occupations that, though small, are necessary for accurate squeeze
estimation. Finally, we will show below that, due to pump-dependent cavity and mechanical
heating, our optimal squeezing generates moderate levels of quantum squeezing. At these
levels of squeezing, the rigorous upper bound of Eq. (5.10) overestimates the quadrature
occupations above the zero-point level.
For these reasons, we take a different approach: we will calculate the quadrature occupations via the scattering matrix techniques outlined in Sec. 1.4. Instead of integrating
over the output spectrum, we instead fit the detected noise spectrum to the output spectrum model, extract out the system parameters, then calculate the squeezed and amplified
quadrature occupations as functions on the system parameters. In the following sections, we
first discuss these techniques for perfectly overlapped and centered pumps. Later, we will

relax these assumptions and include all deleterious effects driven by the electromechanical
interaction: imperfect pump tuning, bad cavity effects from finite sideband resolution, and
sources of calibration error.

5.3

Squeezing models

We will first analyze the squeezing interaction assuming perfectly aligned drive tones and
infinite sideband resolution. Under these assumptions, it is possible to generate relatively
simple analytic models for the microwave spectrum and the quadrature occupations. We
will later relax both of these assumptions and instead develop numerical spectrum models
that will be used for spectrum fitting and quadrature extraction routines.

5.3.1

Ideal pumping with RWA

In this section, we calculate the quadrature occupations assuming ideal pump configuration
and making the rotating wave approximation, and show that the two-tone drive does generate
steady-state mechanical squeezing. To start, we derive all relevant spectrum and occupations
from the scattering framework introduced in Sec. 1.4 and setting δ, ∆ = 0. The complex
transmission is given by Eq. (1.104) and follows
− κR κL (γm − 2iω)
S21 [ω] =
4G2 + (κ − 2iω)(γm − 2iω)

(5.11)

For a given pump configuration with G± independently calibrated from measuring the drive
powers at the detector, the driven response ensures that the transduced sidebands are overlapped and centered in the cavity. We also extract κ in-situ (with identical pump configuration) by fitting the measured spectrum to the model.

The output noise spectrum is given as
SR [ω] =

dteiωt h{X̂i (0), X̂i (t)}i,

th
th
)κnth
(4ω 2 + γm
c + 4G− γm nm + 4G+ γm (nm + 1)
= 4κR
|4G2 + (κ + 2iω)(γm + 2iω)|2

(5.12)
(5.13)

With all the S21 parameters specified, the only fit parameters in the noise model is the
th
cavity bath occupation nth
c and the mechanical bath flux γm nm . We keep the mechanical

bath contribution specified as a flux opposed to separating into rate γm and occupation
nth
m since the flux is the most relevant parameter and we have no sensitivity to γm at these
measurement powers, i.e., γm
γtot , κ.
The mechanical quadrature spectrum is
S̄X̂i [ω] = 4x2zp

th
4(G− ∓ G+ )2 κ(nth
c + 2 ) + (κ + 4ω )γm (nm + 2 )
|4G2 + (κ + 2iω)(γm + 2iω)|2

(5.14)

with associated quadrature occupations
Z ∞

dω
S̄X̂i [ω]
−∞ 2π
th
4(G− ∓ G+ )2 κ(2nth
c + 1) + [4G + κ(κ + γm )]γm (2nm + 1)
= x2zp
(κ + γm )(4G2 + κγm )

hX̂i2 i =

(5.15)
(5.16)

where the quadrature designation i = 1, 2 is associated with ±, respectively, leading to
suppression or enhancement of measurement backaction forces proportional to nth
c . As pre-

dicted, the electromechanical interaction does produce steady-state mechanical squeezing via
reservoir engineering.

5.3.2

General spectrum model

th
th
The above model is sensitive not only to the bath occupations nth
c and ṅm ≡ γm nm , but also

the coupling rates G± , linewidths κ and γm , and detunings δ, ∆. Though we do our best to
calibrate these parameters, we must extract all such parameters from signals that come from

ωm + δ

G−

ω−

ωc

G+

ω+

Figure 5.2: General pump configuration. Here, we include deviations from the ideal squeezing pump configuration in the form of detunings ∆ = (ω+ + ω− )/2 − ωc associated with how well the drives are centered
around the cavity and δ = (ω+ −ω− )/2−ωm associated with how well the motional sidebands are overlapped.

microwave transmission and power detection spectra and hence we must always deal with
numerous sources of measurement and calibration error that can propagate throughout such
measurements. This outlines a general issue with how we make this measurement, there are
eleven total system parameters that can effect the electromechanical squeezing interaction
and we must consider how these parameters influence our results.
For accurate estimation and error analysis, we want a spectrum model that incorporates
all system parameters that affect quadrature estimation, captures relevant system corrections like bad cavity or Kerr effects, and is defined entirely in terms of directly accessible
measurement parameters. Fortuitously, the full scattering framework handles these issues in
a compact fashion. We can define the scattering matrix elements as functions on calibration
parameters and can iteratively expand the matrix to sufficiently large sideband order to
include bad cavity effects or include Kerr cross-terms directly in the scattering matrix.
With arbitrary pump detunings and first-order bad cavity corrections, the analytic forms
of Eqs. (5.13)-(5.16) become unwieldy and unfit for model fitting. After much effort to
develop analytic models, it turns out to be sufficiently fast and much simpler to calculate the
output spectrum via numerical inversion of the scattering matrix at each specified frequency.

5.4

Measurement

5.4.1

Calibration measurements

We begin by performing two measurements that calibrate the pump powers detected at
the output of our measurement chain, P± = gain × ~ω± × κR ∆± n±
p , in terms of enhanced
100

optomechanical coupling rates, G± , as well as the effective intracavity photon levels, ∆± n±
p.
Here, ∆± is a correction factor that modifies the cavity transmission off resonance [16] and
has no significance in the following analysis.
With scanning homodyne detection (i.e., via a driven response), we first measure the
mechanical linewidth, γtot = γm + γop , as we increase the power, P− , of a single pump
red-detuned from the cavity center by ωm . Here, γop = 4G2− /κ is the optically-induced
mechanical damping. For γtot
κ, the mechanical response is a simple Lorentzian dip and

we fit γop vs. P± to obtain a calibration for G2− . As γop becomes comparable to κ, we fit the
transmission data to a strong coupling model with G+ = 0 and δ = 0.

Next, we place two balanced pumps, detuned from cavity center by ±(ωm + 2π × 500 Hz)
and with powers P± , at sufficiently low powers so as not to add any damping or amplification
of the thermal noise, and we measure the integrated mechanical noise power of up- and
down-converted motional sidebands, Pm± , over a range of cryostat temperatures T . Due to
weak temperature and power dependence of κ [17], we monitor the cavity linewidth at each
measurement power or temperature. The results of these calibrations are cast in a linear
form and fit with ordinary least squares to extract the calibration factors a, b− and b+ ,

4G2−
γopt =
=a
P− ,
κ̄
κ̄
κ̄
2
κ 2 P
kB T
= b±
κ̄ P±
κ̄
~ωm
κ

(5.17)
(5.18)

where κ̄ is the cavity linewidth averaged over the respective parameter range. We find
a = (3.25 ± 0.09) × 1016 rad2 s−2 W−1 ,
b− = (3.82 ± 0.14) × 104 rad2 s−2 ,
b+ = (6.84 ± 0.22) × 104 rad2 s−2 .

101

We now are able to formulate the pump-dependent model parameters in terms of P± ,
G2− = a × P− ,

b−
G+ = a
× P+ ,
b+

× P− .
∆− np =
b−
Eq. (5.20) follows from the balancing condition

P−
P+

balanced

(5.19)
(5.20)
(5.21)

b−
b+

Next, we introduce a second, blue-detuned drive and directly measure the pump powers
P± which we use to extract the enhanced optomechanical coupling rates G± via Eqs. (5.195.21). Using an RWA model akin to Eq. (5.11), we fit the transmission spectrum via nonlinear
least squares estimation and extract the frequency of the microwave resonator ωc , the cavity
linewidth κ, the frequency of the mechanical oscillator ωm , and the pump detunings ∆, δ.
The two pump tones are iteratively aligned to overlap the mechanical sidebands at the center
of the cavity to ensure that δ is close to zero.
At this point, we now have a set of calibration parameters acquired in three methods:
either directly measured (P± ), estimated from fits to transmission data (ωc , ωm , κ, ∆, δ), or
extracted from independent calibration measurements (a, b± ). Each parameter represents
an underlying probability distribution that captures the precision of each respective calibration. To simplify the following analysis, we assume the fit parameters are independent and
normally distributed such that each parameter is associated with an underlying probability
density function modeled as a Gaussian distribution with mean and variance given by the
statistical estimators generated via measurement and fitting.

5.4.2

Noise spectrum measurement

Keeping the same two-tone pump configuration from the transmission calibration, we measure the microwave noise spectrum via linear detection. Calibrating out the gain of the

102

Figure 5.3: Squeezing noise spectrum. a. Raw noise spectrum (grey dots) and binned noise data (blue dots)
fit to the squeezing spectrum model (red line). We include a linear noise floor offset (black dotted line) which
we subtract off in later analysis. b. We calculate the data residuals (blue dots) by subtracting off the fit.
We estimate the measurement noise from the sample variance of the residuals over a range far outside the
cavity linewidth (red region).

output amplifier chain, the measured spectrum is given by
S̄out (ω) = S0 (ω) + SR (ω) ,

(5.22)

where SR (ω) is the noise spectrum of the electro-mechanical system and S0 (ω) is the noise
floor of the system. The noise floor is dominated by the noise figure of the cryogenic HEMT
amplifier in addition to smaller power-dependent offsets due to phase noise from the entire
amplifier chain. We spend an equal time interleaving measurements of the pumped and
unpumped noise spectra over the same bandwidth. We subtract off the unpumped floor,
then account for power-dependent amplifier effects by removing a linear floor offset that
we fit over a span roughly seven times greater than the cavity linewidth. The linear offset
matches independent measurements of the phase noise from our room temperature amplifier
with matching pump configuration.

103

5.5

Bayesian parameter estimation and error analysis

Essential to any claim of sub-zero-point squeezing is the error bar for the reported quadrature
occupation. Here, we consider a systematic approach to incorporate the uncertainty from all
the sources of our measurement, including systematic calibration error, measurement noise,
and the uncertainty from fitting the model to a measured noise spectrum. This problem has
been addressed by Bayesian analysis techniques that explicitly incorporate all known sources
of error. In the following, we largely follow the analysis outlined in Ch. 3 of [91]. Our purpose
for using this analysis is to address the issue of estimating error bars from nonlinear fitting
with a fit model that also has uncertainty.
In what follows, we develop statistical estimators for the quadrature occupations, hX1,2
i,

from two sets of measurements: the detected noise spectrum and the system calibrations.
Here, system calibration refers to the combination of initial calibrations (a, b− , b+ ), driven
response data (κ, ∆, δ) and power detection (P− , P+ ). We refer to such parameters as β =
{a, b− , b+ , κ, ∆, δ, P− , P+ }. The only remaining unknowns are the bath contributions, here
denoted as α = {nc , ṅth
m }.

To systematically incorporate the uncertainty from our calibrations and spectrum measurements, we consider the Bayesian posterior distribution
p (α, β|D, I) =

p (D|α, β, I) p (α, β) ,

(5.23)

where D is the observed noise data, I is the set of all assumptions required for this analysis,
Z = p (D) is a normalization constant that is not necessary for sampling of the posterior,
p (D|α, β, I) is the likelihood function, and p (α, β) is the prior distribution for α, β.
The prior distribution captures how well we have confined our calibrations in parameter
space. Assuming all system parameters are independent, the prior simplifies to a product of
single-parameter normal distributions, i.e., p (β) is a product of Gaussian distributions with
mean and variance set by the statistical estimators (ᾱi and σαi , respectively) for each system

104

calibration.
P (α) =

σαi

(α − ᾱi )2
exp −
σα2 i
2π

(5.24)

For the unknowns, p (α) is the product of uninformed Jeffreys priors [91]; these priors are
uniform in log space to be scale-invariant and are set here to span a decade above and below
initial estimates for nc , ṅth
m specified by minimum bound βi,min and maximum bound βi,max :

P (β) =

 Q

i log(βi,max /βi,min ) βi ,

for βi ∈ [βi,min , βi,max ],

 0,

(5.25)

otherwise.

Since we operate in the high cooperativity regime and have no sensitivity to the intrinsic mechanical linewidth, we assume that the calibration parameter γm also follows an usninformed
Jeffrey’s prior.
The likelihood captures how well the data matches the noise spectrum model with specified α, β. We calculate residuals of the detected noise data {Ni } by subtracting off the noise
spectrum model {Si } calculated at frequencies matching the data. Next, we assume the
measurement noise at each measured frequency is independent and Gaussian with identical
variance. The measurement noise σ is directly sampled from noise data over a 150 kHz
window detuned outside cavity center by ±3κ (see Fig. 5.3). Hence, the likelihood is the
product of residual probabilities derived from N (0, σ),
P (D; α, β, I) =

(Ni − Si )2
√ exp −
2σ 2
σ 2π

(5.26)

All together, we construct the posterior distribution as
P (α, β; D, I) ∝ p(D; α, β, I)P (α)P (β).

(5.27)

Since the posterior distribution is difficult to calculate analytically, we instead model the
posterior via an affine-invariant Markov chain Monte Carlo (MCMC) ensemble sampler [92].
We implement this calculation with emcee, an open-source Python package developed in the

105

Figure 5.4: Markov chains generated via emcee. One hundred parallel walkers traverse the multi-dimensional
parameters space over a period of one thousand steps. In steady-state, the walkers generate pseudo-random
chains that accurately sample the Bayesian posterior distribution. As evidenced by the transient relaxation
for many chains, the first five hundred steps (grey region) is discarded to ensure proper burn-in and parameter
estimation is evaluated over the remaining samples.

astronomy community with over three hundred citations since 2012 [93]. With emcee, we
generate a sufficiently large number of of pseudo-random parameter chains (α, β)i sampled
from the posterior distribution. For the calculation, we initialize a hundred walkers and run
for a minimum of a thousand steps. We discard the first half to ensure that the resulting
distributions are steady state (allowing initial transients to relax) but maintain a large enough
sample size to render the Monte Carlo uncertainty negligible. These chains are displayed in
Fig. 5.4 for the pump ratio n+
p /np = 0.4.

In Fig. 5.5, we display the single parameter and pairwise histograms evaluated over the
collection of MCMC chains. The single parameter histograms represent the marginalized distributions for each system parameter. Intriguingly, the pairwise distributions clearly display
any correlations between system parameters.
Finally, we calculate expectation values and 1-σ intervals for nc , ṅth
m and hX1,2 i. For nc

and ṅth
m , we construct the marginalized distributions from their respective Markov chains
and then tabulate the statistical estimators for mean and variance. For the mechanical
106

1.2
1.0

ṅm

0.8
20
1.0
1.0
05
90
0.9
75
0.9
10
γm

1.0 1.0

P−

1.0
0.9
0.9
08

1.0
1.0
02

P+

96
0.9
0.9
90
1.0

1.0
0.9
1.0
1.0

b−

0.9
0.9
1.0
1.0

0.9
0.9

1.2
1.8

2.4

ṅm

γm

P−

P+

b−

Figure 5.5: Triangle plot of single and pairwise parameter distributions. The posterior distribution is
projected into pairwise distributions and single parameter histograms. MCMC sampling techniques provide
direct access to visualize all correlations between system parameters. All parameters are normalized by their
respective estimates acquired from calibration measurements or maximum likelihood estimation. Again, the
first five hundred steps are discarded to ensure MCMC burn-in.

107

1.2

1.8

2.4

3.0

1.0
b+

1.0

0.9

0.9
0.9
1.0
1.0

1.0

0.9

1.0
0.9
90
0.9
96
1.0
02
1.0
08

1.0

0.9

0.90

1.0

0.9

1.2

1.0

0.8

1.2

1.1

1.0

0.9

3.0

0.50

1e3

Frequency (A.U.)

0.85

0.55

0.85

1e3

3.0e4

ṅm

7.5e4

1e3

0.65

X̂12

0.95

1e3

X̂22

7.5e4
ṅm

3.0e4
0.95
X̂12

0.65
15
X̂22

Steps

Figure 5.6: Parameter estimation. The MCMC chains corresponding to the individual fit parameters nth
and ṅth
m sample the marginalized fit parameter distributions. The quadrature occupations are extracted via
function evaluation over the entire set of MCMC chains for all system parameters. From these chains, we
evaluate the mean (solid red line), 1-σ interval (dotted red lines) defined via the sample variance, as well as
the median (solid black lines) and 95% quantiles (dotted black lines).

quadratures, we calculate expectation values for functions of system parameters, f (α, β),
with function evaluation over the entire MCMC ensemble,
hf i =

f (α, β) p (α, β|D, I) dαdβ,

(5.28)

1 X
f (α, β)i .
N i=1

(5.29)

The mean and standard deviation for hX1,2
i are generated via Eq. (5.29) with f (α, β) set

to the mechanical quadrature functions discussed in Sec. 5.3.2. Examples of the quadrature
chains with associated histograms and parameter estimators is presented in Fig. 5.6.

108

5.5.1

Comparison to Monte Carlo calibration simulation

As a consistency check, we also perform a more typical data analysis routine that is based
on nonlinear least square fitting. Instead of incorporating the calibration uncertainty in
the form of Bayesian priors, we will instead use Monte Carlo techniques to simulate a large
set of random calibration parameters α sampled from the multivariate normal distribution
described in Eq. (5.24). For each random set of calibration factors, we generate a new
noise spectrum model and then use maximum likelihood estimation to extract the mean and
variance of the fit parameters β. We repeat this process over the entire population of the
simulated calibration sets and, via function evaluation over the entire sample population of
α and β, calculate the mean and variance of that quadrature occupations. At this point,
the quadrature variance is entirely due to the calibration uncertainty. It is not clear how
to incorporate the fit uncertainty since we have no way of tracking correlations between
the calibration uncertainty in α (incorporated here via Monte Carlo simulation) and fit
uncertainty in β (estimated from least squares parameter estimation). Instead, we assume
here that the fit error, σβi , independently propagates into the quadrature occupations via
standard linear error propagation,

σX̂
= σX̂
1 cal

∂hX̂12 i
∂βi

!2
σβ2i .

(5.30)

At optimal squeezing, this Monte Carlo analysis estimates the quadrature occupation
as hX̂12 i/x2zp = 0.80 ± 0.04, consistent with the Bayesian analysis results. The strength of
the Bayesian analysis is that it does not rely on assumptions about correlations between
calibration and fit uncertainty.

5.6

Results

We now present mechanical squeezing results over the full range of pump power ratios used
in the experiment. At each configuration, we adjust the ratio between red- and blue-drive

109

Figure 5.7: Squeezing results. a. Example noise spectrum and fits for pump ratios n+
p /np = 0.3, 0.4, 0.5,
th
th
0.6 and 0.65 (ordered from blue to red). b. Bath occupation factors nc (yellow) and ṅm (blue). We observe
ratio dependent heating in both the cavity and mechanical bath contributions. c. Quadrature estimation
for hX̂12 i (red) and hX̂22 i (blue). d. We observe quantum squeezing hX̂12 i < x2zp at pump ratios between 0.3
to 0.55. At the lowest point, we cool a single quadrature of the mechanics to 0.80 ± 0.03 times the zero-point
level.

powers while keeping the total pump power fixed at n−
p + np = 1.76 × 10 and repeat the

calibration and measurement routines. Typical noise spectra for a selection of power ratios
th
is presented in Fig. 5.7(a). We then extract the bath occupations {nth
c , ṅm } and quadrature

occupations hX̂1,2
i as shown in Fig. 5.7(b)-(d). Though we observe ratio-dependent heating

for both the mechanical and cavity baths, we achieve mechanical squeezing with quadrature
occupation below the zero-point level for a range of pump configurations.
In Fig. 5.7(c) and (d), we observe optimal squeezing at an intermediate power ratio
consistent with the trends of [83]. We can understand this behavior as follows. As the pump
ratio approaches zero, the pump configuration corresponds to sideband cooling [45] such that
the quadrature occupation is asymptotically limited to the zero-point level. At the other
end, as the pump ratio approaches unity, the pump configuration becomes BAE detection

110

which again ensures single quadrature occupation at or above the zero-point level.
The trend of the quadrature occupation of Fig. 5.7(c) – the gradual decrease in occupation
as the pump ratio increases from zero followed by an abrupt increase for ratios approaching
unity – is similar in appearance to the trend of Eqs. 5.16 derived with zero bath heating.
The main affects of the microwave and mechanical bath heating is to reduce the squeezing
efficiency, modify the optimal cooperativity, and skew the squeezing extrema to lower ratios.
All of these effects will crucially depend on how the bath heating scales with the drive
powers. Keeping the total power fixed, we observe increased bath heating as the red and
blue powers ratio tends to unity (see Fig. 5.7(b)). The source of this heating is not entirely
clear however we believe it is consistent with a nonlinear dielectric composed of two-level
fluctuators [17, 25, 94, 95].
In addition to the issue of bath heating, both the mechanics and cavity also exhibit Kerr
nonlinearities [15, 17] leading to noise squeezing and quadrature-dependent amplification
at sufficiently large drive powers. These effects introduce off-axis correlations between the
mechanical (microwave) quadratures that can potentially alter our spectral analysis. In our
current measurements, we rule out the influence of the Kerr effects in in the following manner.
For the mechanics, we directly observe any quadrature-dependent linewidth narrowing and
broadening, a feature that is directly related to parametric squeezing [77], by performing
single quadrature BAE detection over the full quadrature phase space. At the drive powers
used in our work, we observe no significant mechanical linewidth modulation. For the cavity,
we extract the Kerr factor by directly measuring the four-wave mixing at the relevant pump
powers. We find that the effect is not relevant in our parameter regime.

5.7

Conclusion

In conclusion, we have generated a non-classical state of a macroscopic mechanical resonator [63] and have extracted a single quadrature variance as small as 0.80 times the zeropoint level. Most significantly, our squeezing is limited due to power-dependent heating of
the mechanical and microwave baths. If this heating behavior is indeed associated with
111

a nonlinear dielectric, then we can imagine a few device modifications that could potentially mitigate these effects. Increasing the bare optomechanical coupling would allow us to
reach sufficiently large cooperativity at lower drive powers. Alternatively, we could work to
suppress the influence of the intrinsic bath by increasing the external coupling to the microwave cavity and thereby dilute the contribution of the internal loss, though it is difficult
to understand this behavior without a full microscopic model for TLS-induced bath heating.
Another possibility would be to increase the sideband resolution of our device. Larger pump
detunings could sample a higher frequency range of the 1/f phase noise envelope of the TLS
dielectric noise [96], assuming our mechanical frequency is well below the white noise cut-off
frequency of the dielectric noise spectrum.
In terms of continued work in this area, the next logical step is to verify mechanical
squeezing via quadrature-sensitive mechanical detection, i.e., QND quadrature detection via
BAE. We have already implemented this in our current device by introducing two additional
BAE tones sufficiently detuned from the squeezing pumps [63]. However, we again are limited
by bath heating issues that, with the introduction of BAE tones, degrade the amount of
motional squeezing to above the quantum level. By shifting the BAE tones off of cavity
center on the order of the cavity linewidth, we necessarily filter out our mechanical signals
which significantly reduce the signal to noise ratio of the mechanical signals. Even at the
lowest available BAE pump powers, we observe elevated cavity and mechanical bath heating
that limits the squeezed mechanical quadrature occupation to 1.09 times the zero-point level.
There are numerous ways to overcome this issue of limited BAE sensitivity. One such
technique, implemented elsewhere [90], is to engineer additional cavity modes that are all
coupled to the mechanics via the standard optomechanical interaction. With additional
cavity modes, the squeezing and BAE drive tones can address separate cavity modes and
thereby eliminate any direct correlations between the drives. One can now place all mechanical signals tightly in the center of their respective cavity lineshapes and resolved weak
BAE detection becomes feasible. Alternatively, one could also engineer an improved device
to reach the quantum squeezing regime at lower cooperativities. By squeezing in the weak
coupling regime, one can fit all necessary signals tightly within the cavity linewidth. Assum112

ing the BAE drive powers can be decreased to the extent that bath heating is insignificant,
it is now possible to reach mechanical squeezing and BAE detection at the quantum level.
This last topic is the focus of current work.

113

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