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Quantum Storage of Light Using Nanophotonic Resonators Coupled to Erbium Ion Ensembles
Citation
Craiciu, Ioana
(2020)
Quantum Storage of Light Using Nanophotonic Resonators Coupled to Erbium Ion Ensembles.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/yn6n-7x40.
Abstract
This thesis presents on-chip quantum storage of telecommunication wavelength light using nanophotonic resonators coupled to erbium ions. Storage of light in an optical quantum memory has applications in quantum information and quantum communication. For example, long distance quantum communication using quantum repeater protocols is enabled by quantum memories. Efficient and broadband quantum memories can be made from resonators coupled to ensembles of atoms. Like other rare earth ions, erbium is appealing for quantum applications due to its long optical and hyperfine coherence times in the solid state at low temperatures. However, erbium is unique among rare earth ions in having an optical transition in the telecommunication C band (1540 nm), making it particularly appealing for quantum communication applications. In this work, we use nano-scale resonators coupled to erbium-167 ions in yttrium orthosilicate crystals (
167
Er
3+
:Y
SiO
).
We demonstrate quantum storage in two types of resonators. In a nanobeam photonic crystal resonator milled directly in
167
Er
3+
:Y
SiO
, we show storage of weak coherent states using the atomic frequency comb protocol. The storage fidelity for single photon states is estimated to be at least 93.7% ± 2.4% using decoy state analysis, Storage of up to 10 μs and multimode storage are demonstrated. Using a hybrid amorphous silicon
167
Er
3+
:Y
SiO
resonator and on-chip electrodes, we demonstrate a multifunctional memory using the atomic frequency comb protocol with DC Stark shift control. In addition dynamic control of memory time, Stark shift control allows modifications to the frequency and bandwidth of stored light. We show tuning of the output pulse by ± 20 MHz relative to the input pulse, and broadening of the pulse bandwidth by more than a factor of three. The storage efficiency in both devices was limited to < 1%.
On the way to these results, we describe
167
Er
3+
:Y
SiO
spectroscopy measurements including optical coherence times and hyperfine lifetimes below 1 K, and we estimate the linear DC stark shift along two crystal directions. The design and fabrication of the on-chip resonators is presented. We discuss the limitations to storage time and efficiency, including superhyperfine coupling and resonator parameters, and we outline a path forward for improving the storage efficiency in these types of devices.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
rare earth ions; erbium; quantum memory; nanophotonic resonator; cavity quantum electrodynamics; silicon photonics; quantum light-matter interfaces;
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Faraon, Andrei
Group:
Institute for Quantum Information and Matter, Kavli Nanoscience Institute
Thesis Committee:
Vahala, Kerry J. (chair)
Faraon, Andrei
Painter, Oskar J.
Endres, Manuel A.
Defense Date:
19 May 2020
Funders:
Funding Agency
Grant Number
Air Force Office of Scientific Research
FA9550-15-1-0252
Air Force Office of Scientific Research
FA9550-18-1-0374
Air Force Office of Scientific Research
FA9550-15-1-0029
National Science Foundation
EFRI 1741707
Natural Sciences and Engineering Research Council of Canada
PGSD2-502755-2017
Record Number:
CaltechTHESIS:06012020-134801698
Persistent URL:
DOI:
10.7907/yn6n-7x40
Related URLs:
URL
URL Type
Description
DOI
Article adapted for Chapter 5
Related Item
Also available in Evan Miyazono's Caltech PhD thesis
ORCID:
Author
ORCID
Craiciu, Ioana
0000-0002-8670-0715
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
13759
Collection:
CaltechTHESIS
Deposited By:
Ioana Craiciu
Deposited On:
02 Jun 2020 16:52
Last Modified:
28 Feb 2023 18:28
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Quantum Storage of Light Using Nanophotonic
Resonators Coupled to Erbium Ion Ensembles

A thesis by

Ioana Craiciu

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

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2020
Defended May 19, 2020

Ioana Craiciu
ORCID: 0000-0002-8670-0715

iii

ACKNOWLEDGEMENTS
I would first like to acknowledge the direct contributions of several persons to this
work. Andrei Faraon supervised and helped conceive all experiments.
Evan Miyazono was the lead researcher on the erbium quantum memory project
from before I joined until his graduation in 2017. The setups he built and the
Er3+ :Y2 SiO5 spectroscopy and resonator design work he did laid the groundwork
for the work described in this thesis.
Mi Lei joined this project in 2018. Her contributions to this work were countless and
essential. Mi has participated in nearly all aspects of the experiments described in
Chapters 5 and 6, including simulation, design, nanofabrication, and measurement.
Jake Rochman designed and fabricated the nanobeam resonator used in Chapter 5.
He designed the hybrid resonator used in Chapter 6, and the nanofabrication process
for that resonator. Jake contributed to the efficiency calculations in Chapter 3.
Jonathan Kindem contributed ideas for to experiment methods, and the simulations
in Chapter 6 are adapted from his code. John Bartholomew contributed with rare
earth ions theory and interpretation of spectroscopic data. John Bartholomew,
Andrei Ruskuc, and Hirsh Kamakari built the superconducting magnets. Tian Xie
built the laser frequency locking setup used in Chapter 6. The superhyperfine
calculations were performed with the help of Yan Qi Huan, and based on similar
calculations he performed for a different material.
Next, I would like to express my gratitude to the many who helped me get here.
Thank you to Andrei for taking me on as a grad student. Thank you for giving me
space to discover and make mistakes while also being present to provide guidance.
Thank you for your patience, and the reminders to “breathe and take it slow.”
Thank you to Evan for taking me in as a first-year grad student and teaching me
everything from the very basics. Thank you for your friendship. Thank you to Mi for
everything you’ve given to this project. Without your hard work, your curiosity, and
your valuable insights, nothing we’ve achieved would have been possible. Thank
you to Jake for always being willing to help, and for inspiring me to do a good job
with your high standards. Thank you to Jon for your always sharing ideas, and for
making science look fun. Thank you to John for being a top-notch scientist and
mentor and sharing your extensive knowledge with everyone in the group. Andrei

iv
R. and Tian, thank you for always bringing your science A-game while also making
work fun. Thank you to Dimi Cielecki for your work and enthusiasm. Daniel, Chun,
Hirsh, and Yan Qi, thank you for the helpful discussions during subgroup meetings
and beyond. Thank you to Tian and Amir for the mentorship early on. Thank you
to everyone in the flat-optics subgroup for always keeping me on my toes with your
fast progress and impressive results.
Thank you to my defense and candidacy committee, Kerry Vahala, Andrei Faraon,
Oskar Painter, and Manuel Endres, for your time and your guidance.
Thank you to Chuting for being my officemate for 6 years as well as my friend. Thank
you to my book club friends/dictators who forced me to think of the world outside
rare-earth ions and also fed me snacks. Thank you to Heidi for your friendship and
for keeping me fit through our morning swims. Thank you Phil for your friendship.
Thanks to Andrei and Greg for the Great Physics Bakeoffs. Thank you to my music,
drama and art teachers Kathrin Jakob, Kyle LaLone, Brian Brophy, and Jim Barry
for the creative outlet.
Thank you to all the staff at Caltech who has make it possible for students to learn
and do research: Cecilia Gamboa, Jennifer Blankenship, Christy Jenstad for the
countlessly many things you do to support students; everyone at Caltech Facilities
and everyone at Caltech Dining Services.
Thank you to my professors and mentors at Caltech and the University of Waterloo
and my teachers at St. Mary’s High School, St. Michael’s Jr. High, John Costello
Elementary, Sacred Heard Elementary, all in Calgary, Canada, and Principesa Ileana
School in Bucharest, Romania. I wouldn’t be where I am without you.
Thank you to my partner, Max Jones, for the support, for believing in me and
for valuing my work. Thank you for your advice, for listening to many practice
presentations and for the many proof-readings, including this thesis. Thank you to
my friends from Calgary and Waterloo for being a constant source of inspiration,
love, and goofiness. There’s too many of you to name, but you know who you are.
Thank you to my brother Traian, sister-in-law Ankita and father Daniel Craiciu for
your support. And finally, thank you to my mother, Iuliana Craiciu. Thank you
for always supporting me. Thank you for your fierce belief in my abilities, often
expressed as world-class helicopter parenting. Thank you for forcing me to apply to
Caltech when I was convinced I would never get in. Thank you for teaching me that
hard work is rewarding and that the world is beautiful when one is curious about it.

ABSTRACT
This thesis presents on-chip quantum storage of telecommunication wavelength
light using nanophotonic resonators coupled to erbium ions. Storage of light in
an optical quantum memory has applications in quantum information and quantum
communication. For example, long distance quantum communication using quantum repeater protocols is enabled by quantum memories. Efficient and broadband
quantum memories can be made from resonators coupled to ensembles of atoms.
Like other rare earth ions, erbium is appealing for quantum applications due to its
long optical and hyperfine coherence times in the solid state at low temperatures.
However, erbium is unique among rare earth ions in having an optical transition
in the telecommunication C band (1540 nm), making it particularly appealing for
quantum communication applications. In this work, we use nano-scale resonators
coupled to erbium-167 ions in yttrium orthosilicate crystals (167 Er3+ :Y2 SiO5 ).
We demonstrate quantum storage in two types of resonators. In a nanobeam photonic crystal resonator milled directly in 167 Er3+ :Y2 SiO5 , we show storage of weak
coherent states using the atomic frequency comb protocol. The storage fidelity for
single photon states is estimated to be at least 93.7% ± 2.4% using decoy state
analysis. Storage of up to 10 𝜇s and multimode storage are demonstrated. Using a
hybrid resonator based on amorphous silicon on 167 Er3+ :Y2 SiO5 and on-chip electrodes, we demonstrate a multifunctional memory using the atomic frequency comb
protocol with DC Stark shift control. In addition to dynamic control of memory
time, Stark shift control allows modifications to the frequency and bandwidth of
stored light. We show tuning of the output pulse by ±20 MHz relative to the input
pulse, and broadening of the pulse bandwidth by more than a factor of three. The
storage efficiency in both devices was limited to < 1%.
On the way to these results, we describe 167 Er3+ :Y2 SiO5 spectroscopy measurements
including optical coherence times and hyperfine lifetimes below 1 K, and we estimate
the linear DC Stark shift along two crystal directions. The design and fabrication
of the on-chip resonators is presented. We discuss the limitations to storage time
and efficiency, including superhyperfine coupling and resonator parameters, and we
outline a path forward for improving the storage efficiency in these types of devices.

PUBLISHED CONTENT AND CONTRIBUTIONS

[1] Ioana Craiciu, Mi Lei, Jake Rochman, Jonathan M. Kindem, John G.
Bartholomew, Evan Miyazono, Tian Zhong, Neil Sinclair, and Andrei Faraon.
Nanophotonic quantum storage at telecommunication wavelength. Phys.
Rev. Applied, 12:024062, Aug 2019. doi: 10.1103/PhysRevApplied.12.
024062. URL https://link.aps.org/doi/10.1103/PhysRevApplied.
12.024062. I.C. built experimental setup, participated in measurement and
data processing, and wrote manuscript.
[2] Evan Miyazono*, Ioana Craiciu*, Amir Arbabi, Tian Zhong, and Andrei Faraon.
Coupling erbium dopants in yttrium orthosilicate to silicon photonic resonators
and waveguides. Opt. Express, 25(3):2863–2871, Feb 2017. doi: 10.1364/
OE.25.002863. URL http://www.opticsexpress.org/abstract.cfm?
URI=oe-25-3-2863. *These authors contributed equally to this work. I.C.
participated in setting up experiment, nanofabrication, measurements, and data
processing, and gave input on the manuscript.

vii

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Quantum Internet, Quantum Repeaters, and Quantum Memories
1.2 167 Er3+ :Y2 SiO5 for Optical Quantum Memories . . . . . . . . . . .
1.3 Cavity Quantum Memories Using Nanophotonic Resonators . . . . .
1.4 Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter II: Erbium-167 in Yttrium Orthosilicate . . . . . . . . . . . . . . . .
2.1 System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Optical Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Spectral Holeburning, Hyperfine Levels, and Initialization . . . . . .
2.4 Superhyperfine Broadening . . . . . . . . . . . . . . . . . . . . . .
Chapter III: Atomic Frequency Comb Storage - Theory . . . . . . . . . . . .
3.1 Storing Light in an Atomic Frequency Comb . . . . . . . . . . . . .
3.2 AFC Storage Efficiency in a Cavity . . . . . . . . . . . . . . . . . .
Chapter IV: Nanophotonic Resonators: Design and Fabrication . . . . . . . .
4.1 Y2 SiO5 Nanobeam Resonators . . . . . . . . . . . . . . . . . . . . .
4.2 Hybrid Amorphous Silicon-Y2 SiO5 Resonators . . . . . . . . . . . .
4.3 On-Chip Electrodes for DC Stark Shift Control . . . . . . . . . . . .
Chapter V: Quantum Storage in an 167 Er3+ :Y2 SiO5 Nanobeam . . . . . . . .
5.1 Coupling Between an Ensemble of Ions and a Cavity . . . . . . . . .
5.2 Atomic Frequency Comb Storage . . . . . . . . . . . . . . . . . . .
5.3 Coherent Storage of Time-Bin Qubits . . . . . . . . . . . . . . . . .
5.4 Estimating a Lower Bound on Storage Fidelity . . . . . . . . . . . .
5.5 Overcoming AFC Efficiency Limitation . . . . . . . . . . . . . . . .
5.6 Nanobeam Device Temperature . . . . . . . . . . . . . . . . . . . .
Chapter VI: Dynamic On-chip Control of Stored Light Using the DC Stark
Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 DC Stark Shift in 167 Er3+ :Y2 SiO5 . . . . . . . . . . . . . . . . . . .
6.2 Dynamic Control of Storage Time . . . . . . . . . . . . . . . . . . .
6.3 Frequency Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Bandwidth Broadening . . . . . . . . . . . . . . . . . . . . . . . . .

iii
vi
vi
vi
ix
xi
xii
10
10
13
16
25
28
28
32
35
35
36
40
46
46
48
51
53
56
57
59
60
62
67
71

viii
6.5 Cavity-Ion Coupling and AFC Efficiency in a Hybrid 𝛼Si-Y2 SiO5
Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.6 Improving Efficiency in Hybrid Resonators . . . . . . . . . . . . . . 80
Chapter VII: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Appendix A: Simulation of Hybrid 𝛼Si-167 Er3+ :Y2 SiO5 Resonators . . . . . 101
A.1 Calculating Photonic Crystal Bandgap Using MPB . . . . . . . . . . 101
A.2 Calculating Waveguide Band Diagram Using MPB . . . . . . . . . . 102
A.3 Calculating Reflectance of Mirror With Taper Using Comsol . . . . . 103
Appendix B: Fabrication Process for Electrodes . . . . . . . . . . . . . . . . 105
Appendix C: Measurement Setup for Chapter 5 . . . . . . . . . . . . . . . . 107
Appendix D: Measurement Setups for Chapter 6 . . . . . . . . . . . . . . . . 109
D.1 Measurements with SNSPD . . . . . . . . . . . . . . . . . . . . . . 109
D.2 Heterodyne Detection . . . . . . . . . . . . . . . . . . . . . . . . . 112
D.3 Parts List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

LIST OF ILLUSTRATIONS

Number
Page
1.1 A quantum memory . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 A quantum repeater network . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Y2 SiO5 primitive unit cell . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Simplified energy diagram for 167 Er3+ :Y2 SiO5 . . . . . . . . . . . . 13
2.3 Optical coherence time measurement in nanobeam resonator and bulk
167 Er3+ :Y SiO
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Optical coherence time measurement in hybrid 𝛼Si-167 Er3+ :Y2 SiO5
resonator and bulk 167 Er3+ :Y2 SiO5 . . . . . . . . . . . . . . . . . . 16
2.5 Spectral holeburning schematic . . . . . . . . . . . . . . . . . . . . 17
2.6 Spectral holeburning in an inhomogeneous distribution . . . . . . . . 19
2.7 167 Er3+ :Y2 SiO5 spectral hole lifetime . . . . . . . . . . . . . . . . . 21
2.8 167 Er3+ :Y2 SiO5 hyperfine initialization and comb burning . . . . . . 23
2.9 167 Er3+ :Y2 SiO5 hyperfine initialization . . . . . . . . . . . . . . . . 24
2.10 Superhyperfine side structure in Er3+ :Y2 SiO5 holeburning . . . . . . 26
3.1 Inhomogeneous broadening dephasing on a Bloch sphere . . . . . . . 29
3.2 Atomic frequency comb evolution on a Bloch sphere . . . . . . . . . 30
4.1 Nanobeam resonator simulation and micrograph . . . . . . . . . . . 36
4.2 Hybrid 𝛼Si-167 Er3+ :Y2 SiO5 waveguide simulation . . . . . . . . . . 37
4.3 Hybrid 𝛼Si-167 Er3+ :Y2 SiO5 photonic crystal mirror simulation . . . . 39
4.4 Hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator fabrication . . . . . . . . . . . 40
4.5 Simulations of on-chip electrodes . . . . . . . . . . . . . . . . . . . 42
4.6 Various electrode designs . . . . . . . . . . . . . . . . . . . . . . . 43
4.7 Fabrication of on-chip electrodes . . . . . . . . . . . . . . . . . . . 44
4.8 Hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator with electrodes . . . . . . . . 45
5.1 Reflection spectrum of nanobeam resonator . . . . . . . . . . . . . . 47
5.2 AFC experiment in the nanobeam cavity . . . . . . . . . . . . . . . 49
5.3 AFC storage for 10 𝜇s in the nanobeam resonator . . . . . . . . . . . 50
5.4 Multimode storage in the nanobeam resonator . . . . . . . . . . . . . 50
5.5 Double atomic frequency comb . . . . . . . . . . . . . . . . . . . . 52
5.6 Coherent storage in the nanobeam resonator . . . . . . . . . . . . . . 53
5.7 Fidelity measurement data . . . . . . . . . . . . . . . . . . . . . . . 55

5.8
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
A.1
C.1
D.1
D.2
D.3

Nanobeam device temperature measurement . . . . . . . . . . . . . 58
Stark shift measurement . . . . . . . . . . . . . . . . . . . . . . . . 62
Digital storage time pule sequence . . . . . . . . . . . . . . . . . . . 63
Digitally controlled storage time in hybrid resonator . . . . . . . . . 65
Electric pulse calibration . . . . . . . . . . . . . . . . . . . . . . . . 67
Frequency shift pulse sequence . . . . . . . . . . . . . . . . . . . . 68
AFC storage with frequency shift in hybrid resonator . . . . . . . . . 69
Frequency shift versus applied field . . . . . . . . . . . . . . . . . . 70
Bandwidth broadening pulse sequence . . . . . . . . . . . . . . . . . 71
AFC storage with bandwidth broadening in hybrid resonator . . . . . 74
Bandwidth experiment efficiency . . . . . . . . . . . . . . . . . . . 76
Quadrupole field induced dephasing . . . . . . . . . . . . . . . . . . 76
Cavity-ion coupling in hybrid device . . . . . . . . . . . . . . . . . 78
Cavity-ion coupling in another hybrid device . . . . . . . . . . . . . 79
AFC efficiency in the hybrid device . . . . . . . . . . . . . . . . . . 80
Predicted AFC storage efficiency for hybrid resonators (𝐹 = 5) . . . 82
Predicted AFC storage efficiency for hybrid resonators (𝐹 = 15) . . . 83
Predicted AFC storage efficiency vs. 167 Er3+ doping for hybrid resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3D 𝛼Si-167 Er3+ :Y2 SiO5 photonic crystal mirror COMSOL Simulation 104
Measurement setup for Chapter 5 . . . . . . . . . . . . . . . . . . . 107
Measurement setup for Chapter 6 with SNSPD . . . . . . . . . . . . 109
Picture of measurement setup in Chapter 6 . . . . . . . . . . . . . . 111
Measurement setup for Chapter 6 with heterodyne . . . . . . . . . . 113

LIST OF TABLES
Number
Page
5.1 Storage fidelities in the nanobeam device. . . . . . . . . . . . . . . . 56
D.1 Optical components . . . . . . . . . . . . . . . . . . . . . . . . . . 114
D.2 Electronic components . . . . . . . . . . . . . . . . . . . . . . . . . 114
D.3 Laser frequency locking components . . . . . . . . . . . . . . . . . 114

xii

NOMENCLATURE

AFC. Atomic frequency comb. A quantum memory protocol.
CRIB. Controlled reversible inhomogeneous broadening. A quantum memory
protocol.
FWHM. Full width at half maximum.
SEMM. Stark echo modulation memory. A quantum memory protocol.
SNSPD. Superconducting nanowire single photon detector.
ZEFOZ. Zero first-order Zeeman transition.

Chapter 1

INTRODUCTION
In this work, we present on-chip quantum storage at 1540 nm, in the telecommunication C band, using a nanophotonic resonator. This is the first demonstration
of on-chip quantum storage in the telecom C band and the first demonstration of
quantum storage in 167 Er3+ :Y2 SiO5 . Further, we use on-chip electrodes to realize
a multifunctional device which can change the frequency and bandwidth of stored
light, thus providing capabilities for linear processing of light pulses. On the way
to these results, we present: spectroscopy on 167 Er3+ :Y2 SiO5 in new parameter
regimes; the design and fabrication process for on-chip resonators and electrodes;
the coupling between the nanophotonic resonator and an ensemble of ions and
the effect of this coupling on storage efficiency; measurements with light at the
single-photon level in a dilution fridge at sub-kelvin temperatures. We discuss the
efficiency of storage in these devices, with a focus on limitations including device
design and fabrication, and superhyperfine coupling in 167 Er3+ :Y2 SiO5 . We then
outline a path towards on-chip storage with high efficiency.
In this introductory chapter, we present the motivation for creating on-chip quantum
memories for light, with a focus on application in quantum networks, and we
summarize the current state of the art. At the end of this chapter we outline the
structure of this thesis.
1.1

The Quantum Internet, Quantum Repeaters, and Quantum Memories

A quantum memory for light is a device that stores light coherently for some portion
of time. Figure 1.1 shows the basic principles of a quantum memory. Quantum
memories should be able to store light efficiently for a length of time determined by
the application, and to store a quantum state encoded in light with high fidelity.
Quantum memories enable long distance quantum communication. The transfer of
quantum information over large distances enables secure communication via quantum key distribution [13, 58]. Other applications of long distance quantum communication include improved clock synchronization, distributed quantum computing,
and improved measurements including large baseline optical telescope networks
[104].

Figure 1.1: A quantum memory for light.
Due to their fast travel speed and low rates of decoherence, photons are a natural
choice for transmitting quantum information. However, because quantum information cannot be amplified without introducing noise (the no-cloning theorem [108]),
attenuation is a big problem for long distance quantum communication. Secure
quantum key distribution through optical fibers has been demonstrated over distances up to 509 km [26]. The longest quantum communication link is 2000 km
long between Beijing and Shanghai [111]. However, this link uses 32 trusted relays
which are vulnerable to attacks. Demonstrations of satellite-to-ground quantum key
distribution across 7600 km (using the satellite as a trusted relay) have shown that
long distance optical quantum communication through free space is also promising
[65].
Quantum repeaters enable long distance quantum communication without trusted
relays beyond ∼ 500 km. Figure 1.2 shows a schematic of a quantum repeater
network. The goal of the quantum repeater protocol, first introduced by Briegel
et al. in 1998 [22], is to generate an entangled pair spanning a long distance A-B
quickly and with high fidelity. This enables the transmission of quantum information
over the channel using quantum teleportation. Entanglement distillation can be used
to improve the fidelity of the final entangled state [20, 40]. Many types of quantum
repeater protocols have been proposed. References [92] and [109] review protocols
using atomic ensemble based memories.

Figure 1.2: A quantum repeater network. a) In the simplest form, quantum communication works by sending quantum information over a channel from A to B. b)
In a quantum repeater network, a long distance (A to B) is broken up into shorter
segments. The boxes at the end of each segment represent quantum memories. c)
A close up of two segments in the quantum repeater network. In the first step of a
quantum repeater protocol, entanglement is shared across each segment. Here, this
is accomplished by distributing entangled photon pairs, which are stored in quantum
memories. In the second step, a Bell State measurement is performed on photons
from adjacent memories (2 and 3) to swap entanglement: the photons in memories
1 and 4 are now entangled. Entanglement swapping continues until one entangled
pair remains that spans the distance between A and B.
In order to synchronize the generation of entangled pairs across all segments, and
then to store entangled pairs while swapping and distilling, most repeater protocols
require quantum memories. Optical quantum memories for light have been proposed
and demonstrated in a variety of systems [49] including single emitters such as NV
and SiV centers [14], ensembles of cold atoms in a magneto-optical trap [53], warm
atomic vapors [52], and ensembles of rare earth ions in crystals [47]. For single
atoms or defect centers, the memory typically works by transferring the optical
quantum state to a spin degree of freedom [8, 14, 58]. Many types of ensemblebased memories have been proposed and demonstrated.
The initial proposal for a quantum memory using atomic ensembles was using the
DLCZ protocol, named after its creators [39]. Reference [92] analyzes several quantum repeater protocols using this type of memory. In the intervening years, many
quantum storage protocols using ensembles of atoms have been proposed and realized [49]. Memory protocols including electromagnetically induced transparency
(EIT) [41] and Raman scattering [59] use strong optical control pulses to read and

write information stored on spin transitions of an ensemble. Engineered absorption
memories such as gradient echo memory (GEM) [5], controlled reversible inhomogeneous broadening (CRIB) [60], and atomic frequency comb (AFC) [4] store
information in the optical transitions of an atomic ensemble. These protocols rely on
spectral holeburning or quasi-static electric or magnetic fields to induce rephasing
and emission of light stored in the ensemble. To extend memory times, optical
control pulses can be used to transfer stored information from optical to spin transitions and back, but unlike in EIT and Raman memories, these control pulses do
not coincide with the input or output pulses. Although the simple two-pulse photon
echo involves rephasing and emission of light stored in an atomic ensemble, the
associated population inversion leads to noise that renders this protocol unacceptable for quantum storage [88]. The revival of silenced echo (ROSE) protocol is an
optical-pulse based storage protocol designed to avoid this inversion-induced noise
[33]. References [21, 49] touch on a number of other proposed ensemble memory
protocols.
For quantum memories to be useful in a real quantum repeater, they must be very
efficient and their storage times must be sufficiently long. Most studies of "realistic"
repeater protocols assume quantum memory efficiencies exceeding 90%. As a rule
of thumb, the required memory time is given by the highest level of entanglement
swapping, meaning storage time should equal the time it takes to create the final
entangled pair spanning the distance A-B [92]. It is not trivial to estimate a typical
entanglement rate (the inverse of the time it takes to create the final entangled pair),
as it depends on many details, including the specific repeater scheme used, the
total distance A-B, as well as the memory efficiency and memory time. Wu et
al. studied entanglement rates for the realistic scenario where quantum memories
have an exponentially decaying efficiency characterized by a decay constant. They
predicted entanglement generation rates approaching 1 Hz for a 100 km distance
and memory decay constants as low as 1 ms [109]. Another helpful metric for
evaluating a quantum memory is whether it is superior to an optical fiber delay line
in terms of efficiency vs. storage time. Most quantum memories realized to date are
not [29].
Some progress has been made toward efficient and long-lived quantum memories
for light. The record efficiency for storing coherent pulses is 92% using EIT in
cold cesium atoms [53]. This result was for storing 895 nm light for 1 𝜇s, although
storage for up to ∼ 350 𝜇s was shown with lower efficiency in the same work. For

storing a quantum state of light, the current record efficiency is 85% for storing a
polarization qubit at 795 nm for 1 𝜇s, using EIT in cold rubidium atoms [103]. For
storage in solid state, the record storage efficiency of 76% used a cavity-enhanced
EIT protocol in praseodymium doped Y2 SiO5 (Pr3+ :Y2 SiO5 ) [93]. Coherent pulses
of 606 nm light were stored for 2 𝜇s. Hedges et al. stored coherent pulses in
Pr3+ :Y2 SiO5 for 2 𝜇s at 69% efficiency using the GEM protocol. Another notable
result in solid state quantum memories is the storage of light for up to 1 minute using
EIT in Pr3+ :Y2 SiO5 , albeit with efficiency lower than 1% [48]. These results show
that, despite advancements in the last two decades, a highly efficient and long-lived
quantum memory has yet to be achieved. Even so, there has been progress towards
realizing elementary quantum networks using one or more memories [14, 102, 110].
The quantum storage protocol used in this work is the atomic frequency comb (AFC)
protocol introduced by Afzelius et al. [4]. As detailed in Section 3.1 of this thesis,
this protocol relies on spectral holeburning to shape the absorption spectrum of an
ensemble into a comb. There are several advantages to the AFC protocol:
• For the basic protocol, only optical control is required.
• Optical control pulses are not required during memory readout. Control
pulses during readout are a source of noise.
• The AFC bandwidth can be large, allowing multiplexing in frequency. Multiplexing can significantly improve entanglement generation rates in quantum
repeater networks [96].
There have been many demonstrations of quantum storage using the AFC protocol
since its introduction in 2008 [4, 34]. Storage of coherent pulses for up to 1 second
was realized with the full AFC protocol (spin-wave AFC) in Eu3+ :Y2 SiO5 at 580 nm,
albeit with low efficiency. With cavity enhancement, also in Eu3+ :Y2 SiO5 , storage
efficiencies of up to 52% at 2 𝜇s were achieved using AFC in (12% at 15 𝜇s using
spin-wave AFC) [56]. The multimode capacity of this protocol was demonstrated
in Reference [15], in which over 1000 temporal modes were stored at the same time
in an AFC memory in Tm3+ :YAG (793 nm), and in Reference [97], in which 26
spectral modes were stored simultaneously in a Tm:LiNbO3 waveguide (795 nm).
Most demonstrations of AFC storage have relied on crystals doped with ions such
as europium, thulium, and praseodymium [94]. These ions, which have an even
number of electrons, typically have long-lived hyperfine levels which enable spectral

holeburning, as well as the highly coherent optical and hyperfine transitions. For
ions such as erbium, which has an odd number of electrons, electron spin coupling to
other spins and phonons, as well as superhyperfine interactions, make it difficult to
create atomic frequency combs. AFC storage in the optical transition was realized in
erbium doped Y2 SiO5 [63, 64], erbium doped fibers [91], and erbium doped lithium
niobate waveguides [9], although with efficiency below 1% and short storage times.
This sets the stage for the AFC storage in 167 Er3+ :Y2 SiO5 presented in this work.
Recently, AFC storage with spin-wave was achieved using another rare earth ion
with an unpaired electron, 171 Yb3+ :Y2 SiO5 [23]. In that work, Businger et al.
leveraged the zero first-order Zeeman (ZEFOZ) hyperfine and optical transitions of
171 Yb3+ :Y SiO at zero field. They observed long-lived hyperfine states, as well
as evidence of strongly suppressed superhyperfine coupling, which enabled AFC
spin-wave storage of 979 nm light for up to 1.3 ms.
In addition to storing information, atomic ensembles can also be used to manipulate
input states. The ability to adjust the temporal profile or frequency of stored light
can be useful when quantum memories act as interfaces between multiple emitters,
as in a quantum repeater network. For example, Hosseini et al. used a GEM scheme
to modify the amplitude, bandwidth, and order of stored pulses [51]. In this work,
we use electrodes in parallel or quadrupole configurations to change the memory
time, as well as adjust the frequency and bandwidth of light stored using the AFC
protocol. Lauritzen et al. used a similar scheme, where electric field control was
used to modify the otherwise fixed memory time in an AFC without spin-wave [64].
1.2

167 Er3+ :Y SiO for Optical Quantum Memories

Crystals doped with rare earth ions are an appealing platform for optical quantum
memories because of the ions’ long optical [17] and hyperfine [112] coherence
times. As a solid state platform, it is straightforward to couple rare earth ion doped
crystals to nanoscale resonators and on-chip components [35, 79, 84, 114, 115].
Ions across the lanthanide series are called rare earths despite their abundance in the
earth’s crust (iodine, silver, gold, and cadmium are rarer than most of the rare earths).
The long coherence times of rare earth ions, even in the solid state, stem from their
unique electronic structure, in which the electrons participating in optical transitions
are in the partially occupied 4f shell, which is shielded from the environment by the
electrons in the 5s and 5p shells [66].
Erbium is particularly interesting because it is the only rare earth ion with an

optical transition in the telecommunication C band, at 1540 nm. This is important
for long distance quantum communication applications because standard optical
fiber has the lowest attenuation in the C band, at ∼ 0.17 dB/km. For this reason,
erbium has been extensively studied for quantum memory applications. We have
already mentioned some initial erbium quantum storage demonstrations using AFC,
including Er3+ :Y2 SiO5 [63, 64], erbium doped fibers [91], and lithium niobate
waveguides [9]. The storage efficiencies in these works were limited in part by the
lack of suitable long-lived shelving states in the erbium ions in these hosts. Using
the ROSE protocol, which does not require any state preparation, storage of coherent
optical states in Er3+ :Y2 SiO5 for 16 𝜇s with an efficiency of 42% was shown [32].
For memory protocols requiring spectral holeburning such as AFC, erbium doped
materials pose some challenges. Natural isotopic abundance erbium is 23% 167 Er,
which has nuclear spin 𝐼 = 27 , with the balance comprised entirely of isotopes
with zero nuclear spin, predominantly 166 Er, 168 Er, and 170 Er. This means that
in crystals doped with natural abundance erbium, most erbium ions do not have
long lived hyperfine states in the optical ground state. In the absence of hyperfine
states, spectral holeburning can be achieved using Zeeman (electron spin levels).
In initial studies of holeburning and AFC storage in Er3+ :Y2 SiO5 [46, 62, 63], the
ratio between the Zeeman lifetime and the optical lifetime was found to be too low
for good holeburning.
It is therefore advantageous to use 167 Er doped materials, which do have hyperfine
ground states, for spectral holeburning based memories such as AFC. Rančić et
al. found that long lived spectral holes could exist in 167 Er3+ :Y2 SiO5 , provided the
electron spin was frozen using large fields (> 3 T) and low temperatures (1.4 K)
[86]. In the same work, Rančić et al. measured hyperfine coherence times longer
than 1 s, which would be important for AFC spin-wave storage. In this work, we
use 167 Er3+ :Y2 SiO5 at lower fields (0.4 − 1 T), and also lower temperatures (< 1
K) to achieve spectral holes with measured lifetimes of 30 minutes, and to create
good spectral combs. With these spectral combs, we demonstrated for the first
time quantum storage in 167 Er3+ :Y2 SiO5 and for the first time quantum storage at
telecommunications wavelength in an on-chip resonator.
Another appealing property of 167 Er3+ :Y2 SiO5 is that ZEFOZ hyperfine transitions
have been predicted to exist in this material [75, 85]. Because ZEFOZ transitions are
first-order insensitive to magnetic field, they could have even longer coherence times.
Moreover, the superhyperfine interaction between the erbium electron spin and

neighboring yttrium nuclear spins would be suppressed for these levels. As described
in this work, superhyperfine coupling limits the performance of AFC memories by
limiting the minimum spectral feature. ZEFOZ transitions in 167 Er3+ :Y2 SiO5 were
not explored in this work; we address them again in the Future Directions portion
of this thesis.
In addition to ensemble quantum memories, erbium doped crystals can be used for
a suite of quantum technologies. There have been proposals [8] and demonstrations
[35, 84] using cavity-coupled single erbium ions for quantum technologies, and
a proposal for efficient quantum microwave-to-optical transduction using erbium
ensembles [107].
1.3

Cavity Quantum Memories Using Nanophotonic Resonators

For ensemble quantum memories, a resonator can be used to achieve high efficiency.
Afzelius et al. [3] and Moiseev et al. [80] showed that an ensemble quantum memory
can achieve unity efficiency in the impedance matched condition, when the ensemble absorption rate equals the cavity coupling to free space (characterized by an
ensemble cooperativity of 1). Since these proposals, optical resonators have been
used to improve storage efficiencies to above 50% in rare earth ion doped crystals
[56, 76, 89, 93] and cold atoms [10, 106]. Resonators led to higher efficiencies not
only for the AFC memory protocol [56, 89], but also for storage using EIT [93, 106],
DLCZ [10], and ROSE [76].
This work focuses on integrating rare earth ion ensembles with nano-scale onchip resonators. In addition to the improvement in memory efficiency, nano- and
micro-scale resonators enable miniaturization and on-chip integration. For example, multiple atomic memories on the same chip could be used in an entanglement
distillation protocol [20, 40] as part of a quantum network. In this work, we use
nanobeam resonators milled directly in 167 Er3+ :Y2 SiO5 [30, 79, 114], and amorphous silicon resonators coupled evanescently to the 167 Er3+ ensemble [78]. The
first on-chip cavity quantum memory was demonstrated in our lab in Nd3+ :YVO
[115]. This work followed that result by achieving on-chip quantum storage at
telecommunication wavelengths in a 167 Er3+ :Y2 SiO5 nanophotonic resonator. We
note several studies which have used on-chip waveguides for ensemble quantum
memories [9, 73, 90, 95].

1.4

Structure of this Thesis

Chapter 2 introduces erbium-167 doped yttrium orthosilicate (167 Er3+ :Y2 SiO5 ), the
material used throughout this work to store and manipulate light. Methods used to
initialize the ions and burn atomic frequency combs are introduced. Superhyperfine
coupling is also introduced.
Chapter 3 describes the atomic frequency comb protocol used to store light in this
work. The theoretical efficiency of this protocol in a cavity is discussed.
Chapter 4 describes the design and fabrication of on-chip resonators and on-chip
electrodes used for storage and control in this work. Two types of resonators are
presented: an 167 Er3+ :Y2 SiO5 photonic crystal nanobeam, and a hybrid resonator
using amorphous silicon (𝛼Si) waveguides and photonic crystals on 167 Er3+ :Y2 SiO5 .
Chapter 5 shows on-chip storage of telecom wavelength light in a nanobeam resonator in 167 Er3+ :Y2 SiO5 . The multimode storage capacity and storage fidelity
are presented, as are a discussion of storage efficiency and a path toward higher
efficiency. This chapter is adapted from Reference [30].
Chapter 6 shows a new multifunctional optical quantum memory using a hybrid
𝛼Si-167 Er3+ :Y2 SiO5 resonator and on-chip electrodes. This device combines AFC
storage with DC Stark shift control to dynamically control the memory time, and
apply corrections to the frequency and bandwidth of the stored light.
Chapter 7 concludes this thesis. The results are summarized with a focus on
the potential of 167 Er3+ :Y2 SiO5 and nanophotonic resonators for efficient quantum
memories. Future directions are recommended.

10
Chapter 2

ERBIUM-167 IN YTTRIUM ORTHOSILICATE
In this chapter, we introduce erbium-167 doped yttrium orthosilicate (167 Er3+ :Y2 SiO5 ).
We outline its energy structure, with a focus on the telecom optical transitition and
hyperfine ground state levels. We present optical coherence measurements at temperatures below 100 mK, and moderate magnetic fields (0.4-1 T), and also measure
the hyperfine lifetime under these conditions. We introduce the concept of spectral
holeburning, which is used to create atomic frequency combs. The methods used to
initialize the ions into a few hyperfine states at the beginning of every experiment
are discussed. Finally, we introduce superhyperfine coupling, which is an important
limitation to the performance of the AFC protocol in this material.
2.1

System Overview

For a concise introduction to the energy structure of rare earth ions in crystals beyond
what is described here, I recommend Chapter 3 in John Bartholomew’s thesis [11].
The Hamiltonian describing the energy levels of 167 Er3+ :Y2 SiO5 in an applied
magnetic field is:

𝐻 = 𝐻FI + 𝐻CF + 𝐻𝑒Z + 𝐻HF + 𝐻𝑛Z + 𝐻SHF

(2.1)

where 𝐻FI is the Hamiltonian of a free Er3+ ion including spin-orbit coupling, and
𝐻CF is the crystal field correction from Y2 SiO5 . For 167 Er3+ , the first two energy
levels of 𝐻FI are 4 I 15 and 4 I 13 , separated by ∼ 0.8 eV = 195 THz = 1.5 𝜇m. These
levels are (2𝐽 + 1)-fold degenerate. 𝐻CF lifts the degeneracy of these levels, splitting
each into 𝐽 +1/2 twofold degenerate levels, separated by ∼ 5 meV = 40cm−1 = 1THz
[16]. The lowest two-fold degenerate levels of the 4 I15/2 and 4 I13/2 manifolds, called
𝑍1 and 𝑌1 , respectively, are Zeeman doublets and they behave as spin-1/2 electrons.
The optical transition used in this thesis is between the 𝑍1 and 𝑌1 states. This
transition is very weak, characterized by an oscillator strength of 2 × 10−7 [74]. It
is both weakly electric-dipole allowed due to the crystal field and magnetic-dipole
allowed [16, 67].
The remaining terms in Equation 2.1 form the spin Hamiltonian for the electron and

11
nuclear spins. Equations 2.2 to 2.4 and Equation 2.6 in Section 2.4 describe these
terms.
𝐻𝑒Z is the electron spin Zeeman Hamiltonian characterizing with an external applied
field 𝐵:
𝐻𝑒Z = 𝜇 𝐵 𝐵® · 𝑔ˆ · 𝑆,

(2.2)

where 𝜇 𝐵 = 14.0𝐺𝐻𝑧/𝑇 is the Bohr magneton and 𝑔ˆ is the 2nd rank g-tensor, and 𝑆
is the electron spin operator. Due to the low symmetry of the Y2 SiO5 crystal field,
the 𝑔ˆ tensor is characterized by 6 independent parameters, which are experimentally
determined in Refs. [50, 99]. 𝐼® is the spin operator for the 𝐼 = 7/2 nuclear
spin of 167 Er ions. 167 Er comprises 23% of natural abundance erbium, and the
remaining isotopes all have 𝐼 = 0. 𝐻HF is the hyperfine Hamiltonian characterizing
the interaction between the electron spin and 167 Er nuclear spin:
𝐻HF = 𝐼® · 𝐴ˆ · 𝑆® + 𝐼® · 𝑄ˆ · 𝐼,

(2.3)

where 𝐴ˆ and 𝑄ˆ are also 2nd rank, experimentally determined tensors [44, 50]. This
interaction splits each twofold degenerate electron spin level into 16, with a splitting
on the order of 1 GHz. The second term in Eq. 2.3 describes the quadrupole
interaction and the second-order hyperfine interaction. 𝐻𝑛Z describes the 167 Er3+
nuclear spin Zeeman interaction,
𝐻𝑛Z = 𝜇𝑛 𝑔𝑛,𝐸𝑟 𝐵® · 𝐼,

(2.4)

where 𝜇𝑛 = 7.62 MHz/T is the nuclear magneton and 𝑔𝑛,𝐸𝑟 = −0.1611 is the nuclear
g-factor of 167 Er [98]. This effect leads to a small correction to the energy of the
hyperfine levels and usually ignored. The last term of Eq. 2.1, 𝐻SHF describes the
superhyperfine interaction, as defined in Eq. 2.6 and discussed in Section 2.4.
Figure 2.1 shows a primitive Y2 SiO5 unit cell, containing 8 yttrium atoms, 4 silicon
atoms, and 20 oxygen atoms [1, 70, 81]. The structure of the crystal is based on
SiO4 tetrahedra, YO6 octahedra, and YO7 polyhedra [28]. Erbium can substitute
for either of two inequivalent yttrium crystallographic sites, nominally with equal
probability [16]. Each of these two yttrium sites is repeated four times in the unit cell
with different orientations. This leads to four subclasses for each crystallographic

12
site, related by inversion or 𝐶2 symmetry. For magnetic field interactions, all four
subclasses are equivalent if the magnetic field is applied parallel or perpendicular
to the crystal symmetry axis (the 𝑏 axis) [99].

Figure 2.1: Y2 SiO5 primitive unit cell. Pink spheres represent yttrium atoms; cyan
spheres represent silicon atoms; blue spheres represent oxygen atoms. The size of
the spheres is meaningless. The translucent blue polyhedra indicate the structure of
the crystal (see main text). The unit cell is shown in an arbitrary orientation chosen
to highlight the structure.
The crystal field corrections are different for the optical ground and excited states
ˆ and
(𝑍1 and 𝑌1 ), as well as for the two crystallographic sites. The tensors 𝑔,
ˆ 𝐴,
𝑄ˆ must therefore be measured for each crystallographic site in both the ground
and excited optical states. Usually, these tensors are measured in a right-handed
cartesian coordinate system defined by the optical extinction axes of the crystal, 𝐷 1 ,
𝐷 2 , and 𝑏. 𝑏 is also the crystal symmetry axis [16].
The simplified energy structure of 167 Er3+ :Y2 SiO5 is shown in Figure 2.2. The
energy values are for crystallographic site 2, which has an optical transition at
1539 nm, with a magnetic field applied along the 𝐷 1 Y2 SiO5 axis. The magnitude
® ∼ 210 GHz/T is maximized for this site
of the Zeeman interaction | 𝐵® · 𝑔ˆ · 𝑆|
and this direction [99]. All experiments in this thesis were performed using this
crystallographic site and magnetic field orientation to freeze out the 167 Er3+ electron
spin, for reasons described in Section 2.3.
A note about the crystallographic sites in Er3+ :Y2 SiO5 : in crystallographic studies,
the two sites are labelled as "Y1" and "Y2," where "Y2" is the yttrium site surrounded
by 6 oxygen atoms and "Y1" is the yttrium site surrounded by 7 oxygen atoms

I13/2 (Y)

1539 nm

I15/2 (Z)1

210 GHz/T

Figure 2.2: Simplified energy diagram for 167 Er3+ :Y2 SiO5 (crystallographic site 2)
with a magnetic field applied parallel to the 𝐷 1 crystal axis (adapted from Ref. [30]).
From left to right, the levels shown are: (i) the spin-orbit levels with crystal field
corrections 4 I15/2 (𝑍1 ) and 4 I13/2 (𝑌1 ); (ii) the Zeeman split electron spin doublet in an
applied magnetic field; (iii) the eight-fold hyperfine splitting on the lowest electron
spin level.
[28]. Using spectroscopy and crystal field calculations, Doualan et al. identified
the yttrium site surrounded by 6 oxygen atoms as having a 4 I15/2 (𝑍1 ) to 4 I13/2 (𝑌1 )
transition at 6507 cm−1 (1536.8 nm) and called this "site 1," and they identified
the yttrium site surrounded by 7 oxygen atoms as having a 4 I15/2 (𝑍1 ) to 4 I13/2 (𝑌1 )
transition at 6497 cm−1 (1539.1 nm) and called this "site 2" [37]. This assignment
("site 1" has a transition at ∼ 1536.5 nm and "site 2" has a transition at ∼ 1539 nm)
is the norm in recent spectroscopic studies, for example, References [16, 44, 99]. In
short, in Er3+ :Y2 SiO5 literature, site "Y1" is "site 2," and site "Y2" is "site 1."
2.2

Optical Coherence

One of the key advantages of an 167 Er3+ :Y2 SiO5 system is the long coherence time
of the optical transition. For all-optical storage protocols such as those discussed in
this work, storage time is limited by optical coherence time, and therefore this is an
important metric.
In general, rare earth ions in solids have long coherence times due to their unique
electronic structure in which the electrons in the partially occupied 4f shell are
shielded by the electrons in the 5s and 5p shells [66]. The main source of decoherence
is the interactions between phonons in the crystal and the ∼ 200 THz transitions
which arise from crystal field perturbation on optical transitions. Below 4 K, the
phonon distribution peaks below 100 GHz, so phonon processes at 200 THz are

14
suppressed and optical coherence times can be quite long. For this reason, coherent
experiments involving rare earth ions are performed at cryogenic temperatures.
In rare earth ions with unpaired electron spins such as erbium, the interaction
between Zeeman levels and phonons can also lead to decoherence in hyperfine and
optical transitions. If the phonon distribution at 4 K is peaked at ∼ 50 GHz and
the 167 Er3+ :Y2 SiO5 electron Zeeman splitting is 210 GHz/T (see Figure 2.2), then
magnetic fields must be much higher than 250 mT, and/or the crystal temperature
must be much lower than 4 K, in order to freeze out the electron spin and eliminate
spin-lattice relaxation as a source of decoherence. For example, Böttger et al.
measured optical coherences times of up to 4.38 ms in Er3+ :Y2 SiO5 at 7 T and 1.5 K
[19]. Rančić et al. measured highly coherent hyperfine transitions in 167 Er3+ :Y2 SiO5
at 7 T and 1.4 K [86].
Beyond interactions with phonons, there exist two additional mechanisms affecting
coherence times in 167 Er3+ :Y2 SiO5 , both related to magnetic field noise [17]. The
first is magnetic field noise induced by spin flip-flops between 167 Er3+ electron spins.
Since these flip-flops are energy preserving, they persist even at low temperatures,
unless the electron spins are frozen using a combination of high magnetic field and
low temperatures. The spin flip-flop rate is highly dependent on 167 Er3+ doping
concentration [17, 25]. For this reason, we use dilute 167 Er3+ doping of 50 or 200
ppm. The second decoherence mechanism is magnetic field noise due to electronic
and nuclear spin flips in other species. This can include other inadvertent rare
earth ion dopants or the nuclear spins in Y2 SiO5 itself. Yttrium nuclei are 100%
spin-1/2, silicon nuclei are 4.7% spin-1/2, and oxygen nuclei are 0.04% spin-5/2,
where the percentages represent relative isotopic abundance. The nuclear spins have
very low energy transitions, and cannot be frozen at reasonable magnetic fields and
temperatures. Nuclear spin flips are therefore the limit to coherence times in a bulk
167 Er3+ :Y SiO crystal [17, 86].
In this work, we use moderate magnetic fields (< 1 T) and dilution refrigerator
temperatures (< 100 mK) to freeze the 167 Er3+ electron spin. We thus achieve
long optical coherence times, as well as long hyperfine lifetimes (see Section 2.3).
Figures 2.3 and 2.4 show two-pulse photon echo measurements [2] of the optical
coherence time of site 2 in 167 Er3+ :Y2 SiO5 , in bulk and in nanophotonic devices.
Figure 2.3 shows optical coherence measurements in bulk 167 Er3+ :Y2 SiO5 and in the
nanobeam device used in the experiments in Chapter 5. A magnetic field of 380 mT
was applied parallel to the 𝐷 1 optical axis. The 167 Er3+ concentration was nominally

15
50 ppm, although it was measured by secondary ion mass spectrometry to be 37
ppm. The optical coherence time was measured to be 𝑇2 = 4 × 𝜏(1/𝑒) = 759 ± 41 𝜇s
in bulk at 35 mK. The coherence time of the ions coupled to the nanobeam cavity
was 149 ± 4 𝜇s, about 1/4 the coherence time of the bulk ions at the same dilution
refrigerator temperature. This is due in part to the elevated temperature in the
nanobeam compared to bulk, which was measured to be 317 mK (see Section 5.6).
The coherence time of the bulk ions at 1 K was found to have a similar value of
136 ± 9 𝜇s. Ions coupled to the nanobeam are also closer to the surface, where
coherence times can be lower due to surface charge fluctuations [12]. It is also
possible that the fabrication process caused disorder that affected the coherence
time of the ions. Rare-earth ions coupled to similar nanobeam resonators have been
shown to have coherence times similar to the bulk [113, 115], however, the longer
bulk coherence times measured in the current work allow a much more sensitive
probe of the ions’ environment.

Figure 2.3: Optical coherence time measurement in nanobeam resonator and bulk
167 Er3+ :Y SiO . Two-pulse photon echo counts versus inter-pulse time delay with
exponential fits; the vertical axis is scaled for clarity. Measurements in the bulk
167 Er3+ :Y SiO crystal at 35 mK (black circles and solid black line fit), in bulk
at 940 mK (black squares and dashed black line fit), and in the nanobeam at 42
mK (red circles and solid red line fit). Temperatures were measured at the dilution
refrigerator stage to which the sample was thermally connected.
Figure 2.4 shows a similar experiment in bulk 167 Er3+ :Y2 SiO5 and in the hybrid resonator based on amorphous silicon resonators on 167 Er3+ :Y2 SiO5 (𝛼Si167 Er3+ :Y SiO ) used in Chapter 6. The nominal 167 Er3+ concentration was 200
ppm, although it was measured by secondary ion mass spectrometry to be 135
ppm. A magnetic field of 980 mT was applied parallel to the 𝐷 1 optical axis. The
higher magnetic field was used to suppress erbium electron flip-flops at this higher
concentration. In the bulk 167 Er3+ :Y2 SiO5 crystal, the optical coherence time was

(echo amplitude)2

measured to be 𝑇2 = 1.11 ± 0.15 ms. In the hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator
we measured 𝑇2 = 108 ± 13 𝜇s. The similarity between this 𝑇2 value and the one for
the nanobeam resonator in Fig. 2.3 suggests that proximity to the surface might play
an important factor, since the bulk crystal 𝑇2 values and fabrication processes were
different for the two resonators. The hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator was surrounded by electrodes (see Section 4.3). Applying equal and opposite square pulses
to the electrodes between the first and second optical pulses did not significantly
affect the measured optical coherence time, indicating that electric pulses were not
a source of optical decoherence.
10

10-1

200

400

600

800

1000

[ s]

Figure 2.4: Optical coherence time measurement in hybrid 𝛼Si-167 Er3+ :Y2 SiO5
resonator and bulk 167 Er3+ :Y2 SiO5 . Two-pulse photon echo energy versus interpulse time delay with exponential fits. Measurements in the bulk 167 Er3+ :Y2 SiO5
crystal at 71 mK (black circles and solid black line fit) and in the hybrid resonator
at 90 mK (red circles and solid red fit). For the resonator measurement, the data is
scaled photon counts, measured on an SNSPD. The bulk data was measured using
a heterodyne setup (see Appendix D), and the signal was squared to match the
resonator data, then scaled. Temperatures were measured at the dilution refrigerator
stage to which the sample was thermally connected.

2.3

Spectral Holeburning, Hyperfine Levels, and Initialization

In order to implement the atomic frequency comb protocol, the 167 Er3+ :Y2 SiO5 ions
must be initialized into a long-lived frequency comb. Atomic frequency combs are
atomic absorption profiles which contain transparent (non-absorbing) gaps, called
spectral holes. Creating these long-lived spectral features requires long hyperfine
state lifetimes, which are observed in 167 Er3+ :Y2 SiO5 at low temperatures and
moderate magnetic fields.

17
Spectral Holeburning
Spectral holeburning occurs in systems with optical transitions and long-lived states
in the optical ground state manifold. To create a spectral hole, it is simply required to
move population from one long-lived state to another. In the case of 167 Er3+ :Y2 SiO5 ,
the long-lived states are hyperfine states. Figure 2.5 shows a simple schematic of
holeburning in a 4-level system (2 ground states and 2 excited states). We assume
that the two ground-state levels start with equal population. In the case of an
ensemble, population can be interpreted as number of ions in each state. Pumping
on the "h" transition with a laser leads to population moving to the excited state
and eventually falling to the other ground state. This depletes the lowest ground
state of population, leaving behind a spectral hole "h" when the absorption spectrum
(optical depth vs. frequency) of the 4-level system is probed. The spectrum will
also show a side-hole "s" corresponding to the transition from the depleted ground
state to the other excited state. The other ground state level has more population
than normal, so the transitions connecting the upper ground state level to the two
excited states will be more absorbing, leading to two anti-holes "a1 " and "a2 ."

holeburning spectrum
a2 a1
a1 a2 h s

h s

Figure 2.5: Spectral holeburning schematic. Left hand side shows a 4-level energy
diagram. Black circles represent ion population. Arrows show transitions corresponding to a spectral hole "h," a side-hole "s," and two anti-holes "a1 " and "a2 ". On
the right hand side is a representation of an absorption spectrum after holeburning,
with less absorption in the hole and side-hole, and more absorption in the anti-holes.
Figure 2.5 shows the hole, side-hole and anti-holes that form as a result of population
moving between ground state levels. There are other types of side-holes called
transient holes, which can be observed a short time after a laser pulse has been used
to burn a spectral hole, before population has had time to decay from the optical
excited states (𝑡 < 𝑇1 , where 𝑇1 is the optical lifetime). In the experiments presented
in this work, we wait tens of milliseconds (𝑡 > 3 × 𝑇1 ) after spectral holeburning to

18
make sure that all population has decayed from the excited state to one of the ground
states, so transient holes do not play a role in our spectra.
In rare-earth ion doped crystals, the absorption spectrum of optical transitions can
be quite broad, on the order of tens of megahertz to gigahertz. This inhomogeneous
broadening is a result of the slightly different crystal environment that each ion experiences. Variance in the radii of different isotopes, mismatch between the radii of
the dopant and host, crystal defects, and strain on a crystal can all increase the inhomogeneous broadening [18]. In 167 Er3+ :Y2 SiO5 , we measured the inhomogeneous
broadening to be 150 - 350 MHz.
Figure 2.6 shows the theoretical holeburning spectrum in an inhomogeneously
broadened system. A particular laser frequency "h" will not match the same transition for each ion. Rather, for a four-level system, there are four sets of ions (four
energy level combinations), which have one transition with frequency "h." Set 1 is
the same as in Fig. 2.5. Sets 2 - 4 all have a spectral hole at frequency "h," but the
frequencies of their side-hole and anti-holes are all different, as shown in the spectra
under the level diagrams. When the inhomogeneous linewidth is greater than the
energy splittings in the ground and excited states, all four sets of ions are present in
the inhomogeneous line, and the spectrum will look like the top image in Figure 2.6.
In this case, the holeburning spectrum will always be symmetric about the hole.
The schematic shown in Fig. 2.6 is shown with 4 levels for simplicity. In general
if the inhomogeneous broadening is larger than the frequency span of all ground
states and the span of all excited states, each spectral hole creates 𝑛𝑒 − 1 side-holes
and (𝑛𝑔 − 1) × 𝑛𝑒 anti-holes, where 𝑛𝑒 (𝑛𝑔 ) is the number of excited (ground) state
energy levels.

19
holeburning spectrum
all sets

set 1

set 2

set 3

set 4

a1 a2 h s

a2 a1

Figure 2.6: Spectral holeburning in an inhomogeneous distribution. Top figure
shows the holeburning spectrum in an inhomogeneously broadened ensemble of
ions. The central feature is the spectral hole, and there are also 2 side-holes and 6
anti-holes. Below the spectrum are four level diagrams showing the four sets of ions
in the ensemble which have one transition at frequency "h." Arrows show transitions
corresponding to a spectral hole "h," a side-hole "s," and two anti-holes "a1 " and
"a2 ." Each of these sets has a distinct side structure (side-holes and anti-holes) shown
below the level diagrams. These spectra add up to create the observed holeburning
spectrum at the top of the figure.
Hyperfine State Lifetime
In order for spectral holes to be deep and long lived, the lifetime of the ground state
levels must be much larger than the optical lifetime [46, 62]. In 167 Er3+ :Y2 SiO5 , this
is possible when the ground state levels are hyperfine levels. The optical transition
lifetime is 9 ms [16]. Rančić et al. measured spectral holes in 167 Er3+ :Y2 SiO5
with lifetimes larger than 1 minute at 3 T and 1.4 K [86]. They attributed these
long lived holes to long hyperfine state lifetimes, and found that the hyperfine state
lifetimes were limited by electron spin flips. If the electron spin is frozen using
a combination of high magnetic field and low temperatures, the hyperfine lifetime
is sufficient to allow long-lived spectral features. In this work, we use dilution
refrigerator temperatures (< 100 mK) and moderate magnetic fields (< 1 T) to

20
freeze the electron spin.
Figure 2.7 shows a scan of the inhomogeneously broadened 167 Er3+ :Y2 SiO5 absorption line (black curve) in a bulk crystal with 50 ppm 167 Er3+ doping concentration
(site 2). A magnetic field of 380 mT was applied parallel to the 𝐷 1 crystal axis.
The sample was thermally connected to the dilution refrigerator plate at ∼ 25 mK.
Transitions between individual hyperfine levels (in the lower electron spin branch)
in the optical ground and excited state manifolds are partially resolved. There is
likely some splitting of inequivalent magnetic subclasses due to a misalignment of
the magnetic field and the 𝐷 1 crystal axis (see Section 2.1), contributing to the complex shape of the spectrum. A wide spectral transparency (trench) was created in
this absorption line by optical pumping: a laser was scanned slowly and repeatedly
over a 140 MHz band, which depleted those hyperfine levels with resonant optical
transitions. The population was then allowed to return to its equilibrium distribution while measuring transmission using a room temperature InGaAs photodiode.
Scanning the line to probe the hole depth likely led to an underestimation of T1 by
redistributing population, but this effect was minimized by using low scan power.
The width of the trench was chosen to minimize effects of spectral diffusion on the
T1 measurement. The depth of the spectral feature, or hole depth 𝑑, is a measure
of how much population is missing from the hyperfine levels that were optically
pumped:

𝑑 (𝑡) =

optical depth(𝑡 → ∞) − optical depth(𝑡)
optical depth(𝑡 → ∞)

(2.5)

The maximum 𝑑 (𝑡 = 0) achieved here was 0.9. This was in part due to a noisy
background from detector noise and Fabry-Perot resonances in the setup, which
made it difficult to accurately measure the depth of the hole. It is also possible that
hole depth is limited by the 167 Er3+ isotopic purity.
Figure 2.7b shows the decay in time of the hole depth 𝑑 (𝑡) as measured in the center
of the trench. The fast decay is fit to an exponential with a lifetime of 29 min,
while the slow decay is fit to a lifetime of 6 hours. Following Reference [86], we
interpret these lifetimes as a two-step change in the spectrum: first, the trench fills in
due to spin-spin interactions between 167 Er3+ electron spins, then, at a slower rate,
the population is redistributed among hyperfine levels due to electron spin-lattice
relaxation.
For spectral holeburning memories, it is also important that spectral diffusion be

21
a)

0.7
0 min
4 min
31 min
183 min
1065 min

Optical Depth

0.6
0.5
0.4
0.3
0.2
0.1
-2

-1

Fraction of OD

Detuning (GHz)
0.8
0.6
0.4
0.2

100

150

200

250

Time (min)

Figure 2.7: 167 Er3+ :Y2 SiO5 spectral hole lifetime.
b) Inhomogeneous
167 Er3+ :Y SiO line showing the hyperfine structure and a spectral trench at 0
GHz detuning, shown as a function of time after trench creation. The detuning is
measured from 194814 GHz ± 1 GHz. c) Depth of trench measured as a fraction of
optical depth vs. time (black circles) and fits to two exponential curves (red lines).
small, since it leads to washing out of spectral features over time. Böttger et al.
characterized the spectral diffusion for Er3+ :Y2 SiO5 at various magnetic fields and
temperatures in Reference [17]. We found that below 1 K and at a magnetic field
of 380 mT applied parallel to the 𝐷 1 axis, the spectal diffusion was small but
not insignificant. With a three pulse photon echo measurement in the nanobeam
resonator device (Section 4.1) we estimated the increase to optical linewidth due to
spectral diffusion to be ∼ 130 kHz/s.
Hyperfine Initialization
Due to the hyperfine splitting in 167 Er3+ :Y2 SiO5 , the linewidth of the absorption
feature at 1539 nm is broadened from the inhomogeneous linewidth of a single
optical transition (∼ 150 MHz) to ∼ 1.5 GHz, since the optical transitions from the
various hyperfine levels in the ground state to the various hyperfine levels in the
excited state occur at different frequencies (see Figure 2.2). Because of this, the
optical depth in the center of the line is decreased relative to an otherwise identical
Er3+ :Y2 SiO5 sample using an erbium isotope with zero nuclear spin. Moving
population into fewer hyperfine states can increase this optical depth.

22
At high magnetic fields, where the electron spin Zeeman term is very large compared
to the hyperfine term, the hyperfine levels can be approximately labelled with
|𝑚 𝑠 , 𝑚 𝐼 i quantum numbers (see Figure 2.9b). The quantization axis for the nuclear
spin in this case is the electron spin dipole moment. Since the 𝑔ˆ tensors of the
optical ground and excited states are different, the electron spin dipole moment
in the optical excited state is different from that in the ground state. Because the
quantization axes for the nuclear spin are different, not only are optical transitions
of the type |𝑚 𝑠 = − 12 , 𝑚 𝐼 = − 72 iground to |𝑚 𝑠 = − 12 , 𝑚 𝐼 = − 27 iexcited allowed, but
|𝑚 𝑠 = − 12 , 𝑚 𝐼 = − 27 iground to |𝑚 𝑠 = − 21 , 𝑚 𝐼 = − 52 iexcited transitions are also allowed,
although they are weaker. In what follows, we refer to transitions of the former type
as Δ𝑚 𝐼 = 0 transitions and the latter type as Δ𝑚 𝐼 = +1. Since we always use the
lower electron spin levels (𝑚 𝑠 = − 12 ) in both the optical ground and excited state,
we will drop the 𝑚 𝑠 label.
At high magnetic fields, the entire optical ground state population can be initialized
into one hyperfine state by pumping on all Δ𝑚 𝐼 = +1 or all Δ𝑚 𝐼 = −1 transitions, as
done by Rančić et al. [86]. The idea is that by pumping on all Δ𝑚 𝐼 = +1 transitions,
all ions end up in the 𝑚 𝐼 = | + 72 i ground state level, since there is no Δ𝑚 𝐼 = +1
transition for that ground state level. Similarly, pumping on the Δ𝑚 𝐼 = −1 transition
should initialize the ions into the 𝑚 𝐼 = | − 72 i ground state level. At 380 mT applied
parallel to the 𝐷 1 axis, only a partial initialization can be performed because the
Δ𝑚 𝐼 = ±1 optical transitions are not fully resolved from the Δ𝑚 𝐼 = 0 optical
transitions.
Figure 2.8 shows the 167 Er3+ :Y2 SiO5 1539 nm transition in a bulk crystal, both
with and without initialization, always with a spectral comb created in the center.
Initialization was performed by turning on strong sidebands of an EOM modulator
and sweeping their detuning over a range of ±(0.37 GHz – 0.82 GHz). This was the
full range of the voltage controlled oscillator used to sweep the sidebands. Both 1st
order sidebands were used in the initialization procedure, and the amplitude of the 0th
order sideband was minimized by using strong RF drive power. This procedure was
used at the beginning of every atomic frequency comb storage experiment in Chapter
5. A pulse sequence schematic is shown in Fig. 5.2. We think this procedure moved
population from the higher and lower energy hyperfine states to the middle few
hyperfine states. However, we did not identify the frequencies of the various optical
transitions, so this is not a conclusive assignment. Additionally, there may be some
splitting of inequivalent magnetic subclasses due to a misalignment of the magnetic

a) 1.5

b) 1.5

Optical Depth

field and the 𝐷 1 crystal axis. This would lead to a more complicated spectrum.
The optical depth was improved by a factor of ∼ 3 compared to the uninitialized
case. Sweeping one sideband to pump on only one side of the inhomogeneous line
produced a similar enhancement in optical depth.

0.5

-1

-0.5

Detuning [GHz]

0.5

-1

-0.5

0.5

Detuning [GHz]

Figure 2.8: 167 Er3+ :Y2 SiO5 hyperfine initialization and comb burning in 50 ppm
167 Er3+ :Y SiO sample. Absorption spectrum of the inhomogeneous line in bulk
167 Er3+ :Y SiO at 18 mK and 380 mT (parallel to 𝐷 ) with 40 MHz wide comb
created in the center. a) Without initialization, the peak optical depth of comb is
0.4. b) With initialization into a few hyperfine states before comb creation, the peak
optical depth of comb is 0.9. Gray area shows the extent of the sweep of the EOM
sidebands during the initialization step. The detuning was measured from 194814
GHz ± 1 GHz.
Figure 2.9 shows a similar hyperfine initialization procedure in the 200 ppm
167 Er3+ :Y SiO sample used in experiments in Chapter 6. The sample was at
70 mK (measured at the dilution refrigerator plate) with an applied field of 980 mT
parallel to 𝐷 1 . In this case, the magnetic field along the 𝑏 axis was cancelled using
trim coils, so the spectra of the two magnetically inequivalent subclasses are overlapping. Figure 2.9b shows the predicted splittings between the lowest 8 hyperfine
levels at this field (see Figure 2.2 for a schematic of the other energy levels in the
system).
Before any initialization, the red curve in Fig. 2.9a shows 3 broad peaks, which
we think belong to sets of transitions with Δ𝑚 𝐼 = −1, 0, and 1. These sets of
transitions are not fully resolved, even at a field close to 1 T. Rančić et al. found that
these transitions were fully resolved at 7 T, and used this to perform highly efficient
initialization into only one hyperfine state [86]. For the initialization shown in Fig.
2.9a, the laser frequency was swept back and forth using its piezo drive in the region
shown by the green area for 100 ms. By pumping in the range of the Δ𝑚 𝐼 = −1
transitions, most ions were pumped into the lowest energy hyperfine state 𝑚 = | − 72 i,
which has a Δ𝑚 𝐼 = 0 transition on the left side of the central peak at high fields

24
[86]. However, since the Δ𝑚 𝐼 = −1 and 0 transitions are not fully resolved, this
initialization was only partly effective.
For the experiments in Chapter 6, the initialization procedure was closer to that
shown in Figure 2.8. A fast EOM (10 GHz) was driven around 6 GHz using a VCO.
∼ 650 MHz wide trenches were burned on either side of the atomic frequency comb
(∼ 70 MHz away from the comb edge), which was toward the right hand side of
the absorption profile. We think that the higher frequency trench was pumping on
the Δ𝑚 𝐼 = +1 transition, which put most of the population to the highest energy
hyperfine level 𝑚 = | + 72 i, with its Δ𝑚 𝐼 = 0 on the right hand side of the central
peak. The lower frequency trench was added after experimentally determining that
it improved the AFC storage efficiency. We think this trench was burning population
out of lower hyperfine levels through Δ𝑚 𝐼 = 0 transitions. This would lead to more
efficient initialization into the higher energy hyperfine level because the Δ𝑚 𝐼 = 0
transitions are stronger than Δ𝑚 𝐼 = +1 transitions.

3.5

782 MHz

790 MHz

2.5

797 MHz

803 MHz

1.5

813 MHz
∆m = -1

∆m = +1

835 MHz
879 MHz

0.5
-3

-2

-1

detuning (GHz)

Figure 2.9: 167 Er3+ :Y2 SiO5 hyperfine initialization in 200 ppm sample. a) Red
curve shows absorption spectrum (optical depth) of the inhomogeneous line in bulk
167 Er3+ :Y SiO . Green curve shows the same spectrum, but initialized into a few
hyperfine states. Green shaded area shows frequency range of laser burning for
initialization. The detuning was measured from 194823 GHz ± 1 GHz. b) Lowest
8 hyperfine levels (𝑚 𝑠 = − 21 ) showing splittings for site 2 with a field of 980 mT
parallel to 𝐷 1 , predicted with 𝑔, 𝐴, and 𝑄 tensors from [27].

25
2.4

Superhyperfine Broadening

In Y2 SiO5 , yttrium atoms are a source of nuclear spin interactions and magnetic
noise in this material. There is only one stable isotope of yttrium, 89 Y, which has
nuclear spin number 𝑠 = 1/2. The unpaired Er3+ electron has a large magnetic
dipole moment 𝜇 𝑒 ∼ 𝑔 𝜇ℏ𝐵 characterized by a large Landé factor 𝑔 ∼ 15. This creates
a "frozen core" effect in Er3+ :Y2 SiO5 , where the nuclear transition frequencies
of nearby yttrium atoms are modified by the Er3+ electron spin [45]. This is
called a superhyperfine interaction (cf. the hyperfine interaction, which describes
the interaction of an atom’s electron spin with its own nuclear spin). Due to the low
symmetry of the Y2 SiO5 crystal, the magnetic field generated by the electron spin
dipole moment is highly anisotropic, characterized by a 2nd rank tensor 𝑔ˆ [99]. This
leads to a number of inequivalent electron-nuclear interactions between an erbium
ion and its yttrium neighbors [24]. In this section, we look at the effect of these
interactions on the holeburning spectra in 167 Er3+ :Y2 SiO5 .
Figure 2.10a shows a part of the Y2 SiO5 unit cell with an erbium ion substituting
for a yttrium atom. The four closest yttrium atoms are 3.4 Å to 3.7 Å away from
the Er3+ atom. The superhyperfine Hamiltonian is a sum of nuclear Zeeman and
dipole-dipole interactions [24]:

𝐻SHF =

𝑁𝑌

− 𝜇®𝑌(𝑖) · 𝐵® −

𝜇0 (𝑖)
(𝑖)
𝜇® · 𝜇®𝑒 + 3(𝑟ˆ · 𝜇®𝑌 )(𝑟ˆ · 𝜇®𝑒 ) ,
4𝜋𝑟 3 𝑌

(2.6)

where 𝑁𝑌 is the number of yttrium atoms being considered, 𝜇0 is the vacuum permeability, 𝜇®𝑌 = 𝜇𝑛 𝑔𝑛,𝑌 𝐼® is the yttrium nuclear spin dipole moment, and 𝜇®𝑒 = 𝜇 𝐵 𝑔𝑒,𝐸𝑟
ˆ · 𝑆®
is the erbium electron spin dipole moment. 𝜇𝑛 = 7.62 MHz/T is the nuclear magneton, 𝜇 𝐵 = 14.0 GHz/T is the Bohr magneton, 𝑔𝑛,𝑌 = −0.2748 is the yttrium nuclear
spin g-factor [98], and 𝑔𝑒,𝐸𝑟 is the 2nd rank magnetic g-tensor for the Er3+ :YSO
electron spin, measured for both ground (Z1 ) and excited (Y1 ) optical states in Reference [99]. For the spin-1/2 yttrium nuclear spin and the effective spin-1/2 erbium
electron spin, 𝐼® = 𝑆® = ℏ2 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑧 , where 𝜎𝑖 are the Pauli matrices.
Each neighboring yttrium atom splits the energy levels of the Er3+ electron spin in
two. For large magnetic fields, this splitting is dominated by the nuclear Zeeman
term 𝜇𝑌 𝐵 = 2.1 MHz/T. For small magnetic fields, this is dominated by the dipoledipole interaction, which is on the order of ∼ 0.3 MHz and is different for each
yttrium atom due to the low symmetry of the crystal. If we consider the closest

26
a)

~ 3.5 Å

c)
0.5
-0.5

0.98 T

-1
-4 -2 0 2 4

d)
0.5
-0.5

0.38 T

-1
-4 -2 0 2 4

Figure 2.10: Superhyperfine side structure in Er3+ :Y2 SiO5 holeburning. a) Part of
the Y2 SiO5 unit cell with an erbium ion substituting for a yttrium atom. Blue spheres
represent oxygen atoms. b)-d) anti-holes and side-holes created when a spectral hole
is burned at 0 MHz detuning for various magnetic fields applied parallel to the 𝐷 1
crystal axis for a site 2 Er3+ :Y2 SiO5 ion. 𝑦-axis shows optical depth in arbitrary
units. Insets show a closeup of structure around 0 MHz. b) 𝐵 = 7 T c) 𝐵 = 0.98 T,
the magnetic field applied in Chapter 6 experiments. d) c) 𝐵 = 0.38 T, the magnetic
field applied in Chapter 5 experiments.
four yttrium atoms, each electron spin level is split into 24 = 16 levels. If we look
at electron-spin-preserving optical transitions, this splitting leads to 16 × 16 = 256

27
transitions.
When a spectral hole is burned in 167 Er3+ :Y2 SiO5 , these numerous transitions lead
to a large number of side-holes and anti-holes. As discussed in Section 2.3, if the
inhomogeneous broadening is larger than the frequency span of all ground states
and of all excited states, each spectral hole creates 𝑛𝑒 −1 side-holes, and (𝑛𝑔 −1) ×𝑛𝑒
anti-holes, where 𝑛𝑒 (𝑛𝑔 ) is the number of excited (ground) states.
Figures 2.10b-d show the superhyperfine side hole structure that is expected when a
10 kHz wide hole is burned at zero detuning, considering only the 4 nearest yttrium
spins for a site 2 Er3+ :Y2 SiO5 ion. For this calculation, the superhyperfine energy
levels in the optical ground and excited states were computed by diagonalizing
𝐻 = 𝐻𝑒Z + 𝐻SHF from Equations 2.2 and 2.6. Only transitions connecting the
𝑚 𝑠 = − 12 electron spin states in the ground and excited optical manifolds were
considered. The relative amplitudes of the hole, each side-hole and each antihole were estimated using |h𝜓 𝑒 |𝜓𝑔 i| 2 , where 𝜓𝑔 and 𝜓 𝑒 are the ground and excited
states from which that spectral feature resulted. The assumption is made that the
population burned away from the hole is evenly distributed among the remaining
15 ground state superhyperfine levels. For this calculation, the 167 Er3+ nuclear spin
was not considered. The electron spin dipole h 𝜇®𝑒 i is modified through the hyperfine
interaction with the 167 Er3+ nuclear spin. However, for fields as low as 380 mT
(parallel to the 𝐷 1 crystal axis), the hyperfine interaction ∼ 1 GHz is much smaller
than the electron Zeeman interaction ∼ 210 GHz/T, so including the hyperfine
interation is not expected to have a qualitative effect on the results in Fig. 2.10.
As expected, at high magnetic fields, the side structure is dominated by the nuclear
Zeeman term 𝜇𝑌 𝐵 = 2.1 MHz/T. However, at low to medium magnetic fields, the
Zeeman and dipole-dipole terms are comparable, leading to a forest of side-holes
and anti-holes that lead to an effective hole broadening. At 380 mT, for example,
the entire spectral feature has an effective linewidth of ∼ 2 MHz.

28
Chapter 3

ATOMIC FREQUENCY COMB STORAGE - THEORY
This chapter describes the atomic frequency comb protocol used to store light in
this work [4]. After introducing the protocol, we present the theoretical efficiency
of storage using the AFC protocol and an ensemble of ions in a cavity [3]. When the
ensemble cooperativity is 1 and the cavity is one-sided, known as the impedancematched condition, this protocol can have an efficiency of 100%.
3.1

Storing Light in an Atomic Frequency Comb

When a photon is absorbed by an ensemble of ions, the excitation is distributed
among the ions, creating an entangled Dicke state |Ψi, as shown in Equation 3.1
[4]:

|Ψi =

𝑐 𝑗 𝑒𝑖𝜔 𝑗 𝑡 𝑒 −𝑖𝑘 𝑟®𝑗 0...1 𝑗 ...0𝑁 .

(3.1)

𝑗=1

Each ket in the sum represents one ion absorbing the optical pulse and being
promoted to the excited state |1i, and the rest of the ions remaining in the ground
state |0i. Each ion has a position in the crystal 𝑟®𝑗 and an optical transition frequency
𝜔 𝑗 . Due to inhomogeneous broadening, the state |Ψi rapidly dephases as each ion
accumulates a different phase 𝜙 𝑗 = 𝜔 𝑗 𝑡.
Figure 3.1 presents this dephasing by representing each term in the sum in equation
3.1 as a vector on the equator of the Bloch sphere. Shortly after absorption, at
time 𝑡 ∼ 𝜖, the different terms of the sum dephase. By 𝑡 ∼ Δ inhom
, where Δ inhom is
the linewidth of the inhomogeneously broadened distribution, the ions are mostly
dephased. The coherence can be restored by reversing the phase evolution, for
example by sending in an optical 𝜋 pulse as in two pulse photon echo [2]. However,
this will also create population inversion in the case of a weak input pulse, which
leads to amplification and therefore cannot be used for quantum storage [88]. If no
additional rephasing pulses are sent in, the energy will leave the system as incoherent
photoluminescence.
The atomic frequency comb storage protocol works by limiting which frequencies
𝜔 𝑗 ions are allowed to posses. Specifically, in an ideal comb, the ions can only have

29
b)

Figure 3.1: Schematic of dephasing due to inhomogeneous broadening on a Bloch
sphere. a) Dephasing a short time after absorption. b) Dephasing at 𝑡 ∼ Δ inhom
Details in main text.
frequencies that are multiples of the comb period Δ:
𝜔 𝑗 = 2𝜋Δ 𝑛 𝑗 , 𝑛 𝑗 ∈ Z.

(3.2)

In that case, the Dicke state |Ψi characterizing the ion ensemble after a photon is
absorbed is given by Equation 3.3:

|Ψi =

𝑐 𝑗 𝑒𝑖2𝜋Δ 𝑛 𝑗 𝑡 𝑒 −𝑖𝑘 𝑟®𝑗 0...1 𝑗 ...0𝑁

(3.3)

𝑗=1

Looking at the phase evolution term, when Δ × 𝑡 = 𝑚 where 𝑚 is an integer, the
term is equal to 𝑒𝑖2𝜋𝑚 𝑛 𝑗 = 1 for all ions, indicating a rephasing event. This means
that a chain of rephasing events will occur at 𝑡 = 𝑚
Δ , 𝑚 = 1, 2, 3... where 𝑡 = 0 is the
time when the photon was absorbed.
Figure 3.2 presents the dephasing and first rephasing of a photon absorbed by an
atomic frequency comb (AFC). Each term in the sum in Equation 3.3 is represented
as an arrow on the Bloch sphere. The inset shows a sketch of an AFC with 5 teeth.
Each tooth is not an ideal delta function, but rather a distribution of frequencies,
as in a realistic AFC. Immediately after the photon is absorbed, the ions start to
dephase, but the phase evolution is described by just 5 groups, one for each comb
tooth. At 𝑡 = 4Δ
, the phases of the 5 groups are maximally scrambled, but the phase
evolution continues. By 𝑡 = Δ1 − 𝜖, all groups are close to arriving at the same phase
again: the phase of one group of ions (green) has stayed put, two groups (cyan and
orange) have evolved by 𝜙 = ±2𝜋, and two groups (red and purple) have evolved

30
by 𝜙 = ±2 × 2𝜋, and all groups are rephasing. When all arrows point in the same
direction again at 𝑡 = Δ1 , the ions will spontaneously emit a photon that is identical
in frequency, polarization, and spatial distribution to the one that was absorbed at
𝑡 = 0. This is the output pulse of the AFC memory. Note that the groups of arrows
are now broader than when they started. This results from the finite width 𝛾 of the
comb, and leads to a limit on the efficiency of storage through the term 𝜂 𝑑 [4] (see
Section 3.2).
a)

Figure 3.2: Sketch of atomic frequency comb evolution on a Bloch sphere. a)
Dephasing a short time after absorption (𝑡 = 𝜖). Inset shows a schematic of an
atomic frequency comb (optical depth versus frequency). (b) Dephasing at 𝑡 = 4Δ
c) Shortly before rephasing, at 𝑡 = Δ − 𝜖. The diagram is in the rotating frame of
the ions in the center of the comb (green). Details in main text.
Although the AFC protocol can in principle store any kind of photonic qubit, qubits
encoded in the polarization degree of freedom of light are not compatible with
our implementation. This is because the photonic crystal cavities only support
one polarization of light. In this work, we store time-bin qubits for the form
|𝜓in i = 𝛼|earlyi + 𝑒𝑖𝜙 1 − 𝛼2 |latei. Since AFC is a first-in-first-out storage protocol
[101], the |earlyi and |latei pulses will be emitted in the same order, leading
to an output state identical to the input state in the case of an ideal protocol:
|𝜓out i = 𝛼|earlyi + 𝑒𝑖𝜙 1 − 𝛼2 |latei.
The atomic frequency comb is created using spectral holeburning. To create an
atomic frequency comb in 167 Er3+ :Y2 SiO5 in our experiments, a set of periodic
spectral holes is burned in the absorption line after hyperfine initialization (see
Section 2.3). Each hole is created by pumping on the transitions at that optical
frequency with a laser, until population is moved to other long-lived hyperfine
levels. The teeth of the comb are the remaining absorption peaks between the

31
spectral holes. See Figure 5.2 for pulse sequence and comb scan. Side-holes
and anti-holes from other hyperfine transitions are usually outside of the frequency
window of the comb. Superhyperfine side-holes and anti-holes, however, affect the
comb structure (see Section 2.4). Combs of arbitrary finesse 𝐹 = Δ𝛾 are created by
creating spectral trenches between the comb teeth. These trenches are created by
scanning the laser in frequency as it pumped away population. See Figure 6.2 for
pulse sequence, and Figure 6.3 for a scan of a high finesse comb.
In some cases, the AFC was created using the accumulated AFC method [34], where
the entire frequency comb is created simultaneously by creating an optical frequency
comb and imprinting that onto the ion absorption line. See Figure 5.3 for a pulse
sequence used to generate a comb in this way. The finesse obtained with this method
is always 𝐹 ∼ 2.
AFC with spin-wave
Up to this point, we have described storing light on an optical transition using an
atomic frequency comb. All quantum storage experiments in this thesis were done
according to this procedure, which is usually referred to as "optical AFC". However,
this is only part the AFC protocol as it was originally proposed in Reference [4].
The full protocol, called spin-wave AFC, requires two additional optical pulses and
an empty hyperfine level to which population from the excited state can be coherently
and reversibly transferred. Additionally, a microwave frequency oscillating magnetic
field (∼ 850 MHz) would be needed to rephase the inhomogeneously broadened
spins during storage.
In the AFC with spin-wave protocol, after an input pulse (or input qubit) is stored
on the optical transition, an optical 𝜋 pulse coherently moves population from the
optical excited state to a second long-lived spin state in the optical ground state
manifold. The coherence is then stored between the two ground state levels for
a time that is limited by the spin coherence time 𝑇spin . This time can be much
longer than the optical coherence time. For example, in 167 Er3+ :Y2 SiO5 , hyperfine
coherence times longer than 1 second have been measured [86]. After some desired
wait time 𝑇s < 𝑇spin , another optical 𝜋 pulse moves the population from the second
spin state back to the excited state, and the phase evolution continues until 𝑡 = Δ1 +𝑇s ,
when the ions rephase and the output pulse is emitted. During the storage on the spin
state, microwave pulses are usually used to rephase the inhomogeneously broadened
spin ensemble.

32
3.2

AFC Storage Efficiency in a Cavity

In this work, we use on-chip resonators to enhance the coupling between light and
ensembles of 167 Er3+ :Y2 SiO5 ions. This allows the miniaturization of devices and
can in theory lead to highly efficient storage of light using the atomic frequency
protocol. Afzelius et al. showed that an atomic frequency comb memory can
approach unit efficiency when the ensemble of ions is coupled to a cavity, and that
coupling is characterized by an ensemble cooperativity of 𝐶 = 1 [3]. This is called
an impedance matched cavity quantum memory. A similar result was derived more
generally by Moiseev et al. [80].
The efficiency of AFC storage in a cavity is given by [3, 80, 115]:

4𝜅 in Γcomb
𝜂 𝐴𝐹𝐶 =
(𝜅total + Γcomb + Γbg ) 2

2
𝜂𝑑 ,

(3.4)

where 𝜅in is the rate of cavity coupling through the input/output port, 𝜅total is the total
energy decay rate of the cavity, and 𝜂 𝑑 accounts for dephasing due to the finite width
of comb teeth [4]. Assuming spectral teeth with a Gaussian profile, this dephasing
term equals:
𝜋2
𝜂 𝑑 = exp −
2ln2(Δ/𝛾) 2

(3.5)

Γcomb and Γbg are the absorption rates of the cavity field by the ensemble of ions in
the comb and background, respectively. Background ions are those ions remaining
after optical pumping, with transition frequencies where transparency is desired (i.e.
between the teeth of the comb). Nonzero Γbg results from limitations in spectral
holeburning. Using 𝜂spectral , the fractional optical depth of a spectral hole, Δ, the
comb period, and 𝛾, the width of one comb tooth, these can be estimated as follows:
Γcomb ≈ 𝜂spectral Γions ,
(3.6)
Γbg = 1 − 𝜂spectral Γions ,
(3.7)
2 /(Δ
where Γions = |𝑊 (𝜔 = 𝜔ions )| = 𝜋log2𝑔total
ions /2) is the absorption rate of the
cavity field by the ensemble of ions before comb preparation (see Section 5.1 for
definition of 𝑊 (𝜔)).
We can define an effective AFC cooperativity, 𝐶 0:

33
𝐶0 =

Γcomb + Γbg 𝜂spectral
+ (1 − 𝜂spectral ) 𝐶,
𝜅total /2

(3.8)

where 𝐶 = Γions /(𝜅/2) is the ensemble cooperativity and 𝐹 = Δ𝛾 is the comb finesse.
Rewriting Equation 3.4 gives rise to the following expression for the AFC storage
efficiency:

4𝐶 0 ª®
𝜅in
𝜂 𝐴𝐹𝐶 =
𝜂𝑑 .
0 2®
𝐹 𝜂spectral
− 1 + 1 𝜅total (1 + 𝐶 )

(3.9)

The efficiency is maximized when 𝐶 0 → 1. This is the impedance matched condition. In the case where the atomic frequency comb is burned efficiently (no backin
→ 1, and 𝜂 𝑑 → 1
ground ions, 𝜂spectral → 1) and the cavity loss is minimal, 𝜅𝜅total
(which requires narrow teeth comb teeth), the memory efficiency approaches unity.
Ensemble Cooperativity
The ensemble cooperativity can be estimated from material parameters and simulation values. Consider the cooperativity before any spectral initialization for an
ensemble of ions with a Gaussian distribution of transition frequencies:
Γions 4 𝜋log2𝑔total
𝐶=
𝜅/2
𝜅Δ ions

(3.10)

Note that the factor of 𝜋log2 comes from considering a Gaussian distribution
[36, 79]. To estimate 𝑔total , we first approximate the sum over ion couplings with an
integral:
𝑔total =
𝑔𝑖 ≈
d®
𝑟 𝜌ions 𝑔 2 (®
𝑟)
(3.11)
𝑉YSO

where 𝜌ions is the number density of ions. Using

𝑔𝑖 =

𝜇® · 𝐸® (®
𝑟) ,
2ℏ𝑉mode max 𝜖 (®
𝑟 ) 𝐸 (®
𝑟)

(3.12)

where 𝜇 is the ion optical dipole moment, and

d®
𝑟 𝜖 (®
𝑟 )| 𝐸® (®
𝑟 )| 2
,
𝑉mode =
max 𝜖 (®
𝑟 ) 𝐸® (®
𝑟)

(3.13)

34
𝑔total
can be estimated by:

𝑔total

𝜌ions 𝜔|𝜇| 2
𝛽,
2ℏ𝜖YSO

(3.14)

where
𝛽=

𝑉YSO

d®
𝑟 𝜖YSO | 𝐸® (®
𝑟 ) · 𝜇|
ˆ2

d®
𝑟 𝜖 (®
𝑟 )| 𝐸® (®
𝑟 )| 2

(3.15)

Note that this 𝛽 is very similar to the fraction of the resonator mode energy in
Y2 SiO5 , 𝛽mode , defined in Eq. 4.1. If the electric field component optical mode of
the resonator is mostly aligned with the electric dipole moment of the ions, then
| 𝐸® (®
𝑟 ) · 𝜇|
ˆ ≈ | 𝐸® (®
𝑟 )| and 𝛽 ≈ 𝛽mode . In this case, the ensemble cooperativity is
approximated by:
2 𝜋log2𝜌ions |𝜇| 2 𝛽mode 𝑄
𝐶≈
ℏ𝜖 YSO Δ ions

(3.16)

where 𝑄 = 𝜔/𝜅 is the resonator quality factor. This relationship shows that the
ensemble cooperativity scales linearly with cavity quality factor, and the fraction of
the optical mode in the Y2 SiO5 . It is inversely proportional to the linewidth of the
ensemble absorption spectrum. Note that the ensemble cooperativity does not scale
with the number of ions coupled to the cavity, but rather scales with the density of
ions
ions, 𝜌ions ∼ 𝑉𝑁mode

35
Chapter 4

NANOPHOTONIC RESONATORS: DESIGN AND
FABRICATION
In this chapter, we introduce the optical resonators coupled to 167 Er3+ :Y2 SiO5 ions
which are used in Chapters 5 and 6 to store light with the AFC protocol. We
first introduce the 167 Er3+ :Y2 SiO5 nanobeam resonators which will be used for
quantum storage in Chapter 5. Not much is said about these resonators as they have
been previously described elsewhere. Next, we introduce in more detail the hybrid
amorphous silicon 167 Er3+ :Y2 SiO5 resonators, which will be used in Chapter 6. We
describe their design and fabrication. We end with the design and fabrication of
the electrodes used for Stark shift control in Chapter 6. We show simulations of
the electric field for two configurations, parallel and quadrupole, which enable the
multiple functions of this device.
4.1

Y2 SiO5 Nanobeam Resonators

Figure 4.1 shows the nanoresonator used in the quantum storage experiment described in Chapter 5. Jake Rochman designed and fabricated the resonator, as
described in Reference [114]. The triangular nanobeam photonic crystal cavity
was milled in a Y2 SiO5 crystal from Scientific Materials doped with isotopically purified 167 Er3+ (92% purity) at a nominal concentration of 50 ppm. The
nanobeam was 1.5 𝜇m wide and ∼ 20 𝜇m long. The slots in the nanobeam created a photonic crystal bandgap and the periodic pattern (lattice constant = 590 nm,
groove width = 450 nm) was modified quadratically in the center to create a cavity
mode.

Ez
-1

b)
input &
output

z y

Figure 4.1: Nanobeam resonator simulation and micrograph (adapted from Ref.
[30]). (a) Finite element analysis simulation of the TM cavity mode in the triangular
nanobeam resonator. Red-blue color gradient indicates the electric field component
normal to the surface, 𝐸 𝑧 ; black outline indicates Y2 SiO5 -air interface; yellow arrow
indicates coupling. (b) Scanning electron micrograph of the resonator, showing
input/output coupling through a 45◦ angled slot coupler.
4.2

Hybrid Amorphous Silicon-Y2 SiO5 Resonators

The hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator used for the Stark shift AFC-storage
experiment is described in Chapter 6. The Fabry-Perot type resonator was comprised
of a waveguide with photonic crystal mirrors on either end. The resonators were
designed by Jake Rochman and fabricated by myself and Mi Lei.
Design
Figure 4.2a shows an amorphous silicon (𝛼Si) waveguide on a Y2 SiO5 substrate
(𝑤 = 605 nm, ℎ = 310 nm), and Figure 4.2b shows the fundamental TM mode in
this waveguide, simulated using COMSOL. A fraction of the optical mode penetrates
into the 167 Er3+ :Y2 SiO5 and evanescently couples to 167 Er3+ ions.
The effective mode index for the fundamental TM mode is 𝑛eff = 2.5. The fraction
of the optical mode energy in 167 Er3+ :Y2 SiO5 was found from simulation to be
𝛽mode = 0.1 using:
𝑉YSO

𝛽mode = ∫

d®
𝑟 𝜖YSO | 𝐸® (®
𝑟 )| 2

d®
𝑟 𝜖 (®
𝑟 )| 𝐸® (®
𝑟 )| 2

(4.1)

b)
600

600

300

αSi

y (nm)

Y2SiO5

-300

-600

-600
-600

-300

x (nm)

300

600

-600

-300

300

600

x (nm)

Figure 4.2: Hybrid 𝛼Si-167 Er3+ :Y2 SiO5 waveguide simulation. a) Materials: amorphous silicon (𝑛 = 3.5) in purple, Y2 SiO5 (𝑛 = 1.8) in light blue, surroundings
in white are air (𝑛 = 1). b) Fundamental TM mode in waveguide. Purple-white
gradient shows electric field in the 𝑦-direction, 𝐸 𝑦 .
where 𝐸® (®
𝑟 ) is the electric field, 𝜖 (®
𝑟 ) is the material permittivity (𝜖 YSO is the permittivity of Y2 SiO5 ), and 𝑉 is the simulation volume (𝑉YSO is the Y2 SiO5 volume in
the simulation).
To maximize coupling between the optical mode and the ensemble of ions, 𝛽mode
should be as large as possible. Its role in the efficiency of AFC quantum storage is
discussed in Section 3.2.
Mirrors in the form of 1D photonic crystal (periodic dielectric waveguides [55])
are placed on either side of a 100 𝜇m long waveguide to create an on-chip FabryPerot resonator. As shown in Figure 4.3, elliptical air holes are added to the
𝛼Si-167 Er3+ :Y2 SiO5 waveguide in a periodic pattern to create a photonic crystal
bandgap centered at the design frequency of 𝜈◦ = 195 THz. Figure 4.3a shows the
simulated bandgap of an infinite 1D photonic crystal, along with the dispersion of
the waveguide mode. For the photonic crystal simulation, even 𝑦 symmetry was
imposed to ensure that the first band gap was the fundamental TM mode (in this 3D
simulation the electric field of the TM mode is along the 𝑧 axis). The wavenumber
of the evanescent mode in the photonic crystal is 𝑘 PhC = 2𝜋
𝑎 ◦ , while the wavenumber
2𝜋𝑛eff
of the TM waveguide mode is 𝑘 PhC = 𝜆◦ (given by the intersection of the two red
lines in Figure 4.3a), where 𝜆 ◦ = 𝑐/𝜈◦ and 𝑛eff = 2.5. The difference in 𝑘 values
of the two modes leads to scattering between photonic crystal and waveguide. To
mitigate this scattering, a taper region is introduced in the photonic crystal, where
the hole size and period are both gradually reduced. The purpose of the taper is to
adiabatically change the photonic crystal Bloch mode into the TM waveguide mode.

38
The last hole in the taper is circular with a radius of 𝑟 1, 𝑓 = 𝑟 2, 𝑓 = 25 nm and the last
spacing between holes is 𝑎 𝑓 = 310 nm. A photonic crystal with these parameters
was simulated, and as shown in Fig. 4.3a, one of the modes in its band diagram has
a 𝑘 value at 𝜈◦ = 195 THz very close to that of the waveguide.
Figure 4.3b shows a COMSOL simulation of a finite photonic crystal mirror (20
holes) with tapered regions on either side. A TM waveguide mode is launched
from the port on the left and the reflectance is measured to be 𝑅 = 0.988 from
simulation. To make one side of the resonator more strongly coupled, one mirror
could have fewer holes. In the resonators used in Chapter 6, 30 taper holes (15 on
either side) were used in all resonators, and the coupling side mirror had either 4 or
6 holes excluding the taper, while the second, more reflective, mirror had 30 holes
excluding the taper. Appendix A describes the COMSOL simulation in more detail
and includes the code for the band diagram simulations.
To couple light into the waveguide mode, a grating coupler similar to that in [79] was
used. A scanning electron micrograph of such a coupler is shown in Figure 4.8. The
simulated and measured one-way coupling of the grating coupler was 40% − 50%.
When measuring our devices at cryogenic temperatures, we observed that the resonances shifted by ∼ 14 nm to the blue (lower wavelength) when cooling down from
room temperature to 4 K. Therefore, the target resonance for the fabricated device
at room temperature should actually be at ∼ 193 THz (∼ 1553 nm). A bandgap
centered at this new frequency can be easily achieved with small modifications to the
photonic crystal described in Fig. 4.3, for example increasing the period to 𝑎 ◦ = 380
nm and increasing the hole semi-major axis to 𝑟 1,◦ = 135 nm.
One approach to increase 𝛽mode in a hybrid device is to decrease the height of the
𝛼Si waveguide, which pushes more of the mode into the 167 Er3+ :Y2 SiO5 . Another
approach is to use a slot-mode waveguide [6], where the mode maximum is in an
air gap between two 𝛼Si waveguides. Since the slot mode is mainly in air, the
denominator of Equation 4.1 is smaller, leading to higher 𝛽mode values. During his
SURF, Dimitrie-Calin Cieleki simulated slot mode resonators using slot waveguides
with photonic crystal mirrors for quantum memories. Both thinner waveguides (ℎ =
250 nm) and slot mode waveguides were fabricated, but neither led to improvements
1 Note that in Figure 4.3, the blue area (extended modes in the Y SiO ) on the left applies to

all modes, but the blue area on the right only applies to modes in the photonic crystal with period
𝑎 ◦ = 370 nm, since that crystal’s band diagram is symmetric about 𝑘 = 2𝑎𝜋◦ , whereas the band
diagram for the photonic crystal with period 𝑎 𝑓 = 310 nm is symmetric about 𝑘 = 𝑎2 𝜋𝑓 , and the band
diagram for the waveguide is not symmetric.

Figure 4.3: Hybrid 𝛼Si-167 Er3+ :Y2 SiO5 photonic crystal mirror simulation. a) Band
diagram 𝜔(𝑘) of photonic crystal and waveguide, simulated using the harmonic
mode solver MPB [57]. Connected blue circles: modes of a photonic crystal
comprised of periodic elliptical air holes with semi-major axis 𝑟 1,◦ = 125 nm, semiminor axis 𝑟 2,◦ = 115 nm, period 𝑎 ◦ = 370 nm, in a waveguide ℎ = 310 nm, 𝑤 = 605
nm (𝑟 2,◦ is parallel to the long axis of the waveguide). Blue areas are the light cone
where extended modes propagating in bulk Y2 SiO5 exist, delineated by the light line
𝑘 1. The red dashed line shows the design resonance frequency
in Y2 SiO5 𝜔 = 𝑛YSO
𝜈◦ = 195 THz. The solid red line shows the dispersion of the waveguide (ℎ = 310
nm, 𝑤 = 605 nm) with no air holes. Connected purple dots show one mode of
a photonic crystal created by the last hole in the taper (see (b) and main text for
details). b) 2D slice at 𝑧 = ℎ/2 = 155 nm from a 3D COMSOL simulation of
a photonic crystal mirror with 20 regular-sized holes and 15 taper holes on either
side. The taper hole dimensions are determined by linear interpolation between 𝑎 ◦ ,
𝑟 1,◦ , 𝑟 2,◦ and 𝑎 𝑓 = 310 nm, 𝑟 1, 𝑓 = 𝑟 2, 𝑓 = 25 nm. The purple mode in (a) is for a
photonic crystal with the latter parameters. Purple-white color gradient shows the
out of plane electric field component 𝐸 𝑧 . Solid green (white) background shows air
(𝛼Si) at 𝑧 = ℎ/2 = 155 nm.
over the ℎ = 310 nm design. As detailed in Section 3.2, high AFC efficiency requires
an ensemble cooperativity 𝐶 > 1 (𝐶 0 = 1), which is greater than cooperativities
achieved so far for nanophotonic resonators coupled to 167 Er3+ :Y2 SiO5 . 𝐶 depends
linearly on both the cavity quality factor 𝑄 and 𝛽mode (see Equation 3.16). Any
increase in 𝛽mode achieved in fabricated thinner or slot mode devices was offset by
an equal or greater decrease in measured 𝑄. The reason for this is that the photonic

40
crystal mirror designs for either thinner or slot waveguides were more sensitive to
fabrication errors than those for the ℎ = 310 nm waveguide.
Fabrication
The fabrication process for the hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonators, outlined in
Figure 4.4, was similar to the process described in References [77, 79]. A Y2 SiO5
boule with nominal 167 Er3+ doping of 200 ppm was custom-grown by Scientific
Materials (FLIR) and diced and polished by Brand Laser Optics & Mfg. The crystal
was grown using the Czochralski method, with 167 Er3+ incorporated in the melt.
The 167 Er3+ concentration was measured to be 135 ppm using secondary ion mass
spectrometry. Each Y2 SiO5 piece had dimensions (8.5 mm × 7.5 mm × 0.5 mm)
along the (𝑏 × 𝐷 2 × 𝐷 1 ) crystal axes. 310 nm of amorphous silicon (𝛼Si) was
deposited using plasma-enhanced chemical vapor deposition (PECVD) using 5%
SiH4 in Ar at 200 ◦ C (Fig. 4.4i). 10 nm of Ti was evaporated as a charge conduction
layer for electron beam lithography. A negative resist was spun onto the Y2 SiO5
chip (300 nm of flowable oxide diluted in MIBK) and patterned using electron beam
lithography using a 100 kV, 300 pA beam (Fig. 4.4ii). The fluid oxide was developed
using tetramethylammonium hydroxide (TMAH). The pattern was transferred to the
𝛼Si using inductively coupled plasma reactive ion etching (ICP-RIE) with SF6 /C4 F8
chemistry (Fig. 4.4iii). The fluid oxide was removed with dilute HF. Scanning
electron micrographs of fabricated resonators are shown in Figure 4.8.
(i) Deposit 𝛼-silicon

(ii) Electron beam lithography

(iii) Reactive ion etching

resist
𝛼Si
YSO

𝛼Si
YSO

Figure 4.4: Hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator fabrication. Details in main text.
4.3

On-Chip Electrodes for DC Stark Shift Control

On chip electrodes were integrated with the hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator
to enable dynamic control of the AFC output pulses. This section describes the
design and fabrication of on-chip electrodes used in the experiments in Chapter 6.
The electrodes were designed and fabricted by Mi Lei and myself.

41
Design
For the control sequences in Chapter 6, two different electric field distributions were
desired: (1) A constant distribution 𝐸 𝑦 (𝑥) = 𝐸 𝑦 as in a parallel plate capacitor, (2)
A linearly varying distribution 𝐸 𝑦 (𝑥) = 𝑥𝐸 𝑦,◦ as in a quadrupole electrode configuration. Electrodes in both parallel [7, 73] and quadrupole [5, 64, 72] configurations
have been used previously in rare earth ion storage experiments using protocols such
as AFC-CRIB [64], GEM [5, 72] or SEMM [7]. The four-electrode design used in
this work was chosen to meet the following criteria: (1) Electric field applied over
100 𝜇m length of resonator. (2) Operate in one "parallel" configuration with high
homogeneity of field. (3) Operate in a second "quadrupole" configuration with a
field that approximates a linear gradient. (4) Provide a high electric field 𝐸 𝑦 with
low applied potential Δ𝑉.
The electrode design is shown in Figure 4.5a. Four circles (𝑟 = 35 𝜇m) provide an
electric field with a linear gradient in the quadrupole configuration, while the four
rectangles (20 𝜇m × 60 𝜇m) connected to the circles create a more homogeneous
field in the parallel configuration. The center-to-center distance between the circles
are 𝑑 𝑦 = 90 𝜇m and 𝑑𝑥 = 150 𝜇m. Each of the 4 electrodes can be biased individually
using wires that are not shown in the figure. Figures 4.5b-f show results from a
3D COMSOL Electrostatics simulation. Figures 4.5c and 4.5d show the potential
𝑉 in the parallel and quadrupole configurations. Fig. 4.5b shows the electric field
distribution for the two cases. The dashed lines show the ideal homogeneous and
linear gradient electric field distributions.
Figures 4.5e and 4.5f show the proportion of ions that experience each electric field
d𝐸
𝜌ions (𝐸 𝑦 ) ∼ ( d𝑥𝑦 ) −1 in the two configurations. 𝜌ions (𝐸 𝑦 ) is a probability distribution
from which the electric field which each ion experiences is sampled, normalized
∫∞
such that −∞ d𝐸 𝜌ions (𝐸) = 1. This assumes a uniform distribution of ions along
the 𝑥 axis, which is a reasonable assumption for uniformly doped 167 Er3+ :Y2 SiO5
crystals. The goal in the parallel configuration is to apply one electric field to all
ions, which is represented by a delta function in Fig. 4.5e (dashed line). The goal
in the quadrupole configuration is to apply a different frequency to each ion, as
represented by the uniform distribution in Fig. 4.5f (dashed line).

42
a)

Figure 4.5: Simulations of on-chip electrodes. a) Gold electrodes on Y2 SiO5 . Black
line denotes position of 𝛼Si-167 Er3+ :Y2 SiO5 resonator (𝑦 = 0, 𝑧 = 0, −50 𝜇m < 𝑥 <
50 𝜇m). b) Simulated electric field distribution at 𝑦 = 0, 𝑧 = 0 (solid lines) and
ideal electric field distribution (dashed lines) for the two biasing configurations:
parallel (green) and quadrupole (orange). The bias applied to each electrode is ±1𝑉.
c) Parallel biasing configuration: potential 𝑉 in red-white-blue gradient. Green
line indicates position of resonator. d) Quadrupole biasing configuration: potential
−1 < 𝑉 < 1 in red-white-blue gradient. Orange line indicates position of resonator.
e) and f) Proportion of ions experiencing every electric field value 𝜌ions (𝐸 𝑦 ) in the
parallel (e) and quadrupole (f) configurations. 𝜌ions (𝐸 𝑦 ) from simulated field profile
in (b) (solid lines) and ideal (dashed lines).

43
The TM optical mode of the resonator only penetrates a few hundred nanometers
(1/𝑒 ∼ 150 nm), while the 1/𝑒 point of the electric field in the 𝑧 direction is
tens of microns. Therefore the electric field does not change significantly in the 𝑧
direction, so only values at 𝑧 = 0 are shown. Likewise, there was no significant
variation in electric field in the 𝑦 direction over the 605 nm width of the optical
resonator. However, the optical mode is expected to penetrate significantly into
the photonic crystal mirrors as seen in Figure 4.3b. The electric field 𝐸 𝑦 (𝑥) was
therefore simulated for −56 𝜇m < 𝑥 < 56 𝜇m for Figures 4.5b,e and f. The value of
6 𝜇m as an effective penetration depth of the optical mode was decided empirically
as discussed in Section 6.4.
Other electrode geometries are possible. Figure 4.6 compares the electric field
distributions for the final design (from Fig. 4.5), a similar design without the rectangular electrodes, and a third design without rectangular electrodes where the circular
electrodes are evenly spaced at 𝑑𝑥 = 𝑑 𝑦 = 150 𝜇m. The latter design meets criteria
(1-3), but the electric field magnitude achieved is 4 times lower than the final design.
Bringing the electrodes closer together, as in the second design, increases the maximum field achieved, but leads to large inhomogeneity in the parallel configuration
(see Fig. 4.6a). Figure 4.6b also shows that adding the rectangular electrodes does
not significantly affect the quadrupole configuration electric field profile.
a)

Figure 4.6: Various electrode designs. Electric field profile at 𝑦 = 0, 𝑧 = 0 for the
parallel (a) and quadrupole (b) biasing configurations of three electrode designs.
Details in main text.
Fabrication
On-chip electrodes were added after the hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonators were
fabricated. Figure 4.7 shows the fabrication process for the electrodes. A ∼ 450 nm

44
layer of positive resist (PMMA) was spun on the Y2 SiO5 chip with 𝛼Si resonators
already patterned. Aligned electron beam lithography was used to pattern the
electrodes, with a 100 kV, 100 nA beam (Fig. 4.7i). Alignment was made possible
by an array of 20 𝜇m × 20 𝜇m gold markers, which were patterned before the
resonators using the same procedure as in Fig. 4.7. Both the resonator pattern and
electrode pattern were aligned to the same marker grid. After developing the PMMA
resist using an MIBK-IPA mixture, a 10 nm Ti adhesion layer was evaporated onto
the chip, followed by a 100 nm gold film (Fig. 4.7ii). The chip was soaked in acetone
to dissolve the PMMA and lift off the excess gold film, leaving behind the patterned
electrodes (Fig. 4.7iii). Appendix B describes the fabrication steps for the on-chip
electrodes in more detail.
(i) Electron beam lithography

resist

YSO

(ii) Evaporate gold
gold

YSO

(iii) Lift-off
gold

𝛼Si

YSO

Figure 4.7: Fabrication of on-chip electrodes. Details in main text.
Figure 4.8 shows a completed hybrid 𝛼Si-167 Er3+ :Y2 SiO5 device with a closeup
of a photonic crystal mirror and a grating coupler. Up to 81 such devices were
patterned on the same 𝐷 1 -cut Y2 SiO5 chip, with resonators aligned either to the 𝑏
or 𝐷 2 crystal axes. 10 𝜇m wires, shown in Fig. 4.8a, connected the electrodes to
300 𝜇m × 300 𝜇m contact pads, which were connected via aluminium wire bonds
to SMP connectors on a custom-made PCB board (see Appendix D).

50 µm

5 µm

Figure 4.8: Hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator with electrodes. a) Optical micrograph of a hybrid 𝛼Si-167 Er3+ :Y2 SiO5 with electrodes. b) Scanning electron
micrograph of an 𝛼Si photonic crystal mirror, shown here on a sapphire substrate
(𝑛Sapphire = 𝑛YSO = 1.8). c) Scanning electron micrograph of an 𝛼Si grating coupler
on sapphire. Imaged at a 40◦ angle.

46
Chapter 5

QUANTUM STORAGE IN AN 167 Er3+ :Y2 SiO5 NANOBEAM
In this chapter, we demonstrate on-chip quantum storage of telecommunication light
at the single photon level. The results were first presented in Reference [30], and this
chapter is an expanded version of that manuscript. We used a nanophotonic crystal
resonator milled directly in 167 Er3+ doped Y2 SiO5 (167 Er3+ :Y2 SiO5 ) to couple to
an ensemble of erbium ions and realize quantum storage using the AFC protocol
[4]. The resonator, which was introduced in Section 4.1, increased the absorption of
light by the ion ensemble, allowing on-chip implementation of the memory protocol
[3]. By working in a dilution refrigerator and using permanent magnets to apply
a field of 380 mT, we accessed a regime in which the ions have optical coherence
times of ∼ 150 𝜇s and long-lived spin states to allow spectral tailoring. For a storage
time of 165 ns, we achieved an efficiency of 0.2%, with lower efficiencies for longer
storage times, up to 10 𝜇s. We demonstrated storage of multiple temporal modes
and measured a high fidelity of storage, exceeding the classical limit. We then
identified the limits on the storage efficiency and proposed avenues for overcoming
them to achieve an efficient 167 Er3+ :Y2 SiO5 quantum memory for light. Appendix
C describes experimental setup used for measurements in this chapter.
5.1

Coupling Between an Ensemble of Ions and a Cavity

The coupling between an ensemble of absorbers and a cavity can be characterized
by measuring the reflectance curve of the cavity-ion system. Following Reference
[36], we fit the reflectance curve to the following model:

𝑅 = 𝛼1 (1 − 𝛼 𝑓 ) + 𝛼 𝑓 𝑒

𝑖𝜃 𝑓

𝑖𝜅 in
+ 𝛼2 (5.1)
𝜔 − 𝜔cavity + 𝑖 2 + 𝑊 (𝜔, 𝑔total , Δ ions , 𝜔ions )

where 𝛼1,2 are amplitude and background fit parameters, 𝛼 𝑓 𝑒𝑖𝜃 𝑓 accounts for Fano
interference (both 𝛼 𝑓 and 𝜃 𝑓 are fit parameters), 𝜅 is the total cavity energy decay
rate, 𝜅in is the coupling rate through the input/output port, and 𝜔cavity is the cavity
resonance frequency. 𝑊 (𝜔, 𝑔total , Δ ions , 𝜔ions ) is the absorption rate of the cavity
Í 𝑔𝑖2
field by the ensemble of ions, 𝑊 ∼ 𝑖 𝜔−𝜔
, where 𝑔𝑖 is the coupling between one ion
and the cavity [3, 36]. We approximated the irregular shape of the inhomogeneously

47
and hyperfine broadened optical transition as a Gaussian, and used the expression
for 𝑊 from Reference [36]:

𝑊 =𝑖

𝜋log2𝑔total

Δ ions /2

!#
2#
𝑖 log2(𝜔 − 𝜔ions )
𝜔 − 𝜔ions
1 − erf −
× exp −log2
Δ ions /2
Δ ions /2
(5.2)

where Δ ions is the ensemble transition’s linewidth, 𝜔ions its the center, and 𝑔total
Í 2
𝑖 𝑔𝑖 . Finally, the ensemble cooperativity was computed using Equation 3.10,
where Γions = |𝑊 (0)| [79].

Figure 5.1 shows the reflection spectrum of the nanobeam cavity, which has a
measured loaded quality factor of 7 × 103 . The cavity was tuned onto resonance
with the 1539 nm transition of the 167 Er3+ ions by freezing nitrogen gas onto the
nanobeam at cryogenic temperatures [82]. The coupling of the ensemble of ions to
the cavity is seen as a peak in the cavity reflection dip. The inset shows a close-up of
the ion-cavity coupling (in black). The ensemble cooperativity was estimated from
a fit to this curve to be 0.1.
200

Counts/103

190

180
30
-5

170

160
150
140
130
-20

-10

Detuning (GHz)
Figure 5.1: Reflection spectrum of nanobeam resonator when tuned on resonance
to the 1539 nm 167 Er3+ :Y2 SiO5 transition (adapted from Ref. [30]). Detuning
is measured from 194816 GHz ± 2 GHz. Inset shows a close-up of ion coupling
before (black) and after (red) partial hyperfine initialization. Circles are data points;
solid black and dashed red lines are fits to theory (see main text for details).
For high efficiency storage using ions coupled to a cavity, the ensemble cooperativity
should equal one (see Section 3.2). An increased ensemble cooperativity of 0.3 (inset
of Fig. 5.1, in red), was obtained using a partial hyperfine initialization procedure

48
(see Section 2.3). To extract the cooperativity of coupling between a resonator and
an ensemble of 167 Er3+ ions, each cavity reflection spectrum shown in the inset of
Fig. 5.1 was fit using Equation 5.1. 𝜅 = 27.3 GHz and 𝜅𝜅in = 0.21 were measured
from reflectance curves where the cavity was detuned from the 167 Er3+ transition.
For the case with no initialization, the fit yielded: 𝜔cavity − 𝜔ions = 2𝜋 × 2.5 GHz,
𝑔total = 2𝜋 × 0.79 GHz, Δ ions = 2𝜋 × 1.4 GHz, 𝐶 = 0.1.
For the case with initialization, the fit yielded: 𝜔cavity − 𝜔ions = 2𝜋 × 1.5 GHz,
𝑔total = 2𝜋 × 0.70 GHz, Δ ions = 2𝜋 × 0.36 GHz, 𝐶 = 0.3.
5.2

Atomic Frequency Comb Storage

The nanobeam device was used to demonstrate quantum optical storage using the
AFC protocol [4]. In this protocol, a pulse of light that is absorbed by an atomic
frequency comb with an inter-tooth spacing of Δ is stored for 𝑡 = Δ1 . Frequency
selective optical pumping was used to create a comb within the inhomogeneous
linewidth, as shown in Fig. 5.2a. Figure 5.2b shows a schematic of the protocol.
First, a long pulse with strong frequency modulated sidebands was used for partial
hyperfine initialization (see Section 2.3). The next 15 pulses, repeated 𝑛pump = 20
times, created the comb: the laser frequency was swept through 15 values, separated
by Δ = 6.1 MHz, to optically pump away ions and create 15 spectral transparencies.
The following 𝑛input = 60 pulses were zero-detuning weak coherent states which
were stored in the frequency comb. The full experiment was repeated ∼ 104
times. As shown in Fig. 5.2c, 60 ns wide pulses with an average photon number
of 𝑛¯ = 0.60 ± 0.09 were stored for 165 ns with an efficiency of 0.2%. The storage
efficiency was limited by the ensemble cooperativity of the device (see Section 3.2).
Coherent pulses could be stored in the device for up to 10 𝜇s, although with a
lower efficiency of 10−5 , as shown in Fig. 5.3. Here, as for all storage times longer
than 165 ns, we used an accumulated AFC method [34] to create the comb (see
Section 3.1). As shown in the inset of Fig. 5.3, weak pairs of pulses separated
by 𝑡storage = 10 𝜇s were repeatedly sent into the cavity. The Fourier transform
of each pulse pair is a frequency comb, which imprinted onto the 167 Er3+ :Y2 SiO5
inhomogeneous line to create the AFC. This procedure utilized laser frequency
stabilization during comb creation, which enabled the creation of fine-toothed AFCs
required for longer storage. For the 165 ns storage, where a coarser AFC is practical,
the procedure shown in Fig. 5.2b with no laser frequency stabilization led to higher
efficiencies by creating a more consistent comb over the entire bandwidth. This is

Counts/10

a)
60
55
50

-50

-25

npump

ninput

Detuning (MHz)

b)
amp.

freq.

Counts/103

40
20

×100

100

200

300

400

Time (ns)

Figure 5.2: AFC experiment in the nanobeam cavity (adapted from Ref. [30]). (a) A
section of the resonator reflection spectrum, showing an atomic frequency comb in
the center of the inhomogeneously broadened 167 Er3+ :Y2 SiO5 transition. Detuning
is measured from 194814.2 GHz ± 0.1 GHz. The apparent slope of the comb is due
to its center frequency not being precisely aligned to the cavity resonance, leading
to a dispersive shape. (b) Schematic of AFC pulse sequence showing amplitude
(yellow) and frequency (purple) modulation of the laser (pulses not to scale, see
main text for detail). (c) AFC storage: the input pulse (red dashed line) was partially
absorbed by the comb; an output pulse was emitted at time 𝑡 = 1/Δ = 165 ns (black
line, ×100). The black line also shows the partially reflected input pulse (𝑡 = 0) and
a smaller second output pulse at 𝑡 = 330 ns.
because the accumulated AFC has a sinc function envelope. The storage efficiency
at 10𝜇s was limited by residual laser frequency jitter and by superhyperfine coupling
to the yttrium ions in Y2 SiO5 . Superhyperfine coupling limits the narrowest spectral
feature to ∼ 2 MHz at this field (see Section 2.4). Since this exceeds the period of
the comb needed for this storage time (Δ = 1/𝑡storage = 0.1 MHz), the resulting AFC
will have a lower contrast, leading to lower storage efficiency.
The AFC protocol is capable of storing multiple temporal modes [4]. Ten coherent
pulses were stored in this device, as shown in Fig. 5.4. The AFC in Figure 5.2a
has a bandwidth of ∼ 90 MHz (see Fig. 5.2a) which can accommodate storage

npump
...

ninput

×20 000

Figure 5.3: AFC storage for 10 𝜇s in the nanobeam resonator (adapted from Ref.
[30]). Red dashed line shows the input pulse. Black line shows the partially reflected
input pulse and the output pulse (×20 000). The reflected input pulse appears small
due to detector saturation. Inset shows a schematic of the pulse sequence following
hyperfine initialization. Pairs of comb preparation pulses 10 𝜇s apart were repeated
𝑛pump = 10 000 times, followed by input pulses 20 ns wide, repeated 𝑛input = 10
times.
in multiple frequency modes [97]. Multiplexing in time or frequency can significantly improve entanglement generation rates in quantum repeater networks [96].
An inhomogeneous linewidth of 150 MHz limits the bandwidth of storage in this
system. Although there exist methods to increase this linewidth [18, 105], the bandwidth cannot be increased much further before being limited by overlapping optical
transitions from other hyperfine levels.

×1000

Figure 5.4: Multimode storage in the nanobeam resonator (adapted from Ref. [30]):
ten 20 ns wide input pulses (reflection off cavity shown) and the corresponding 10
output pulses (×1000) from a Δ = 500 kHz AFC.

51
5.3

Coherent Storage of Time-Bin Qubits

In quantum storage protocols, the phase of the stored state must be preserved. A
double AFC was used as an interferometer in order to characterize the coherence
of the storage process [34]. Two overlapping AFCs with tooth spacing Δ 1 , Δ 2 and
with frequency detuning 𝛿1 , 𝛿2 , were created, as shown in Figure 5.5. The two
combs lead to two output pulses for every input pulse, at times Δ11 , Δ12 . The two
AFCs were created by alternately burning away population between the teeth of each
comb. Low power burn pulses were used and the relative burn powers and number
of repeats (npump ) were calibrated such that the amplitudes of the two output pulses
were maximized and equal to one another. The discrepancy between the data and
the fit is in part due to photon counting uncertainty ( 𝑁counts ∼ 350), and in part
due to superhyperfine side-hole and anti-hole structure (see Section 2.4).
For a given input pulse, the relative phase between the two output pulses 𝜙rel is given
by:

𝛿2
𝛿1
𝜙rel = 2𝜋
Δ2 Δ1

(5.3)

where 𝛿1 (𝛿2 ) is the detuning of the first (second) comb, measured relative to the
frequency of the input pulse [4].
Next, we consider the case where a time bin qubit is stored by the double AFC. The
input state is a superposition of weak coherent states |𝜓in i = √1 (|earlyi + |latei),
where the |earlyi pulse is absorbed at time 𝑡early , and the |latei pulse is absorbed at
time 𝑡late . When absorbed by the double AFC, this input will be mapped to a total
of four output pulses at times ( Δ11 + 𝑡early ), ( Δ11 + 𝑡 late ), ( Δ12 + 𝑡early ), ( Δ12 + 𝑡late ). By
appropriately selecting the time interval between the early and late input pulses, two
of the four output pulses were made to overlap ( Δ11 + 𝑡late = Δ12 + 𝑡early ). Depending
on 𝜙rel , these pulses either constructively or destructively interfered, as shown in
the inset of Fig. 5.6. The double AFC therefore leads to a phase-dependent output
amplitude, much like an interferometer.
Using an input state with mean photon number 𝑛¯ = 0.6 ± 0.09 and sweeping 𝜙rel
via the detuning 𝛿2 , the interference fringe shown in Fig. 5.6 was obtained (see
caption for details). The measured visibility of 91.2% ± 3.4% demonstrates the high
degree of coherence of this on-chip storage process. The visibility was limited
by
Δ2
the 12 counts in the total destructive interference case 𝛿2 = 2 → 𝜙rel = 𝜋 . These
were due in part to imperfect cancellation of the two overlapping output pulses,

52
a)

Figure 5.5: Double atomic frequency comb, measured by scanning the resonator
reflection spectrum. Red circles with connecting lines represent counts with the
minimum value subtracted. Black lines are fits to a sum of cosine functions: each
of the two combs with finesse 𝐹 ∼ 2 is approximated by a cosine function. For both
(a) and (b), Δ 1 = 5.0 MHz, 𝛿1 = 0 MHz, and Δ 2 = 3.4 MHz. (a) 𝛿2 = 0 → 𝜙rel = 0,
(b) 𝛿2 = 0 → 𝜙rel = 𝜋.
resulting from the slightly different efficiencies of storage in the two AFCs, and
in part to a dark count rate of 18.5 Hz, leading to a baseline of 7 counts. The
dark-count-subtracted visibility is 97.0% ± 3.6%.

1 This value is corrected from Reference [30]

53
250

Counts

200
150
100
50

-0.5

0.5

1.5

Detuning (MHz)

Figure 5.6: Coherent storage in the nanobeam resonator (adapted from Ref. [30]).
Visbility curve was acquired using a double comb experiment with Δ 1 = 5.0 MHz 1,
Δ 2 = 3.4 MHz, 𝛿1 = 0 MHz. The detuning of the second comb was swept from
𝛿2 = −0.2 MHz to 𝛿2 = 2.2 MHz, and the intensity of the two central overlapping
output pulses was measured.
Black circles show the sum of counts in the overlapping
pulse region with 𝑁counts uncertainty bars. Red line shows a least squares fit to a
sinusoid. Inset shows the four output pulses (middle two overlapping) in the case
of the maximally constructive (dashed black line) and maximally destructive (solid
red line) interference.
5.4

Estimating a Lower Bound on Storage Fidelity

In the absence of a single photon source, a lower bound on the storage fidelity of a
single photon input state can be found using weak coherent states and decoy state
analysis [68, 97]. Using this method, we estimated a lower bound for the fidelity of
storing single photon time bin states, 𝐹 (𝑛=1) ≥ 93.7% ± 2.4%, which exceeds the
classical limit of 𝐹 = 2/3.
In the decoy state method, a time bin state 𝜓 with a mean photon number 𝑛¯ is stored
using the AFC protocol, and the fidelity 𝐹𝜓( 𝑛)
of storage is measured as
𝐹𝜓( 𝑛)

𝑁𝜓
𝑁𝜓 + 𝑁 𝜙⊥𝜓

(5.4)

where 𝑁𝜓 (𝑁 𝜙⊥𝜓 ) is the number of photons measured in the output time bin corresponding to 𝜓 (𝜙 ⊥ 𝜓), and 𝜙 ⊥ 𝜓 denotes the state orthogonal to 𝜓. The gain of
the output, 𝑄 𝜓( 𝑛)
is also estimated using,
𝑄 𝜓( 𝑛)
= 𝑁𝜓 + 𝑁 𝜙⊥𝜓 .

(5.5)

54
are measured for mean photon numbers 𝑛¯ 1 and 𝑛¯ 2 , where 𝑛¯ 1 < 𝑛¯ 2 ,
and 𝑄 𝜓( 𝑛)
𝐹𝜓( 𝑛)
and 𝑛¯ 2 < 1.

The lower bound on the fidelity of storing a one-photon input state 𝐹𝜓(𝑛=1,𝐿) is then
computed using:
𝐹𝜓(𝑛=1,𝐿) = 1 −

𝐸 𝜓( 𝑛¯ 1 ) 𝑄 ( 𝑛¯ 1 ) exp 𝑛¯ 1 − 𝐸 (0)𝑌 (0)
𝑌 (𝑛=1,𝐿) 𝑛¯ 1

(5.6)

where
𝐸 𝜓( 𝑛)
= 1 − 𝐹𝜓( 𝑛)

(5.7)

is the error rate of storing a state 𝜓 with mean photon number 𝑛,
¯ and

𝑛¯ 2 𝑛¯ 22 − 𝑛¯ 21 (0)
𝑛¯ 2
( 𝑛¯ 1 ) 𝑛¯ 1
( 𝑛¯ 2 ) 𝑛¯ 2 1
(𝑛=1,𝐿)
(0)
𝑄 𝑒 −𝑄 𝑒 2 −
= max 𝑌 ,
𝑛¯ 2 𝑛¯ 1 − 𝑛¯ 21
𝑛¯ 2
𝑛¯ 22

!)
(5.8)

is the lower bound on the detection yield for the storage of a single photon state (see
Reference [68]). 𝑌 (0) = 𝑄 (𝑛=0) is the yield when the input state is vacuum, equal to
the dark counts in both output time bins. The superscripts denote photon number
and whether the value is a lower bound (𝐿). 𝐸 (0) = 𝐸 (𝑛=0) is the vacuum error rate,
which is 0.5 by definition [68].
In order to obtain an average fidelity bound for all possible time bin states, the fideli|earlyi+|latei
|earlyi−|latei
ties for storing time bin states |earlyi, |latei, |+i =
and |−i =
were measured for input photon numbers 𝑛¯ 1 = 0.30 and 𝑛¯ 2 = 0.60. The input pulses
defining the |earlyi and |latei basis were 60 ns wide and 90 ns apart. A double AFC
was used for measurements of all states with the memory times associated with the
two combs being 𝑡1 = 200 ns and 𝑡2 = 290 ns 2, such that 𝑡2 − 𝑡1 = 90 ns. Figure 5.7
shows the input time bins centered at 𝑡early = 0 ns and 𝑡late = 90 ns, and the output
time bins centered at 200 ns, 290 ns, and 380 ns. Of the three output time bins, the
first and third were used for measuring 𝐹early and 𝐹late , while the second time bin
was used for measuring 𝐹+ and 𝐹− , as shown in Figure 5.7.
Following Equation 5.6, 𝐹+(𝑛=1,𝐿) was computed using:
𝐹+(𝑛=1,𝐿) = 1 −
2 The 𝑡

( 𝑛¯ 1 )
(0)
𝐸 +( 𝑛¯ 1 ) 𝑄 +/−
exp𝑛¯ 1 − 𝐸 (0)𝑌+/−
(𝑛=1,𝐿)
𝑌+/−
𝑛¯ 1

1 and 𝑡 2 values are corrected from Reference [30]

(5.9)

Counts

a) 15

Counts

10
-100
15

100

200

300

400

500

600

100

200

300

400

500

600

100

200

300

400

500

600

200

300

400

500

600

10
-100
15
10
-100
15
10

-100

100

Time (ns)

Figure 5.7: Part of the raw data used in the fidelity measurement. 𝑛¯ = 0.6 input
(left, cut off) and output pulses from double comb. Input pulses are time bin states
a) early, b) late, c) +, d) −. In all figures, blue (red) dash-dot line represents the 60
ns time bins counted as 𝑁𝜓 (𝑁 𝜙⊥𝜓 ). Time resolution is 1 ns. The same data set,
shown in a) and b), was used for both the early and late fidelity calculations, since
absolute time is irrelevant. In this data set, a small pulse 100 ns after the read pulse
can be seen. The origin of this pulse was unclear, but it disappeared in the absence
of the double comb. In c) and d), the black curve represents data taken with comb
detunings 𝛿1 = 𝛿2 = 0 (𝜙rel = 0), while the gray curve represents data taken with
𝛿1 = 0, 𝛿2 = Δ22 (𝜙rel = 𝜋).
( 𝑛¯ 1 )
(0)
(𝑛=1,𝐿)
with similar equations for the other three states. 𝑄 +/−
, 𝑌+/−
, and 𝑌+/−
are averaged
over the |+i and |−i fidelity measurements.
(𝑛=1,𝐿)
The lower bound on the fidelity of storing an arbitrary single photon state, 𝐹average
93.7% ± 2.4%, was then computed as follows:

(𝑛=1,𝐿)
(𝑛=1,𝐿)
1 © 𝐹early + 𝐹late
ª 2
(𝑛=1,𝐿)
𝐹average =
®+

𝐹+(𝑛=1,𝐿) + 𝐹−(𝑛=1,𝐿)

(5.10)

Table 5.1 summarizes the measured fidelity values for storing weak coherent states,

56
as well as the estimated single photon storage fidelities used in Equation 5.10. The
uncertainties are calculated based on 𝑁photon standard deviation on all 𝑁𝜓 values
due to Poissonian statistics of photon counting and the uncertainty, estimated to be
15%, of the mean input photon numbers, 𝑛.
( 𝑛)
( 𝑛)
( 𝑛)
𝑛)
Input photon number
2 𝐹early + 𝐹late
2 𝐹+ + 𝐹−
𝑛¯ = 0.60 ± 0.09
𝑛¯ = 0.30 ± 0.05
𝑛=0
𝑛=1

89.04% ± 1.34%
91.90% ± 1.32%
82.59% ± 1.80%
90.75% ± 1.84%
50%
50%
(𝑛=1,𝐿)
(𝑛=1,𝐿)
(𝑛=1,𝐿)
(𝑛=1,𝐿)
early
late
89.85% ± 1.97%

95.59% ± 3.37%

Table 5.1: Storage fidelities in the nanobeam device.
Similar to the visibility curve discussed in Section 5.3, the measured fidelity was
limited in part by dark counts and in part by the double comb protocol being an
imperfect interferometer. The dark counts limited fidelity bound was estimated to
be ∼ 96.5%.
5.5

Overcoming AFC Efficiency Limitation

While the storage presented here was limited in efficiency, a nanophotonic cavity
coupled to 167 Er3+ ions in Y2 SiO5 promises to be an efficient quantum storage
system. The main limitations to the storage efficiency in this work were a low
ensemble cooperativity of 0.3 and loss from the optical nanobeam cavity. The
cooperativity can be increased using higher 167 Er3+ doping and better hyperfine
initialization, which would require increasing the applied magnetic field [86] or
changing its angle. A higher intrinsic quality factor resonator would serve to both
increase cooperativity and decrease cavity loss. For example, using a Y2 SiO5 crystal
with 200 ppm 167 Er3+ doping, optimal hyperfine initialization, and a resonator with
an intrinsic quality factor of 2 × 106 , the theoretical efficiency of the AFC quantum
storage is 90%. Mature silicon nanofabrication technology can be leveraged to
achieve this goal by using a silicon resonator evanescently coupled to 167 Er3+ ions
in Y2 SiO5 [35, 79]. With this efficiency level and a storage time of 10 𝜇s, the device
would outperform a delay line composed of standard telecommunication fiber [29],
an important benchmark on the way to achieving a quantum memory suitable for
scalable quantum networks.
With the optical AFC protocol alone, it will be difficult to achieve efficient storage

57
for this duration due to superhyperfine coupling. However, the AFC spin-wave
protocol, where the stored information is reversibly transferred from the optical to
the hyperfine manifold [4], would enable storage longer than 10 𝜇s without the same
requirements for narrow spectral features, as well as enabling on-demand recall.
Following the analysis above, the subsequent set of devices, which are used in the
next chapter (Chapter 6), used a higher doping of 167 Er3+ ions, and resonators based
on amorphous silicon waveguides on 167 Er3+ :Y2 SiO5 . The storage of efficiency of
those devices was also limited. Section 6.6 discusses a path to making devices with
higher storage efficiency using the AFC protocol in more detail.
5.6

Nanobeam Device Temperature

Due to poor thermal conduction at low temperatures in insulating materials such
as Y2 SiO5 and the small cross section of the nanobeam, the device was warmer
than its ∼ 25 mK surroundings when optical pulses were coupled in. The device
temperature was estimated via the 167 Er3+ electron spin temperature [61], which
was computed from the ratio between the lower and upper electron spin populations
𝑁 |↑i
− ℏ𝜔
= 𝑒 𝑘𝐵 𝑇 .
in the optical ground manifold using 𝑁 |↓i
Under an applied field of 380 mT parallel to 𝐷 1 , the electron spin in the optical
ground state was frozen for any temperature under ∼ 500 mK, enabling the long
hyperfine lifetimes required for AFC storage. To be sensitive to lower temperatures,
the electron spin population measurements were performed with a lower magnetic
field of 110 mT (parallel to the 𝐷 1 axis of the crystal), leading to an electron Zeeman
splitting of 𝜔 = 2𝜋 × 23 GHz, where the upper electron spin state had detectable
population down to ∼ 250 mK. Because the Zeeman splitting was still considerably
greater than the hyperfine splitting, one can consider two electron spin states, |↓i
and |↑i, each split into eight by the hyperfine interaction. The population in the two
electron spin states was measured via the electron-spin-preserving optical transitions
|↓i to |↓i and |↑i to |↑i. The nanobeam was tuned such that these transitions were
both resonant with the cavity, and photoluminescence (PL) was collected as a
function of frequency, as shown in Fig. 5.8a. 𝑁 |↑i /𝑁 |↓i was extracted from the
area ratio of the two transitions. Figure 5.8b shows the electron spin temperatures
computed from these ratios for different dilution refrigerator temperatures. The
inset in Fig. 5.8b shows the electron spin temperature measured under input power
conditions identical to two experiments: 317 mK ± 49 mK for the T2 measurement
in the nanobeam (Fig. 2.3) and 413 mK ± 24 mK for the 165 ns storage experiment

Normalized PL

1.5
0.5
-4

-2

b)
Electron Spin Temperature [K]

Detuning (GHz)
0.8
0.6

0.6

0.4

0.2
0.2

0.2

0.4

0.6

0.03

0.8

0.04

Refrigerator Temperature [K]

Figure 5.8: Nanobeam device temperature measurement (adapted from Ref. [30]).
a) Photoluminescence (PL) from the nanobeam device as a function of detuning at
three refrigerator temperatures: 720 mK (gray squares), 385 mK (black diamonds),
37 mK (red circles). Detuning was measured from 194810 ± 0.1 GHz. PL was
collected after a 500 𝜇s resonant pulse at 0.3 pW (estimated power in nanobeam).
Background counts were subtracted, and each curve was normalized and offset for
clarity. Solid lines are fits to a sum of two Gaussians with equal widths and center
frequencies 3.2 GHz apart. The |↓i to |↓i transition is at the higher frequency. b)
Electron spin temperatures (EST) computed from the PL data in (a), as a function of
refrigerator temperature. Dashed gray line indicates where the two temperatures are
equal. The inset shows a closeup of the EST measurement at 37 mK (black circle), the
EST estimated during the T2 measurement (green diamond), and the 165 ns storage
experiment (blue square). To estimate the latter two temperatures, the same pattern of
laser pulses as in the actual experiments was sent to the nanobeam, at 0.3 nW and 0.02
nW, respectively, and PL was collected after
√ the pulses. Error bars are propagated
standard deviations from photon counting ( 𝑁counts ). In all measurements, the laser
frequency was slowly modulated within each optical transition to prevent hyperfine
holeburning.
in Fig. 5.2. Assuming the electron spin was in thermal equilibrium with the device,
we therefore estimated the temperature of our device during experiments to be ∼400
mK.

59
Chapter 6

DYNAMIC ON-CHIP CONTROL OF STORED LIGHT USING
THE DC STARK SHIFT
In this chapter, we use on-chip electrodes to realize a new multifunctional device
which can not only store light, but also modify its frequency and bandwidth. The
ability to adjust the temporal profile or frequency of stored light can be useful when
quantum memories act as interfaces between multiple emitters, as in a quantum
repeater network. We demonstrate dynamic control of memory time in a digital
fashion with storage times from 50 ns to 400 ns. By shifting the frequency of
the 167 Er3+ ions during emission, we change the frequency of stored light by ±20
MHz relative to the input frequency. Using a quadrupole electrode configuration,
the bandwidth of stored light is increased by over a factor of three, from 5.7 MHz
(input) to 18 MHz (output).
The quantum memory protocol used in this chapter is atomic frequency comb with
added dynamic control. The dynamic control is enabled by the optical DC Stark
® the
shift. When a rare earth ion in a crystal interacts with a DC electric field 𝐸,
optical transition frequency changes according to:
1 ® ˆ ®
· 𝐿 · 𝐸,
𝛿 𝑓 = − 𝛿𝜇

(6.1)

® = 𝜇®e − 𝜇®g is the difference between
where 𝐿ˆ is the local field correction tensor and 𝛿𝜇
the electric permanent dipole moments in the optical excited state ( 𝜇®e ) and optical
ground state ( 𝜇®g ) [69]. For a non-centrosymmetric site such as the yttrium sites in
Y2 SiO5 for which Er3+ ions substitute, the above linear stark shift term dominates,
although a quadratic Stark shift term also exists.
A DC Stark shift enables dynamic control of light stored in an AFC by allowing
changes to the optical transition frequency of each ion in the excited Dicke state
[4]. Modifying Equation 3.3 from Section 3.1, we can see the effect of this dynamic
control:

|Ψi =

𝑗=1

𝑐 𝑗 𝑒𝑖 ( 𝜔 𝑗 +2𝜋 𝛿 𝑓 𝑗 (𝑡) ) 𝑡 𝑒 −𝑖𝑘 𝑟®𝑗 0...1 𝑗 ...0𝑁 .

(6.2)

60
The Stark shift of each ion 𝛿 𝑓 𝑗 (𝑡) can be varied over time by applying a slowly
varying electric field, where slow is defined relative to optical frequencies. In this
section, we use the electrodes described in Section 4.3 to control the frequencies
of 167 Er3+ :Y2 SiO5 ions. The DC Stark shift can be the same for all ions, as when
electrodes are biased in a parallel configuration, or different for every ion, as when
electrodes are biased in a quadrupole configuration.
Section 6.1 discusses the Stark shift in 167 Er3+ :Y2 SiO5 . Sections 6.2-6.4 show
three different ways in which Stark shift control can be used to improve and add
functionality to an atomic frequency comb storage protocol. Section 6.5 includes
some supporting experiments for this chapter. Lastly, Section 6.6 describes strategies
for improving the efficiency of storage in the devices used in this chapter.
For all experiments conducted in this chapter, the temperature of the mixing chamber plate to which the sample was attached was ∼ 70 mK. A magnetic field of
0.98 T parallel to the 𝐷 1 axis of the Y2 SiO5 crystal was applied with a solenoid
superconducting electromagnet. Any magnetic field along the crystal 𝑏 axis was
cancelled using a set of trim-coils on either side of the solenoid. For more details
of the experimental setup, see Appendix D. All experiments are performed on
crystallographic site 2 of 167 Er3+ :Y2 SiO5 , which has a zero field optical transition
frequency at 194756 GHz (𝜆◦ = 1538.85 nm).
6.1

DC Stark Shift in 167 Er3+ :Y2 SiO5

The DC Stark shift described in Equation 6.1 is dependent on the orientation of
® [42]. Without knowing 𝛿𝜇
® or 𝐿,
ˆ it is possible
the applied field relative to 𝛿𝜇
to empirically characterize the Stark shift induced by an electric field applied in a
particular direction using the stark shift parameter 𝑠𝑛ˆ , where 𝑛ˆ specifies the direction:

𝛿 𝑓 = 𝑠𝑛ˆ 𝐸 𝑛ˆ .

(6.3)

As introduced in Section 2.1 there are four subclasses of 167 Er3+ :Y2 SiO5 sites
with different orientations for each crystallographic site. Focusing on site 2, this
® Therefore, in an ensemble
means that there are four different orientations for 𝛿𝜇.
of 167 Er3+ :Y2 SiO5 ions, four different Stark shifts will be observed for an electric
field applied in an arbitrary direction. When the applied electric field is parallel
or perpendicular to the 𝑏 axis, the Stark shifts of the four subclasses are pair-wise
degenerate, such that there are only two equal and opposite DC Stark shifts 𝛿 𝑓± =

61
±𝑠𝐸. In this thesis, all electric fields are applied either parallel or perpendicular to
the 𝑏 axis, so from here on, I will refer to two 167 Er3+ subclasses for simplicity.
The Stark shift for 167 Er3+ :Y2 SiO5 was estimated using holeburning spectroscopy in
an ensemble of ions coupled to a hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator, for electric
fields parallel to the crystal 𝑏 and 𝐷 2 axes. For this measurement, on-chip parallel
plate electrodes on either side of the resonator were used to apply a homogeneous
electric field to all ions. These electrodes were comprised of two gold rectangles
on either side of the resonator (not the electrode geometry described in Section
4.3). The rectangular electrodes were much longer than the resonator (300 𝜇m).
The distance between the electrodes was 90 𝜇m (𝑏 measurement) or 20 𝜇m (𝐷 2
measurement).
To measure the Stark shift, a spectral hole was first burned in the 167 Er3+ :Y2 SiO5
ensemble using a long laser pulse. After a wait time of 10 ms, the hole and
surrounding inhomogeneous line were scanned in frequency space using the laser
piezo drive. A constant electric field was applied during the scan. Figure 6.1a shows
the scans of the spectral hole as a function of the electric field applied parallel to
the 𝑏 axis of Y2 SiO5 . As the field increases, the spectral hole splits into two holes
corresponding to the two subclasses.
Figure 6.1b shows the central frequency of each hole as a function of applied electric
field for fields oriented parallel to the 𝑏 and 𝐷 2 crystal axes. By fitting 𝛿 𝑓± = ±𝑠𝑥 𝐸 𝑥 ,
where 𝑥 is a crystal axis direction, to the centers of the holes, the Stark shift parameter
was estimated to be 𝑠 𝑏 = 15.0 ± 0.9 kHz/(V/cm) for electric fields applied parallel to
the 𝑏 axis, and 𝑠 𝐷 2 = 3.0 ± 0.3 kHz/(V/cm) for electric fields applied parallel to the
𝐷 2 axis. Two separate devices were used in these measurements. The error bars are
95% confidence intervals from the line fit to the data, and do no take into account
any misalignment of the resonator/electrode pattern to the crystal axes, which we
estimate to be . 5◦ . The misalignment of the electrodes to the resonator is limited by
the precision of the electron beam aligned write and is less than 0.1◦ . The measured
DC Stark shift is also dependent on our calibration of the electric field applied to the
ions. For this calibration, the average electric field applied along the resonator for a
given bias on the electrodes was simulated in COMSOL as in Chapter 4.3. The bias
applied to the electrodes by the function generator was measured assuming a high
impedance load, because the electrodes act as open circuits for DC signals.
For the experiments described in the rest of this chapter, we used a hybrid 𝛼Si167 Er3+ :Y SiO resonator with the electrodes described in Section 4.3. The elec2

(a)

(b)

Figure 6.1: Stark shift measurement for electric fields aligned to 𝑏 and 𝐷 2 axes in
167 Er3+ :Y SiO . (a) Holeburning spectra with different electric field applied along
the crystal 𝑏 axis. Red lines are double Gaussian fits to the holes. (b) Linear fit to
the centers of spectral holes as a function of applied electric field along the 𝐷 2 (red)
and 𝑏 (blue) axes.
trodes and resonator were aligned such that the applied electric field 𝐸 𝑦 (𝑥) was
parallel to the 𝑏-axis of the Y2 SiO5 crystal. Using a similar procedure to the holeburning experiment described above, with the electrodes being biased in a parallel
(homogeneous field) configuration, the Stark shift parameter in this device was measured to be 𝑠 𝑏 = 11.6±0.2 kHz/(V/cm). There is a difference of ∼ 30% between this
𝑠 𝑏 value and the value measured with the parallel electrodes (𝑠 𝑏 = 15 kHz/(V/cm)).
As this device was on a different Y2 SiO5 substrate than the parallel plate device
used to measure the Stark shift 𝑠 𝑏 above, it is possible that the difference in these
two values is due to a difference in alignment of electrodes to crystal axes.
6.2

Dynamic Control of Storage Time

An electric field can be used to control the storage time of an atomic frequency comb
memory by preventing emission during an AFC rephasing event using electric-fieldinduced dephasing, then reversing the electric field to emit on a subsequent AFC
rephasing. Similar protocols have been used in rare-earth ion doped crystals, using
optical [64] and spin [7] transitions.
The pulse sequence used to achieve dynamic control of AFC storage is shown in
Figure 6.2. Not shown in the figure is the hyperfine initialization, which is performed

63
before every experiment (see Section 2.3). First, an AFC with periodicity Δ is created
using spectral holeburning (see Section 3.1). The comb burning step is repeated
𝑛comb times. A sketch of the comb is shown above the pulse sequence. After waiting
tens of milliseconds until photoluminescence from comb creation is mostly gone,
an input pulse is sent into the resonator, as shown by the red square pulse. Shown
in light red are possible emissions corresponding to rephasing events of the AFC
at times 𝑡 = 𝑚
Δ , where 𝑚 is an integer. Without electric field control, the largest
emission, which we consider the output of the memory, would be at 𝑡 = Δ1 (𝑚 = 1).
The figure shows instead an emission at 𝑡 = Δ3 (𝑚 = 3) in red. The choice of time
bin in which emission occurs is enabled by two equal and opposite electric field
pulses, one before 𝑡 = Δ1 , and one immediately before the desired emission.

V electrodes
(parallel)

ncomb

laser amp.
(input & echo)

laser freq.

Figure 6.2: Schematic representation of a digital storage time AFC experiment
(pulse sequence not to scale). The 𝑥-axis represents time. Sequence is described in
main text.
The sketches of the comb indicate the effect of the two electrical pulses. Here the
electrodes are biased in a parallel configuration, so we can assume all 167 Er3+ ions
experience roughly the same electric field. There are two subclasses of 167 Er3+
ions which experience an equal and opposite frequency shift when an electric
field is applied: 𝛿 𝑓± = ±𝑠 · 𝐸 (see Section 6.1). The combs above the electrical
pulses in Fig. 6.2 illustrate the subclass splitting. During the an electric pulse,
the phase accumulated by the two subclasses of ions in the rotating frame is 𝜙± =
2𝜋𝑡pulse 𝛿 𝑓± = ±2𝜋𝛿𝑡 pulse 𝑠𝐸. For the first electrical pulse, the "killing" pulse, the
values of 𝛿𝑡 pulse and 𝐸 are chosen such that the difference in phase accumulated

64
by the two subclasses of ions is 𝛿𝜙 = 𝜙+ − 𝜙− = 𝜋. A 𝜋 phase shift between two
ensembles of excited ions in a spectral comb leads to a suppression of emission
at the time of AFC rephasing [30, 34]. This condition continues until the phase
difference is eliminated, preventing emission for any AFC rephasing event. The
second pulse, the "reviving pulse" is equal in time, but opposite in electric field,
such that the new accumulated phase difference 𝛿𝜙 = −𝜋 cancels out the first. The
first AFC rephasing to follow the reviving electrical pulse will therefore lead to the
emission of stored light. By choosing the location in time of the second pulse, the
memory time can therefore be dynamically set to be any integer multiple of Δ1 .
As shown in Figure 6.3, digitally controlled AFC storage was demonstrated using
the protocol described in Fig. 6.2. Figure 6.3a shows the atomic frequency comb
with periodicity Δ = 20 MHz (measured 19.7 ± 0.1 MHz), which will lead to a
rephasing event every 𝑡 = Δ1 = 50 ns. The teeth of the comb are each fit to a Gaussian
function. The amplitude and background of each tooth is fit individually, while the
tooth width 𝛾 and period Δ are parameters common to all fits. The finesse of the
comb 𝐹 = Δ𝛾 is estimated from the fit to be 𝐹 = 7.22 ± 0.49 (𝛾 = 2.7 MHz). This
is likely an underestimate of the true finesse value due to a scan power that was too
large. When scanning the comb, population can be removed from the tops of the
teeth, leading to an artificially small finesse measurement. For example, the dotted
gray line in Fig. 6.3a shows the same comb but with a finesse of 𝐹 = 12.2 (𝛾 = 1.6
MHz, discussion on Fig. 6.3c explains the choice of 12.2).
Figure 6.3b shows the partly reflected input and the output for various values of 𝑚.
We consider the largest output pulse on each line (in blue) to be the memory output.
The first electric pulse was a 10 ns long pulse with amplitude 2.0 kV/cm centered
at 25 ns. The second electric pulse was 10 ns long with an opposite amplitude of
-2.0 kV/cm, and its center position was varied as 𝑡pulse 2 = 25 ns + (𝑚 − 1) × 50 ns
to allow the emission at 𝑡 = 𝑚
Δ . For the 𝑚 = 1 case, there were no electric pulses
applied. As this figure shows, the memory time can be dynamically controlled in a
digital fashion, 𝑡memory = 𝑚
Δ by choosing the position of the second electric pulse.
The presence of multiple smaller pulses following the first and largest output pulse is
a feature of the high finesse and low efficiency of the memory. Following Reference
[3], and using the notation from Section 3.2, the amplitude of the first output pulse
from an AFC memory is:

b) 7000

m=2

5000

m=3

-60

c)
normalized counts

m=1

6000

0.5

-40

-20

counts

Cavity R (arb.u.)

detuning (MHz)

4000

m=4

3000

m=5

2000

m=6

0.1

m=7

1000
0.01

m=8

100

200

300

400

t (ns)

Figure 6.3: Digitally controlled storage time in hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator. (a) Atomic frequency comb used in this experiment. The comb is created
as in the first part of Fig. 6.2, then scanned by slowly varying the frequency of the
laser using the piezo drive. Circles are data points, solid black lines are Gaussian
fits (Δ = 19.7 MHz, 𝛾 = 2.7 MHz). Dotted gray line shows curves with the same fit
parameters, except for a narrower tooth width, 𝛾 = 1.6 MHz. Detuning is measured
from 194822 GHz. (b) Storage and emission of light at different times 𝑡memory = 𝑚
Δ.
Partly reflected input pulse is shown in grey; memory output is shown in blue. Subsequent emissions (green to red) are discussed in the main text. (c) Energy emitted
in output pulse for each value of 𝑚. Black data points represent the total counts in
the 𝑡 = 𝑚
Δ time bin when all previous emissions are suppressed (blue output pulses
in (b)). Grey data points represent total counts in the 𝑡 = 𝑚
Δ time bin when previous
emissions√are not suppressed (blue-red pulses on line "𝑚 = 1" in (b)). Error bars
represent 𝑁counts . Solid line is the theoretical dephasing function for a comb with
Gaussian teeth, fitting only to the comb finesse.

𝜅in 4𝐶 0
𝜋2 1
Eout 𝑡 =
exp −
Ein (𝑡 = 0) .
=−
𝜅 (1 + 𝐶 0) 2
4ln2 𝐹 2

(6.4)

Any subsequent emission will depend on both the input pulse amplitude and the amplitude of preceding emissions. For example, the amplitude of the second emission
is [31]:

𝜅 in 4𝐶 0
𝜋 2 22
Eout 𝑡 =
=−
exp −
Ein (𝑡 = 0)
𝜅 (1 + 𝐶 0) 2
4ln2 𝐹 2
𝜅in 8𝐶 02
𝜋2 2
exp −
Ein (𝑡 = 0) .
𝜅 (1 + 𝐶 0) 3
4ln2 𝐹 2

(6.5)

66
The first term in Eq. 6.5 corresponds to the rephasing of the input at 𝑡 = 0. This
term is similar to Eq. 6.4 for the first emission, with the exception of the exponential
dephasing term, which now has a factor of 22 because the time between input in
rephasing is twice as long. The second term in Eq. 6.5 corresponds to light that is
absorbed at 𝑡 = 0, emitted and reabsorbed during the first rephasing at 𝑡 = Δ1 , then
emitted again at 𝑡 = Δ2 [31]. Because of the opposite sign between the two terms,
the second emission becomes smaller as the first emission becomes larger. In the
case of an impedance matched memory (𝐶 0 → 1) and a high finesse (𝐹 → ∞), the
two terms are equal and opposite and the emission at 𝑡 = Δ2 is suppressed entirely.
This holds also for subsequent emissions at 𝑡 = 𝑚
Δ . Therefore, for a highly efficient,
impedance matched quantum memory, only one emission at 𝑡 = Δ1 is present. Using
electric pulses, this single emission can be moved to any position 𝑡 = 𝑚
Δ.
If all previous emissions are efficiently suppressed by electric-field induced dephasing, the efficiency of the 𝑚 𝑡ℎ emission is limited by the comb finesse, as indicated
by the dephasing term in Equation 6.6 [64]:
𝜋2 𝑚2
𝑚 𝜅in 4𝐶 0
exp −
𝜂AFC,𝑚 𝑡 =
𝜅 (1 + 𝐶 0) 2
2ln2 𝐹 2

(6.6)

Figure 6.3c shows the energy emitted in the 𝑚 𝑡ℎ time bin for 𝑡memory = 𝑚
Δ . This
corresponds to the total counts in the blue pulses in each
2curve
in Fig. 6.3b. The data
𝜋 𝑚2
is fit using the dephasing term from Eq. 6.6: exp − 2ln2 𝐹 2 , since the impedance
matching prefactor will be identical given the same resonator and spectral comb.
The 𝑚 = 1 data point is excluded from the fit. This data point is believed to be
smaller than expected due to the ∼ 100 ns dead time of the SNSPD detector after
the input pulse. A comb finesse of 𝐹 = 12.2 ± 0.2 (𝛾 = 1.6 MHz) is extracted from
this fit. With this finesse, the 1/𝑒 point for digital storage time is 200 ns (𝑚 = 4). To
improve on this storage time, the finesse must be increased while keeping the comb
period Δ the same, meaning a decrease in tooth width 𝛾. This is not possible for
this current material due to superhyperfine broadening (see Section 2.4). The grey
data points in Fig. 6.3c show the total counts in the 𝑚 𝑡ℎ time bin when the previous
output pulses are not suppressed using electric pulses. This data is included to
clearly indicate that the 𝑚 𝑡ℎ pulse is larger if previous emissions are suppressed.
Calibrating Electric Pulses
The measured Stark shift can be used to determine the amplitude and length of elec𝜙kill
tric field pulses needed to kill and revive emission in this protocol: 𝐸 kill = 4𝜋𝑠𝛿𝑡
pulse

67
4𝑠𝛿𝑡 pulse . In this device, the Stark shift was measured to be 11.6 kHz/(V/cm), so for a

20 ns pulse, the 𝐸 kill = (−𝐸 revive ) = 1.1 kV/cm. To confirm this, a calibration curve
was obtained, as shown in Figure 6.4. To acquire this curve, an AFC was created,
and an electrical pulse with a fixed time of 20 ns but varying amplitude was placed
between the optical input pulse and the first AFC rephasing at 𝑡 = Δ1 . The sum of
counts in the output time bin at 𝑡 = Δ1 was measured. As the electric field amplitude
is varied, the relative phase shift between the subclasses 𝛿𝜙 changes, leading to a
sinusoidal variation in the output pulse. The minimum output counts were found to
be at 4.3 Vpp, corresponding to 0.96 kV/cm. The discrepancy between this value
and the expected value of 1.1 kV/cm is thought to be due to imperfect pulse-width
control and frequency dependent power transmission through the transmission chain
from the function generator to electrodes. Note that for a field of 2 × 0.96 kV/cm,
corresponding to a phase difference of 2𝜋, the output emission is again maximized,
although the amplitude is lower than the zero-field case. This is because of spatial
inhomogeneity in the electric field, which means not all ions can experience a ±𝜋
phase shift for the same electric pulse. A similar calibration was performed for the
10 ns pulses used in Figure 6.3.
E field (V/cm)
1000

500

1000

1500

2000

sum counts

800
600
400
200

Vpp

Figure 6.4: Calibrating the amplitude of "killing" or "reviving" pulses in a parallel
configuration of the quadrupole/parallel electrodes. Pulses are rectangular with
width 20 ns, inserted between the input and first emission of an AFC experiment.
Data points are sum counts in a time bin centered at 𝑡 = Δ1 . Solid line is a fit to a
cosine function multiplied by an exponential decay.

6.3

Frequency Shifting

With DC Stark shift control, the frequency of light stored in an AFC can be dynamically modified during emission. This is achieved using the same 𝛼Si-167 Er3+ :Y2 SiO5

68
device with electrodes (see Section 4.3), and biasing the electrodes in the parallel
configuration, such that the atomic frequency comb is shifted in frequency during
the emission of a pulse. This requires that one of the two inequivalent 167 Er3+
subclasses (see Section 6.1) be eliminated from the spectral window, leaving only
ions which experience a positive start shift, 𝛿 𝑓+ = +𝑠 𝑏 𝐸. The choice of subclass
with positive versus negative Stark shift is arbitrary.
Figure 6.5 shows a schematic of the frequency shift protocol. A two-part comb
burning procedure is used to eliminate one of the two subclasses of 167 Er3+ ions.
With the first burning step, a normal AFC containing both subclasses is created,
shown in the sketch above the pulse sequence. The center frequency of this comb
is defined as 𝑓comb = 0. For the second burning step, the two subclasses are split
using a parallel electric field. Specifically, an electric field is chosen such that
𝛿 𝑓± = ±Δ/4, where Δ is the period of the comb. The frequency offset between the
two subclasses is therefore Δ/2, which allows optimal burning of just one subclass
as it leads to the maximum splitting between the teeth of two combs with period Δ.
The second burn step uses the same laser pulses as for the first comb, but this time
centered around 𝑓comb = Δ/4, which allows ions with a positive shift 𝛿 𝑓+ = +𝑠𝐸 to
remain, and burns away ions with a negative shift 𝛿 𝑓− = −𝑠𝐸. When the electric
field is turned off, what remains is an AFC centered at 𝑓comb = 0 with only one
subclass. As shown in the sketch above the pulse sequence, this comb has half the
amplitude of a normal AFC, since it contains half the ions.

ncomb
V electrodes
(parallel)
laser amp.
(input & echo)
laser freq.

Figure 6.5: Schematic representation of a frequency shifting AFC experiment (pulse
sequence not to scale). The 𝑥-axis represents time. Sequence is described in main
text.
After waiting for tens of milliseconds for any photoluminescence from the comb
burning to be gone, a read pulse is sent in with frequency 𝑓input = 𝑓comb = 0, as

69
shown on the right side of Fig. 6.5. The rephasing of the AFC causes an emission
at 𝑡 = Δ1 . During this emission, an electric field pulse with amplitude 𝐸 pulse applied
in the parallel configuration will lead to a uniform shift of all ion frequencies, as
shown in the comb schematic above the pulse sequence. The ions will emit at a
frequency 𝑓output = 𝑓comb = +𝑠 · 𝐸 pulse .
Figure 6.6 shows frequency shifting of light stored with an AFC. The AFC had a
width of 72.5 MHz, a finesse of 𝐹 ∼ 2, and a period Δ = 5 MHz, leading to a storage
time of 𝑡 = Δ1 = 200 ns. To detect the frequency shift of the output, a heterodyne
measurement is used. The AFC output pulse was beat with a local oscillator with
Δ 𝑓heterodyne = 100 MHz and measured, then a Fourier transform was performed on
the data and 100 MHz was subtracted from the 𝑥-axis (see Appendix D).
a)

0.6
0.4

-40 -20 0

20 40

echo detuning (MHz)

output detunig (MHz)

0.8

0.2

FFT output (arb.u.)

0.8
0.6
0.4
0.2

-40 -20 0

20 40

echo detuning (MHz)

20
10
-10
-20
-1500 -1000 -500

500 1000 1500

electric field (V/cm)

Figure 6.6: AFC storage with frequency shift in hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator. (a-b) Two examples of AFC output pulses without (blue) and with (green)
a frequency shift. Filled in area is a Gaussian fit to the data; circles represent data
points. (a) A frequency shift of 13 MHz. (b) A frequency shift of 20 MHz. (c) The
frequency of AFC emission as a function of electric field applied during emission.
Circles are the centers of the Gaussian fits shown in (a-b), error bars are 95% confidence intervals for those fits. The solid line is a linear fit to the data, yielding a
slope of 𝑚 = 14.3 kHz/(V/cm).
Figures 6.6a-b show an AFC output pulse with and without frequency shifting.
Figure 6.6c shows the trend of output frequency versus 𝐸 pulse . The solid line is a
linear fit to the data. The slope of 𝑚 = 14.3 kHz/(V/cm) does not exactly match
the 11.6 kHz/(V/cm) DC Stark shift estimated in this device. The source of error is
likely our calibration of the electric field (𝑥-axis in Fig. 6.6c). In this experiment, the
electric pulses were distorted, meaning they were not constant over the entire output
pulse. The cause of this distortion is known: the DC coupled power splitter we
used between the function generator and electrodes, combined with the impedance
mismatch between the source and electrodes, led to many reflections of the electric

70
pulses. This was an issue for the frequency splitting experiment only.

subclass 1

subclass 2

+36.25

output
FWHM

-36.25

Figure 6.7: Frequency shift versus applied field in the experiment from Fig. 6.6. This
figure explains one of the reasons for a decreased output amplitude at high fields.
In addition to the curve from Fig. 6.6c, there are four sets of bands. The central
blue band shows the expected frequency shift of the output pulse (or, equivalently,
the ions which absorbed the input pulse). The extent of this band represents the
FWHM of the input and output pulses. The green bands sloping upward represent
ions outside of the comb width of 72.5 MHz, which have not been initialized into a
comb. The red bands sloping downwards are ions of the second subclass, which also
have not been initialized into a comb. These ions experience the opposite frequency
shift 𝛿 𝑓− = −𝑚 · 𝐸 pulse . At the highest fields (positive or negative), the comb ions
that absorbed the input pulse are overlapping with background ions of the second
subclass. This would lead to partial absorption of the output pulse by the second
subclass of ions, and therefore a decreased storage efficiency.
The decrease in output amplitude with frequency shift that is evident in Figures 6.6ab is still under investigation. We believe there are three reasons for the decrease
in efficiency with increasing frequency shift: (1) imperfect removal of the second
subclass of ions from the comb, (2) overlap with the ions of the second subclass
on either side of the comb (see Fig. 6.7), and (3) electric pulses varying in time
(due to distortion) leading to a time-varying frequency shift. Simulations (method
discussed in Section 6.4) confirmed that both (1) and (2) lead to a decrease in AFC
output pulse for a nonzero frequency shift. All of these limitations are surmountable.
Frequency shifting experiments with improved subclass elimination and larger comb
bandwidths would elucidate if there are any other effects leading to a decreased
output amplitude with frequency shift.

71
6.4

Bandwidth Broadening

The DC Stark shift enables dynamic control over the bandwidth of stored pulses.
The bandwidth of emitted pulses is proportional to the bandwidth of ions storing
the excitation, which can be modified using an electric field that varies across the
ion ensemble.
Figure 6.8 shows a schematic of a bandwidth broadening experiment. In the first
part of the sequence, a normal AFC is created (see Section 3.1). Next, after waiting
tens of milliseconds, an input pulse is sent into the device and an AFC rephasing
leads to an output pulse at 𝑡 = Δ1 . Electric pulses are applied during the input and
output optical pulses, and also during the wait time, with the electrodes biased in
a quadrupole configuration (see Section 4.3). The electric pulse during the input
optical pulse slightly broadens the distribution of ions comprising the comb, as
shown in the sketch above the pulse sequence. With quadrupole biasing, each ion
along the resonator’s ∼ 100 𝜇m length experiences a different electric field, which
causes the ion distribution to broaden. The second electric pulse is used to optimize
the electric field rephasing for this protocol, which will be explained in more detail
shortly. During the optical output pulse, the third and most important electric pulse
broadens the comb. The higher the amplitude of this electric pulse, the broader the
ion distribution during emission. When the broadened ions emit the output pulse,
the optical pulse will be broader in frequency, and therefore narrower in the time
domain, as indicated in the optical trace of the pulse sequence.

ncomb
V electrodes
(quadrupole)
laser amp.
(input & echo)
laser freq.

Figure 6.8: Schematic representation of bandwidth broadening AFC experiment
(pulse sequence not to scale). The 𝑥-axis represents time. Sequence is described in
main text.

72
The reason there are three electric pulses is to ensure proper rephasing of the
ensemble. To explain this, we will go through the case where there is only one
electric pulse, two electric pulses, and finally three electric pulses.
1. One electric pulse: To broaden the ion distribution during emission, only
the electric pulse at 𝑡 = Δ1 is needed. However, if there is only one electric
pulse, the erbium ensemble would not successfully rephase at 𝑡 = Δ1 and the
emission would be suppressed, since the electric field gradient leads each ion
in the ensemble to accumulate a different phase.
2. Two electric pulses: To compensate for the dephasing caused by the electric
field gradient, an electric pulse with an equal and opposite area could be used
during absorption, at 𝑡 = 0. Then the phase accumulated by each ion due
to the electric field during the pulse at 𝑡 = 0, would be cancelled out by the
phase accumulated during the second electric pulse at 𝑡 = Δ1 . There is one
complication with this method: when an optical pulse is emitted by an AFC,
the temporal profile of the output pulse is identical to the input. AFC storage
is therefore called a first-in-first-out protocol [101]. Conversely, rephasing
that happens as a result of two equal and opposite gradient electric fields leads
to a reversal of the temporal profile of the stored pulse. Protocols that use only
reversal of electric field gradients such as CRIB, are therefore first-in-last-out
protocols [51]. Since, in the current situation, we require both electric field
gradient and AFC rephasing to occur at the same time, this leads to a loss of
efficiency, as the temporal profile of the stored pulse cannot be both reversed
and not reversed anywhere except at its center.
3. Three pulses: To solve this problem, we need the rephasing due to the electric
field gradient to occur such that the output pulse profile is first-in-first-out.
This can be done by using three electric pulses with equal area, with the
middle pulse having an opposite sign from the other two. Hosseini et al.
explain this condition well in Fig. 2 of Reference [51]. The rephasing of the
AFC and the rephasing due to the electric field gradient are still not perfectly
matched in this case, as explained in the discussion on Figure 6.10, but the
output pulse amplitude should be larger than in the two pulse case.
To predict the bandwidth broadening for a given electric field, a simple model can
be used as given by Equation 6.7:

73
max
max
𝑤 output ≈ 𝑤 input + 2𝛼𝑠 𝐸 output
− 𝐸 input

(6.7)

where 𝑤 input (𝑤 output ) is the FWHM in frequency of the input (output) pulse, 𝑠 is the
max (𝐸 max )
Stark shift parameter, 𝛼 is an empirically determined parameter, and 𝐸 input
output
is the maximum electric field applied during input (output) optical pulses. This
is valid for 𝑤 input smaller than the bandwidth of the AFC. For an approximately
linear electric field gradient, the gradient applied during the input pulse would be
max
2𝐸 input
d𝐸
d𝑥
𝐿 , where 𝐿 is the length of the resonator. The intuition behind this equation
is as follows: 𝑤 input is the bandwidth of both the input pulse and of the group of ions
that is excited by this pulse. When the input is absorbed, there is a gradient induced
max at every frequency point in the comb. When the output pulse
broadening ∼ 2𝑠𝐸 input
max − 𝐸 max at every point. The
is emitted, there is an extra broadening of ∼ 2𝑠 𝐸 output
input
factor of 𝛼 accounts for the nontrivial way in which the local broadening leads to a
broadening across the entire excited ion distribution, which depends on the spatial
field distribution 𝐸 𝑦 (𝑥), the structure of the AFC, and the frequency profile of the
input pulse.

Figure 6.9 shows a bandwidth changing experiment. The 90-tooth AFC had a
width of 143.7 MHz, a finesse of 𝐹 ∼ 2, and a period of Δ = 1.6 MHz, leading
to a storage time of 𝑡 = Δ1 = 630 ns. Figure 6.9a shows AFC storage with no
bandwidth broadening (top) and maximum achieved broadening (bottom). The
broadening in frequency space leads to a narrowing of the output pulse in time.
From fits of the output pulses to Gaussians, the temporal FWHMs (Δ𝑡) of input and
output pulses are extracted and converted to bandwidth or frequency FWHMs (Δ 𝑓 )
4log2
using: Δ 𝑓 = 2𝜋 (Δ𝑡) −1 . This relationship follows from the fact that the Fourier
transform of a Gaussian pulse in the time domain is a Gaussian in the frequency
domain with 2𝜋𝜎 𝑓 = 𝜎𝑡−1 , and FWHM = 2 log2𝜎 for a Gaussian pulse with
− 𝑡

standard deviation 𝜎, 𝑓 (𝑡) = 𝑒 2𝜎2 . Note that the FWHM defined here is for the
squared amplitude of the optical pulse | 𝑓 (𝑡)| 2 , which is proportional to the number
max .
of counts. Figure 6.9b shows the trend of output bandwidth as a function of 𝐸 output
The simple linear model described in Equation 6.7 does not completely capture the
effect of the electric field gradient on the output bandwidth. To confirm that the
trend observed in the data is expected for this AFC, input pulse, and electric field
distribution 𝐸 𝑦 (𝑥), a simulation of the experiment was performed by numerically
integrating the time-evolution equations of the atoms and cavity, as described in
the next section. The simulation data reproduces the trend in FWHM vs. field,

74
a)
1500

counts

×10-3

1000
500
×10-3

200

400

600

800

time (ns)

b)
output FWHM (MHz)

20
15
10
500

1000

1500

2000

2500

max electric field (V/cm)

Figure 6.9: AFC storage with bandwidth broadening in 𝛼Si-167 Er3+ :Y2 SiO5 hybrid resonator. (a) Two AFC experiments with electric field gradient pulses. The
partially reflected input pulse with FWHM 77.4 ns (5.7 MHz FWHM in the frequency domain) is shown in both traces at 𝑡 = 0, demagnified by a factor of 103 .
The output pulse is at 𝑡 = 630 ns. Circles are photon counts, colored area is a
Gaussian fit from which widths are extracted. The top trace shows the case withmax = 𝐸 max = 0.67 kV/cm, where the width of the
out bandwidth broadening: 𝐸 output
input
output is 77.1 ± 2.0 ns (5.7 ± 0.1 MHz). The bottom trace shows the maximum
max = 4𝐸 max = 2.8 kV/cm, where the width of the output
bandwidth broadening,𝐸 output
input
is 24.3 ± 0.5 ns (18.1 ± 0.4 MHz). Insets show a schematic of the pulse sequences.
max . In all cases, 𝐸 max = 0.67
(b) Bandwidth of pulses as a function of the 𝐸 output
input
kV/cm. Filled circles are FWHM data, error bars are 95% confidence intervals from
Gaussian fits. Solid line is a fit to Equation 6.7, from which 𝛼 = 0.24 is extracted
(𝑠 = 11.6 kHz/(V/cm)). Unfilled circles are QLE simulation data (see main text for
details).
although it underestimates the broadening slightly. This discrepancy is thought to
be due to the way we modelled the extend of the optical mode along the length of the
resonator. In the real device, the optical mode penetrates the photonic crystal and
exponentially decays (see Figure 4.3b), whereas in the simulation, we assume a cutoff
at an effective optical mode penetration depth of 6 𝜇m beyond the smallest taper
hole. This affects the electric field profile experienced by the ions in the simulation
(see Figures 4.5b and 4.5f), which in turn affects the predicted broadening. The

75
effective optical penetration depth of 𝑥 eff = 6 𝜇m was found by coarsely sweeping
𝑥 eff in 1 𝜇m increments in this simulation, and choosing the value 𝑥 eff for which the
simulation best matched the data.
Figure 6.10 shows the same output pulses as Figure 6.9a, but overlaid for comparison.
Although the widths of the pulses are different, their heights are the same. This
means that the energy of the output (proportional to the area of the pulse) is lower
when the bandwidth is broadened. Also shown is an output pulse from the same
AFC in the case where no electric pulses are used. The amplitude in the no-field
max = 𝐸 max ), which indicates that the
case is the same as in the equal field case (𝐸 output
input
presence of electric pulses is not causing some dephasing or loss. Rather, the energy
of the broadened output pulse is lower because the AFC rephasing and rephasing
due to the electric field gradient do not occur on the same timescale. The AFC
rephasing always happens over the timescale determined by the input pulse, while
the rephasing due to the electric field happens over a timescale that is inversely
proportional to the strength of the applied field. Fig. 6.10 also shows an output
max = 0). In this case, we
pulse where only the first two electric pulses are used (𝐸 output
expect the dephasing induced by the first two electric pulses to prevent the emission
of an output. The presence of some counts at 𝑡 = Δ1 indicates that the electric field
gradient experienced by the ions is not ideal. To characterize this, we next look at
the ability of a single gradient electric field pulse to prevent an AFC emission in this
device.
Figure 6.11 shows the effect of a single electric gradient pulse placed between the
input and output of an AFC with period Δ = 5 MHz. This is the same as having
only the second pulse in the sequence above. The figure shows the logarithm of
the summed counts in the output time bin at 𝑡 = Δ1 . Because the electric field
gradient induces dephasing between the ions, the AFC output should be suppressed.
A decay that can be fit with two timescales is observed. We believe this is due to
the non-linear electric field variation across the resonator (see Section 4.3), which
results in some ions experiencing a high gradient, and some ions experiencing a low
gradient. Increasing 𝐸 max beyond 2.8 kV/(V/cm), or designing electrodes with a
more linear field profile 𝐸 𝑦 (𝑥) should lead to a better suppression of emission.

76
350
300

counts

250
200
150
100
50
500

600

700

800

time (ns)

Figure 6.10: Bandwidth experiment efficiency. Output pulses from the experiment
max = 𝐸 max
in Fig. 6.9 in the case where: no electric field is applied (black), 𝐸 output
input
max
max
max
(blue), 𝐸 output = 4𝐸 input (red), and where 𝐸 output = 0 (green). Circles represent
counts, solid lines are Gaussian fits.
10

log(counts)

500

1000

1500

2000

2500

3000

max E field (V/cm)

Figure 6.11: Quadrupole field induced dephasing. With electrodes biased in
quadrupole configuration, rectangular electric field pulses with width 100 ns were
inserted between the input and first emission of an AFC experiment. Data points are
sum counts in a time bin centered at 𝑡 = Δ1 = 200 ns. Solid line is a fit to a double
exponential decay. The fast decay rate was 3.6 larger than the slow rate. Note: this
data was taken using a different device on the same chip.
Simulation of the Cavity-Ion Ensemble Time Evolution
Simulations of the cavity-ion system were performed to confirm qualitative observations in Section 6.3 and trends in the data in Section 6.4. The code was adapted
from one created by Jonathan M. Kindem. These simulations involved numerically
solving the following equations of motion for the cavity field 𝑎 and a number 𝑛 of
ions represented semi-classically as field operators 𝑏 [36]:

77
𝛾
𝑏¤ 𝑖 (𝑡) = −
+ 𝑖Δ𝜔𝑖 (𝑡) 𝑏𝑖 (𝑡) − 𝑔𝑎(𝑡),
𝜅
𝑎(𝑡)
¤ =−
+ 𝑖Δ𝜔𝑎 𝑎(𝑡) +
𝑔𝑏𝑖 (𝑡) − 𝜅 in 𝑎 in (𝑡),
𝑖=1

(6.8)
(6.9)

where 𝛾 is the ion excitation decay rate, 𝑔 is the ion-cavity coupling, 𝜅 is the
total cavity energy decay rate, 𝜅in is the coupling rate of the input field to the
cavity (i.e. decay rate through one mirror), and Δ𝜔𝑎 is the cavity detuning (the
equations are solved in the rotating frame). Δ𝜔𝑖 (𝑡) is the detuning of each ion,
which can vary in time as a function of applied electric field at the location of the
ion, Δ𝜔𝑖 (𝑡) = Δ𝜔𝑖,0 ± 𝑠𝐸 (𝑥𝑖 , 𝑡), where Δ𝜔𝑖,0 is the detuning of each ion in the
absence of an applied electric field (arising from inhomogeneous broadening) and
the ± sign depends on which subclass the ion is in (see Section 6.1).
The cavity field is coupled to external fields as described by input-output formalism.
The reflected field 𝑎 r is related to the cavity field by:
𝑎 r = 𝑎 in + 𝜅in 𝑎.

(6.10)

The initial conditions are 𝑎(0) = 0, 𝑏𝑖 (0) = 0. 𝑎 in (𝑡) describes the time-varying
input pulse and |𝑎 out | 2 =|𝑎 r | 2 represents the measured output intensity.
For the simulation, a system of 𝑛 + 1 differential equations (Equations 6.8 and 6.9)
are numerically solved using 𝑁 𝐷𝑆𝑜𝑙𝑣𝑒 in Mathematica. To keep the number of
equations to a reasonable size, the number of ions simulated 𝑛 ∼ 104 is significantly
smaller than the true number of ions coupled to the cavity ∼ 107 . To accurately
represent the absorbing power of the ions, 𝑔 in the simulation is chosen such
that 𝑔total
= 𝑛𝑔 2 , where 𝑔total is measured from the cavity reflectance curve. The
time-independent frequency distribution of the ions (frequency comb with 1 or 2
subclasses) is described as a continuous distribution, and 𝑛 values of Δ𝜔𝑖,0 are
sampled from it. For a parallel electrode experiment, a time dependent scalar
±𝑠𝐸 (𝑥𝑖 , 𝑡) is added to all ion detunings. For a quadrupole electrode experiment,
𝐸 (𝑥𝑖 , 𝑡) is sampled from a probability distribution 𝜌ions (𝐸 𝑦 ) (see Section 4.3) and
varied in amplitude and time to represent each electric pulse.
6.5

Cavity-Ion Coupling and AFC Efficiency in a Hybrid 𝛼Si-Y2 SiO5 Device

In this section, we describe cavity-ion coupling and the AFC storage efficiency
in hybrid 𝛼Si-167 Er3+ :Y2 SiO5 devices. As discussed in Section 3.2, the coupling

78
rate between the cavity and the ensemble of ions in large part determines the AFC
storage efficiency in a device. Figure 6.12 shows the measured ion-cavity coupling
for the device used in this chapter. The cavity was tuned onto resonance with the
1539 nm transition of the 167 Er3+ ions by freezing nitrogen gas onto the nanobeam
at cryogenic temperatures [82]. In our setup, it was difficult to control the rate of gas
deposition, so usually the cavity would end up too far to the red (high wavelength)
after nitrogen gas tuning. To get the cavity back on resonance, we evaporated some
of the nitrogen by heating up the cavity for short periods of time (a few seconds)
with high laser power on the order of 100 mW.
Using Equation 5.1, the data in Fig. 6.12 was fit to a model and an ensemble
cooperativity was extracted. An estimated 𝜅𝜅in of 0.3 was used in the fit, and the
following values were extracted from the fit: 𝜅 = 2𝜋 × 7.3 GHz (corresponding to a
quality factor of 27×103 ) along with a 𝑔total = 2𝜋×0.64 GHz, ion distribution FWHM
= 2𝜋 × 1.05 GHz, cooperativity 𝐶 = 0.32. Using partial hyperfine initialization as
described in Section 5.2, this cooperativity was improved by a factor of ∼ 4 before
AFCs were created in each experiment.

Reflectance

0.9
0.8
0.7
0.6
0.5
0.4
0.3

-6

-4

-2

Detuning (GHz)

Figure 6.12: Cavity-ion coupling in the hybrid 𝛼Si-167 Er3+ :Y2 SiO5 device. Green
and red circles are normalized cavity reflectance data. Solid black line is a fit to
green data points. Detuning was measured from 194822 GHz.
Figure 6.13 shows the results of partial hyperfine initialization on a different device
(with a lower quality factor) on the same chip. The measured quality factor of the
device was 18 × 103 , and from the fit, the cooperativity without initialization was
found to be 𝐶 = 0.13, and the cooperativity with initialization was found to be
𝐶 = 0.53.
We can use Equations 3.8 and 3.9 to predict the AFC storage efficiency in the device
used for the experiments described in this chapter. With a pre-AFC cooperativity

79
a)

b)
0.75

Reflectance

0.75

0.7

0.65

0.6
-4

-2

Detuning (GHz)

0.7

0.65

0.6
-4

-2

Detuning (GHz)

Figure 6.13: Cavity-ion coupling in another hybrid device. Green and red circles
are normalized cavity reflectance data. Solid black line is a fit to green data points.
(a) Close-up scan of cavity-ion coupling. (b) Cavity-ion coupling after initialization
into a few hyperfine states by burning a trench to the right of the main Δ𝑚 𝐼 = 0
peak. Detuning is measured from 194822 GHz.
of 0.32 × 4 = 1.28 (including a factor of 4 from hyperfine initialization), assuming
a comb finesse of 𝐹 = 2, 𝜅𝜅in of 0.3, and assuming perfect comb initialization, the
expected AFC storage efficiency is 1.4%. The efficiency varies with comb finesse,
and will therefore vary with the choice of AFC storage time. This is because the
comb tooth width is fixed to 𝛾 ∼ 1 MHz by superhyperfine broadening, so the comb
finesse 𝐹 = Δ/𝛾 is not independent of comb period Δ.
Figure 6.14 shows an AFC experiment with a relatively short storage time of 200
ns and a storage efficiency of 0.25%. The efficiency of storage was estimated from
the ratio between the output and input pulse areas. The input pulse area was first
divided by two to account for the fact that half the energy of the input pulse is far
off-resonant. This is because a phase EOM is used to control the input frequency,
and only the positive sideband is resonant with the cavity and ions. The measured
efficiency was lower than the predicted value, likely due to an overestimate of the 𝜅𝜅in
factor or due to imperfect comb initialization, leading to a background of absorbing
ions.

counts

1500

1000

500
×10-2

100

200

300

400

500

time (ns)

Figure 6.14: AFC efficiency in the hybrid device. Data points are counts, colored
areas are fits to Gaussian functions. At 𝑡 = 0 is a reflected input pulse, measured off
resonance at 194695.3 GHz. At 𝑡 = Δ1 = 200 ns is an AFC output pulse. The pulse
and AFC were both centered at 194822.0 GHz. The storage efficiency of the AFC
is 0.25% (see main text).
6.6

Improving Efficiency in Hybrid Resonators

To increase the efficiency of AFC storage in these devices, the most important device
parameters are the quality factor 𝑄 = 𝜅total
and the intrinsic quality factor 𝑄 𝑖 = 𝜅𝜔𝑖 ,
where we define 𝜅𝑖 to be loss into any channel other than the input/output channel.
The quality factors are related by:

𝑄 = 𝑄 𝑖 (1 −

𝜅in

(6.11)

where 𝜅in is the coupling rate through the input/output mirror and 𝜅𝜅in is the parameter
introduced in Section 3.2 for characterizing how one-sided a cavity is. To make the
𝑄 smaller while keeping 𝑄 𝑖 the same, one has to make one mirror of the cavity (the
input/output mirror) less reflective.
It is essential for the intrinsic quality factor 𝑄 𝑖 to be as high as possible, because
loss will always reduce the storage efficiency. The total 𝑄 of the cavity, on the other
hand, has an optimum value. This is because of the impedance matching condition,
which demands that the effective cooperativity of the AFC must be 𝐶 0 = 1, where
𝐶 0 ∼ 𝐶𝐹 , 𝐶 is the ensemble cooperativity before comb burning and 𝐹 is the finesse
of the comb.
From Section 3.2, we rewrite Equation 3.16:
2 𝜋log2𝜌ions |𝜇| 2 𝛽mode 𝑄
𝐶≈
ℏ𝜖YSO Δ ions
Using 𝜌ions = 135 ppm, the dipole moment from Reference [74], and the measured
inhomogeneous linewidth of Δ ions = 150 MHz, we can compute the expected

81
cooperativity using Equation 3.16 for different values of 𝑄 and 𝛽mode . Then using
Equations 3.8 and 3.9, we can compute the theoretical efficiency of AFC storage.
Figure 6.15 shows the predicted AFC efficiencies as a function of intrinsic quality
factor, given a finesse of 𝐹 = 5, two values of 𝛽mode , and several total quality factors
𝑄. There are a few features to note. First, the efficiency is limited to 76% by the
dephasing term (Eq. 3.2), which is only a function of finesse. With superhyperfine
broadening limiting the width of teeth to 𝛾 & 1 MHz, a higher finesse means
a shorter memory time, since 𝑡optical storage = Δ1 = 𝐹𝛾
. Therefore, for a finesse of
𝐹 = 5, the memory time is limited to 𝑡optical storage . 200 ns. Taking into account that
two optical 𝜋 pulses need to fit within the span of 𝑡 optical storage in order to implement
AFC with spin-wave (see Section 3.1), we cannot make 𝑡optical storage much shorter.
Therefore, it seems that 𝐹 = 5 is an upper limit to the comb finesse for spin-wave
AFC storage in 167 Er3+ :Y2 SiO5 .
Another feature to note is that in Fig. 6.15b, the highest efficiency is achieved for a
quality factor of 𝑄 = 40 × 103 , and increasing the quality factor to 𝑄 = 100 × 103
actually decreases the efficiency. A higher total quality factor is not necessarily
better, because of the impedance matching condition, which requires the ensemble
cooperativity be 1. On the other hand, the intrinsic quality factor should be as high
as possible to minimize loss. Finally, by comparing Fig. 6.15a and Fig. 6.15b, it is
evident that the requirements for 𝛽mode and 𝑄 are related by 𝐶 ∼ 𝛽mode × 𝑄: with a
higher 𝛽mode value, a lower 𝑄 yields the optimal efficiency.
If we focus on using these types of devices for demonstrations in the near term,
Figure 6.15 shows the resonator parameters required to achieve a storage efficiency
of 50%. For resonators with the same fraction of optical mode in 167 Er3+ :Y2 SiO5
as those presented in this work (𝛽mode = 0.1), the quality factor required would be
𝑄 = 100 × 103 , with an intrinsic quality factor of 𝑄 𝑖 = 500 × 103 . If resonators with
𝛽mode = 0.25 are used instead, the requirements for the quality factor are slightly
lower: 𝑄 = 40 × 103 , with an intrinsic quality factor of 𝑄 𝑖 = 220 × 103 .
One possibility to overcome this superhyperfine limit on efficiency is to find an
applied magnetic field for which both the optical ground and excited states are at
ZEFOZ points, in order to eliminate superhyperfine coupling. ZEFOZ points are
predicted to exist for the hyperfine levels in this material [75, 85]. However, it is
not clear whether a magnetic field can be found such that hyperfine levels in both
the optical ground and excited states are first-order insensitive to magnetic fields.
Additionally, under the new magnetic field, hyperfine levels in the ground state

82
a)
β=0.1
75%

50%

10%

b)
β=0.25
75%

50%

10%

Figure 6.15: Predicted AFC storage efficiency for hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonators (𝐹 = 5). a) AFC efficiency versus intrisic quality factor of device 𝑄 𝑖
for different total quality factors: 𝑄 = 10 × 103 (purple), 𝑄 = 40 × 103 (blue),
𝑄 = 100 × 103 (green). The fraction of the optical mode in the 167 Er3+ :Y2 SiO5 is
assumed to be 𝛽mode = 0.1. b) AFC efficiency versus intrisic quality factor 𝑄 𝑖 with
𝛽mode = 0.25. Colors represent the same 𝑄 values as in (a). The 𝑄 values were
chosen such that at least one 𝑄 value led to the optimal efficiency in each subfigure.
would still need to be long-lived to enable spectral holeburning. Another possibility
is to look for a different host material for 167 Er3+ ions, one which does not have
nuclear spins and therefore no superhyperfine coupling. If we assume one of these
two solutions, then we can increase the finesse value beyond 𝐹 = 5. We can see from
the dephasing term (Eq. 3.2) that a finesse larger than 𝐹 = 8 would be necessary for
storage efficiencies exceeding 90%.
Assuming a solution for the superhyperfine-coupling limit to comb burning, Figure
6.16 shows the predicted efficiencies for hybrid resonators using a comb finesse of

83
𝐹 = 15. Note that the quality factors are larger than those in Figure 6.15, because
an increase in finesse must be met with an increase in ensemble cooperativity such
that 𝐶 0 ∼ 𝐶𝐹 ∼ 1. With this finesse, a storage efficiency of greater than 90% can be
achieved, for example with 𝛽mode = 0.25, 𝑄 = 100 × 103 , and 𝑄 𝑖 = 3 × 106 .
a)
90%
β=0.1
50%

10%

b)
90%

β=0.25
50%

10%

Figure 6.16: Predicted AFC storage efficiency for hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonators (𝐹 = 15). a) AFC efficiency versus intrisic quality factor of device 𝑄 𝑖
for different total quality factors: 𝑄 = 50 × 103 (purple), 𝑄 = 100 × 103 (blue),
𝑄 = 250 × 103 (green). The fraction of the optical mode in the 167 Er3+ :Y2 SiO5 is
assumed to be 𝛽mode = 0.1. b) AFC efficiency versus intrisic quality factor 𝑄 𝑖 with
𝛽mode = 0.25. Colors represent the same 𝑄 values as in (a). The 𝑄 values were
chosen such that at least one 𝑄 value led to the optimal efficiency in each subfigure.
We finally look at the effect of increasing the 167 Er3+ ion concentration. As expected
from Equation 3.16, the ion concentration increases the cooperativity. Therefore,
increasing the concentration can make a lower quality factor device more efficient. For example, Figure 6.17 shows the predicted efficiency for a resonator with

84
𝑄 = 10 × 103 . If the concentration is increased to 𝜌ions = 1000 ppm, a 50%
efficient mirror can be achieved for intrinsic quality factors as low as 60 × 103 ,
which can be easily achieved. However, it is not known whether good coherence
properties will be possible in such a highly doped sample, due to 167 Er3+ -167 Er3+
interactions. Further, this calculation does not take into account any additional
inhomogeneous broadening that occurs when the doping concentration is increased
(see Reference [18]). Any increase in inhomogeneous broadening would offset the
ions
effect of increased concentration on cooperativity due to the Δ𝜌inhom
term in Equation
3.16.
a)
β=0.1
75%

50%

10%

Figure 6.17: Predicted AFC storage efficiency vs. 167 Er3+ doping for hybrid 𝛼Si167 Er3+ :Y SiO resonators (𝐹 = 5). a) AFC efficiency versus intrisic quality factor
of device 𝑄 𝑖 for different 167 Er3+ ion concentrations: 𝜌ions = 135 ppm (purple),

𝜌ions = 200 ppm (red), 𝜌ions = 1000 ppm (orange). The fraction of the optical
mode in the 167 Er3+ :Y2 SiO5 is assumed to be 𝛽mode = 0.1, and the quality factor is
𝑄 = 10 × 103 .

85
Chapter 7

CONCLUSION
7.1

Summary

In this work, we demonstrated on-chip quantum storage of telecommunication wavelength light in two types of on-chip devices.
In a nanobeam photonic crystal resonator milled directly in 167 Er3+ :Y2 SiO5 , we
showed storage of coherent states at the quantum level using the AFC protocol.
Using a double AFC procedure and decoy state analysis, we measured a storage
fidelity for single photon states of at least 93.7% ± 2.4%. Storage of multiple
temporal modes was demonstrated. The longest storage time achieved was 10 𝜇s,
limited by superhyperfince coupling. Storage time in this experiment was fixed by
the AFC period Δ, and could not be dynamically changed.
In a hybrid 𝛼Si-167 Er3+ :Y2 SiO5 resonator, we again used the AFC protocol to store
light. Four on-chip electrodes, which could be biased in either a quadrupole or a
parallel configuration, surrounded the resonator, and provided dynamic control of
the ion ensemble through the DC Stark shift. This control was used to dynamically
change the memory time 𝑡memory = 𝑚
Δ from Δ = 50 ns to Δ = 400 ns. By shifting
the frequency of the 167 Er3+ ions during emission, we changed the frequency of
stored light by ±20 MHz relative to the input frequency. Then, using a quadrupole
electrode configuration to broaden the 167 Er3+ ion distribution, the bandwidth of
stored light was changed from 5.7 MHz (input) to up to 18 MHz (output).
On the way to these results, we performed spectroscopy on 167 Er3+ :Y2 SiO5 in new
parameter regimes, estimated the linear Stark shift parameter in this material (in 2
directions), and analyzed the effects of superhyperfine coupling on storage using the
AFC protocol. Despite careful analysis of the AFC storage efficiency in a cavity
and efforts in device design and fabrication, the storage efficiency in both of these
devices was less than 1%. We outlined a path forward for improving the storage
efficiency in these types of devices.

86
7.2

Future Directions

Toward an Efficient and Long-Lived Quantum Memory
Technical progress is needed to make a more efficient on-chip cavity quantum
memory in 167 Er3+ :Y2 SiO5 . The storage efficiency in this work for short memory
times was ∼ 0.3%, while the efficiency most commonly used for simulating realistic
near term quantum networks is 90%. The efficiency in this work was mainly
limited by a low ensemble cooperativity and loss from the cavity as described by the
in
parameter 𝜅𝜅total
. Resonators with higher intrinsic quality factors are required. For a
hybrid resonator design, the fraction of the mode in the Y2 SiO5 , 𝛽mode can also be
improved from its current value of 0.1. The cooperativity could also be improved
by increasing the 167 Er3+ concentration. However, there is a limit to how far the
concentration can be beneficially increased.
In addition to the above considerations, it must be mentioned that the total efficiency
of storage includes the coupling into and out of the cavity. The one-way coupling
efficiency in this work was ∼ 20 − 50% depending on the type of coupler used. Once
the intrinsic memory efficiency is significantly improved, the coupling efficiency
should also be increased. For example, tapered fibers [54] could be used to obtain
coupling efficiencies exceeding 95% [43, 100].
The memory time must also be increased. For a quantum memory to start to
be useful in long distance quantum repeater networks, storage time of & 1 ms is
necessary [87, 109]. Superhyperfine coupling in this material limits efficient storage
on the optical transition using the AFC protocol to . 200 ns. Spin-wave AFC could
extend the memory time to the hyperfine coherence time, which was measured to
be longer than 1 s [86]. However, it appears that superhyperfine coupling will also
limit the efficiency of storage using a spin-wave AFC protocol, as detailed in Section
6.6. The effect that superhyperfine splitting has on storage using hyperfine levels
will also need to be studied. In the sole demonstration of AFC with spin-wave in a
rare earth ion with an unpaired electron (171 Yb3+ :Y2 SiO5 ), the authors used ZEFOZ
transitions for which superhyperfine coupling is strongly suppressed [14]. ZEFOZ
hyperfine transitions have been predicted to exist in 167 Er3+ :Y2 SiO5 [75, 85], and
could be leveraged for a spin-wave AFC in this material.
167 Er3+ :Y SiO
167 Er3+ :Y SiO has attracted some excitement due to its highly coherent telecommu2

nication wavelength optical transition [19] and highly coherent hyperfine transitions

87
at microwave frequency [86]. In addition to work using 167 Er3+ :Y2 SiO5 for on-chip
quantum memories, there have been recent proposals for using 167 Er3+ :Y2 SiO5 for
other quantum applications. Asadi et al. proposed a quantum repeater networks
with nodes comprised of two single 167 Er3+ :Y2 SiO5 ions coupled to a cavity [8].
Rakonjac et al. presented measurements of long hyperfine coherence times near zero
field [85] and the same group proposed using 167 Er3+ :Y2 SiO5 ions for a microwave
quantum memory [107].
Due to the low symmetry of Y2 SiO5 and due to the high nuclear spin number of
167 Er3+ , characterizing the hyperfine levels in this material is not a trivial task.
There are twelve 2nd -rank tensors (𝑔, 𝐴, and 𝑄, for 2 cystallographic sites, for
optical ground and excited states) describing the Zeeman and hyperfine states in
167 Er3+ :Y SiO . Value for some of these tensors have been known since the 2000’s
[27, 44, 99], and Horvath et al. recently measured the hyperfine tensors in the excited
state of site 1 [50]. The excited state hyperfine tensors 𝐴 and 𝑄 for crystallographic
site 2 have yet to be measured. Measuring these tensors would be beneficial to continuing the work described in this thesis. For example, knowing the level structure
of both excited and ground state manifolds would allow a better understanding of
hyperfine initialization and spectral side-structure during comb burning.
For AFC quantum memories, 167 Er3+ :Y2 SiO5 is a good material because it holeburns
well at low temperature and moderate field, and because of its long optical and
hyperfine coherence times. Additionally, the permanent dipole moment of the
ground and excited states in 167 Er3+ :Y2 SiO5 result in a linear DC Stark shift, which
was leveraged for dynamic control in this work. However, the strength of the
4 I to 4 I transition in 167 Er3+ :Y SiO , characterized by an oscillator strength of
13
15
−7
2 × 10 [74], is weak even when compared to other rare earth ion doped materials.
The superhyperfine coupling of the 167 Er3+ electron spin to yttrium nuclei in this
material also presents serious complications for AFC quantum storage. By limiting
the width of the teeth of the atomic frequency comb, superhyperfine splitting limits
the efficiency of storage for all but the shortest storage times.
If the goal is to create an efficient quantum memory at telecom wavelengths using
rare earth ions, erbium is the only choice. However, it is possible that erbium ions
could have better properties in a crystal other than Y2 SiO5 . Such a crystal would
ideally have no nuclear spin. Studies of new erbium-doped materials such as the
recent work by Phenicie et al. would be beneficial in this regard [83].

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101
Appendix A

SIMULATION OF HYBRID 𝛼Si-167 Er3+ :Y2 SiO5 RESONATORS
This appendix details how to simulate the bandgap and reflectance of an 𝛼Si167 Er3+ :Y SiO photonic crystal mirror, as was done in Figure 4.3. The bandgaps
(Fig. 4.3a) were computed using the harmonic mode solver MPB [57]. The reflectance of the photonic crystal mirror with tapers (Fig. 4.3b) was simulated in
COMSOL Multiphysics.
A.1

Calculating Photonic Crystal Bandgap Using MPB

This is the code used to obtain the photonic crystal mirror bandgaps. The code is in
Scheme, compatible with the Scheme user interface of MPB (a Python interface is
also available). To find TM modes, we solve for modes with an even symmetry in
the 𝑦 direction.
(set-param! resolution 30)

; pixels/a

(define-param a 0.370)
(define-param d1 0.230)
(define-param d2 0.250)
(define-param h 0.310)
(define-param w 0.605)

; units of um
; units of um
; units of um
; units of um
; units of um

(set! d1 (/ d1 a))
(set! d2 (/ d2 a))
(set! h (/ h a))
(set! w (/ w a))
(define-param sc-x 1)
(define-param sc-y 4)
(define-param sc-z 4)

; units of "a"
; units of "a"
; units of "a"
; units of "a"
; cell depth
; cell width
; cell height

(set! geometry-lattice (make lattice (size sc-x sc-y sc-z)))
(define-param nSi 3.50)
(define-param nYSO 1.8)
(define Si (make dielectric (index nSi)))
(define YSO (make dielectric (index nYSO)))

102
(set! geometry (list
(make block (center 0 0 0) (size infinity w h) (material Si))
(make ellipsoid (center 0 0 0) (size d1 d2 infinity) (material air))
(make block (center 0 0 (* 0.25 (+ sc-z h)))
(size infinity infinity (* 0.5 (- sc-z h))) (material YSO))
))
(set! k-points (list (vector3 0.25 0 0)
(vector3 0.5 0 0)))
(define-param num-kpoints 53)
(set! k-points (interpolate num-kpoints k-points))
(set-param! num-bands 4)
;TM mode
(run-yeven)

To run the above code, save it as a .𝑐𝑡𝑙 file, then run it from the Terminal using the
following command:
mpb photonic_crystal_bandgap.ctl | tee "output.out"
grep yevenfreqs: "output.out" |cut -d , -f3,7- |sed 1d > "output.dat"

To plot the bands ( 𝑓 vs. 𝑘), import the .𝑑𝑎𝑡 file into MATLAB (or any other plotting
𝑘𝑎
, where
software). The first column of the data is the wavenumber data in units of 2𝜋
𝑘 is the wavenumber and 𝑎 is the unit cell dimension along the axis of propagation.
The following columns of the data are frequency, in units of 𝑓𝑐𝑎 , where 𝑓 is the
frequency, and 𝑐 is the speed of light.
A.2

Calculating Waveguide Band Diagram Using MPB

For the waveguide band diagram, a similar code is used, but the unit cell dimention
in the direction of propagation is ∞ instead of 𝑎. To run the code and plot the output
data, use the same procedure as detailed in the previous section.
(set-param! resolution 30)
(define-param a 0.370)
(define-param h 0.310)
(define-param w 0.605)

; pixels/a
; units of um
; units of um
; units of um

103
(set! h (/ h a))
(set! w (/ w a))
(define-param sc-y 4)
(define-param sc-z 4)

; units of "a"
; units of "a"
; cell width
; cell height

(set! geometry-lattice (make lattice (size no-size sc-y sc-z)))
(define-param nSi 3.50)
(define-param nYSO 1.8)
(define Si (make dielectric (index nSi)))
(define YSO (make dielectric (index nYSO)))
(set! geometry (list
(make block (center 0 0 0) (size infinity w h) (material Si))
(make block (center 0 0 (* 0.25 (+ sc-z h)))
(size infinity infinity (* 0.5 (- sc-z h))) (material YSO))
))
(set! k-points (list (vector3 0 0 0)
(vector3 1 0 0)))
(define-param num-kpoints 53)
(set! k-points (interpolate num-kpoints k-points))
(set-param! num-bands 4)
;TM mode
(run-yeven)

A.3

Calculating Reflectance of Mirror With Taper Using Comsol

The reflectance of the mirror with tapers was simulated in COMSOL Multiphysics
5.4 using the Electromagnetic Waves, Frequency Domain package. The photonic
crystal mirror was simulated in 3D, as shown in Figure A.1. Ports 1 and 2 were
Numeric ports. A three step Study was used: (1) Boundary Mode Analysis at
Port 1, (2) Boundary Mode Analysis at Port 2, and (3) Frequency Domain. For
the Boundary Mode Analyses, modes with effective index 𝑛eff = 2.5 were found,
matching the TM mode in the waveguide (see Fig. 4.2). The Reflectance value was
found from the Frequency Domain step.

104

Figure A.1: 3D photonic crystal mirror COMSOL Simulation

105
Appendix B

FABRICATION PROCESS FOR ELECTRODES
Before electrode fabrication:
• Start with a Y2 SiO5 chip
• Add 20 𝜇m × 20 𝜇m gold alignment markers, using the same procedure as for
depositing and patterning electrodes, described below.
• Deposit, pattern and etch amorphous silicon resonators, aligned to the gold
markers.
Deposit and pattern gold electrodes, as follows:
Clean Y2 SiO5 substrate:
• Rinse chip with each of the following for 30 s: acetone, IPA, DI water
• Dry chip with N2 , then bake on hot plate at 100 ◦ C for 1 minutes
Spin 950PMMA A5 positive resist:
• Cover ∼ 2/3 of chip area with resist
• Spin: 2500 rpm/20 acc/60 s
• Bake on hot plate at 180 ◦ C for 1 minute
• Allow chip to cool off for 2 minutes
Spin aquaSAVE antistatic agent on top of resist:
• Cover entire chip area
• Spin: 1500 rpm/10 acc/60 s
• Bake on hot plate at 70 ◦ C for 5 minutes

106
Aligned electron beam write (Raith EBPG 5200):
• Mount chip onto a holder with movable clips, and use dummy chip to set
height
• Pattern electrodes using aligned e-beam write, using gold alignment markers
• Dose: 1000 𝜇C/cm2 ; current: up to 100 nA
Develop resist:
• Dip the chip in DI water momentarily to remove aquaSAVE
• Dip the chip in 3:1 IPA:MIBK solution for 30 s
• Immediately rinse off with a constant stream of IPA for 30 s
• Dry chip with N2
Deposit gold (Lesker Labline Ebeam Evaporator):
• Mount chip onto a glass slide using Kapton tape and mount slide in evaporator
• Deposit 10 nm Ti (adhesion layer) and 100 nm gold, both at 1 Å/s
Gold lift-off:
• Place chip in acetone for 2 hours
• Rinse chip while submerged with a jet of acetone for 2 minutes
• Rinse in IPA and DI water for 30 s each
• Dry chip with N2

107
Appendix C

MEASUREMENT SETUP FOR CHAPTER 5
Here we describe the measurement setup used in Chapter 5. Figure C.1 shows a
schematic of the setup. For details on the components used, see Section D.3. Some
components used in this experiment differ, and will be noted below.
The 167 Er3+ :Y2 SiO5 chip was mounted on a gold-coated copper plate with indium
metal, and the plate was secured atop a stack of attocube 𝑥𝑦𝑧 nanopositioners. The
copper plate was thermally linked using a copper braid to the mixing chamber plate
of a BlueFors dilution fridge, which had a temperature of 𝑇 ∼ 25 mK.
A fiber-coupled tunable external-cavity diode laser was used to probe the nanobeam
device and implement the AFC storage protocol.
dilution fridge
MEMS
polarization
control

SNSPD
~ 100 mK
circulator

ND filters

sample holder
~ 25 mK

to wavemeter

to laser
locking

EOM
AOM
AOM

LASER
Figure C.1: Measurement setup for Chapter 5 (adapted from Ref. [30]). Details in
main text.
One percent of the laser light was directed to a wavemeter for measurement. Another
∼ 1% of the light was picked off and sent to a locking setup, in which the laser
frequency was stabilized by locking to a home-built fiber cavity using the PoundDrever-Hall technique [38]. The fiber cavity was made by splicing two ends of a
4×4 fiber-optic 99/1 splitter together. It had a linewidth of 4 MHz and a free spectral

108
range of 230 MHz. Temperature and vibration stabilization was accomplished by
placing the fiber cavity inside a box of sand, and placing that box inside a foam-lined
box. Two Newport LB1005 servo controller boxes were used to feed back to the
laser current and piezo voltage.
The laser light was directed to the sample through two acousto-optical modulators
(AOMs) for pulse shaping. The AOMs were driven by RF sources with variable
power. RF switches between the drivers and AOMs were used to block any leakage
current from the drivers.
An electro-optic phase modulator (EOM) was used to control the phase of the light
or to add strong sidebands for hyperfine initialization. To drive the EOM, one of two
RF sources was used: (1) a voltage controlled oscillator (Minicircuits ZX95-1300+)
amplified using a JDSU H301-1210 amplifer or (2) the pulsed output of the arbitrary
wave generator (AWG). A Minicircuits ZASWA-2-50DR+ switch toggled between
the two RF sources as needed. Source (1) was used for hyperfine initialization
before comb burning. Source (2) was used for adding a 𝜋 phased shift to the time
bin qubit when needed. Source (2) was also used with a 0 bias at all other times in
the experiment (no phase modulation).
Neutral density (ND) filters and polarization paddles provided attenuation and
polarization control, respectively. A fiber optic circulator directed light to the
167 Er3+ :Y SiO crystal located inside a dilution refrigerator. An aspheric lens pair
focused light from an optical fiber onto the angled coupler of the resonator. The
coupling into the device was optimized by moving the 𝑥𝑦𝑧 nanopositioners.
Light from the resonator was directed by the circulator onto a superconducting
nanowire single photon detector (SNSPD) at ∼ 100 mK [71]. Strong initialization
pulses were prevented from reaching the SNSPD by a micro electro-mechanical
switch (MEMS). A magnetic field B = 380 mT was applied to the sample, parallel
to the 𝐷 1 crystal axis, using two cylindrical permanent magnets.
A tuning line (1/4" copper pipe) pointing at the sample was used to spray nitrogen
gas onto the resonator to tune its resonance frequency.
A 4-channel AWG with 8 digital markers was used to control and synchronize the
AOM drivers, the VCO (EOM), and the MEMS switch. For details and MATLAB
code see Reference [77].

109
Appendix D

MEASUREMENT SETUPS FOR CHAPTER 6
This section describes the measurement setup used in Chapter 6. For almost all
measurements in this chapter, the setup shown in Figure D.1 was used. For the
frequency shift experiment, a frequency sensitive measurement was required, so the
setup was modified to allow for heterodyne detection, as shown in Figure D.3.
D.1

Measurements with SNSPD

Figure D.1 shows a schematic of the setup. For details on the components used, see
Section D.3. Figure D.2 shows a picture of the setup.
The setup is very similar to the one described in Appendix C.
dilution fridge
MEMS
SNSPD
~ 100 mK

polarization
control
circulator
ND filters

sample holder
~ 70 mK
to wavemeter

to laser
locking

EOM
AOM
AOM

LASER
Figure D.1: Measurement setup for Chapter 6 with SNSPD. Details in main text.
The 167 Er3+ :Y2 SiO5 chip was mounted on a copper chuck using PELCO conductive
silver paint. A custom PCB board with 4 SMP connectors was secured on top of
the chuck. Aluminum wire bonds provided electrical contact between gold contact
pads on the chip and the PCB board (see inset of Fig. D.2). The copper chuck was
thermally linked to the mixing chamber plate of a BlueFors dilution fridge via a

110
copper rod screwed into a copper assembly. The temperature of the mixing chamber
plate was 70 − 90 mK.
A fiber-coupled tunable external-cavity diode laser was used to probe the nanobeam
device and implement the AFC storage protocol.
One percent of the laser light was directed to a wavemeter for measurement. Another
∼ 1% of the light was picked off and sent to a locking setup, in which the laser
frequency was stabilized by locking to a Stable Laser Systems cavity using the
Pound-Drever-Hall technique [38]. Details of the locking setup are included in
Table D.3.
The laser light was directed to the sample through two acousto-optical modulators
(AOMs) for pulse shaping. The AOMs were driven by RF sources with variable
power. RF switches between the drivers and AOMs were used to block any leakage
current from the drivers.
An electro-optic phase modulator (EOM) was used to add strong sidebands to the
laser frequency. An amplified voltage controlled oscillator (HMC-C029) was used
to drive the EOM at frequencies between 5 and 8 GHz. Only the negative first
order sideband was resonant with 167 Er3+ :Y2 SiO5 transitions. The two first-order
sidebands were not equal in size: the negative one was larger and was measured to
contain ∼ 50% of the optical energy. The amplitude of the negative sideband was
not perfectly constant over the tuning range.
Neutral density (ND) filters and polarization paddles provided attenuation and
polarization control, respectively. A fiber optic circulator directed light to the
167 Er3+ :Y SiO crystal located inside a dilution refrigerator. An aspheric lens pair,
mounted on a stack of attocube 𝑥𝑦𝑧 nanopositioners, focused light from an optical
fiber onto the grating coupler of the resonator. The coupling into the device was
optimized by moving the 𝑥𝑦𝑧 nanopositioners.
Light from the resonator was directed by the circulator onto a superconducting
nanowire single photon detector (SNSPD) at ∼ 100 mK [71]. Strong initialization
pulses were prevented from reaching the SNSPD by a micro electro-mechanical
switch (MEMS). A magnetic field B = 980 mT was applied to the sample, parallel
to the 𝐷 1 crystal axis, using a home-build superconducting electromagnet. Trim
coils on either side of the electromagnet cancelled out stray magnetic field along
the crystal 𝑏 axis. The stray field was likely caused by misalignment of the solenoid
symmetry axis with the crystal 𝐷 1 axis, and we estimated it to be on the order of 30

111
mT before cancellation.
A tuning line (1/4" copper pipe) pointing at the sample was used to spray nitrogen
gas onto the resonator to tune its resonance frequency.
A 4-channel AWG with 8 digital markers was used to control and synchronize the
AOM drivers, the VCO (EOM), and the MEMS switch. For details and MATLAB
code see Reference [77].
A 2-channel waveform generator (10 Vpp) was used to generate the electric pulses
for DC Stark shift control. The output of each channel was split in 2 using a
Minicircuits ZSC-2-1+ power splitter, for a total of 4 channels. For the frequency
shift measurement, a DC coupled power splitter (Minicircuits ZFRSC-2050+) was
used instead. Electric pulses were sent to the chip (PCB board with SMP connectors)
via a coaxial chain with "0 dB" attenuators at every stage of the dilution fridge.

Figure D.2: Picture of measurement setup in Chapter 6. Visible are: the copper
assembly connected to the mixing chamber plate (gold); the attocube nanopositioners with wiring; the optical fiber and the 1/2" lens tube containing the lens pair for
coupling; the solenoid electromagnet; two magnet trim coils to the left and right
of the solenoid; and the copper tuning line pointing at the sample. Not visible are
the sample and the SMA cables behind the sample. The inset on the bottom left
shows an old sample mounted on a copper chuck and electrically connected with
wire-bonds to a custom PCB board with 4 SMP connectors.

112
D.2

Heterodyne Detection

This section presents the heterodyne measurement setup used for the frequency shift
measurement in Chapter 6 (Figure 6.6) and for the T2 measurement in Fig. 2.4.
In a heterodyne measurement, a small signal 𝐸 s 𝑒𝑖𝜔s 𝑡 is combined with a large local
oscillator 𝐸 LO 𝑒𝑖𝜔LO 𝑡 on a photodetector, leading to an electronic signal at the beatnote frequency 𝐸 (𝑡) = 𝐸 s 𝐸 LO cos ((𝜔s − 𝜔LO )𝑡). The local oscillator 𝐸 LO boosts
the small signal amplitude to 𝐸 s 𝐸 LO , making it detectable using a photodetector
and oscilloscope. Because the electronic signal from the photodetector varies
LO
sinusoidally with frequency 𝑓het = 𝜔s −𝜔
2𝜋 , and 𝜔LO is known, the frequency of
the small signal 𝜔s can also be measured using this method, by taking the Fourier
transform of the photodetector signal. This is how the frequency of the output pulse
is measured in Chapter 6.
Figure D.3 shows the heterodyne measurement setup. The laser, wavemeter, laser
locking and EOM are all the same as in the previous section. After the EOM, the
signal is split using a fiber-optic splitter into the signal arm and the local oscillator
(LO) arm. Only the 100 MHz AOM is used in this case, and only on the signal arm.
This AOM sets the heterodyne frequency 𝑓het = 100 MHz, as the signal arm and local
oscillator arm differ by 100 MHz (as we have said, the signal frequency can vary –
in this case it will vary around 100 MHz). The signal arm goes through additional
attenuation, described below, then through the circulator and to the sample. The
output signal from the sample goes through the circulator and is combined with the
local oscillator with another fiber-optic splitter. Both signals are then measured on
a photodetector (photoreceiver). The photodetector signal is amplified using two
amplifiers and measured on an oscilloscope (see Table D.2 for part names).
The relative phase between the local oscillator and signal is not stable in this
configuration. For this reason, there is significant jitter on the 100 MHz sinusoidal
signal, and averaging this signal for improved signal-to-noise would yield zero. For
this reason, individual frames are saved from the oscilloscope using the FastFrame
feature, then the absolute value of the Fourier transform of the signal is computed.
These spectra are then averaged together for improved signal-to-noise.
Coming back to the signal arm, the reason there are two paths after the AOM is
because the input signal that probes the AFC still needs to be quite strong (∼ 0.12
mW) in order to create an output detectable using our heterodyne setup (the local
oscillator power is ∼ 0.5 mW). However, this input power level is too strong to
create the atomic frequency comb, and the extinction ratio of the AOM is too small

113
dilution fridge

circulator

photoreceiver

10%

90%
99%

input

sample holder
~ 70 mK
ND filters

to laser locking

polarization
control

MEMS
AOM

(100 MHz)

to wavemeter

LASER

50%
1%

50%

EOM

Figure D.3: Measurement setup for Chapter 6 with Heterodyne detection. Details
in main text.
to supply both sufficiently high input power and sufficiently low comb burning
power. For this reason, there are two paths that split and recombine on the signal
arm: the strong "input" arm, and the weak comb burning arm. The MEMS switch
sends light through either of the two paths during the experiment as needed.
D.3

Parts List

Tables D.1-D.3 list the part numbers used in the measurement setups.

114
Part
laser
wavemeter
AOM 1
AOM 2
EOM
fiber-coupled splitters
neutral density filters
polarization paddles
circulator
lens pair
MEMS
SNSPD
photoreceiver

Type
Toptical CTL, 1490 nm - 1580 nm, 50 mW
Bristol 671A
Brimrose AMM-100-20-25-1536-2FP (100 MHz)
Brimrose IPM-500-100-5-1536-2FP (500 MHz)
iXBlue MPZ-LN-10-P-P-FA-FA (10 GHz)
Thorlabs or Oz Optics (PM before modulators)
Thorlabs or Oz Optics (SM after modulators)
Thorlabs (C coated)
Thorlabs (SM)
Thorlabs (SM)
Thorlabs C230260P-C
Sercalo SW1x2_9N
WSi from NASA JPL
Newport 1611 (1 GHz)

Table D.1: Optical components

Part
arbitrary wave generator
waveform generator
voltage controlled oscillator
amplifier 1
amplifier 2
oscilloscope
RF driver (AOM 1)
RF driver 2 (AOM 2)
RF switch x2 (for AOM drivers)

Type
AWG5014
Agilent Keysight 33622A
Analog Devices HMC-C029
Minicircuits ZFL-1000LN+
Minicircuits ZX60-P103LN+
Tektronix TDS7014
Brimrose FFA-100-B1-F0.5 (100 MHz)
Brimrose FFA-500-B1-F1 (500 MHz)
ZASWA-2-50DR+

Table D.2: Electronic components

Part
EOM (PDH)
EOM (offset)
laser servo electronics
stable cavity
function generator
RF signal generator
photodetector

Type
iXBlue MPX-LN-0.1-P-P-FA-FA (150 MHz)
iXBlue MPZ-LN-10-P-P-FA-FA (10 GHz)
Vescent D2-125
Stable Laser Systems (custom)
SRS DS345 (PDH EOM)
WindFreak SynthNV (offset EOM)
Thorlabs PDA05CF2

Table D.3: Laser frequency locking components