Enhancing the scan rate for axion dark matter:
Quantum noise evasion and maximally informative analysis
byDaniel A. Palken
B.A., Bowdoin College, 2014
M.S., University of Colorado, 2017

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2020

This thesis entitled:
Enhancing the scan rate for axion dark matter:
Quantum noise evasion and maximally informative analysis
written by Daniel A. Palken
has been approved for the Department of Physics

Konrad Lehnert

Eric Cornell

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Palken, Daniel A. (Ph.D., Physics)
Enhancing the scan rate for axion dark matter:
Quantum noise evasion and maximally informative analysis
Thesis directed by Prof. Konrad Lehnert

Dark matter axions, particles whose hypothesized existence could resolve two of the largest
outstanding mysteries in physics, are made difficult to detect by their extremely feeble coupling
to ordinary matter. The axion haloscope, first realized experimentally three decades ago, remains
among the most viable detection platforms, but even today’s leading technology would optimistically require many millennia to scan commonly targeted portions of axion parameter space. Today’s
axion direct detection community is searching therefore not just for the elusive particles, but fortechnologies and innovations that will permit more efficient searches. In this thesis, I present twosuch innovations. First, the resource of quantum squeezing sequesters the noise of a haloscopemeasurement into unmeasured observables, improving the sensitivity bandwidth and hence thescan rate of the detector, its primary figure of merit. Second, valuable information pertinent to
the existence of the axion is discarded by the standard hypothesis testing framework used to look
for axions. An alternative, Bayesian analysis utilizes the full information content of the haloscopemeasurement, yielding a tangible speedup at zero operational or hardware cost. Together, these
innovations are used improve threefold the scan rate of a dark matter search performed by the
Haloscope At Yale Sensitive To Axion Cold dark matter (HAYSTAC) experiment.

iv

Dedication

In loving memory of Marcia Linsey

v

Acknowledgements

It is no overstatement to say that none of the work in this thesis would have been possible
without the support, mentorship, collaboration, and friendship of the many wonderful people whohave come into my life since moving to Colorado, or were already a fixture in it beforehand.
The foundation upon which my ability to get up each morning and work towards somethinglarger than myself is built are my family and friends. To my mother and father, Judy and Rich
Palken, an infinity of thank yous for all you have done and continue to do to support my education(and to dad in particular: I am sorry I never built the warp-drive you wanted; we will have to findanother way to teach the aliens about Moe, Larry, and Curly before they get to meet you). To mybrother Greg: thank you for always reminding me of the joy of stepping outside for a breath of
fresh air, and a quick spin up a series of mountain passes. To Peyton, Sam, James, and my other
non-Colorado friends who have been a part of my life long before graduate school and will continue
to be so long after it: I appreciate the asymmetry in who has flown out to visit whom these past
several years, and the subdued chuckle (or maybe two) we’ve shared on such occasions. To Vivian,
Ursula, and Melinda, I am delighted our paths intersected for a time in a second location, and to
John, I look forward to a third. To the many close friends I have made during my time in Colorado— Adam, Bryce, Karl, Kelsey, Kristina, Lindsay, Marissa, and Steven, to name a subset who are
not singled out below — thank you for all the adventures, board games, meals, and conversations.
To the many volunteers I have worked with and mentors I have had at the Citizens’ Climate Lobby,
thank you for all you have done for me, and continue to do every day for others.
It has now been nearly a decade since I first took the introductory astronomy course with

vi
Professor Thomas Baumgarte at Bowdoin College with which set me on the path of pursuing aphysics degree. That ambition could well have been derailed by my engrossment, that very samesemester, in an English literature course modernity taught by Professor Ann Kibbie. Instead,my professors at Bowdoin encouraged me to pursue both, and the humanities taught me a deep
appreciation for the sciences which is among the greatest gifts I’ve ever received. In addition tothat, my other physics professors at Bowdoin, Dale Syphers, Mark Battle, Stephen Naculich, andmy undergraduate advisor Madeleine Msall, taught me more than a bit of physics as well. More
importantly, they taught me how to learn what I did not already know.
The physics I have learned since, and the experiments I have lent a hand to in my time at CU
Boulder and JILA, featured a supporting cast layers deep which constantly lifted me up and putme squarely in a position where I could not but succeed. It has been a joy to work with the talented
and scrupulous members of the HAYSTAC collaboration towards the goal of detecting dark matter.
Within JILA, The Lehnert Lab made for an environment where quantum circuits were fun, and so
was eating lunch together most days at the C4C. Will Kindel, Adam Reed, Jeremie Viennot, Ben
Chapman, Pete Burns, Xizheng Ma, Lucas Sletten, Eric Rosenthal, Max Urmey, and Rob Delaney
featured especially prominently throughout my time in there, and you all have my sincere thanks.
In addition to colleagues from my own generation, I have had the opportunity to work withand be informally advised by one-of-a-kind physicists who have been at it for considerably longer(though you would know as much only from speaking to, and certainly not looking at them). Reina
Maruyama brings a broad knowledge of dark matter detection to our team. Karl van Bibber knowsmuch about dark matter axions, which we both very much hope exist, but he is all the while anexistence-proof that optimism and good spirit should be thought of as properties of the journey,
destination come when it will. Eric Cornell absorbs new information with a coupling rate suggestiveof a 2π error, and it was one of the great delights of my time in JILA to get to work with him on
an unforeseen project of unforeseen scope, all for unseen dark matter. In addition, I extend myfurther thanks to Eric, as well as to Manny Knill, Kevin Stenson, and Vanja Dukic, for taking thetime to read and engage with my work as members of my thesis committee.

vii
The core group of colleagues I collaborated with include Tanya Roussy of the Cornell lab,
who proved a kindred spirit in ruthlessly applying Bayes’ theorem to look for the missing parts ofour universe, and Kelly Wurtz, who arrived in the Lehnert Lab knowing nothing about quantum
circuits, and left less than two years later having taught me how they could be used to improve
tenfold upon one of the two central ideas of this thesis.
And then there are Kelly Backes, Max Malnou, and Ben Brubaker. Winning the lottery
turns out to be underrated; I can attest to this, having done so now on three occasions.
Max Malnou joined the Lehnert Lab at a time when I was still finding my place in both groupand collaboration. I had a passion for learning from books, but struggled to make the connection
to practical application. I credit Max with changing that. He proved an adept and patient mentor
and colleague, and modeled what I strove to become as a physicist. By the end of his time at JILA,
the two of us shared a brain on many matters, and knew how to complement the work style andstrengths of each other on what remained. It was an absolute joy and privilege having a partnerto collaborate so closely with on such fascinating projects.
The small group at Yale which runs the axion haloscope around which all the work in thisthesis revolves could well have chosen any of a number of people to step into the very large shoesof their departing force of nature of a graduate student. I very much doubt any of those choices
would have been better suited to the task at hand than Kelly Backes. Kelly involves herself in every
detail, grand or minutial, of her experiment. For a project like ours with so many moving parts,
this is as difficult as it is essential. Her diligence and eagerness are in equal measure contagious,and she is a joy to sit in an office with and bang one’s head against a wall for hours on end — an
activity known to cause headaches in most other contexts.
The shoes that Kelly now comfortably wears were previously filled by Ben Brubaker. Ben,
where do I begin? We knew each other, though not terribly well, my first few years of graduate
school: you were busy building from the ground up the experiment I would inherit a role on for
my own research. It was when you came to JILA to work in the Lehnert Lab for your postdoctoral research that our more direct partnership began to flourish. Speaking very conservatively,

viii
you’ve taught me the vast majority of what I know (and also some that I don’t know) about dark
matter, and you encouraged and facilitated a dive into statistics which has permanently changed
my thinking in ways that extend well beyond the applications at hand. In a few short years, you’ve
also read and reread more dubious grammatical constructions (including this one) that I’ve written
into more scientific documents (including this one) than I can ever rightly thank you for. In doingso, you’ve helped me clarify the underlying thoughts for myself and others, and made me a betterscientist as part of the bargain.
Lastly, to my advisor, Konrad Lehnert: your dedication to and stamina for scientific inquiry
set a standard that everyone who works with you admires and attempts to emulate. I’ve seen you
exhibit integrity when there was risk in doing so, and put the ambitions and goals of your students
and post-docs ahead of your own, somehow without ever sacrificing on the quality of work thatcomes out of the lab. You have a keen sense of what matters and what does not, in both the physics
world and the human one, and it all shows in how you choose to allocate your time. Based on the
resources that have flowed to myself and others in the lab these past years, I am pretty sure you
must be writing a lot of grants, but I would be hard pressed to say when (Wednesday mornings,
perhaps?), given the seemingly uninterrupted meeting schedule you’ve constructed around all of thedifferent physics and physicists you oversee. That simultaneous attention to detail and to people
has made the Lehnert Lab a place where I have not only gotten to contribute to science, but haveenjoyed doing so.

Contents

Chapter
1 Thesis structure

1

2 Dark matter, axions, and haloscopes

5

2.1

Evidence of dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.1

Galactic velocity dispersion in the Coma Cluster . . . . . . . . . . . . . . . .

6

2.1.2

Galactic rotation curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.3

Further arguments for and against dark matter . . . . . . . . . . . . . . . . .

8

2.2The not-so-miraculous WIMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

The QCD axion solution to dark matter . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4

2.5

2.3.1

The strong CP problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2

The Peccei-Quinn mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3

Axions as dark matter candidates . . . . . . . . . . . . . . . . . . . . . . . . . 14

Detectable features of axionic dark matter . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1

Axion parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2

Local axion dark matter density . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.3

Axion lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

The axion haloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1

Previous haloscope results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.2

Haloscope scan rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.3

Improving the scan rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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3 Josephson parametric amplifiers
3.1

3.2

3.3

3.4

3.5

29

The Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1

Josephson inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.2

Josephson energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

The DC SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1

Critical current and inductance . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.2

Geometrical inductance and flux-trapping . . . . . . . . . . . . . . . . . . . . 37

Circuit quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1

LC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.2

Lagrangian and Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.3Classical equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.4

Quantization and ladder operators . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.5

Heisenberg equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Lumped-element Josephson parametric amplifier . . . . . . . . . . . . . . . . . . . . 42
3.4.1

Basic TKC circuit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.2

TKC Lagrangian and Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.3

Kerr Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.4

Input-output theory derivation of gain . . . . . . . . . . . . . . . . . . . . . . 50

3.4.5

Gain-bandwidth product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.6

Single-quadrature amplification, deamplification, and squeezing . . . . . . . . 56

3.4.7

Derivation of flux-pumped JPA gain . . . . . . . . . . . . . . . . . . . . . . . 58

Validity of the Duffing approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.1

The Qp condition for a single-junction JPA . . . . . . . . . . . . . . . . . . . 63

3.5.2

Significance of the participation ratio . . . . . . . . . . . . . . . . . . . . . . . 65

3.5.3

From junction to array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5.4

Series RLC and general Thévenin case . . . . . . . . . . . . . . . . . . . . . . 66

3.5.5

Parallel RLC and general Norton case . . . . . . . . . . . . . . . . . . . . . . 68

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3.6

3.7

3.5.6

The Qp condition for an array-based JPA . . . . . . . . . . . . . . . . . . . . 69

3.5.7

Independence of the validity of the Duffing approximation from the nonlinearity 70

3.5.8

Modified condition for the Duffing approximation . . . . . . . . . . . . . . . . 71

Current-pumped JPA implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6.1

SQUID array specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.6.2

TKC specifications and simulations . . . . . . . . . . . . . . . . . . . . . . . . 74

3.6.3

Characterization measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Flux-pumped JPA implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.7.1

Flux-pumped JPA design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.7.2

Characterization measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4 Optimal current-pumped Josephson parametric amplifier operation for vacuum squeezing

95

4.1

Signal transformation in phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.2

Coherent state deamplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3

Optimizing vacuum squeezing with a current-pumped JPA . . . . . . . . . . . . . . . 103

4.4

Experimental details and calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.5

4.4.1

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.4.2

Inter-JPA loss calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.4.3

Added noise calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Concluding remarks on current-pumped JPAs optimized for squeezing . . . . . . . . 110

5 Squeezed vacuum used to accelerate the search for a weak classical tone
5.1

112

Theory of the squeezed state receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.1

Lossless calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.1.2

Calculation with transmission losses . . . . . . . . . . . . . . . . . . . . . . . 123

5.1.3

Equivalence between single- and double-quadrature amplification . . . . . . . 126

5.2

Model for the axion field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3

Enhanced signal visibility with squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 132

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5.4

Mock-haloscope experiment: accelerated search for a weak classical tone . . . . . . . 135

5.5

Experimental details and calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5.1

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.5.2

Phase-locking of SQ and AMP pumps . . . . . . . . . . . . . . . . . . . . . . 140

5.5.3

Intra-SSR loss measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.6

Complementary measurements of the microwave tone’s improvement in resolution . . 143

5.7

Processing of the SSR spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.8

Concluding remarks on the SSR for enhancing the scan rate of real haloscope searches148

6 Axion haloscope processing and analysis

149

6.1

Haloscope processing overview

6.2

Axion search analysis considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2.1

6.3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Distinguishing features of axion and haloscope searches . . . . . . . . . . . . 156

Frequentist threshold framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.3.1

Framework details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.3.2

Aggregate exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.4

p-value framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.5

Look-elsewhere effect: implications for detection and exclusion . . . . . . . . . . . . 167
6.5.1

Frequentist frameworks and the look-elsewhere effect . . . . . . . . . . . . . . 172

7 A maximally informative axion dark matter analysis
7.1

7.2

174

Bayesian power-measured framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.1.1

Updated belief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.1.2

Aggregation and the look-elsewhere effect . . . . . . . . . . . . . . . . . . . . 178

7.1.3

Choice of priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.1.4

Correlations between grand spectrum bins . . . . . . . . . . . . . . . . . . . . 183

7.1.5

Practical departures from Gaussianity . . . . . . . . . . . . . . . . . . . . . . 184

Effects of data processing on information content . . . . . . . . . . . . . . . . . . . . 186

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7.3

7.4

7.5

Comparison of analysis frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.3.1

Bayesian threshold 1 framework. . . . . . . . . . . . . . . . . . . . . . . . . 190

7.3.2

Bayesian threshold 2 framework. . . . . . . . . . . . . . . . . . . . . . . . . 192

7.3.3

Outcome distributions for Bayesian analyses . . . . . . . . . . . . . . . . . . . 195

Test using HAYSTAC Phase I dataset . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.4.1

Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.4.2

Least-unlikely candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

7.4.3

Additional features of the Phase I dataset . . . . . . . . . . . . . . . . . . . . 204

Concluding remarks on statistical analysis frameworks for axion searches . . . . . . . 205

8 A quantum-enhanced search for dark matter axions

207

8.1

Setup and operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

8.2

Data processing and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

8.3

8.4

8.2.1

Subspectra and raw spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8.2.2

Processed spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

8.2.3

Rescaled spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.2.4

Combined spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

8.2.5

Grand spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.3.1

JILA analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

8.3.2

Official HAYSTAC Phase II results and conclusions . . . . . . . . . . . . . . 227

Calibration measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
8.4.1

Squeezing and cavity spectral density measurements . . . . . . . . . . . . . . 231

8.4.2

Added noise measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

9 Search for axion-like dark matter with the JILA eEDM experiment

234

9.1

ALP sensitivity of the JILA eEDM dataset . . . . . . . . . . . . . . . . . . . . . . . 236

9.2

Spectral content of the JILA eEDM dataset . . . . . . . . . . . . . . . . . . . . . . . 238

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9.3

Quantifying the hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
9.3.1

No-ALP hypothesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

9.3.2

ALP hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

9.3.3

Coherent ALP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

9.3.4

Partially coherent ALP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

9.3.5

Finite sample time correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

9.4

ALP exclusion results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

9.5

Prospects for future JILA eEDM ALP searches . . . . . . . . . . . . . . . . . . . . . 254

10 Conclusion and outlook
10.1 Conclusion

256. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

10.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Bibliography

264

Appendix
A NIST trilayer fabrication process

290

B Resolution to the flux-tuning problem for flux-pumped JPAs

293

C Effect of improper tuning rod thermalization on scan rate297

D List of Abbreviations

301

Tables

Table
5.1

Representative physical values used to estimate axion port parameters . . . . . . . . 130

5.2

Model parameter values calculated using the physical values from Table 5.1 . . . . . 132

7.1

Sample set of probabilities used to estimate prior axion hypothesis probability . . . . 182

7.2

Comparison of statistical analysis frameworks . . . . . . . . . . . . . . . . . . . . . . 194

9.1

ALP coherence for high and low frequency ALP analyses and HAYSTAC Phase II . 250

C.1 Theoretical parameters for the HAYSTAC Phase II haloscope with and without a
hot rod and/or squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Figures

Figure2.1

Galactic rotation curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

Bullet Cluster collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3

WIMP parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4

Axion-photon parameter space prior to this thesis

2.5

Virialized axion lineshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6

Electric and magnetic fields in simple and tunable cylindrical cavities . . . . . . . . . 24

2.7

Axion-photon exclusion obtained using haloscopes . . . . . . . . . . . . . . . . . . . 25

3.1

Representations of a Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2

DC SQUID circuit representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3

Vector potential integration path for a SQUID . . . . . . . . . . . . . . . . . . . . . 34

3.4

Total versus externally applied flux for a SQUID with different geometrical inductances 38

3.5

Simple LC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6

Circuit schematic and IOT model for a TKC . . . . . . . . . . . . . . . . . . . . . . 44

3.7

Relationship between TKC and JPA . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8

Duffing oscillator response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.9

Relation between one- and two-quadrature gain . . . . . . . . . . . . . . . . . . . . . 57

. . . . . . . . . . . . . . . . . . . 18

3.10 PNL and SNL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.11 Simple embedding circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

xvii
3.12 Series inductor algebra showing the equivalence between stray and array linear inductances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.13 Parallel inductor algebra showing the equivalence between stray and array linear
inductances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.14 Duffing and higher-order oscillator responses at different quality factors . . . . . . . 72
3.15 Josephson junction critical current measurements . . . . . . . . . . . . . . . . . . . . 75
3.16 Superconducting niobium wiring layout for a pair of SQUIDs . . . . . . . . . . . . . 76
3.17 SEM images of a TKC for a current-pumped JPA . . . . . . . . . . . . . . . . . . . . 76
3.18 Layouts and design parameters for current-pumped TKCs . . . . . . . . . . . . . . . 77
3.19 Simulated responses and parameter values for current-pumped JPAs . . . . . . . . . 78
3.20 JPA configuration for current pumping . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.21 Resonance frequency measurements for current-pumped JPAs . . . . . . . . . . . . . 80
3.22 Flux tuning of current-pumped JPA resonance with a DC current . . . . . . . . . . . 81
3.23 Gain and bandwidth measurements for a current-pumped JPA . . . . . . . . . . . . 82
3.24 Discarded flux-pumped JPA designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.25 Final flux-pumped JPA design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.26 EM simulations of a flux-pumped JPA . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.27 Flux-pumped JPA scattering parameter simulations . . . . . . . . . . . . . . . . . . 88
3.28 Flux-pumped JPA gain simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.29 JPA configuration for flux pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.30 SEM images of a TKC for a flux-pumped JPA. . . . . . . . . . . . . . . . . . . . . 90

3.31 Photo of a mounted flux-pumped JPA . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.32 Flux tuning of flux-pumped JPA resonance with a DC current. . . . . . . . . . . . 92

3.33 Flux-pumped JPA gain curves showing tunability between and within different coiltunings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.34 Flux-pumped JPA gains at multiple operating points . . . . . . . . . . . . . . . . . . 94
3.35 Power-handling measurements for flux- and current-pumped JPA operation . . . . . 94

xviii
4.1

Theory of distortion in a JPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2

Experimental setup for coherent tone distortion and vacuum squeezing . . . . . . . . 100

4.3

Dependence of coherent tone distortion upon amplifier operating point . . . . . . . . 101

4.4

Dependence of deamplification upon amplifier operating point . . . . . . . . . . . . . 102

4.5

Vacuum squeezing with current-pumped JPAs . . . . . . . . . . . . . . . . . . . . . . 104

4.6

Dependence of vacuum squeezing on operating point . . . . . . . . . . . . . . . . . . 105

4.7

Full experimental setup used for both coherent tone deamplification and squeezing . 107

4.8

VTS noise calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1

Schematic of the SSR and cavity and illustration of the benefit of squeezing . . . . . 115

5.2

Theoretical scan rate enhancement for idealized and realistic haloscope models . . . 118

5.3

Equivalent modelling of the fictitious axion port

5.4

Photos of the Al cavity used to test the SSR

5.5

Squeezing achieved in the SSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.6

Visibility improvement from squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.7

Data processing and results from many trials of the mock-haloscope experiment . . . 137

5.8

Full experimental setup used for the mock-haloscope experiment . . . . . . . . . . . 141

5.9

Intra-SSR transmission loss calibration measurement . . . . . . . . . . . . . . . . . . 143. . . . . . . . . . . . . . . . . . . . 130. . . . . . . . . . . . . . . . . . . . . . 133

5.10 Complementary measurements of signal visibility in the SSR . . . . . . . . . . . . . 144
6.1

Gaussian probability distributions for no-axion and axion distributions . . . . . . . . 155

6.2

Illustrative grand spectrum data representation for standard haloscope subject to an
FT analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.3

Effective correction to the required SNR from the look-elsewhere effect . . . . . . . . 171

7.1

Four possible outcomes for a two-scan FT protocol . . . . . . . . . . . . . . . . . . . 192

7.2

Outcome probability distributions for total and aggregate prior updates . . . . . . . 197

7.3

HAYSTAC Phase I BPM reanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 201

xix
8.1

Simplified HAYSTAC Phase II experimental schematic . . . . . . . . . . . . . . . . . 209

8.2

HAYSTAC cavity photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

8.3

Photos of the LD250 BlueFors dilution refrigerator used for HAYSTAC Phase II . . 210

8.4

Field specifications for the solenoid magnet used in HAYSTAC . . . . . . . . . . . . 211

8.5

Mixing chamber stage layout and photo of magnetic shielding . . . . . . . . . . . . . 212

8.6

Measurement of JPA tunability with magnetic field on and off . . . . . . . . . . . . . 213

8.7

Squeezing measured in HAYSTAC Phase II . . . . . . . . . . . . . . . . . . . . . . . 213

8.8

Illustration of the SSR-equipped haloscope showing the transformation of the vacuum
state in quadrature space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

8.9

Advantage conferred by squeezing calculated with parameters used in HAYSTAC . . 215

8.10 Raw spectrum cuts for HAYSTAC Phase II initial scan . . . . . . . . . . . . . . . . . 217
8.11 HAYSTAC Phase II raw spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.12 Individualized SG filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.13 HAYSTAC Phase II processed spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.14 Rescaling visibility and individual rescaled spectrum . . . . . . . . . . . . . . . . . . 222
8.15 HAYSTAC Phase II initial scan combined spectrum . . . . . . . . . . . . . . . . . . 224
8.16 HAYSTAC Phase II initial scan grand spectrum . . . . . . . . . . . . . . . . . . . . 225
8.17 Results of the JILA analysis of the HAYSTAC Phase II dataset . . . . . . . . . . . . 226
8.18 Final HAYSTAC Phase II axion exclusion . . . . . . . . . . . . . . . . . . . . . . . . 228
8.19 Model used for HAYSTAC Phase II noise calibration measurements. . . . . . . . . 229

8.20 Sample calibration measurements to determine cavity spectral density and squeezer
gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.1

JILA eEDM dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

9.2

LSSA of the JILA eEDM dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

9.3

Phase sensitivity of the JILA eEDM dataset . . . . . . . . . . . . . . . . . . . . . . . 242

9.4

Construction of the no-ALP and ALP distributions in quadrature space . . . . . . . 245

xx
9.5

Effect of ALP field decoherence on the ALP-only distribution variance . . . . . . . . 250

9.6

Effect of finite sample time on the ALP-only distribution variance . . . . . . . . . . 251

9.7

ALP exclusion results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

9.8

Astrophysical and laboratory limits on the ALP-gluon interaction . . . . . . . . . . . 254

10.1 Axion namesake-inspired representation of the SSR . . . . . . . . . . . . . . . . . . . 256
10.2 Impact on scan rate of decreasing haloscope noise over time . . . . . . . . . . . . . . 258
10.3 Concept for future quantum enhancement to haloscope scan rate . . . . . . . . . . . 262
A.1 NIST trilayer fabrication process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
B.1 Location of cuts made with a FIB on dummy SQUIDs . . . . . . . . . . . . . . . . . 293
B.2 Bidirectional flux-tuning sweep of FIB-cut JPA showing fixed tuning problem . . . . 294
B.3 Gain map and traces for a FIB-cut JPA . . . . . . . . . . . . . . . . . . . . . . . . . 295
C.1 Theoretical signal and noise spectral densities with and without hot rod and squeezing298
C.2 Theoretical axion visibilities with and without hot rod and squeezing . . . . . . . . . 299
C.3 Theoretical scan rates with and without hot rod and squeezing . . . . . . . . . . . . 300

Chapter 1

Thesis structure

We do not know what constitutes the cosmos, and it may take us a very long time to findout. The problem is that of dark matter, and the difficulty of directly detecting it. This thesis sets
out to speed up the search for dark matter axions through the use of quantum measurement andstatistical analysis.
I begin in Ch. 2 with a motivating discussion of the problem of dark matter, the axion as apotential solution, and the axion haloscope as the primary device used in efforts to detect it. In
Sec. 2.1, I overview some of the most prominent evidence for dark matter. Section 2.2 discusses theweakly interacting massive particle (WIMP),1 a once highly promising dark matter candidate now
somewhat disfavored by experimental null results. Section 2.3 then turns to the candidate mostcentral to this thesis, the axion of quantum chromodynamics. This section and the ensuing Sec. 2.4
discuss those properties that qualify axions as dark matter candidates, and those pertinent todetecting their signature experimentally, respectively. In Sec. 2.5, I introduce the axion haloscope.
The chapter concludes with a quantitative discussion of what limits the performance of axion
haloscopes, as parametrized by their scan rates — the speeds at which they can move throughparameter space in search of dark matter axions.
Chapter 3 introduces Josephson parametric amplifiers (JPAs), devices that can create andprocess squeezed states of microwave light, which will ultimately enable me to enhance the scanrate of an axion haloscope. This chapter beings in Sec. 3.1 with a discussion of the Josephson
1

A full list of acronyms and other abbreviations appearing in this thesis can be found in Appx. D.

2junction, whose special properties will be used to enable the noiseless amplification and squeezing
later in this thesis. I derive in Sec. 3.2 key properties of pairs of Josephson junctions known assuperconducting quantum interference devices (SQUIDs). After Sec. 3.3 provides the language ofcircuit quantization, Sec. 3.4 combines the building blocks developed thus far into a JPA. I derive
how JPAs achieve gain and squeezing, before turning in Sec. 3.5 to a discussion of the validity of theDuffing approximation, a key simplification made in the JPA gain derivation. The implications of
this discussion are important for understanding the physics of array-based JPAs as a partial remedy
to the problem of limited power handling. Finally, turning our attention to the devices themselves,
Secs. 3.6 and 3.7 discuss the design and characterization of JPAs intended to be operated withdifferent types of pump tones. The current-pumped JPAs of Sec. 3.6 will be central to Ch. 4, while
the flux-pumped JPAs of Sec. 3.7 will feature in Chs. 5 and 8. The fabrication process, performed
at NIST Boulder, is discussed in Appx. A, and the resolution to a flux-tuning problem that plagued
our flux-pumped JPAs is discussed in Appx. B.
Chapter 4 presents a technique to improve the power handling of current-pumped JPAs used
for squeezing. I begin in Sec. 4.1 with a theoretical treatment of how a JPA transforms a signal
in phase space, drawing on results from Ch. 3. Section 4.2 details an experiment performed withcoherent states of the microwave field whose amplitudes are comparable to vacuum fluctuations.
The results of this experiment suggest that fragility to fluctuations in the power of a JPA’s pumptone can be traded off for improved squeezing. In Sec 4.3, I show in a second experiment performedon vacuum states that this result indeed improves the squeezing that can be delivered to a physicalsystem such as an axion haloscope. Section 4.4 describes experimental details and calibration
measurements for the transmission losses and added noise in the setup used throughout this section.
The improvement achieved in this chapter, summarized in Sec. 4.5, is limited by presently achievable
degrees of squeezing (in turn limited by transmission losses within the system), suggesting that thesuperior options for parametric amplification are the flux-pumped JPAs used in ensuing chapters.
Chapter 5 modifies the experimental architecture of Ch. 4 from one for testing squeezing, to
one for using it: the integration of two JPAs into an axion haloscope-like setup forms a squeezed

3state receiver (SSR) apparatus capable of exploiting the uncertainty principle of quantum mechanics
to enhance the search rate for an axion-like signal. Section 5.1 uses input-output theory to show that
the SSR works by eliciting a favorable tradeoff between peak haloscope sensitivity and bandwidth,and includes a detailed theoretical calculation with experimental parameters used throughout thechapter. Section 5.2 mathematically models how the axion field interacts with a haloscope. Sections
5.3 and 5.4 report on a pair of experiments to demonstrate the efficacy of the SSR. In Sec. 5.3, aprobe tone much larger than an axion is used to characterize the squeezing over the bandwidth of
the SSR. Section 5.4 describes a complete mock-haloscope experiment, injecting and detecting anaxion-like tone of unknown frequency with and without the assistance of squeezing. Section 5.5
discusses protocols and calibrations common to the two experiments in this chapter. Section 5.6
then discusses complementary measurements to those taken in Sec. 5.3, and Sec. 5.7 describes thedata processing used in Sec. 5.4. The chapter concludes in Sec. 5.8. Discussion then shifts fromquantum to classical information processing, before ultimately returning to a real axion search, inwhich both tools are used advantageously.
Chapter 6 presents data processing and statistical analysis techniques commonly used inaxion haloscopes. Section 6.1 overviews the processing protocol of the Haloscope At Yale Sensitive
To Axion Cold dark matter (HAYSTAC) experiment, and Sec. 6.2 introduces some considerations
pertinent to axion search analyses. Sections 6.3 and 6.4 describe frequentist frameworks commonlyused to test for axions. In relation to these frameworks and that introduced in the next chapter,
Sec. 6.5 discusses the role of the look-elsewhere effect in axion searches.
In Ch. 7, I introduce the Bayesian power-measured (BPM) statistical framework, a novel
analysis designed for the particulars of axion direct detection, and demonstrate its utility on existing
HAYSTAC data. Section 7.1 introduces the framework and describes key features pertinent to axion
detection. Section 7.2 then evaluates the existing haloscope data processing protocols, and Sec. 7.3
contextualizes the BPM framework alongside the frequentist framework of Sec. 6.3, showing in what
limits the two may be compared. Section 7.4 then applies the framework to the HAYSTAC Phase
I dataset, achieving deeper reach into axion parameter space at zero hardware or operation cost.

4
The discussion of statistical frameworks for axion detection concludes in Sec. 7.5.
Chapter 8 combines the innovations of Chs. 5 and 7 to deliver a threefold combined speedup
to the scan rate of Phase II of the HAYSTAC experiment. Section 8.1 describes the setup andbasic operation of the experiment, Sec. 8.2 presents the data processing and analysis of the Phase
II dataset, and Sec. 8.3 discusses squeezed state receiver operation and presents the data processing and results. We observe no signature of the axion, reporting exclusion results obtained atroughly three times the scan rate achievable without the innovations presented in this thesis. Section 8.4 discusses calibration measurements and protocols, and an issue with improper tuning rod
thermalization during the data run is explored in Appx. C.
Lastly, Ch. 9 uses the BPM framework of Ch. 7 for a separate application, searching for thesignature of axion-like particles (ALPs) in data from the JILA electron electric dipole moment(eEDM) experiment as much as 17 orders of magnitude lower in mass than the axions targeted in
Ch. 8. Section 9.1 discusses how the JILA eEDM experiment may be sensitive to ALPs. Section 9.2
then uses a lease-squares spectral analysis to extract spectral content of the measurement. Section
9.3 formulates and refines the competing ALP and no-ALP hypotheses, which are tested subject
to the measured data, with results reported in Sec. 9.4 and prospects for futures searches discussed
in Sec. 9.5.
Chapter 10 concludes in Sec. 10.1, and in Sec. 10.2 discuss future prospects for speeding up
dark matter axion searches both by using quantum measurement and by taking full advantage ofthe analysis framework presented in this thesis.

Chapter 2

Dark matter, axions, and haloscopes

Arguments pointing to the presence of more gravitationally interacting mass out in spacethan is indicated by visible light date back to Lord Kelvin in the late 19th century [1]. The
ensuing hundred-plus years produced ever more compelling data points in support of the dark
matter hypothesis: that a substantial fraction of the universe’s matter density, presently estimatedat 85% [2], does not strongly interact with electromagnetism (or else we would see it, and it would
not be “dark”), and is not made of particles from within the Standard Model of particle physics.
Theoretical predictions and experimental evidence have each played integral roles in the history ofdark matter, motivating promising theories and ruling some of them out. A dark matter candidate
is viable insofar as it is well motivated by known physical phenomena, and not strongly disfavored byexisting experimental evidence. Perhaps the dark matter candidate that best fits this description
is the axion particle predicted to resolve an outstanding mystery in quantum chromodynamics(QCD), the study of the strong nuclear force.
This chapter presents an overview of the history of our understanding of dark matter. Startingwith a broad focus, I address some of the key pieces of evidence for dark matter. Zooming in, I
touch upon some of the most historically popular candidates to constitute the dark matter, andin particular the QCD axion. This chapter ends with a discussion of axion haloscopes, detectors
designed to observe dark matter axions in the galactic halo by first converting them to microwave
photons, and the present limitations upon their performance.

6

2.1

Evidence of dark matter
The case for dark matter has been built over the past century from sources ranging from thebehavior of bodies within galaxies, to the interaction of galaxies, to measurements of the cosmos.
This section discusses a few prominent examples: observations of the velocity dispersion of galaxies
within clusters and of the speed of orbiting bodies within galaxies, which each provided some of
the most compelling early evidence of dark matter. Evidence from the collision of two galactic
clusters has strengthened the case for dark matter over competing theories, and, to date, the mostprecise measurements of dark matter density come from the cosmic microwave background (CMB)
radiation [3]. The broader history of dark matter is a rich and fascinating topic, and the interested
reader is encouraged to consult Ref. [4] for a more comprehensive overview or Ref. [5] for a nuanced
examination of some of the sociological forces that drove physicists to the problem.

2.1.1

Galactic velocity dispersion in the Coma Cluster

Fritz Zwicky had one of the most impactful early insights into the composition of our universe
when in the 1930’s he first applied the virial theorem to the motion of luminous bodies within the
Coma Cluster of galaxies [6, 7]. The virial theorem,
2hT i = −nhV i,(2.1)

relates the total time-averaged kinetic (T ) and potential (V ) energies of a system bound by apotential of the form V ∝ r−n . For the gravitational potential energy
VG = −

Gm1 m2r(2.2)

between point masses m1 and m2 a distance r apart, n = 1. Zwicky estimated hT i based on
the observed amount of luminous matter within the cluster. He found an order-of-magnitude
discrepancy between the velocity dispersion of galaxies implied by Eq. (2.1) and the actual, observed
values.

7
Zwicky’s results implied a total abundance of matter ∼ 500 times greater than just its luminous component. This result turns out to be incorrect in magnitude, but still remarkable for itsprescience.2 It would be decades before dark matter became fully recognized as one of the central,
unexplained riddles of modern physics.

2.1.2

Galactic rotation curves

Fritz Zwicky’s early estimates of the dark matter density of the Coma Cluster had relied
upon the velocity dispersion of galaxies, pointedly eschewing the possibility of inferring the massof individual galaxies from their rotation curves: how fast luminous matter orbits the galactic
center as a function of radial distance. Zwicky wrote: “It is not possible to derive the masses of[galaxies] from observed rotations, without the use of additional information” [4,7]. The additional
information Zwicky was referring to included the internal damping of motion within galaxies fromthe interaction of massive bodies. A few years later, however, Subrahmanyan Chandrasekhar
showed that these viscous interactions were effectively negligible [9], opening up another window
into gauging dark matter density.
Galactic rotation curves provide a very straightforward test of whether the matter distributionof a galaxy is as it appears — that is, proportional to the luminous matter. A simple Newtonian
calculation (see e.g. Example 5.2 of Ref. [10]) shows that for a typical galaxy where the galactic
mass M appears to be clumped tightly at the center, the orbital velocity vorb should rise initiallywithin the galaxy, and then fall off outside of it with radial distance r asrvorb =

GM
.
r

(2.3)

The dashed line in Fig. 2.1a shows what this would look like for the Andromeda galaxy. Spectrographic measurements (solid line) such as those taken by Vera Rubin and Kent Ford proved
inconsistent with this, revealing a much flatter curve [11]. Measurements obtained using the 21
2

In particular, Zwicky was using a now-obsolete value [8] for the Hubble constant, describing the rate at which
the universe is expanding, to calculate the velocities of galaxies at a distance. An updated value [2] returns a morereasonable answer, and one still in line with the Zwicky’s basic conclusion.

8
cm line of hydrogen [12], Fig. 2.1b (left column), provided a proxy for the amount of luminous
matter out at larger radii, where there are not many stars. While the luminous matter was indeed
clumped towards the center across several observed galaxies, the rotation curves (right column)
stayed remarkably flat past the point where most of the matter was contained. The flat rotation
curves were suggestive of a dark matter halo around galaxies, much broader in extent than the
directly observable cores of luminous matter [13].

Figure 2.1: (a) Measurements of the orbital speed of the Andromeda galaxy (solid line) versus
distance away from the galactic center shown against a simple model assuming only luminous
matter, Eq. (2.3). (b) 21 cm line measurements (left column) of the hydrogen density withingalaxies with flat rotation curves (right column). 80% of the hydrogen lies within R80 . Panel (a)reproduced from Ref. [10]. Panel (b) reproduced from Ref. [12].

2.1.3

Further arguments for and against dark matterArguments for dark matter have proliferated since the 1970’s, and have disfavored manyotherwise plausible alternatives. For instance, it turns out that more matter exists as ionizedhydrogen between galaxies than as stars within them [14]. Ionized hydrogen does not emit atthe 21 cm line used in Sec. 2.1.2, instead emitting X-rays. This opens up the possibility that the“missing” matter revealed by galactic velocities could simply be made up of baryonic matter3 not
3Baryonic matter is the name given to normal (i.e. non-dark) matter. It is understood to also include matter notcomposed of baryons (but still described by the Standard Model). While baryons are not the only type of “baryonic”

9
luminous at the frequencies being observed.
However, precise measurements of the anisotropy of the CMB weigh in heavily against thisand other alternative explanations. The CMB radiation spectrum still contains the imprint of early
universe physics, including the ratio between normal and dark matter. In essence, the galactic
structure formation that we observe imprinted upon the anisotropy of the CMB could only havearisen from a very precise ratio of dark-to-normal matter, and is completely incompatible with thenonexistence of non-baryonic matter. These calculations come from a ΛCDM model accounting
for normal matter, cold dark matter (CDM), and the cosmological constant (Λ) governing theuniverse’s expansion [15].
One further popular class of alternatives to the dark matter hypothesis deserves remarking
upon; these are theories of modified Newtonian dynamics (MOND), or modified gravity. First
posited by Mordehai Milgrom in 1983 [16, 17], MOND posits that the gravitational force FG may
actually behave differently than the simple
FG =Gm1 m2r2(2.4)

corresponding to Eq. (2.2). Of course, where local gravitational forces are strong, the modified force
must reduce to Eq. (2.4). It is not hard to write down a functional form, such as
FG,MOND =Gm1 m2 µ aa0 r2

(2.5)

with this property. Here, thinking of m1 as the mass of a galaxy and m2 as the mass of a star at someradius r from its center, µ is a function of m2 ’s acceleration a relative to some new fundamental
constant a0 such that µ(a/a0  1) → 1. Accounting for the flat rotation curves of Fig. 2.1 is assimple as positing that µ(a/a0  1) → a/a0 .
One of several strong cases against MOND and for dark matter (in addition to the CMB)
comes from observations taken in the early 2000’s of a colliding pair of galaxy clusters [18,19]. The
Bullet Cluster is remarkable for being observed post-collision with a larger cluster of galaxies. An
matter, they do account for the vast majority of the baryonic matter density in the universe.

10
image of this collision is shown in Fig. 2.2. The figure highlights two regions of interest. First,
X-ray emission from the galaxy clusters is shown in pink overlay. The X-rays are sourced from thelarge quantities of inter-galactic gases which account for the majority of the baryonic matter. The
galaxy clusters also bend the path of light sourced from behind them, in a phenomenon known
as gravitational lensing. The lensing effect reveals the location of the gravitationally attracting
matter, shown in blue overlay. The key feature of Fig. 2.2 is that for both clusters the blue andpink regions are spatially separated. This is easy to account for with dark matter: the hot, ionizedgas between galaxies is slowed as it interacts electromagnetically with that of the other cluster,while the dark matter sails through unimpeded. Though there remain a few voices of dissent [20],the weight of evidence from the Bullet Cluster seems to favor the dark matter explanation overtheories of modified gravity [21].

Figure 2.2: Optical image of the Bullet Cluster collision with overlaid X-ray and lensing map.
Sources of X-ray emission within the clusters indicate the location of baryonic matter (pink). The
collision slowed down the intra-cluster gasses, spatially separating them from the dark matterregion (blue), which has been reconstructed from gravitational lensing studies. The separationof the luminous matter from the center of gravity is strong evidence for dark matter. Figure
reproduced from Ref. [22].

The evidence discussed above represents just a small sampling of some of the most historically11
important advances in our understanding of the dark matter problem, and are by no means acomprehensive survey thereof. Moving forward, we will ask what dark could be composed of, andwhat its essential properties are.

2.2

The not-so-miraculous WIMPThe fields of particle physics and astrophysics motivate (or in some cases merely allow)

particles or objects that interact only weakly (or strongly, but from our perspective occasionally)
with normal matter, and could plausibly populate much of space. Some of the most popular earlycandidates such as massive astrophysical compact halo objects (MACHOs) have been ruled out
by observational studies [23, 24]. Others, including weakly4 interacting massive particles (WIMPs)[25] have had meaningful fractions of formerly favored parameter space excluded by laboratory
experiments.
WIMPs are the lightest stable particles predicted by supersymmetry (SUSY) theories, and are
well motivated from both theoretical and experimental perspectives [26]. Their popularity traces
to two sources. First is the “WIMP miracle,” the fact that the number of WIMPs expected tohave survived until the present day should decrease with their self-annihilation cross-section hσvi,
predicted to be of order 3×10−26 cm3 /s to account the present dark matter density5 — and this crosssection is the right scale for a weakly interacting particle with mass mχ ∼ 100 GeV/c2 . Second,
we know how to detect them. WIMPs are predicted to scatter off of baryonic matter, producing
sound [28], light, and/or charge signals in physical detectors, which often seek to correlate eventsin two of these three categories to veto backgrounds.
These experiments have been successful in sweeping out much of the WIMP parameter space
shown in Fig. 2.3 [30–32]. The WIMP-baryon scattering cross-sections originally predicted by the
WIMP miracle have been ruled out, weakening the case for WIMPs. Additionally, as detectors
become more and more sensitive, they approach the “neutrino floor” (orange shaded region in
4“Weakly” in this acronym refers to the weak nuclear force, and not the feeble coupling to normal matter, thoughneither meaning would be a mischaracterization.
5
2
So long as mχ >∼ 10 GeV/c , the predicted value for hσvi is not strongly dependent on mχ [27].

12Figure 2.3: Weakly interacting massive particle (WIMP) parameter space showing spin-independent(SI) WIMP-nucleon cross section vs. WIMP mass. Experimental exclusion results are rapidly closing in on the neutrino floor (orange shaded region), where coherent scattering of neutrinos will
make it difficult to detect a WIMP dark matter signal. See Ref. [29] for references to experimental
results. Figure reproduced from Ref. [29].

Fig. 2.3), where ubiquitous neutrino scattering events will provide a formidable background [33].
WIMP direct detection in the presence of neutrino-scattering may still be achievable [34], butwill prove more difficult. The theoretical and experimental cases for WIMPs, while by no means
inviable [35], are thus each less compelling than they seemed in the 1970’s and 1980’s, and, as such,the case for competing hypotheses is strengthened.

2.3

The QCD axion solution to dark matter
A particularly compelling candidate for dark matter is the QCD axion. We will see that

the QCD axion, like the WIMP, solves an outstanding mystery of modern physics and possesses
key attributes of dark matter. Unlike the WIMP, experimental evidence has not yet weighed instrongly on the most plausible parameter space.

13
2.3.1

The strong CP problem

The QCD axion was “invented” to resolve a mystery seemingly unrelated to that of dark
matter: why the strong nuclear force seems to obey charge-parity (CP ) symmetry [36, 37].6 The
problem was vexing because CP symmetry is known not to be a good symmetry of nature. CP
violation was first observed in neutral kaon decays [39], and it took almost four decades before it
was observed outside of kaon physics, in B mesons [40]. Recently, long-baseline measurements ofneutrino oscillations observed CP violation in elementary particles for the first time [41]. But what
all of these occurrences have in common is that they are governed by the electroweak interaction.
In the strong sector, CP violation has never been observed, and not for a lack of sensitivity
to it. CP violation in the strong interaction would be directly observable via the measurementof a nonzero neutron electric dipole moment (nEDM) dn . In particular, dn scales as a parameterθ̄ arising in part from QCD.7 However, the most precise nEDM measurement to date of dn =(0.0 ± 1.1) × 10−26 ecm [43] accommodates θ̄ < 10−10 , whereas its being an otherwise unconstrained
angle suggests it should be O(1).
This part-in-1010 fine-tuning problem has several physically allowed explanations, the mostplausible of which predicts QCD axion particles as a byproduct. First, it could truly just be
random chance. But this is, by construction, unlikely, and there is no anthropic necessity for θ̄to be zero [42]. Second, CP symmetry could actually be an exact symmetry of nature that isspontaneously broken [42]. But explanations of this sort tend to either be ruled out by existingmeasurement [29] or require fine-tuning of their own atop elaborate theoretical mechanisms [44–46].
Third, one of the quarks being massless would make dn independent of θ̄, but current measurements
2of the up quark (the lightest quark) place its bare mass at 2.2+0.5−0.4 MeV/c [29], inconsistent with
6

CP symmetry exists if, upon flipping the signs of all the charges in the universe and inverting the positions of
all particles in space about some point, a careful observer cannot tell the difference between the flipped universe andthe original. Charge-parity-time (CP T ) symmetry, which is symmetry under a CP operation plus a reversal of thearrow of time, is believed to be a good symmetry of nature [38]. Therefore, CP violation implies T violation: theuniverse does not obey the same laws of physics with the tape run backwards.
7
Another contribution to θ̄ actually comes from a Yukawa term in the electroweak interaction, making the socalled “strong” CP problem a problem for the whole Standard Model: θ̄ = θQCD + θYukawa . For a far more detailed
explanation of the attendant physics, see Ref. [42].

14zero. Finally, there is the Peccei-Quinn (PQ) solution to the strong CP problem.

2.3.2

The Peccei-Quinn mechanism

In 1977, Roberto Peccei and Helen Quinn proposed a theory in which θ̄ is promoted from astatic parameter to a dynamical field, causing it to naturally relax to 0 and elegantly cleaning up
the strong CP problem [36,37].8 This cleanliness-promoting feature gave rise to the name “axion,”after the dish detergent [48]. Axions are pseudo-Goldstone bosons9 that arise as an immediate
consequence of the PQ mechanism [49, 50]. It was not long before particle physicists realized thatsufficiently light axions have key features resembling dark matter. This main results of this thesis
will pertain to axions of mass ma ∼ 10 µeV/c2 .

2.3.3

Axions as dark matter candidates

The light axions we will focus on are bosonic, stable on cosmological timescales, and weaklyinteracting. They also may have a production mechanism capable of making up in numbers what
the individual particles lack in mass, while allowing for them to be nonrelativistic. We will see that
all of these features together make axions a promising dark matter candidate.
Any sufficiently light dark matter candidate must be bosonic. This is because fitting enoughfermions into a galaxy to account for observed dark matter densities gives the highest energy
particles a Fermi velocity greater than galactic escape velocities. Their low masses [29] are one
way we know that neutrinos, for example, do not constitute dark matter [42]. Axions are indeedbosons, meeting this criterion.
To be viable, a dark matter candidate must also have been cold — that is, nonrelativistic— at the time of galactic structure formation. In particular, baryonic matter only begins to form
large-scale structure after the decoupling of radiation from protons and electrons, which occurs
8

For a delightful analogy illustrating the PQ solution to the strong CP problem, see Ref. [47].
While Goldstone bosons are the massless excitations of a flat potential resembling the circular mode of travel atthe bottom of a wine-bottle, axions are pseudo-Goldstone bosons because their wine bottle-potential is tilted. The
mass of a particle scales with the square root of the curvature of this potential, so axions have nonzero, but still
relatively low, masses.
915
when the universe is ∼ 380, 000 years old. In a universe with only baryonic matter, this would
not allow enough time for the structure we observe to have formed [42]. Dark matter, meanwhile,
can in principle begin clumping once the universe become matter dominated, at ∼ 50, 000 years.
However, lighter particles usually have a higher velocity. The equipartition theorem gives the
root-mean-square (RMS) velocity of particles of mass m in an ideal gas at temperature T asrvRMS =

3kB T
,
m

(2.6)

where kB is the Boltzmann constant. The faster velocity gives rise to a longer free-streaming
length, the characteristic distance that the particles travel before being slowed by cosmic expansion.
Structure in matter density fluctuations in the early universe at smaller spatial scales than this freestreaming length is wiped out. Dark matter with primordial RMS velocities comparable to the speedof light, vRMS ∼ c, precludes the structure needed to form galaxies through the local clumping ofmatter beginning at ∼ 50, 000 years. “Hot,” or relativistic, dark matter has accordingly been shown
through simulations to form galaxies very slowly, producing a universe different than the one wefind ourselves in [51]. If the dark matter were baryonic, it would be in equilibrium with radiationand thus hot.
Axions, despite likely being even lighter than neutrinos, can be cold if they were produced
athermally, that is: not in thermal equilibrium with their early-universe surroundings. The T in
Eq. (2.6) is no longer required to be the universe’s temperature at the time of structure formation,and light dark matter is allowed provided it is sufficiently abundant. In 1983, several independentgroups of theorists [52–54] showed how PQ mechanism may account for dark matter by athermally
producing axions. Furthermore, this mechanism produces disproportionately more axions the lighter
ma is.10
We have seen that axions could have been produced in large quantities as cold bosons in
10

A key determinant of the number of axions produced derives from whether a spontaneous symmetry breaking at
the heart of the PQ mechanism occurred before or after the early universe underwent inflation. If symmetry breaking
occurs before inflation, the quantity of axions produced is in a sense much more random: it scales as the absolutevalue of random angle going from −π to π. If symmetry breaking occurs afterwards, many independently chosen
2values are averaged over, eliminating that source of uncertainty. Sufficiently low-mass QCD axions (ma <∼ 10µeV/c )require symmetry breaking before inflation, plus some fine-tuning for the angle.

16
the early universe; what remains to qualify them as a dark matter candidate is their interaction
with normal matter. Axions are predicted to interact weakly with most forms of normal matter.
Though they have no electromagnetic charge, their dominant interaction is with electromagnetism
via a two-photon coupling in the universe’s Lagrangian,L ⊃ gaγγ |E · B|a,(2.7)

where E and B are the local electric and magnetic field vectors, respectively, and a is the axion
field amplitude. We will henceforth reparametrize the axion-photon coupling gaγγ as an order-unity
dimensionless number
gγ =

gaγγ
ma c2



πΛ2
α


,

(2.8)

where α ≈ 1/137 is the fine structure constant and Λ = 77.6 MeV encodes the dependence of
the axion mass on hadronic physics. We have normalized out the linear scaling with rest energy
ma c2 , because QCD axion couplings to normal matter are proportionate to ma .11 The valuethat gγ takes is model-dependent, with the benchmark Kim-Shifman-Vainshtein-Zakharov (KSVZ;
gγKSVZ = −0.97) [55, 56] and Dine-Fischler-Srednicki-Zhitnitsky (DFSZ; gγDSFZ = 0.36) couplings[57,58] serving as more and less optimistic targets, respectively, for detection experiments.12 These
values for gγ imply an extremely long lifetime for the axion with respect to its two-photon decay
channel: axions below the ∼ 10 eV mass scale are stable on cosmological timescales [42]. Moreover,
for any axion in the QCD band [59] containing both KSVZ and DFSZ models, gγ is sufficiently
small that our nonobservance of axions thus far is not meaningful evidence against their existence.
Rather, if QCD axions — which have all the essential properties of dark matter — in fact constitutedark matter, it will take a concerted effort to find them.
11

More precisely, gaγγ and ma both scale inversely as the energy scale fa at which PQ symmetry-breaking occurs.
This energy scale dwarfs those achieved by physicists in high energy experiments, including the 13 TeV record at theLarge Hadron Collider. The largeness of fa is why the axion is predicted to be both so light and so weakly coupled.
12
Physically, gγ can take positive or negative values, but since only gγ2 contributes to the axion signal power, I will
henceforth use gγ to stand in for |gγ | throughout this thesis.

17

2.4

Detectable features of axionic dark matter
The properties discussed above qualify axions as dark matter candidates; now I will discuss

properties pertinent to their detection. A simple yet well motivated and actionable view of axions
holds that they exist in a bounded, “1.1-dimensional” parameter space of mass ma and couplinggγ with known velocity distribution and density throughout the cosmos. To start, let us consider
the axion-photon parameter space in which this thesis will primarily work.

2.4.1

Axion parameter space

Figure 2.4 shows a portion of axion-photon parameter space which has attracted attention
in recent years for its experimental accessibility [60]. Most of the plot (white and yellow regions)has not been experimentally probed. The diagonal yellow region represents the QCD axion band,13
with the KSVZ line having unit slope in the space of gaγγ and ma . While non-QCD axions, also
known as axion-like (ALPs),14 of very high couplings would have imprinted a signal in devices
such as helioscopes (excluded blue region) [61], the only demonstrated method for reaching QCD
couplings is with axion haloscopes (excluded green regions), discussed below.
The parameter space between the dashed, black lines at 10 µeV/c2 and 1 meV/c2 in Fig. 2.4
is theoretically favored by a cosmological and an astrophysical argument.

Axions below themOC ∼ 10 µeV/c2 scale in models in which PQ symmetry-breaking occurs after inflation tend
to be overproduced to the point of “overclosing” the universe: that is, more dark matter is predicted than is observed, disfavoring lower masses [52]. In the event that inflation did occur first,
QCD axions are allowed both below and above mOC , making it a somewhat soft bound.
The upper bound of mSN ∼ 1 meV/c2 is set by the ∼ 10 s temporal duration of the observed
neutrino burst of supernova 1987A [63]. The production of axions in the exploding star should
scale with their coupling to normal matter, which in turn scales as ma . Sufficiently heavy axions
13

Different physicists draw the boundary of the QCD band slightly differently, though everyone agrees it encompasses both the KSVZ and DFSZ lines.
14
ALPs, which will feature in Ch. 9, do not solve the strong CP problem. Since there are no strong bounds on
the ALP-photon coupling, it is also possible that ALPs could lie below the QCD band, which would make them very
difficult to detect.

18

Figure 2.4: A portion of dark matter axion-photon parameter space prior to the work discussed inthis thesis. The white and diagonal yellow regions are unprobed, with the diagonal yellow region
highlighting the theoretically favored QCD band. Haloscope exclusion is shown in green, withother excluded regions shaded different colors. For a thorough discussion of the existing constraints
within axion parameter space, see Ref. [60]. The black, dashed lines show the 10 µeV/c2 –1 meV/c2
range of axion masses favored by overclosure arguments and supernova 1987A bounds discussed in
the text. The orange, dashed lines enclose the 1–10 GHz range of axion mass-converted frequencies
νa , Eq. (2.9), of particular interest to this thesis. See Ref. [29] for references to experimental results.
Figure adapted from Ref. [62].

would have made off with much of the star’s energy, leaving fewer neutrinos to lengthen the burst
duration by colliding with each other. This is considered by many a more concrete bound than thatof overclosure, but it is conceivable that modelers have misunderstood the origin of the observed
neutrinos, which would undo the argument [64]. But even if one is equally drawn to the adjacentregions, the two decade window of roughly 10 µeV/c2 –1 meV/c2 remains well-motivated parameter
space.
Zooming farther into the 10 µeV/c2 –1 meV/c2 meV window, arguments abound for specific favored or disfavored mass ranges [65–68]. Of particular relevance to the work presented in this thesis,
Klaer and Moore (26.2±3.4 µeV/c2 ) [69] and Buschmann, Foster, and Safdi (25.2±11.0 µeV/c2 ) [70]have independently predicted the axion to lie in overlapping regions around 25 µeV/c2 . Near and19
below these masses, the axion acts more like an oscillating field at with axion rest-mass converted
frequency (henceforth “axion frequency”)
νa =

ma c2
.
h

(2.9)

It is beyond the scope of this work to adjudicate between theoretical models, except by contributing
to the experimental toolkit and the body of axion direct detection data that will enable us to answer
the question directly.

2.4.2

Local axion dark matter density

In order to discuss how axions may be detected, two final features of cold dark matter bear
examining; the first is the local dark matter density ρDM . In this work and throughout the axion
exclusion literature, experimental constraints such as those in Fig. 2.4 are typically placed on thequantity gγ , or equivalently gaγγ via Eq. (2.8). Implicit in these constraints are two assumptions:that axions saturate 100% of the local dark matter density (ρa = ρDM ) and that the local dark
matter density takes a specific value, typically quoted as ρDM = 0.45 GeV/cm3 . The observablecombination that shows up in experiments is generally ρa gγ2 , so it is a trivial matter to scale any
constraint placed on gγ to assume a different value of ρa . For consistency with the literature, inthis thesis we will use ρa = 0.45 GeV/cm3 unless otherwise stated.15

2.4.3

Axion lineshape

The other feature of axionic dark matter that will prove relevant to axion detection is thetotal energy (Etot ), or equivalently frequency ν = Etot /h, of the axion particles.16 Here, we will
not want to think of ν as a constant, but rather a random variable: a quantity that can take arange of values specified by a probability density function fν (ν), or equivalently its integral, the15
WIMP dark matter experiments have used the more conservative value ρDM = 0.3 GeV/cm3 , also consistent
with measurement [71]. Though the primary goal is to detect dark matter, the secondary goal of certifying that it
does not exist in a specified coupling range does seem more naturally served by the more conservative value. In Ch. 9,
for consistency with the ALP literature, we will use 0.40 GeV/cm3 .
16
Thinking of axions as particles will be helpful here. The results will work just as well for measurements thattake advantage of the wave-like properties of axions.

20
cumulative distribution function Fν (ν) =Rν00−∞ fν (ν )dν .

The frequency distribution is obtained

through approximating the total energy as the rest energy plus the nonrelativistic kinetic correction:
1
Etot ≈ ma c2 + ma v 2 .
2(2.10)For cold dark matter, the dimensionless velocity β = v/c is O(10−3 ), meaning the nonrelativistic
term is of order 106 times greater than the first relativistic correction 3ma v 4 /8c2 to Eq. (2.10),
making this an excellent approximation. Determining the frequency distribution then amounts todetermining the axion frequency νa and the velocity distribution fv (v) of axion particles. The axion
frequency is a free parameter of our search, and assumed to have a single, constant value, hencefν (ν) → fν (ν|νa ). The velocity distribution is model-dependent, and in this thesis we will stickto the standard halo model (SHM), which describes virialized axions as observed from the moving
laboratory frame of our solar system.17
The virial theorem, Eq. (2.1), relates the mean kinetic energy hT i = ma hv 2 i/2 and gravitational potential energy hVG i of axionic dark matter in the Milky Way Galaxy, allowing us towrite down fv (v) and ultimately fν (ν) in the galactic rest frame. The galactic rest frame is definedas the frame in which our galaxy has no net angular momentum. In it, the velocity components{vx , vy , vz } of axions are independent, identically distributed Gaussian random variables with mean
zero and variance σv2x = hv 2 i/3: {vx , vy , vz } ∼ N (0, σv2x ). Summing the velocity component random variables in quadrature yields the Maxwell-Boltzmann (MB, or Maxwellian) distribution for
the velocity amplitude of particles in an ideal gas:"   #
4 v2
v 2,
fv,MB (v) = √ 3 exp −vc
π vc(2.11)√
where vc = σvx / 2 ≈ 220 km/s is the modal velocity of the distribution. The corresponding MB
frequency distribution obtained using Eq. (2.10) isr  3 r
  22 c
1 ν − νac
ν − νafν,MB (ν|νa ) = 4
exp −2
π vc
νa
νa
vc2
νa

(2.12)

17
The techniques discussed in this work are applicable independent of axion lineshape, and may be readily adapted
as new evidence refines our knowledge of the dark matter halo [72, 73], or to test alternative lineshapes [74, 75].

21
for νa ≥ ν and 0 otherwise. The axion frequencies would be MB-distributed according to Eq. (2.12)
if our solar system were stationary with respect to the galactic rest frame. Instead, we move atvs ≈ 232 km/s through a sea of axions in the galactic rest frame. The resulting SHM velocity andfrequency distributions, results which will be of direct use to us in Chs. 8 and 9, are
" 
 #



v
v + vs 2
4vvs
fv,SHM (v) = √
exp −
−1
exp
vc
vc2
πvc vs(2.13)

and2
fν,SHM (ν|νa ) = √
π



c2
vc vs



1
sinh
νa



vs c
vc2

r

ν − νa8νa"
exp −2



c
vc

2

ν − νa
−
νa



vs
vc

2 #
,

(2.14)

again only for ν ≥ νa [75, 76]. The lineshapes, Eqs. (2.12) and (2.14), are plotted in Fig. 2.5. The
primary effect of vs > 0 is to broaden18 fν,SHM (blue) relative to fν,MB (green), but in either casethe fractional linewidth ∆a ∼ νa 10−6 (equivalently, the quality factor of the axion is of order amillion: Qa ∼ 106 ).19
This concludes our brief overview of axion theory. For a far more thorough discussion ofmuch of what was covered here, the interested reader is encouraged to consult Ref. [42]. We are
now ready to discuss how it is that QCD axions may be detected.

2.5

The axion haloscope
The axion haloscope was first proposed by Pierre Sikivie in the 1980’s to detect dark matteraxions in the galactic halo by converting them to photons20 in a microwave cavity using the inverse
Primakoff effect and a large static magnetic field B = B0 ẑ [85–87]. The cavity delivers a Q-factor
enhancement so long as the electric field E of at least one of its modes overlaps with ẑ so as toalign with B, Eq. (2.7). The prototypical haloscope uses a right circular cylindrical cavity with
18

Narrower lineshapes such as those given by Eq. (2.12) or Ref. [74] are in principle easier to detect because of
their higher signal-to-noise ratios [77].
19
Further corrections come from the orbital and rotation motion of the earth around the sun and its axis, respectively. For virialized axions these corrections are negligible [76, 78].
20
What is more precisely meant by “photons” unless otherwise indicated is “electromagnetic energy.” While theenergy could in principle be measured in the number basis [79–82], at the frequencies where this thesis focuses, it
is more efficient to measure the continuous quadrature observables [83, 84]. It is nonetheless common to refer toaxion-induced excesses in a haloscope cavity as photons.

22

Figure 2.5: Virialized axion lineshapes fν (ν|νa ) for νa = 4 GHz. The Maxwell-Boltzmann (MB)spectral distribution, Eq. (2.12) (green), is narrower than the standard halo model (SHM) distribution, Eq. (2.14) (blue), which accounts for the motion of the sun in the galactic rest frame.
The means µ and standard deviations σ of the two distributions are indicated by dots and bars,
respectively.

mode indices m, n, and l21 for which the spatial overlap associated with a given cavity mode isparametrized as the form factor
2
E(x) · ẑ d3 xVCmnl = R,
2 3
V ε(x)|E(x)| d x

R

(2.15)

where V is the volume of the cavity and ε(x) is the position (x)-dependent permittivity.22 Figure
2.6a shows the field directions (ignoring amplitude) of transverse magnetic (TM), transverse electric(TE) and transverse electromagnetic (TEM) modes for simple cylindrical cavities. For a cylindricalcavity, the only modes that produce a nonvanishing form factor are the TM0n0 modes [88, 92].
However, an unperturbed cylindrical cavity is of limited use for axion detection because its TM010
mode — the one with the highest form factor and lowest density of “intruder” modes insensitive to
21

These correspond respectively to the m, n, and p indices in Sec. 8.7 of Ref. [88].
For the haloscope central to this thesis, ε(x) = ε0 , the vacuum permittivity, everywhere. Some haloscopeexperiments have taken advantage of the dielectric properties of liquid helium [89] or dielectric rods [90, 91] to tune
their cavity frequencies.
22

23
the axion — can only probe around a fixed resonant frequency,23 and broadening its resonance toachieve a larger bandwidth incurs disproportionate losses in sensitivity [95]. The usual solution is
to insert a metal and/or dielectric tuning rod coaligned with the vertical axis ẑ of the cavity. Fora metal rod, rotation away from the center of the cavity, shown in Fig. 2.6b, tunes the TM0n0 -likemodes to lower frequencies.24

2.5.1

Previous haloscope results

Axion haloscopes have been successfully running since the late 1987, but have as yet seen
no evidence of the axion. Figure 2.7 shows in greater detail than Fig. 2.4 the region of parameter
space that has been partially excluded by haloscopes.25 The earliest experiments were carriedout at Brookhaven National Laboratory in collaboration with the University of Rochester andFermilab [97], and at the University of Florida [98]. The Axion Dark Matter Experiment (ADMX)
came online in 1998 [89], and between then and 2016 [99] was the only operational haloscope,
primarily probing parameter space at KSVZ sensitivity.
In 2017, the Haloscope At Yale Sensitive To Axion CDM (HAYSTAC) became the second
active haloscope [90, 106], sensitive to higher couplings gγ and and operating at higher frequencies
νa than ADMX. We will see in Sec. 2.5.2 that going deep in coupling and high in frequencies arecomparably difficult propositions. As of early 2020, the division of labor in the axion detection
field featured one experiment actively pursuing each of these objectives, with ADMX sensitive to∼DFSZ axions around 660 MHz [91], and HAYSTAC searching for ∼ 3-KSVZ axions around 5.7
23
While most axion haloscopes have only used the TM010 mode, the ADMX Sidecar and ORGAN experiments
recently demonstrated use of the TM020 mode to collect axion-sensitive data [93,94]. Using the TM020 mode (or evenhigher modes) affords sensitivity to higher frequencies with a much higher cavity volume V than would be achievablewith the TM010 mode of a smaller cavity. However, the tradeoff is not as favorable as it may appear: the electricfield undergoes 1 1/2 oscillations over the height of the cylinder, and the form factor integral, Eq. (2.15), ensures thatthe positive and negative half of the first oscillation cancel out, reducing the form factor roughly threefold as thevolume grows by the same amount. Furthermore, around higher TM0n0 mode frequencies there is a higher numberof unwanted crossings with axion-insensitive cavity modes. The real utility of higher modes is that in principle they
can be read out simultaneously, opening up parallel data streams at no cost to the first.
24
The metal rod has the side effect of reducing the available cavity volume used for axion conversion. The effectis not always a small one: in HAYSTAC, the rod occupies roughly 1/4 of the cavity volume [96], reducing theexperiment’s main figure of merit, its scan-rate, roughly twofold.
25
Excluding with a haloscope at a given frequency and coupling entails excluding all stronger couplings at thesame frequency.

24

Figure 2.6: (a) Electric and magnetic field directions for simple cylindrical cavity geometries. Only
the subset of transverse magnetic (TM) modes whose electric fields integrate to a nonzero value
along the vertical direction (aligned with the magnetic field) are useful for axion detection. (b)Simulated electric field magnitudes for the TM010 -like mode of a cylindrical cavity with a metal
tuning rod at different angles. The changing mode structure tunes the resonance frequency andalters the form factor C010 , Eq. (2.15). Figure adapted from Ref. [62].

GHz [107].
Additionally, the CAPP-8TB experiment has reported its first results at 1.6 GHz [108], the
ADMX Sidecar experiment has probed parameter space above the QCD band at a range of higher
frequencies [93], and the QUAX-aγ [109] and ORGAN [94] experiments have reported early results
at much higher frequencies, but with limited sensitivity. Concurrent with these developments, otherhaloscope-like platforms have been [110–115] coming online or are poised to do so [116, 117].26
26

These experiments all look for axions on the assumption that they constitute dark matter. However, axions could
very well exist and not constitute any meaningful portion of dark matter. In that event, “light shining through walls”(LSW) experiments that attempt to first create and then detect axions [118–120] (upper yellow region in Fig. 2.4),
axion helioscopes which look for axions produced in the sun [61, 121, 122] (blue region in Fig. 2.4), or searches for

25

Figure 2.7: Axion-photon exclusion obtained using haloscopes prior to the main work of this thesis.
The regions labeled RBF (pink) and UF (blue) represent early results insensitive to the QCD axion(yellow band) obtained at Brookhaven National Laboratory [97, 100] and University of Florida[98, 101], respectively. ADMX (light red) [89, 91, 93, 99, 102–105], HAYSTAC (green) [106, 107], and
CAPP (red) [108] have all reached the QCD band.

The collective null result of Fig. 2.7 only extends over a minute fraction of the parameter
space axions could plausibly inhabit. Nevertheless, there is real cause for optimism: haloscope
searches have sped up roughly one millionfold between the first search carried in 1987 [97] and the
major results of the past few years [91, 105–107]. Such comparisons are made between the scanrates of axion haloscopes.

2.5.2

Haloscope scan rate

Scan rate R is the primary figure merit for axion haloscopes:27 it quantifies how rapidly ahaloscope can scan through parameter space at some fixed gγ . It has the curious dimensions ofaxion-mediated spin forces [123, 124] may be the best hopes to resolve the strong CP problem.
27
This is admittedly a somewhat subjective assessment, as it is hard to compare the more objective scan rate withother important haloscope traits, such as affordability, ease of operation, density of intruder modes, and accessible
parameter space. Furthermore, because haloscopes spend nonnegligible amounts of time rescanning potential signals
at individual frequencies, the maximum single-frequency sensitivity, which can be different for two haloscopes of thesame scan rate, is an important secondary figure of merit. For the purpose of this thesis, I will treat improving scanrate as the ultimate goal, paying attention where appropriate to other important objectives. This simple framing isvery useful, and can be made to encompass some of the others figures of merit mentioned above.

26
frequency per time, and is proportionate to several key parameters [42]:! 4 2 2
gγ4
B0 V Cmnl Q0
ρ2a Qa
R∝
,
2
Λ8
Nsys
SNR02

(2.16)

The first set of parentheses consists of scan properties chosen by the experimentalist: what signalto-noise power ratio (SNR) is desired at what coupling gγ for the experiment. The prime on the
SNR is to denote a modification that owes to the look-elsewhere effect, and is derived in Sec. 6.5.
We will see that as a rough approximation, the denominator of the first term can be thought of as
being
SNR2 ∼ SNR02 + 2 log(NI ),(2.17)

where NI is the number of “independent” frequencies being probed for the axion. The second setof parentheses comprises several physical parameters of axion dark matter. The final set contains
experimental parameters: the magnetic field B0 , cavity volume V , form factor Cmnl , and unloaded
cavity quality factor Q0 28 should be maximized,29 while the total system noise Nsys (in units ofphotons, or quanta) should me minimized. Improving a haloscope’s scan rate through hardware istypically synonymous with improving one of these parameters.30

2.5.3

Improving the scan rate

The millionfold improvement in scan rate accrued over the last three decades owes predominately to dramatic advances in just one of the experimental parameters in the final parentheticalof Eq. (2.16): namely, the total system noise Nsys . Cylindrical cavity volumes change asV ∝ ν −ζ28

(2.18)

Equation (2.16) assumes Q0 to be smaller than Qa , as has been the case for all haloscopes to date. Improving
Q0 beyond Qa while operating in the standard manner does not improve scan rate farther, as the coherence time of
the axion becomes the limiting factor.
29
We will see in Ch. 5 that another parameter of importance to the scan rate is how strongly the cavity is coupled
to its readout chain. This parameter is in practice easy to adjust to its optimal value, so for now we leave it out of
Eq. (2.16), which only includes quantities which pose fundamental or practical limitations to scan rate.
30
Scan rate can be improved in other ways as well. Advances in analysis techniques [75, 78, 125, 126] that allow for
more efficient use of information will be a central focus of this thesis. Additionally, improvements to acquisition live
time can come through either improved software or hardware. Optimizing scan strategy remains a largely unexploredavenue by which total scan time might be reduced [95, 126].

27
to accommodate one half-wavelength at the frequency of interest, where ζ ∈ [2, 3] can be as low as
2 if experimentalists are able to tolerate the linear growth in TE mode density with cavity aspectratio (cavity length/radius). There has been no significant innovation that has broken this linkage
without a commensurate sacrifice to the form factor (Footnote 23). The unloaded quality factor,
which is limited by the anomalous skin effect [127] for normal metals, changes as
Q0 ∝ ν −2/3 .

(2.19)

It could be improved substantially by switching to partially superconducting cavities, but thepresence of the large magnetic field makes benefiting from this approach extremely difficult [109].
The magnetic field itself, despite the promising R ∝ B04 dependence of Eq. (2.16), is today still
within 25–50% of its 1987 value of 6 T [91,107,108]. The system noise Nsys , however, has improveda few hundredfold between 1987 [97] and 2018 [107].
The improvement to system noise owes to two separate innovations: the use of colder thermal environments culminating in dilution refrigerators to house the haloscopes, and the continued
improvement of amplifier technology [104], culminating in Josephson parametric amplifiers capableof adding near the 1/2-photon required by the uncertainty principle [128]. In 2017, HAYSTAC
arrived at both of these termini [90, 106] suggesting that a new approach would be needed if noise
performance is to continue to improve. This thesis sets out in part to deliver that approach, which
will involve circumventing the “quantum limit” on system noise.31
Continuing to improve haloscope scan rates is essential in light of the daunting amount ofparameter space that remains unscanned. Scan rate optimistically scales with frequency asR ∝ ν −14/3
for single-mode haloscopes.32
31(2.20)

This scaling comes from the cancellation of several frequency-

There is an analogy to make to Moore’s law, the observation that the number of transistors on computer
chips has tended to grow exponentially over time. As transistors approach atomic size scales, the trend cannot
possibly continue without a transition to quantum computation. Similarly, it is the neighboring subfield of quantum
measurement that may carry axion haloscopes beyond their version of tiny transistors: the uncertainty principle (inreality, most computational tasks do not benefit from quantum computing, so the outlook for axion haloscopes —which only exist to do one task — is in a sense brighter).
32
Haloscopes are furthermore known to achieve 75% of the limit on integrated axion sensitivity set by the Bode-Fano28
dependencies of the signal and noise in axion haloscopes, leaving just the dependence of the cavity
volume and quality factor on frequency, Eqs.(2.18) and (2.19) [42]. It optimistically assumes ζ = 2
in Eq. (2.18), and neglects the look-elsewhere effect correction to Eq. (2.16) discussed in Sec. 6.5.
Equation (2.20) implies that today’s leading haloscopes [91, 107] would take 20 millennia to scan
only the 1–10 GHz frequency decade at DFSZ sensitivity, generously assuming noise at the quantum limit and 100% live time. To emphasize just how optimistic even this 20 millennia figure
is, consider a more pessimistic assessment: using only noise levels achieved before this work, amore realistic live time of 10%, and ζ = 3 in Eq. (2.18), a less optimistic dark matter density ofρa = 0.3 GeV/cm3 (Footnote 15), and aiming for roughly gγDFSZ /10 to probe the entire QCD band
(yellow region in Fig. 2.4) yields ∼ 200 trillion years for the 1–10 GHz decade!
This thesis focuses on two distinct but compatible innovations to speed up detection efforts.
One is the first use of the quantum measurement techniques to enhance a search for new particle
phenomena. In particular, I use squeezed states of microwave light to avoid the standard quantum
limit on noise in an axion haloscope, and speed up the scan rate of the HAYSTAC experiment nearly
twofold. The second innovation is the development of a novel Bayesian analysis framework tailoredto the problem of dark matter axion detection. I apply this technique to both axion haloscope data
and to data in a search for ALPs as much as 17 orders of magnitude lower in mass. Together, these
two innovations deliver a roughly threefold scan rate enhancement to the HAYSTAC experiment.33

criterion [129, 130], making them near-optimal vehicles for single-mode, linear measurements of the axion field [95].
For an abridged discussion of the results in Ref. [95], see Ref. [131].
33
On its own, a threefold scan rate enhancement is insufficient to solve a 20,000+ year problem. The real utilityof the advances discussed in this work is that they naturally invite further innovation in both quantum measurement
and scan strategy. The threefold scan rate enhancement achieved herein can additionally be combined with future
improvements to live times, magnets [132], and cavities [92, 109, 133]. While 20,000 is a large number of years,
log3 (20000) is not so large a number of comparable scan rate enhancements needed to make the problem quite
tractable.

Chapter 3

Josephson parametric amplifiers

Two superconductors sandwiching a thin layer of insulator form a Josephson junction [134,135], the only dissipationless, nonlinear circuit element available at microwave frequencies [136].
Those three properties have enabled physicists to build nearly lossless parametric amplifiers, such as
the Josephson parametric amplifier (JPA), capable of manipulating quantum states of microwave
light [137–153]. JPAs and related devices have received particular attention in recent years fortheir usefulness across a range of quantum optics and quantum information processing applications
using superconducting circuits [154–170]. In this thesis we will ultimately use them to improveaxion haloscopes beyond the quantum limit [128]. This chapter will explore the physics of JPAs,
how they achieve amplification and squeezing, and their relevant design considerations. We will also
shed light on one of the core approximations used to model JPAs, a discussion with implications
for improving power handling in future designs. We will end by examining the specific JPAs usedthroughout this thesis.

3.1

The Josephson junction
The Josephson junction, shown schematically in Fig. 3.1 is the element at the heart of manysuperconducting circuits. It consists of two superconductors separated by an insulating layer(Fig. 3.1a) which Cooper pairs of electrons may tunnel through, forming a current. It is characterized by a critical current Ic , equal to the maximum superconducting current that can flow
through it. The critical current that a junction will have when superconducting can be determined30
from its room temperature resistance Rs via the Ambegaokar-Baratoff relation [171],π∆(T )
Ic =
tanh
2eRn



∆(T )
2kB T


,

(3.1)

where ∆(T ) is the temperature (T )-dependent superconducting energy gap of the material. The
dynamics of the junction are determined by the phase drop
δ = θ1 − θ2

(3.2)

of the macroscopic superconducting wavefunction across it. For a more complete introduction to
Josephson physics, see Refs. [136, 172].
We will take as a starting point the two equations34 relating this phase drop to the voltage
V across and the current I through the junction:
I(t) = Ic sin δ(t)V (t) =

Φ0
δ̇(t),
2π

(3.3)
(3.4)

where Φ0 = h/2e ≈ 2.07 × 10−15 Wb is the magnetic flux quantum, h is Planck’s constant, e is the
positron charge, and the dot denotes a time (t) derivative. Instead of working primarily with thevoltage and current, we will often find it more useful to refer to their time integrals, the flux andcharge variables
Z tΦ(t) =−∞
Z t

Q(t) =

V (t0 )dt0

(3.5)

I(t0 )dt0 .

(3.6)

−∞

Equivalently,
V (t) = Φ̇(t)(3.7)

I(t) = Q̇(t).

(3.8)

We see from Eqs. (3.4) and (3.7) that the Φ = φ0 δ, where the reduced flux quantum35 φ0 = Φ0 /2π =
~/2e.
34
35

These are known as the Josephson relations.
In superconductivity, the flux through a closed superconducting loop is quantized in units of Φ0 .

31

Figure 3.1: (a) A superconductor-insulator-superconductor (SIS) junction forms a Josephson junction. We will treat the Josephson junction as a two-node electrical object. (b) Circuit representationof and conventions used for the Josephson junction. The junction is symbolically represented as an‘X.’ It is specified by its critical current Ic , which relates the current I flowing through it to thephase drop δ across the junction. The voltage V across the junction is proportional to the time
derivative of the phase drop.

3.1.1

Josephson inductance

The Josephson junction behaves as a nonlinear inductor.36 To understand the junction as an
inductor, we begin by combining Eqs. (3.3) and (3.4) to relate the voltage to the time derivative of
the current:,
V =

LJ0 ˙
I,
cos δ

(3.9)

φ0
Ic

(3.10)

where
LJ0 =

is a constant describing the junction with dimensions of inductance.
The Josephson inductance can be defined in two different ways: either as the absolute value
36
The Josephson junction can also be modeled as having a self-capacitance, giving rise to a resonance frequency
known as the plasma frequency of the junction, and nonzero loss [136]. For our applications, these two effects can beneglected.

32of the flux-to-current ratio37
LJ =Φ
|arcsin(I/Ic )|
,
= LJ0
I
I/Ic

(3.11)or through Eq. (3.9) as the voltage-to-time derivative of current ratio,
LJ2 =

V
LJ0
=p
.
I˙
1 − (I/Ic )2(3.12)

Both definitions are valid, and which we use will depend on the context [173].

3.1.2

Josephson energy

An energy can also be associated with the Josephson junction. This is the energy that it
takes to build up a certain amount of current flowing through the junction, or equivalently, a certain
phase drop across the junction. We know from Eq. (3.3) that applying a current to the junction
sets a phase drop across it and from Eq. (3.4) that phase drop’s temporal behavior sets a voltage
across the junction. Taking energy to be the time integral of power, we haveZ
EJ (t) =I(t0 )V (t0 )dt0 = EJ0 (1 − cos δ),(3.13)

where
EJ0 = φ0 Ic(3.14)

has dimensions of energy. We have taken advantage of the fact that energy is only defined up toan additive constant to ensure a minimum value of 0 when no current is flowing.

3.2

The DC SQUID
We have now seen how the Josephson junction acts as a dissipationless, nonlinear inductance.

In this section, we will see that a pair of Josephson junctions in parallel maintains all of theseessential features, while making the inductance tunable with an externally applied magnetic flux.
37

I will leave out the explicit time dependence in contexts where it is less important to emphasize.

33
The DC superconducting quantum interference device (SQUID) consists of two Josephson
junctions Ic1 and Ic2 in parallel, shown schematically in Fig. 3.2.38 In order to derive key propertiesof the SQUID, it will be useful to work with
I = I1 + I2

(3.15)

I1 − I2,
2(3.16)

Icirc =

respectively the total and circulating combinations of the currents I1 and I2 through the junctions.
The voltage across the SQUID is V , and a magnetic field Btot may penetrate the plane of the loop
as a magnetic flux
Z
Btot · da,Φtot =(3.17)awhere a is the area of the loop and da is the orthogonal-pointing surface element.
To understand how this object functions as an effective tunable junction, it is necessary todelve into its internal structure. I will treat just the case of two identical junctions
Ic1 = Ic2 = Ic

(3.18)

in a symmetrical loop with geometrical inductance Lg , which will be used throughout this work.

3.2.1

Critical current and inductance

The junction phase drops δ1 and δ2 are related by the flux penetrating the SQUID loop. To
prove this, consider the integration path illustrated in Fig. 3.3 and the four labelled points — one
above and below each junction. The magnetic vector potential A is obtained from the gradient of
the phase of the superconducting wave function via the relation [172]A = −φ0 ∇θ.

(3.19)

Therefore,
Z B

Z D∇θ · dl +

A
38

C

1
∇θ · dl = −
φ0

Z B

Z D
A · dl +A
A · dl .

(3.20)C

The radio frequency (RF) SQUID is the other popular SQUID configuration. It consists of a superconducting
loop with a junction in just one arm [172]. This thesis only relies upon the DC SQUID, so henceforth I will drop theDC prefix.

34

Figure 3.2: Circuit representation of and conventions used for the direct current (DC) superconducting quantum interference device (SQUID). The SQUID consists of two Josephson junctions
with critical currents Ic1 and Ic2 on opposing arms of a superconducting loop. The direction of
positive current flow is given by the arrows. The total current (I, green) and circulating current (Icirc , blue) are defined from the currents I1 (orange) and I2 (purple) through the junctions,Eq. (3.3). The voltage across the SQUID is V . The magnetic flux Φtot penetrating the loop arises
from any externally applied flux and from Icirc in conjunction with the geometrical inductance Lgof the loop.

Figure 3.3: A useful integration path (orange dashed line) for the magnetic vector potential arounda SQUID loop. Labels above and below each junction show the integration limits used in Sec. 3.2.1.

35
Recognizing the sums of integrals on both sides of the equation as the full loop integrals minus theother segments, we have
I

Z C

Z A

1
∇θ · dl −∇θ · dl −∇θ · dl = −
φ0
B
D

I

1
A · dl +
φ0

Z C

Z A
A · dl +BA · dl .

(3.21)

D

The phase at any point on the circuit must be single-valued, so we label the first integral −2πn,where n is an integer. The next two integrals, by the fundamental theorem of calculus, reduce tothe differences of the integration bounds. The first term on the right, using B = ∇ × A with B
being the magnetic field local to the loop, is −Φtot /φ0 . Equation (3.21) now reduces to1− (θC − θB ) − (θA − θD ) −
φ0

Z C

Z A
A · dl +BA · dl = 2πn −

D

Φtot
.
φ0

(3.22)

In gauge-invariant form, the phase drops are expressed as
Z A
1
A · dlδ1 = θA − θD +
φ0 D
Z B
1
δ2 = θB − θC +A · dl,φ0 C(3.23)(3.24)

reducing Eq. (3.22) toδ2 = δ1 −Φtot+ 2πn.
φ0(3.25)

This relation will allow us to eliminate δ2 and work with the enclosed flux Φtot , a quantity we can
influence more readily in the lab.
Using Eq. (3.4) for either junction, we can calculate the inductance of the SQUID. AssumingΦtot is time-independent, we see that

h
i
tot
φ0 / 2Ic cos πΦΦ0
V = I˙ rhi
1 − 2Ic cos

πΦtot
Φ0

2

(3.26)

Comparing to Eq. (3.12), we see that the SQUID behaves exactly as a Josephson junction with
flux-dependent critical currentπΦtot
IS (Φtot ) = 2Ic cos
Φ0
,(3.27)

36
where the absolute value ensures positivity. This result makes intuitive sense: the factor of two
owes to having two paths in parallel for the current to take. The effect of an applied flux is tocreate a circulating current, reducing the amount of total current that can flow through the SQUID
before one of the junctions reaches its critical current. The SQUID inductance and energy39 are
analogous to Eqs. (3.11) and (3.13), respectively:
arcsin[I/IS (Φtot )]I/IS (Φtot )Φ,ESQ (t) = ES (Φtot ) 1 − cosφ0

LSQ (Φtot ) = LS (Φtot )(3.28)(3.29)

where Φ is the flux drop40 across the SQUID, Eq. (3.5), andLS (Φtot ) =φ0IS (Φtot )(3.30)ES (Φtot ) = φ0 IS (Φtot ).

(3.31)IS0 = 2Ic

(3.32)

φ0
IS0(3.33)We also define

LS0 =ES0 = φ0 IS0(3.34)

as the maximum value for the SQUID critical current, and linear terms for inductance and energy
when no flux threads the loop, respectively. Before we proceed to treat the SQUID as a Josephson junction with a flux-tunable critical current, we must consider the effect of a nonnegligible
geometrical inductance.
39

This is the energy associated with the total current flowing through the SQUID, and does not include any energy
stored in the junctions by a circulating current.
40
Though they have the same dimensions, the flux drop across (Φ) and penetrating (Φtot ) the SQUID loop are
distinct physical quantities. The former is defined as the time integral of the voltage across the SQUID, whereas thelatter is a flux in the more familiar sense of a field penetrating an area.

37
3.2.2

Geometrical inductance and flux-trapping

The effect of nonzero geometrical inductance Lg in the SQUID loop, Fig. 3.2, is to modify
the amount of flux threading the loop from the externally applied value Φext to a new value41
Φtot = Φext + Lg Icirc .

(3.35)The modification to the external flux will be expressed as a transcendental equation, and giverise to a design rule for SQUIDs whose Φtot we wish to keep as a single-valued function of Φext .
Combining Eqs. (3.16) and (3.25), we obtain
 






Φtot
Φtot
Φtot
Ic
sin δ1 −− sin δ1 = −Ic sin πcos δ1 − π.
Icirc =
2φ0
Φ0
Φ0(3.36)

Eliminating δ1 in favor of the total current I, a quantity we can impose experimentally, yields

s

2
Φtot
I
Icirc = −Ic sin π
1−.
(3.37)Φ0IS (Φtot )Substituting Eq. (3.37) into Eq. (3.35) gives us an expression relating the applied and total fluxes:


2
s
Φtot
I
Φtot = Φext − Lg Ic sin π
1−.
(3.38)Φ0IS (Φtot )This is our transcendental equation.
When
∂Φext
< 0,
∂Φtot

(3.39)

Φtot takes on multiple values for a given Φext , as illustrated in Fig. 3.4. Here, the SQUID becomeshysteretic, and enters the “flux-trapping” regime, which is undesirable for controlling it. To quantify
at what value of Lg flux-trapping sets in, we will consider the simple case where I = 042 so thatΦtot
Φext = Φtot + Lg Ic sin π,Φ041(3.40)For the symmetric SQUID loop, Eq. (3.18), total current I will not directly produce any additional magnetic
flux, as the two branches of the SQUID will contribute flux equally and oppositely.
42
Without this assumption the math gets considerably more opaque. The condition we derive with I = 0 will
suffice for us as a design rule.

38

Figure 3.4: Total versus externally applied flux for a SQUID with different geometrical inductances,Eq. (3.42). When βL > 2 (purple and brown curves) some values of Φext become associated withmultiple values of Φtot , and the SQUID becomes hysteretic.

which differentiates to
∂Φext
βL
=1+
cos
∂Φtot
2



πΦtot
Φ0


,

(3.41)

where
βL =

Lg
LJ0

(3.42)

dimensionlessly parametrizes the geometrical inductance. To avoid flux-trapping, Eq. (3.39), we
requireβL ≤ 2.

(3.43)

So long as we adhere to Eq. (3.43), a SQUID functions as a flux-tunable Josephson junction,
which in turn acts like a nonlinear inductor. In order to build circuits out of SQUIDs that can process quantum mechanical signals, we will next need to develop the language of circuit quantization.

39

3.3

Circuit quantization
The simple harmonic oscillator (SHO) is a canonical introductory example for students new

to quantum mechanics [174]. Though it is often introduced in a mechanical context — as a mass ona spring or hanging from a pendulum — the physics applies just as well to electrical oscillators. We
will first study the instructive example of a simple resonant circuit, and then apply the technique toa circuit which incorporates the SQUIDs of Sec. 3.2 in order to perform parametric amplification.

3.3.1

LC circuit

The circuit of Fig. 3.5 consists of an inductor L and capacitor C, which alternatingly store
magnetic and electric energy, respectively, as
1 Φ2
2 L
1
EC = C Φ̇2 .
2
EL =(3.44)(3.45)We have written everything in terms of a single generalized coordinate: the flux across the elements
Φ, Eq. (3.5), and its time derivative. The inductively and capacitively stored energies will beanalogous to the potential and kinetic energies of a mechanical oscillator.43

Figure 3.5: Simple LC circuit. The inductor L stores magnetic energy and the capacitor C stores
electric energy. Together, they form a resonator as the energy sloshes from one circuit element to
the other.

43
In the analogy to a mass m on a spring k, the capacitance C corresponds to m, and the inductance L corresponds
to 1/k.

40
3.3.2

Lagrangian and Hamiltonian

By writing down the Lagrangian of the system, we will be able to obtain the Hamiltonian,which compactly stores the information on the time evolution of the system. From Eqs. (3.44) and(3.45), we can write down the Lagrangian of the circuit in Fig. 3.5:
1
1 Φ2
L = C Φ̇2 −.
2
2 L(3.46)The momentum conjugate to Φ is obtained as∂L= C Φ̇ = Q.
∂ Φ̇

(3.47)

Physically, it is the charge on one capacitor plate, Eq. (3.6). To obtain the Hamiltonian from the
Lagrangian, we perform a Legendre transformation:
H(Φ, Q) = Φ̇1 Q2 1 Φ 2
∂L
+
,
−L=
2 C
2 L
∂ Φ̇

(3.48)

which is the sum of the circuit element energies.

3.3.3

Classical equations of motion

Before quantizing the system, we can check our understanding by writing down the Hamilton’s
equations of motion from Eq. (3.48):
dΦ
∂H
Q
=
=
dt
∂Q
C
dQ
∂H
Φ
=−
=−
dt
∂Φ
L

(3.49)(3.50)

The first equation, using Eq. (3.5), is simply the constitutive equation for the capacitor, commonly
writtenC=

Q.
V(3.51)

Using the definition for the conjugate momentum, Eq. (3.47), the second equation yields the equation of motion for a SHO,ΦC Φ̈ = − ,L(3.52)

41
which resonates at
ωLC = √

3.3.4

1
.
LC

(3.53)

Quantization and ladder operators

By analogy with the position and momentum of a mechanical oscillator, we promote Φ and
Q to Hermitian operators Φ̂ and Q̂, giving a Hamiltonian operator
Ĥ =1 Q̂2 1 Φ̂2
+
.
2 C
2 L

(3.54)

These new operators satisfy the canonical commutation relationship[Φ̂, Q̂] = i~,(3.55)and can be expressed in terms of the ladder operators441Φ̂ + iZLC Q̂
2~ZLC


1
Φ̂ − iZLC Q̂ ,Â† = √
2~ZLC
Â = √(3.56)(3.57)

where
r

L
C

(3.58)[Â, Â† ] = 1(3.59)ZLC =is the impedance of the resonator andfollows from Eq. (3.55). The inverse relationships to Eqs. (3.56) and (3.57) arer~ZLC †Φ̂ =(Â + Â)2
r~
Q̂ = i(Â† − Â).
2ZLC

(3.60)(3.61)In terms of the ladder operators, the Hamiltonian operator, Eq. (3.54), takes the familiar form




1
1
†
Ĥ = ~ωLC Â Â += ~ωLC n̂ +,22

(3.62)The ladder operators are commonly referred to as the creation, or raising, operator A† , and the annihilation, or
lowering, operator A for how they act upon energy eigenstates of the Hamiltonian [174].
44

42
where n̂ = Â† Â is the number operator, which acts on an energy eigenstate |ni asn̂ |ni = n |ni(3.63)

to return the number n of photons with energy ~ωLC occupying the resonator. The +1/2-photon
term in Eq. (3.62) is the ground state energy of the oscillator.45

3.3.5

Heisenberg equation of motion

The Hamiltonian, Eq. (3.54) or Eq. (3.62), is nothing more than a suitcase for compactly
carrying around the time-evolution behavior of a nondissipative system such as our LC circuit. To
unpack the time dependence, we use the Heisenberg equation of motion for the time evolution ofan operator Ô [175]:46dÔi= [Ĥ, Ô].
dt~

(3.64)For the example of our simple LC circuit, this yieldsdÂ= −iωLC Â.
dt(3.65)

If Â were a complex field amplitude, Eq. (3.65) says that it would simply evolve in phase withoutchanging amplitude, as is expected for an undriven, undamped resonator.
This concludes our introductory discussion of circuit quantization. We will take the languageof this section and apply it to more complex devices made out of the components of this andprevious sections to model their behavior.

3.4

Lumped-element Josephson parametric amplifier
Parametric amplification is a process that will allow us to both amplify (as the name suggests)

and also deamplify quantum observables [137, 176]. It is achieved by modulating a resonator suchas that modeled in Fig. 3.6 at approximately twice its resonance frequency. Equation (3.28), which
45

The +1/2 will correspond to the quantum noise in systems we are interested in measuring.
This equation works in the Heisenberg picture of quantum mechanics, in which operators evolve. It assumes
that Ô contains no explicit time dependence in the Schrödinger picture.
46

43
Taylor expands in the I/IS  1 limit to"1
LSQ (Φtot ) = LS (Φtot ) 1 +6IIS (Φtot )

2


+O

4 !#
I
IS (Φtot )(3.66)

suggests two possible ways of doing this with a microwave resonator containing SQUIDs. First,
the leading I 2 -dependence of the inductance provides a natural frequency-doubling: a current
oscillating at frequency ω will vary the inductance at 2ω. Second, the flux-dependent SQUID
inductance LS (Φtot ) can be directly modulated at 2ω. These two options correspond to applying a single, monochromatic current-pump tone (henceforth, current pumping)47 [180, 181] or asingle, monochromatic magnetic flux-pump tone (henceforth, flux pumping) [141, 182–186] to the
SQUID(s) in the resonator.
The resonators we will work with exist in the lumped-element regime [187], wherein their
spatial extent is much smaller than a wavelength of light at any frequency of interest.48 In particular,
the 4 GHz < ν < 15 GHz frequency window that this thesis operates within corresponds towavelengths
λ=

vTL
ν

(3.67)

of 1.3–5.0 cm, where vTL ≈ 2c/3 is the approximate speed of light in our transmission line geometries. Our devices, for comparison, will be sub-mm in length.
Figure 3.6a shows the resonator we will be working with: a lumped-element tunable Kerr
circuit (TKC), which lies at the heart of our JPAs [188]. In this thesis, “JPA” refers to a two-port
device that consists of a TKC plus (at least) one microwave circulator to separate incoming andoutgoing signals, as shown in Fig. 3.7. It is common in the literature, however, to refer to the
JPA as the one-port reflection device. At the center of the TKC is a resonator formed out ofa capacitor C and an inductive series array of N identical, balanced SQUIDs each with critical
47

There is a second way to pump using current: by applying two pump tones roughly equally detuned about
the central frequency of the resonator [177, 178]. Both of these current-pumping methods correspond to a four-wave
mixing process, whereas flux pumping uses three-wave mixing. In this thesis, current pumping will just refer to theuse of a single tone. For a comparison of some aspects of all three pumping schemes, see Ref. [179].
48
It is common for JPAs to use both lumped-element [143, 145, 147, 188] and distributed [139, 141, 148] resonator
designs.

44

Figure 3.6: (a) Circuit schematic for a tunable Kerr circuit (TKC). The TKC (within the dashedorange box) consists of a series array of N SQUIDs with critical currents IS in series with ageometrical inductance LG , all shunted by a capacitance C and coupled to the external environment
via a smaller coupling capacitor Cc . The TKC is here shown biased by a Thévenin equivalent
voltage source V e−iωt with impedance Z0 . (b) Circuit symbol used to represent the TKC in larger
schematics. The TKC will be used as an amplifier in reflection. (c) Input-output theory model of
the TKC as a resonant mode Â (blue) with a resonance ωTKC and Kerr nonlinearity K. The mode
couples at power decay rate γe to input and output propagating modes Âin and Âout , which aredecomposed into classical pump (α) and quantum signal (â) components.

current IS (Φtot ).49 The SQUIDs are modeled in series with any geometrical inductance LG .50
Finally, the TKC includes a smaller coupling capacitor Cc in order to control the coupling rate to
the external environment. Here, that environment is represented as a Thévenin equivalent voltage
source applying a stimulus at some frequency ω through an impedance Z0 . For us, Z0 = 50 Ω,
standard for microwave equipment. Starting in Sec. 3.4.1, we will first derive some basic propertiesof the TKC by analyzing the circuit diagram. In Sec. 3.4.2 we will obtain the Hamiltonian, which
we approximate as that of a Duffing (or Kerr) oscillator in Sec. 3.4.3. Finally, in Secs. 3.4.4–3.4.7
we will use input-output theory (IOT) [189, 190] to understand its capabilities for amplification,
deamplification, and squeezing.
49

We are assumed not to be in the flux-trapping regime of Sec. 3.2.2, such that a magnetic field Φext applied
uniformly across the SQUIDs gives rise to the same field Φtot in each loop via Eq. (3.35).
50
The SQUID loop geometrical inductance Lg of Eq. (3.35) contributes to the TKC series inductance LG .

45

Figure 3.7: Relationship between TKC (Fig. 3.6b) and Josephson parametric amplifier (JPA; triangle). The JPA is represented with the usual triangular symbol for amplifiers with input (output)
port on the left (right). Inside (in gray), it consists of a microwave circulator, which routes propagating waves in the sense indicated by the arrow on it, and the TKC, Fig. 3.6b, which processes signalsthat reflect off of it. Not shown is the line used to provide the pump tone, as the implementationdiffers between current- and flux-pumped designs.

3.4.1

Basic TKC circuit analysis

Basic properties of the circuit — its resonance frequency and external quality factor — canbe obtained from considering the impedances of and energies stored or dissipated within the circuit
elements. Here, we will treat the series array of SQUIDs, whose linear inductance components add
like simple linear inductors, as a single inductance
La = N LS ,(3.68)

where LS comes from Eq. (3.28) and I have dropped the dependence on Φtot for notational convenience. We further combine the inductive array with the geometrical inductance as
LaG = La + LG .

(3.69)

Since the coupling capacitor is in series with a reactive combination of circuit elements(Fig. 3.6a), resonance will occur when the series impedance51
Zseries = ZCc + ZLaG C
51(3.70)

Throughout this section, we assume a weakly coupled resonator Cc  C, with the absolute values of the
impedances of L and C near resonance roughly equal to Z0 . This is necessary to use the condition for a series
resonance.

46
vanishes, where
ZLaG C =ZCc =LaG /C
1iωLaG + iωC(3.71)

1.
iωCc(3.72)

This occurs at
ωTKC = √1
LaG CΣ

,(3.73)

whereCΣ = C + Cc .

(3.74)The external quality factor52 Qe quantifies what amount of the energy stored in the resonatordecays out to the external environment Z0 with each period of oscillation as
Qe = ωTKC

Estored,Pout(3.75)

where Estored and Pout are the energy stored in and power decaying out of the resonator on resonance. For the TKC, using Eqs. (3.44) and (3.45) for the energy stored in inductors and capacitors,
p
 
 2
L/C C 2C≈.
(3.76)
Qe ≈
Z0
Cc
Cc
To obtain the first equality, we have used the approximation C  Cc , which is valid for the JPAs
discussed in this thesis. In practice, we will design ZLC ≈ Z0 , as trying to push the inductor or capacitor impedance magnitude far from this value can give rise to parasitic reactances, causing themto no longer act as intended. The second equality therefore provides a reasonable approximation
for the external quality factor.
Finally, the external power decay rate of the resonator 
ωTKC
Z 0 Cc 2
γe =
,
≈
Qe
L C

(3.77)

corresponds to the 2π times the width (in Hz) of the Lorentzian response profile of the resonator.
With these useful quantities in hand, we will now obtain the Hamiltonian in order to study thenonlinearity and the dynamics of the system.
52

For our purposes, it will be a reasonable approximation to model the TKC as having no internal loss, so theinternal quality factor is treated as infinite. A simple way to model internal loss would be with a resistor in parallelwith the capacitor and array-plus-inductor.

47
3.4.2

TKC Lagrangian and Hamiltonian

The quantization procedure of Sec. 3.3 requires us to write down the energies of the elements
in Fig. 3.6a in terms of a generalized coordinate and its derivative. For simplicity, we will set
LG = 0 and return to its significance in Sec. 3.5 in relation to SQUID arrays. We will also neglect
the voltage source and its impedance, and add in the external coupling rate of Sec. 3.4.1 by handin Sec. 3.4.4. Our coordinate will be the flux drop Φa across the array, which is the sum of the N
identical flux drops ΦSQ across the individual SQUIDs:53
Φa = N ΦSQ .

(3.78)The magnetic energy Em associated with the array, using Eq. (3.29), is thereforeΦaEm = Ea 1 − cos,N φ0(3.79)

Ea = N ES(3.80)

where

via Eq. (3.31).
The electric energy Ee associated with the capacitors is given for each capacitor by Eq. (3.45).
Kirchhoff’s voltage law requires that voltage across Cc be equal and opposite to that across C,
making Φ̇2Cc = Φ̇2C = Φ̇2a and1
Ee = CΣ Φ̇2a .
2(3.81)

We can now obtain the Hamiltonian of the TKC by first obtaining the Lagrangian:


1
Φa
2
LTKC (Φa , Φ̇a ) = CΣ Φ̇a − Ea 1 − cos
.
2
N φ0

(3.82)The momentum conjugate to Φa is given by∂LTKC= CΣ Φ̇a = CVC − Cc VCc = Q∆ ,
∂ Φ̇a
53

(3.83)We are here assuming that the SQUID array has no internal dynamics of interest. If the SQUIDs are nearly
identical, are embedded in similar surroundings to one another, are not in the flux-trapping regime, and begin withthe same initial conditions, this is a reasonable assumption.

48
where the second equality uses Eq. (3.7) and the last equality interprets the quantity via Eq. (3.51) as
the difference in charge on the “positive” plates of the capacitors as denoted in Fig. 3.6a. Performing
the Legendre transformation of Eq. (3.48) yields the classical Hamiltonian,


Φa
1 Q2∆
HTKC (Φa , Q∆ ) =+ Ea 1 − cos
.
2 CΣ
N φ0
3.4.3

(3.84)

Kerr Hamiltonian

Rather than dealing with the full cosine dependence of the Hamiltonian, Eq. (3.84), we canTaylor expand for Φa /N φ0  1, equivalent to ΦSQ  φ0 for the flux drop across each SQUID or
I  Ic for the current through each junction. Performing this expansion, we obtain



 !
Q2∆
Ea
Φa 4
Φa 2
1
HTKC (Φa , Q∆ ) ≈
+
−
,
2CΣ
2
N φ0
12 N φ0(3.85)

where we have kept only the first term beyond the bare Hamiltonian:Q2∆
E a Φa 2
H0 (Φa , Q∆ ) =+.
2CΣ
2 N φ0(3.86)

Quadratic terms (Φ2a , Q2∆ ) in the Hamiltonian become linear in the equations of motion, andcomparing to Eq. (3.48) shows that H0 is indeed the Hamiltonian for a linear LC resonator, with
C → CΣ and
L → La = N

φ0
,
Is

(3.87)

exactly as we would expect for N linear inductances given by Eq, (3.30) in series.
Equation (3.85) will prove to be a Kerr Hamiltonian [191–194] of the form
HK = ~ω0 A∗ A + ~K ∗ ∗A A AA,2(3.88)

where A and its complex conjugate A∗ are the classical analogues54 of the ladder operators A andA† such that the number of photons worth of energy in the resonator isn = |A|2 .
54(3.89)

By this point, we could have quantized the system as in Sec. 3.3, but the Kerr constant can be calculatedclassically, with quantum mechanics providing an obvious motivation for factors of ~ that appear.

49
The Kerr Hamiltonian encodes the dynamics of the Duffing oscillator [195–197], with quartic terms
in the Hamiltonian corresponding to cubic terms in the equation of motion. The Kerr constant
K effectively shifts the resonance of a Duffing oscillator by K/2 for every quantum of energy
in the resonator. If K is positive (negative), the resonance shifts up (down), and in a mechanicalsystem the effect can be thought of as a nonlinear “hardening” (“softening”) of the spring constant.
Following Sec. 3.3.4, we can write down the expressions for A and A∗ :1
(Φa + iZTKC Q∆ )2~ZTKC
1A∗ = √(Φa − iZTKC Q∆ ) ,2~ZTKC
A= √(3.90)(3.91)

wherer
ZTKC =N φ0
CΣ I S(3.92)

has dimensions of impedance and the inverse relations arer~ZTKC ∗(A + A)
2r~(A∗ − A).
Q∆ = i2ZTKC
Φa =(3.93)(3.94)

Plugging Eqs. (3.93) and (3.94) into Eq. (3.85),


1
~2
2
1 4
2
2
∗2
∗4
HTKC ≈ ~ωTKC n −n + n(A + A ) + (A + A ) .
16 N 2 φ20 CΣ
3
6(3.95)The rotating wave approximation (RWA) allows us to ignore terms with unequal factors of A andA∗ on the grounds that for dynamics about resonance, they will oscillate rapidly, averaging to zero.
We identify our resonator’s Kerr constant from Eqs. (3.88) and (3.95) as
K=−

2
~ωTKC
~
=
−
.
8N φ0 IS
8N 2 φ20 CΣ

(3.96)The negative sign indicates that the array nonlinearity acts as a soft spring, pulling resonance
down. The second expression is more useful from a design standpoint. It illustrates how K may bechanged while keeping ωTKC constant. For example, we could increase both N and Ic (and hence
IS ) by some factor m, which would decrease K as m2 without changing ωTKC . Evidently, adding

50
more SQUIDs decreases the nonlinearity: physically, the flux drop across each individual SQUID
gets linearly smaller with N while the array energy, Eq. (3.79), does not depend on the flux drop
to linear order [198].
We have seen that the TKC circuit of Fig. 3.6a can behave as a Duffing oscillator, or equivalently, a resonator with a Kerr nonlinearity. In doing so, we have connected the parameters ωTKC ,Qe , γe , and K, which determine the behavior of the device, to the design parameters Ic , N , C,
Cc , and LG and the experimental knob Φtot . Next, we will see one way in which such a device candeliver parametric gain.

3.4.4

Input-output theory derivation of gain

JPAs amplify small signals at their input by “stealing” some of the power from a much larger55
pump tone. Here, we will derive the gain, deamplification, and squeezing of a current-pumped JPA
using IOT. The goal of IOT, as the name suggests, is to write down equations for the output fields
in terms of the input fields which do not include the internal degrees of freedom of the system.
This derivation closely follows that of Ref. [148]. For a more detailed derivation, see Ref. [180].
We take as our starting point the IOT model of Fig. 3.6c. The cavity is modelled with thequantized Kerr HamiltonianĤK = ~ω0 Â∗ Â + ~K † †Â Â ÂÂ,2(3.97)

whose classical counterpart is Eq. (3.88). Input and output fields56
Âin = (αin + âin )e−iωp t(3.98)Âout = (αout + âout )e−iωp t ,(3.99)

where αin and αout are respectively the incoming and outgoing strengths of a classical pump field at
frequency ωp and âin and âout are the corresponding quantum signals,57 are coupled to the cavity
55

As a rule of thumb for maintaining a sufficiently “stiff” pump tone for current pumping in practice, that pumptone should be at least 20 dB larger than the amplified signal [188].
56
All fields are for now functions of time t, where I will leave the functional dependence out for notational
convenience where it is not needed.
57
The pump can be treated as classical because the effect of its quantum fluctuations within the resonator can bemade much smaller than those of the signal by weakly coupling in a large tone to the shared signal/pump line.

51
at power decay rate58 γe , Eq. (3.77). Factoring out the trivial time dependence e−iωp t allows us towork in the rotating frame of the pump. The input and output fields obey the boundary condition√

γ e Â = Âin − Âout ,(3.100)and standard bosonic commutation relations subject to a Markov approximation [137, 189]. The
cavity mode Â is decomposed into pump (α) and signal (â) components as in Eqs. (3.98) and (3.99):Â = (α + â)e−iωp t .

(3.101)

We follow the standard IOT approach [189, 190]. Using Eq. (3.64), we will write down the
Heisenberg equation of motion for Â in Eq. (3.97). The Heisenberg equation of motion only models
the closed system, however. To model the interaction with the incoming and outgoing fields,
we must add on the terms modeling dissipation and associated fluctuation by hand, forming theHeisenberg-Langevin equation of motion. The Heisenberg-Langevin equation is then solved in thesteady state for large, classical input and output fields αin and αout corresponding to our pump
tone. We use the solution to linearize around the small input and output fields âin and âout , ending
with an equation for âout in terms of only input fields.
Applying Eq. (3.64) to Eq. (3.97) gives the Heisenberg equation for the internal dynamics ofthe system:dÂ
= −iωTKC Â − iK Â† ÂÂ.
dt(3.102)Inserting the coupling terms by hand gives the Heisenberg-Langevin equationdÂγe√= −iωTKC Â − iK Â† ÂÂ − Â + γe Âin ,dt
2(3.103)

which, together with Eq. (3.100), governs the full open system.
We first solve for the response to a large, classical toneÂin → αin e−iωp t .
58(3.104)The power decay rate is the inverse of the characteristic time for the energy in the resonator to deplete. Since
coupling is reciprocal, it also parametrizes how quickly an external drive adds energy to the cavity. It is equal totwice the field decay rate, and corresponds to the full-width at half-max of the Lorentzian response profile of thecavity. The field decay rate therefore corresponds to the half-width at half-max.

52
We are interested in the steady-state solution, where
dα
= 0.
dt

(3.105)

Plugging Eq. (3.104) and its analogues for α and αout into Eq. (3.102) subject to Eq. (3.105) yields
√

γe
α.
2

(3.106)γe nα − 2∆Kn2α + K 2 n3α ,κn2α,in = ∆2 +2

(3.107)∆ = ωp − ωTKC

(3.108)γe αin = i(ωTKC − ωp )α + iK|α|2 α +Multiplying Eq. (3.106) by its complex conjugate gives

wherenα,in = |αin |2(3.109)nα = |α|2(3.110)are respectively the detuning of of the pump from the bare resonance, the incoming number ofpump photons, and the number of photons from the pump in the resonator. Equation (3.106)
will be of considerable use to us in Ch. 4, without need to even directly incorporate the quantum
mechanical signal operators.
For now, we recast Eq. (3.107) in terms of the dimensionless quantities∆
γe|α̃in |2 K
ξ=
γe
nα
ñ =
|α̃in |2
δ=

(3.111)(3.112)(3.113)

as
2



1
2
ñ − 1 = 0,
ξ ñ − 2δξñ + δ +4(3.114)αin
α̃in = √γe(3.115)

2 3

where

53
In Eqs. (3.111), (3.112), (3.113), and (3.115), δ is the dimensionless drive detuning from resonance,59 ξ is the dimensionless nonlinearity (or dimensionless drive power, depending on what one
is interested in), ñ is the dimensionless resonator occupancy, and α̃in is the dimensionless drive
amplitude.
The cubic Eq. (3.114) specifies a critical point {δc , ξc } where
∂δ
∂2δ
= 0.
=
∂ ñ
∂ ñ2

(3.116)The critical drive detuning and power are given by√321
ξc = − √ ,3 3δc = −(3.117)(3.118)and the value of ñ at the critical point is
ñc = 3.

(3.119)For a portion of operation space below δc and above ξc there are three real solutions to ñ. Twoof these — the highest and lowest values for ñ — will be stable solutions, and the third unstable.
Figure 3.8 shows the response of the Duffing oscillator to sub- and super-critical drives. Whilethe bifurcation physics (magenta and black curves in Fig. 3.8) can also be used to read out feeble
signals by, for example, correlating the state of a qubit with the high/low amplitude stable statesof the oscillator [138], we will not be interested in it here.60 Instead, we will work just short of thecritical point (typically between the green and light blue curves in Fig. 3.8), where the nonlinear
response of the JPA is sharp but single-valued.
With the response to a large pump tone in hand, we now solve for the response to the full
input field Eq. (3.101). Equation (3.102) gives
dâ 
γe 
√
= i∆ − 2iKnα +â − iKnα â† + γ e âin ,dt
2
59(3.120)This is an entirely separate quantity from the phase drop δ defined in Eq. (3.2) at the beginning of this chapter.
The flux-pumped JPAs of Sec. 3.7 likewise have useful bifurcation physics. Reference [170] uses a closely-related
design to measure a superconducting qubit without the need for a ferrite circulator prior to amplification.
60

54

Figure 3.8: Duffing oscillator response showing internal amplitude α relative to its value at thecritical point αc versus pump frequency ωp (absent any signal tone) relative to TKC resonant
frequency ωTKC . The different curves correspond to different drive powers linearly spaced from
ξ = 0.25ξc (blue) to ξ = 1.5ξc (black). Beyond the critical point, the response becomes multivalued. Figure adapted from Ref. [173].

where we have thrown out terms higher than linear order in â. Using the conventionrb(t) =γe2πZ ∞˜ et ˜˜ −i∆γd∆eb(∆)(3.121)−∞

to examine the Fourier domain behavior in terms of the dimensionless detuning
˜ = ω − ωp ,∆γeEq. (3.120) becomes



1
˜
˜
 âin (∆)  √ i(−δ + 2ξñ − ∆) + 2
 = γe ˜−iξñe−2iφâ†in (−∆)

(3.122)˜
  â(∆) 

,
1
†
˜
˜
i(δ − 2ξñ − ∆) + 2â (−∆)
ξñe2iφ(3.123)where φ is the phase of the pump-induced field α = |α|eiφ inside the cavity. If we choose to regard
˜ as the signal frequency, −∆˜ is referred to as the idler frequency. Inverting Eq. (3.123) and using∆
the boundary condition, Eq. (3.100), we end up with our final Fourier domain relationship between
the incoming and outgoing signal fields:
˜ = gs (∆)â˜ in (∆)
˜ + gi (∆)â˜ † (−∆),˜âout (∆)
in(3.124)

55
where
˜
˜ = 1 − i(δ − 2ξñ − ∆) + 1/2
gs (∆)˜ − λ− )(i∆˜ − λ+ )(i∆ξñe2iφ˜ =gi (∆)˜ − λ− )(i∆˜ − λ+ )(i∆(3.125)(3.126)

and
λ± =

1 p
± (ξñ)2 − (δ − 2ξñ)2 .
2(3.127)˜ and gi (∆)˜ relate the output field
The frequency-dependent signal and idler amplitude gains gs (∆)˜ and to it’s conjugate at −∆.
˜ The signal and idler power gains61 areto the input field at ∆˜ = |gs (∆)|˜ 2
Gs (∆)(3.128)˜ = |gi (∆)|
˜ 2 = Gs (∆)˜ −1
Gi (∆)(3.129)˜ equally spaced above
The Kerr nonlinearity has mapped the behavior of tones input at both ±∆,and below the pump, to a single output frequency. In other words, the JPA’s output at a given
signal frequency ω cares not only about the signal input at ω, but about the input at the idler
frequency, 2ωp −ω. Since the reverse is true for the output at 2ωp −ω, there is a spectral ambiguity inthe output:62 a signal at a given frequency could have originated at either relevant input frequency.

3.4.5

Gain-bandwidth product

Another important property of the JPA is its preservation of the amplitude gain-bandwidth
product (GBP). The signal and idler gains are both symmetric about the pump frequency,
˜ = Gs (−∆)˜
Gs (∆)(3.130)˜ = Gi (−∆)˜
Gi (∆)(3.131)

61
Both the signal and idler gains are phase-insensitive, though not phase-preserving, as the idler gain is phaseconjugating. The signal gain is also referred to as the “direct gain.” It amplifies both quadratures of the microwave
field equally, so where that property is important to distinguish from single-quadrature gain, (which is a linear
combination of signal and idler gains, Sec. 3.4.6), I will refer to it as the “two-quadrature gain.” Additionally, though
the signal/direct/two-quadrature gain is a frequency-dependent quantity per Eq. (3.128), in some contexts it will be
convenient to refer to the maximum value Gs (0) as simply the signal/direct/two-quadrature gain.
62
The −1 in Eq. (3.129) means the contributions are not strictly identical, but for typical Gs >∼ 100, the differenceis quite negligible to amplification. For squeezing, however, it is important: without it, squeezing would be infinite
at any gain.

56
with full-width at half-max JPA bandwidth
B = 2λ+ .

(3.132)p
2
Gs (0)B ≈ √ γe .
3(3.133)

Near the critical point, Eqs. (3.117)–(3.119),

The quality factor of the resonant circuit in Fig. 3.6a therefore sets the bandwidth (for a given gain)of the Kerr resonator in Fig. 3.6c, independent of nonlinearity.

3.4.6

Single-quadrature amplification, deamplification, and squeezing

We can readily see that Eq. (3.124) gives rise to squeezing. Identifying
˜ = cosh r
gs (∆)(3.134)˜ = eiθ sinh r,
gi (∆)(3.135)˜ iθ sinh r,
˜ = Ŝ † (reiθ )âin (∆)˜ Ŝ(reiθ ) = âin (∆)˜ cosh r − â† (−∆)eâout (∆)
in(3.136)

Eq. (3.124) can be recast as

whereupon we see Ŝ(reiθ ) to be the unitary squeezing operator [193, 194]. If we define the Fourier
domain quadrature operators for the signal as˜ = √1 â(∆)˜ + â† (−∆)
˜
X̂(∆)
2


1
˜ =√
˜ − â† (−∆)
˜ ,
â(∆)
Ŷ (∆)
2i(3.137)(3.138)˜ for an input vacuum state |0i
and set θ = 0, the squeezing operator reduces the variance of X̂(∆)˜ by e2r . For general θ, the same applies
below its vacuum level by e−2r and amplifies that of Ŷ (∆)
to a pair of generalized (i.e. rotated) quadratures.
The two-quadrature gain Gs and the single-quadrature gain or deamplification63 G1Q parametrizedby θ are related by the equation
G1Q (Gs , θ) = (2Gs − 1) + 2
63p
Gs (Gs − 1) cos(2θ),(3.139)Deamplification is a more general term than squeezing for what the JPA does to the quadrature out of phasewith the pump. When the input state has its variance deamplified below vacuum, it is said to be squeezed.

57

Figure 3.9: (a) Single-quadrature gain G1Q plotted as a function of pump-probe phase difference
for a two-quadrature gain Gs = 20. (b) Closeup of (a) for G1Q < 1, where deamplification occurs.

plotted in Fig. 3.9a. Figure 3.9b shows the region below G1Q = 1, where deamplification occurs.
Maximum single-quadrature amplification (deamplification) is achieved at θ = {0, π} ({π/2, 3π/2}),
where in the high signal gain (Gs  1) limit
G1Q (Gs → ∞, 0) → 4Gs

1
π
→
.
G1Q Gs → ∞,2
4Gs

(3.140)(3.141)

Because of the phase-dependent behavior of single-quadrature gain, this mode of operation is also
referred to as phase-sensitive, and conversely two-quadrature gain occurs with phase-insensitive
operation.
When performing phase-insensitive amplification, the JPA must add at least one half quantum of noise spectral density distributed equally over both quadratures, the minimum allowed

58
by quantum mechanics for two-quadrature measurement. [128, 199, 200]. This noise traces to thevacuum fluctuations of the idler mode,64 now mapped to the signal output. In phase-sensitive
operation, the amplifier in principle does not have to add any noise along the amplified quadrature,making the JPA an excellent tool for measuring feeble signals.
This concludes our discussion of current-pumped JPA amplification, deamplification, andsqueezing. In Sec. 3.4.7, we will see that flux pumping can give rise to the same phenomena,
without explicit reliance on the Kerr nonlinearity.

3.4.7

Derivation of flux-pumped JPA gain

Relative to current-pumped JPAs, flux-pumped JPAs perform a more idealized operation
wherein the inductance of the SQUID(s) can be thought of as simply varying at the pump frequency.
When that frequency is twice the resonant frequency, parametric amplification is realized [141,182],
as in the current-pumped case treated in Sec. 3.4.4. This section closely follows the derivation in
Appx. A of Ref. [201], with the goal of obtaining the signal and idler gains equivalent to Eqs. (3.128)
and (3.129).
We begin with the Hamiltonian of Eq. (3.86), which we recast in terms of the TKC resonant
frequency ωTKC , Eq. (3.73), as
H0 (Φa , Q∆ ) =

Q2∆
CΣ 2
+
ω
Φ2 .
2CΣ
2 TKC a(3.142)

We will modulate the resonance as
ωTKC → ωTKC [1 + Λ cos(ωp t)] ,(3.143)where Λ  1 and ωp are the pump amplitude and frequency, respectively.65 Neglecting the Λ2term, Eq. (3.143) modifies the Hamiltonian as
H0 (Φa , Q∆ ) =
64

Q2∆
CΣ 2
+
ω
[1 +  cos(ωp t)] Φ2a ,
2CΣ
2 TKC

(3.144)That is, it corresponds to the +1/2 in the idler mode’s harmonic oscillator Hamiltonian (Footnote 45).
In practice, Eq. (3.143) is the dominant contribution from varying the array inductance, Eq (3.87), at ωp . In
particular, a time-varying Φext gives rise to a time-varying Φtot via Eq. (3.38), which varies the SQUID criticalcurrents, Eq. (3.27), in order to vary the array inductance and with it ωTKC .
65

59
where
 = 2Λ.

(3.145)The corresponding equation of motion of the resonator field operator Â, where we havepromoted the canonical variables to operators as in Sec. 3.3.4, is given by
dÂ 
γe 
√
= −iωTKC −Â − 2iωTKC  cos(ωp t)(Â + Â† ) + γe Âin ,dt
2(3.146)

where we have ignored the ground state energy term and used the operator-equivalents of Eqs. (3.90)
and (3.91) for the field operator and its conjugate. We rewrite Eq. (3.146) asdÂ+ W Â + V Â† + F̂ ,dt

(3.147)

where
W (t) = iωTKC +γe+ 2iωTKC  cos(ωp t)
2

V (t) = 2iωTKC  cos(ωp t)
F̂ (t) =√γe Âin (t)(3.148)(3.149)(3.150)show the time dependencies of each term explicitly. From Eq. (3.147), we obtain the three equationsdÂ†= −W ∗ Â† − V ∗ Â + F̂ †
dt
!
1
dÂ
†
Â =
F̂ −
− W Â
Vdt
d2 Â dWdÂ dv †dÂ†dF̂
+Â + W+ Â + V
=
.
2
dt
dt
dt
dt
dt
dt
Together Eqs. (3.151)–(3.153) yield




d2 Â
1 dV dÂ
dW
W dV
∗
2
2
+
W
+
W
−
+
−
+
|W
|
−
|V
|
Â
dt2
V dt
dt
dt
V dt


dF̂
1 dV
=
+ W∗ −
F̂ − V F̂ † .
dt
V dt(3.151)(3.152)(3.153)(3.154)

We now specify to the case of a classical resonator field and signal input
Â → α(3.155)Âin → Eeiωs t ,(3.156)

60
which will be sufficient to derive the amplification properties of the flux-pumped JPAs. In Eq. (3.156),
E and ωs are the signal amplitude and frequency, respectively. Performing an RWA by letting terms
rotating rapidly relative to ωp ≈ 2ωTKC and ωs ≈ ωTKC average away, Eq. (3.154) becomes

 2
ωp
γe
d2 α
dα
γe
2
2
)
+
iω
α
+
[κ
+
iω
]
+
+
ω
(1
−

−
p
p
TKC
dt2
dt
4
ωTKC
2
(3.157)hγi√√e−iωs t∗ −i(ωp −ωs )t= γe E+ i(ωp − ωs − ωTKC ) e− iωTKC  γe E e.
2
To solve Eq. (3.157), we write down the homogeneous solution, obtained by setting the righthand side to zero, and the particular solution. The homogeneous solution takes the form
αh = C+ eη+ t + C− eη− t ,(3.158)where
γe + iωp± ωTKCη± = −
2

s



2 −

ωp
−1
2ωTKC

2
.

(3.159)The particular solution isZ
αp = −y+f y−dt + y−
D

Z

f y+
dt,
D

(3.160)where
y± = eη± t(3.161)D = (η− − η+ )e(η− +η+ )thγ
i√ef = γe E+ i(ωp − ωs − ωTKC ) e−iωs t − iωTKC  γe E ∗ e−i(ωp −ωs )t .
2

(3.162)αp = As e−iωs t + Ai e−i(ωp −ωs )t ,(3.164)(3.163)

Hence

where
√



γe E γ2e + i(ωp − ωs − ωTKC )  γeAs =  γe
2 2
2 + i(ωTKC − ωs )2 + i(ωp − ωs − ωTKC ) −  ωTKC√
iωTKC  γe E ∗  γe
Ai =  γe.
2 2
2 + i(ωTKC − ωs )2 + i(ωp − ωs − ωTKC ) −  ωTKC

(3.165)(3.166)

The full solution to Eq. (3.157) is
α = αh + αp .

(3.167)

61
For the homogeneous solution αh to die out, the real parts of both η+ and η− need to be
negative. While Re(η− ) < 0 always, Re(η+ ) < 0 occurs when
s
2

ωp
γe
2
−1 .
> ωTKC  −2
2ωTKC(3.168)

We define the critical value
c =

γe
1
=
,
2ωTKC
2Qe

(3.169)where the last equality uses Eq. (3.77). For pumping at twice the resonant frequency, ωp = 2ωTKC ,c is is seen via Eq. (3.145) to be twice the modulation amplitude of the resonance frequency where
the flux-pumped JPA response diverges. This corresponds to modulating the TKC resonance,
Eq. (3.143), by a fractional amount
Λ≈

1
4Qe

(3.170)in order to achieve high gain.
Operating below the critical point,66 the steady state solution to Eq. (3.157) is αp , Eq. (3.164).
Using our boundary condition, Eq. (3.100), the classical output signal is given by
Aout = (E −√γe As ) e−iωs t −√γe Ai e−(ωp −ωs )t .

(3.171)For the non-degenerate case where ωp 6= 2ωs , Eq. (3.171) corresponds to signal and idler power
gains
 γe

2
√
+ i(ωp − ωs − ωTKC )γ e As 22  γe
Gs = 1 −= 1 − γe  γe
(3.172)2 2
E
2 + i(ωTKC − ωs )2 + i(ωp − ωs − ωTKC ) −  ωTKC√
Gi =γ e Ai 2γ ω  γe e TKC=  γe∗2 2E
2 + i(ωTKC − ωs )2 + i(ωp − ωs − ωTKC ) −  ωTKC

2.

(3.173)

These gains are the analogues of Eqs. (3.128) and (3.129) for the current-pumped Gs and Gi in
Sec. 3.4.4. As in Secs. 3.4.5 and 3.4.6, the flux-pumped JPA is has a GBP set by its external
coupling rate γe , and can be used to amplify or deamplify one quadrature of an input signal. In
66
For a derivation of the bifucatory properties of the flux-pumped JPA operated above its critical point, see theSupplemental Material of Ref. [170].

62
Chs. 5 and 8, we will see them used experimentally to squeeze and amplify vacuum in the serviceof speeding up dark matter axion detection efforts.
Having now derived the key amplification properties of both current- and flux-pumped JPAs,
we will return in Sec. 3.5 to the current-pumped case in order to examine the role of stray inductanceand the use of SQUID arrays (Fig. 3.6a) in the TKC design. The ensuing discussion will informour design of current- as well as flux-pumped JPAs in Secs. 3.6 and 3.7, respectively.

3.5

Validity of the Duffing approximation
There is cause to question the validity of the Duffing approximation [195–197] for describing

the physics of array-based, current-pumped JPAs (Henceforth, array-based JPAs).67 Recall thatin the Duffing (or Kerr) approximation shortened the full Hamiltonian, Eq. (3.84), to that of alinear LC resonator plus the first nonlinear correction due to the SQUID array, Eq. (3.85). That
assumption was predicated on small excitations (in terms of flux relative to a flux-quantum, or
equivalently current relative to critical current) of the SQUIDs. In their 2007 paper, Manucharyan
et al. [181] derive the requirements on circuits used to embed Josephson junctions in order to createparametric amplifiers. Their central requirement for the validity of the Duffing approximation isviolated by our array-based JPAs. At stake is not whether array-based JPAs really work: theoreticaland experimental evidence indicate strongly that they do [148, 163, 188, 198]. Rather, the questionis whether they work in spite of violating the Duffing approximation, or subject to it. This section
presents the requirement of Ref. [181], which applies directly to single-junction68 JPAs in thepresence of geometrical inductances, extends it to array-based JPAs, and finally shows that it isincorrect, even for single-junction JPAs. In its place, we will be left with a modified version of theviolated condition of Ref. [181] which our array-based JPAs are seen not to violate.
67

The key point here is we are assuming the oscillator is operated near the critical point, as JPAs typically are(Sec. 3.4.4), in order to inquire if the Duffing approximation adequately describes them in that limit. The Duffing
approximation is of course perfectly valid for low-amplitude drives.
68
The difference between series arrays of junctions and series arrays of SQUIDs is immaterial to this section. For
both simplicity and proximity to Ref. [181], this section works with junctions and arrays thereof.

63
3.5.1

The Qp condition for a single-junction JPA

Manucharyan et al. [181] analyze a single Josephson junction embedded in an arbitrary linear
microwave network, shown in a parallel and series configuration in Fig. 3.10. For a junction withcritical current Ic , the constitutive equation, Eq. (3.3), expands as
"
#
 
Φ
Φ
1 Φ 2
I = Ic sin= Ic
1−+ ··· ,
φ0
φ0
6 φ0

(3.174)

where the first nonlinear term is the Duffing term.

Figure 3.10: (a) The Josephson junction is represented up through its first nonlinear term as an
ideal inductor in parallel with a parallel nonlinear element (PNL) governed by Eq. (3.175). (b)The embedding environment for the PNL is modeled by an admittance Y (ω) with drive and noise
current sources represented by their Norton equivalents. (c), (d), and Eq. (3.176) are the series
nonlinear element (SNL)/Thévenin equivalent versions of (a), (b), and Eq. (3.175), respectively.
Panels (b) and (d) reproduced from Ref. [181].

The relevant procedure is as follows: first, a decision is made regarding the use of the parallel
nonlinear element (PNL) versus the series nonlinear element (SNL), shown in Fig. 3.10a and c,
respectively. Using the PNL (SNL) requires modeling the drive as its Norton (Thévenin) equivalentcircuit, as in Fig. 3.10b(d). The linear portion of Josephson inductance LJ0 , Eq. (3.10), is folded
into the equivalent admittance Y (ω) (impedance Z(ω)). The cubic constitutive equations for the64
PNL and SNL follow directly from Eq. (3.174):
IPNL (t) = −ΦSNL (t) =1
6LJ0 φ20

Φ3PNL (t)

L3J0 3I(t).
6φ20 SNL(3.175)(3.176)

With a model chosen, the next step is to define a participation ratio69 p ∈ [0, 1], which characterizes the amount of Josephson inductance relative to total linear inductance in the embedding
circuit (in which LJ0 is now included). In Fig. 3.6a, additional linear inductance is represented by
LG . Reference [181] gives simple examples for the cases of a parallel and a series RLC circuit,shown in Fig. 3.11a and b, respectively. Calling the linear inductance Lp for the parallel case or
Ls for the series case, pp = Lp /LJ0 for the parallel case and ps = LJ0 /Ls for the series case. The
other relevant parameter to the analysis is the embedding circuit quality factor Q.

Figure 3.11: (a) Current-driven parallel RLC circuit with a PNL. (b) Voltage-driven series RLC
circuit with an SNL. (a) and (b) correspond to simple cases of Fig. 3.10b and d, respectively. Figure
reproduced from Ref. [181].

Reference [181] analyzes the simple and general cases, and concludes that
Qp  1,69(3.177)The subscript-less p is used to denote the participation ratio for statements that refer to both the series andparallel cases.

65
is the necessary condition to keep the maximum current through the junction well below Ic , andhence to satisfy the Duffing approximation.

3.5.2

Significance of the participation ratio

The embedding circuit machinery relies upon the notion that the linear portion of the Josephson inductance can be wrapped up into a linear inductance apart from the junction itself without
any change to the physics. The Josephson inductance as it appears in the participation ratio is there
not merely as a quantification of the linear inductance that the junction carries with it, but as aproxy for the nonlinear inductance that is always associated with that particular linear inductance.
This, then, is the significance of the ratio of Josephson linear inductance-to-non-Josephson linear
inductance: the Josephson inductance always carries with it, in fixed ratio, a nonlinear inductance,
while any stray linear inductance does not.

3.5.3

From junction to array

The ratio of nonlinear-to-linear inductance changes when the junction is swapped out for aseries junction array. In order to faithfully map the participation ratio from a single junction toan equivalent array (meaning one with the same linear inductance, which will therefore give thesame bare resonance), we must first determine the composition of that equivalent array. Then we
will use the fact that we can change what linear inductance we choose to call Josephson or stray70
in order to restore the nonlinear-to-linear inductance ratio of the single-junction case, whence theparticipation ratio is immediately obtained. We will follow this procedure for two cases: the series
RLC case (which generalizes to the general Thévenin/impedance case, Fig. 3.10b) and the parallel
RLC case (likewise the general Norton/admittance case, Fig. 3.10d). The result is the same in bothcases: the participation ratio decreases quadratically with N .
The constitutive equation of a series array of N junctions (each of critical current Ic0 andlinear Josephson inductance L0J = φ0 /Ic0 ) differs fundamentally from that of a single junction or70

The non-Josephson embedding inductance need not be undesirable, as the word “stray” might imply.

66
parallel array, as N infiltrates the sine:71
"
#

2
Φ
Φ
1
Φ
a
a
a
I = Ic0 sin= Ic01−+ ··· ,
N φ0
N φ0
6 N φ0(3.178)where I is the current through and Φa the flux across the array. The nonlinearity has been reduced
relative to the linear Josephson inductance by a factor of N 2 .

3.5.4

Series RLC and general Thévenin caseIn the case of a series RLC embedding circuit, we are concerned with the stray inductanceLs,stray together with the linear and nonlinear inductances of the array.72 We begin by solving
Eq. (3.178) for Φa to third order in I, obtainingI1
Φa = N φ0 arcsin 0 = N L0J0 I + N L0J0
Ic
6



I
Ic0

2
I + ···(3.179)Dividing by I gives the linear (La0 ) and first nonlinear (Ls,nl ) inductance terms:
La0 = N L0J0
1
Ls,nl = N L0J0
6

(3.180)


I
Ic0

2
.

(3.181)

As discussed in Sec. 3.5.2, in order for the participation ratio to be a meaningful proxy for the
nonlinear inductance, we need to ensure that the linear and nonlinear array inductances maintaina fixed ratio rs :
Ls,nl
1
rs =
=
La0
6



I
Ic0

2
.

(3.182)For the array and junction to be equivalent, they must have the same linear inductance. We
therefore require
La0 = N L0J0 = LJ0 .
71(3.183)In the case of a parallel array, such as the simple case of a SQUID, adding junctions in parallel increases thecritical current, Eq. (3.32), something that in principle can be done with a different physical junction, Eq. (3.1).
Adding junctions in series, conversely, is like changing the flux quantum: the constitutive equation of a series arraycannot be achieved with any single junction; as we will see, however, it can be mimicked by a single junction plus
some linear inductance.
72
If we take there to be an inductive term, which we label Ls,stray , somewhere in the general Thévenin case,Fig. 3.10d, the following analysis will apply equally to the more general case.

67
Since φ0 is a constant of nature, the only way to satisfy Eq. (3.183) is with Ic0 = N Ic . Written interms of Ic , Eq. (3.182) becomes
1
rs =
6N 2



I
Ic

2
.

(3.184)We see that Ic0 ’s dependence on N , brought about to ensure equal array and single-junction linear
inductances, implies an N -dependent inductance ratio. The Josephson inductance is now no longer
the proxy for nonlinear inductance it was in the single-junction case, causing the participation ratio
to no longer carry the same meaning. We will restore the constancy of this ratio for the series andthen the parallel embedding circuit case with some algebra incorporating stray linear inductances.

Figure 3.12: Series inductor algebra showing the equivalence between stray and array linear inductances. The total series inductance of an N -Josephson junction array and a stray series inductanceLs,stray (top left) is represented to first nonlinear order by writing the array’s inductance in terms of
its linear and first nonlinear contributions (top right). An equivalent configuration (bottom right),
consisting of the same total linear inductance and nonlinear-to-linear array inductance ratio rs , isachieved when going from array to single-junction by increasing the value of the stray inductance.

To do so, we start by equating the array of junctions with a linear and nonlinear inductor, as
shown in Fig. 3.12. Then we use the physical indistinguishability of the array’s linear inductanceand the stray linear inductance to move some of the former over to the latter such that the total
linear inductance is preserved and the ratio of nonlinear-to-Josephson inductance is modified tomatch the N = 1 case. The two effective linear inductances become:La0 → Leff
s,a =

1
La0
N2

Ls,stray → Leffs,stray = Ls,stray +(3.185)
N2 − 1La0 .
N2(3.186)

68
The sum of linear inductances is preserved, and the ratio
Ls,nl1
rs → rseff = eff =
6
Ls,a0



I
Ic

2
(3.187)is N -independent, as desired.
The array inductance is now a proxy for the nonlinear inductance in just the manner thatthe single-junction inductance is in the simpler case. Hence, Ref. [181]’s conclusions pertaining top will hold exactly if we redefine ps :
ps =

Leff
Leff
LJ0
LJ0
ps
s,a0
s,a0
=
→ peff
=
=
= 2,
s
eff
eff
eff
Ls
LJ0 + Ls,strayN
Ls
Ls,a0 + Ls,stray(3.188)where Ls and Ls,eff are the total linear series inductance and are equal to one another. We conclude
that replacing a single junction with an equally linearly inductive array of N junctions reduces theparticipation ratio by a factor of N 2 in the series embedding case.

3.5.5

Parallel RLC and general Norton case

Figure 3.13: Parallel inductor algebra showing the equivalence between stray and array linear
inductances. The parallel case reasons analogously to the series case of Fig. 3.12: the array (left) isexpressed as a linear inductor La0 and a nonlinear inductor Lp,nl (center). The linear inductanceis increased to restore the ratio rp to its N = 1 value when going from array to single-junction.
Doing so necessitates altering the stray inductance (right).

The reasoning for the parallel case, shown in Fig. 3.13, is analogous to the series case. To
first nonlinear order, the array is represented by a linear inductance La0 = N L0J0 (where, as before,

69
N L0J0 = LJ0 → Ic0 = N Ic ) in parallel with a nonlinear inductance Lp,nl = rp La0 , with the ratio rp
given byrp = 6N

2



φ0
Φa

2
.

(3.189)As such, La0 needs to be increased by N 2 :2
La0 → Leffp,a0 = N La0 ,(3.190)in order to restore the N = 1 ratio and give the participation ratio its original meaning of Sec. 3.5.2.
To preserve the parallel linear inductance, we also require
Lp,stray → Leffp,stray =N 2 La0 Lp,stray,(N 2 − 1)Lp,stray + N 2 La0(3.191)yielding

pp =

Lp
=
LJ0

1
1
La0 + Lp,stray

Lp,a0



−1
→ peff
p =

Leff
p

=
eff

1
Leff
p,a0

La0

+ Leff1−1p,stray

Leff
p,a0

=

pp
,
N2

(3.192)the same result as in the series embedding case.

3.5.6

The Qp condition for an array-based JPA

For both the series and parallel embedding circuits, Eqs. (3.188) and (3.192) indicate thatEq. (3.177) becomes
Qp  N 2 ,(3.193)where p is the original participation ratio defined between the linear Josephson (more generally,array) and total series or parallel linear inductance. Typical JPAs with N ∼ 10 junctions (orSQUIDs) and Q ∼ 50 fail to meet this condition, yet remain functional.
We will see that array-based JPAs work because the participation ratio in fact has no placein what would otherwise be the Qp-condition. In concert with the findings above, this implies thatstray inductance can, like SQUID arrays, be used to increase power handling [148], but potentiallywithout the adverse consequences of nonuniform junction or SQUID loop fabrication [173].

70
3.5.7

Independence of the validity of the Duffing approximation from the nonlinearity

The Duffing approximation analysis in Ref. [181] maps back to the equation for a damped,
driven Duffing oscillator,Ẍ +ω0Ẋ + ω02 X(1 − νX 2 ) = A cos ωd t,
Q(3.194)

with X the coordinate, ω0 the bare resonance frequency, ν the strength of the nonlinearity, andthe right-hand side a drive tone of amplitude A and frequency ωd . For an electrical system, theparticipation ratio p maps directly onto the nonlinearity. For example, in the case of the parallel
RLC circuit with a single junction, ν corresponds to p/6.
Let us consider the effect of changing ν (equivalently p) on the solution of Eq. (3.194) and onthat of the fully nonlinear equation from which it comes,
Ẍ +

 √ 
ω0
ω2
Ẋ + √ 0 sin X 6ν = A cos ωd t.
Q6ν

(3.195)The sine in Eq. (3.195) can be expanded, and an arbitrary number n of nonlinear terms kept:n

X (−6)mω0Ẍ + Ẋ + ω02 X(X 2 ν)m = A cos ωd t.
Q(2m + 1)!

(3.196)m=0Equation (3.194) corresponds to n = 1, and Eq. (3.195) to n = ∞.
Inspection reveals that, if Eq. (3.196) for arbitrary n and ν has solution Xnν (t), then thepequation for n and ν 0 will be solved by ν/ν 0 Xnν (t), provided that the drive amplitude is likewisepadjusted, A → ν/ν 0 A. This corresponds to driving the JPA at a fixed percentage of its criticalpower.73
Therefore, the ratio of any two solutions for arbitrary n1 and n2 and a given nonlinearity ν,Znν1 ,n2 (t) ≡Xnν1 (t),Xnν2 (t)(3.197)

remains unaffected if the nonlinearity is changed to ν 0 . The significance of this statement stems
ν (t) directly quantifies the divergence of the Duffing approximation from thefrom the fact that Z1,∞
73

This can also be seen in Eq. (3.114), wherein the Kerr constant only enters alongside the drive strength [148].

71
full solution: it other words, it is measures how valid the Duffing approximation is. The fact that
ν (t) is independent of ν tells us that the nonlinearity does not affect the validity of the Duffing
Z1,∞

approximation for JPAs.74 So the Duffing approximation may be however bad or good it is, but
its correctness does not depend on changes to the nonlinearity.

3.5.8

Modified condition for the Duffing approximation

Since Q does not ultimately affect the dynamics of the system, Eq. (3.195) or Eq. (3.196), in
the same fashion as ν does (its situation alongside p in the Qp product, Eq. (3.177) or (3.193), being
therefore somewhat incidental in this sense), there is no reason that the Duffing approximation’svalidity for JPA operation should be independent of Q. Numerical tests which confirm the predictionof independence of Duffing validity from ν reveal that the Duffing approximation does break down
as Q approaches unity from above. The phase response of a reflected tone bout is shown near thecritical point in Fig. 3.14 for different orders of approximation n = {1, 2, ∞} and values of Q off of alossless, array-based JPA. The array of N = 20 junctions easily violates the modified Qp condition,
Eq. (3.193), for all values of Q shown. Nevertheless, the agreement between the Duffing system(n = 1) and the full sinusoidal solution (n = ∞) is quite good at high Q. This confirms that theparticipation ratio plays no role in determining the validity of the Duffing approximation for JPA
operation, but that Q still does. Thus,
Q  1,(3.198)independent of geometrical inductance or number of junctions, is the relevant predictor of Duffing approximation validity for JPA operation. The presence of stray inductance or of Josephson
junction arrays have no bearing.
This concludes our detour into the theory of the Duffing oscillator. The independence of thevalidity of the Duffing approximation for JPAs from the degree of nonlinearity in the system explains
why no divergence from the Duffing results has been noted for array-based JPAs in the literature.
74

More generally, it does not affect the agreement of any two levels of approximation, parametrized by different
values for n1 and n2 .

72

Figure 3.14: Duffing (n = 1) and higher-order (n = {2, ∞}) oscillator responses for different
quality factors Q. The system modeled is an array-based Josephson parametric amplifier with
N = 20 Josephson junctions in series. The phase response of a reflected output tone arg(bout ) near
the critical frequency and amplitude (bin = 0.98bin,c ) contains all of the meaningful information,
as there is no internal loss in the system. The agreement within a trio of curves (darker shadeindicates higher n) of a given color (corresponding to a given Q) indicates how accurate the Duffing
approximation is. The Duffing approximation is more accurate at high Q. Trios of curves arevertically offset for visual clarity. Note that with the horizontal axis expanded, all curves are seen
to undergo a full 2π phase drop as expected for a tone reflected off of an internally lossless resonance.

Array-based JPAs that meet Eq. (3.198) can thus confidently take advantage of Duffing oscillator
physics in their design and operation [148]. In addition, geometric inductance, LG in Fig. 3.6a, canbe used to achieve higher dynamic range without violating the Duffing approximation.75
75

The work of thesis takes advantage of the equivalence between stray inductance and arrays in order to use theDuffing approximation, but not to install additional geometrical inductance. Nevertheless, the tradeoff between thecompactness (and broader tunability, when SQUIDs are used) of arrays, and the more idealized behavior of single
junctions with linear inductances, is of potential value for future work.

73

3.6

Current-pumped JPA implementation
Key results in this thesis rely on the use of current-pumped as well as flux-pumped JPAs.76

This section discusses important design considerations and basic characterization measurements forthe current-pumped JPAs used in Ch. 4, and the next section does the same for the flux-pumped
JPAs used in Chs. 5 and 8. The specifications of our SQUIDs, common to both current- and fluxpumped JPAs, is included in this section. The JPAs used in this thesis were designed using Xic
Graphical Input Editor from Whiteley Research, Inc and KLayout open source software.

3.6.1

SQUID array specifications

Our JPAs were fabricated at NIST Boulder using the Nb-AlOx -Nb trilayer process [202, 203]
discussed in Appx. A. The parameters that define our SQUIDs are the junction critical currents Ic ,the geometric inductance Lg of the SQUID loops, and the area a of the SQUID loops for magnetic
flux to penetrate.
The 2.5 µm × 2.5 µm junctions were designed with a nominal critical current density of 0.8µA/µm2 , for a critical current of 5 µA. In practice, our junctions tended to come out within
40% of this value across all fabrication runs.77 A typical set of measurements taken on one of
our flux-pumped JPA chips is shown in Fig. 3.15. The room temperature resistance measurementsof Fig. 3.15a were performed on individual junctions situated within test structures. The layout
for a test structure, which contains four leads in order to cancel out non-junction resistances, isshown in the inset of Fig. 3.15a. The critical currents for the same five junctions, Fig. 3.15b, areobtained from the room temperature resistances via the Ambegaokar-Baratoff relation, Eq. (3.1),
with 2∆/e = 3.0 mV. For the five junctions on this chip, Ic = 5.99 ± 0.12 µA. The ≈ 20%
absolute error from the nominal value corresponds to a ≈ 10% change in the same direction in
the resonance frequency, Eq. (3.73). This imprecision is tolerable given the frequency-tunability
76

In addition to their uses in the work presented this thesis, the JPAs discussed in this thesis have helped enable
a range of experiments performed in the Lehnert Lab at JILA [165, 168–170].
77
The most variation occurred between different fabrications (and hence wafers). The next most variation in Ic
occurred between chips within the same wafer, and the least within a single chip.

74of the devices. The much smaller 2% intra-chip variation is important for achieving good device
performance [173].78
Figure 3.16 shows the geometry of one of our SQUIDs, from which the geometrical inductanceand SQUID area can be estimated. The circumference of the region ringed in by the superconductor
is 20 µm. Using the rule of thumb that a 1 µm length of line corresponds to a 1.2 pH inductance,
Lg ∼ 24 pH. The nominal Ic = 5 µA critical currents imply LJ0 = 66 pH and LS0 = 33 pH by
Eqs. (3.10) and (3.33), respectively. Thus, βL ∼ 0.36 via Eq. (3.42), better than a factor of fiveclear of being in the flux-trapping regime, Eq. (3.43).
The area a of the SQUID loop should really be thought of as an effective area consisting of
the 19 µm2 superconductor-enclosed region, plus a fraction of order 1/2 of the cell’s superconducting surface area. This is because an externally applied flux that would otherwise penetrate thesuperconductor will be moved off of it by the Meissner effect [172]. Knowing a very precisely is notessential, because we can within reason compensate for lower a with higher bias currents and/or
mutual inductances between the SQUID loop and whatever we are using to couple flux into it. So
a rough estimate of a ∼ 50 µm2 will suffice.

3.6.2

TKC specifications and simulations

Our current-pumped TKCs consist of a capacitor C-shunted N -SQUID array capacitively
coupled (Cc ) to an external environment. We do not deliberately add any stray inductance LG ,and what little is there is of secondary importance to the Josephson inductance. Figure 3.17
shows two scanning electron microscope (SEM) images of a device. The capacitors C and Cc
are implemented with interdigitated-finger geometries, which avoid some of the dielectric loss ofparallel plate geometries at the cost of compactness. The main capacitor C actually consists of two
capacitors C/2 in a symmetric, parallel configuration.
To ensure broad frequency coverage, we designed eight different JPAs with bare resonance78

In light of the observation that Ic variation grew with the physical separation of the junctions (Footnote 77),
the fact that the junctions in the test structure are much farther apart than those in a given array suggests that thevariation within an array will be less than or equal to the variation measured for different test structures on a single
chip.

75

Figure 3.15: (a) Room temperature Josephson junction resistance measurements for five different
junctions on a typical chip. The inset shows the layout for the five test structure used to performthe measurements. The four leads allow us to cancel out lead resistances when measuring thejunction’s resistance. Different patterns indicate different electrical layers, discussed in Appx. A(solid = base electrode, striped = counter electrode). The 2.5 µm × 2.5 µm junctions are too small
to see at this scale, but are situated where the two thin lines at the center of each test structure
cross. (b) Critical currents for the same five junctions. The critical currents are obtained from theroom temperature resistances using the Ambegaokar-Baratoff relation, Eq. (3.1).

frequencies logarithmically spaced between 4 and 12 GHz. The layouts and the target resonancefrequencies are shown in Fig. 3.18. The downward-frequency tunability of each device was designed
to overlap with at least one device below it. Parameter values were chosen in accordance with

76

Figure 3.16: Superconducting niobium wiring layout for a pair of SQUIDs. The cell shown here
corresponds to Fig. A.1e. Dimensions useful for estimating the effective area of the SQUID loop
and its geometrical inductance are provided. Blue (pink) corresponds to base electrode (wiring 1)
layer discussed in Appx. A.

Figure 3.17: Scanning electron microscope images of a TKC with false-color highlighting the circuit
elements. The capacitor C (blue) is implemented as a pair of interdigitated finger capacitors C/2
in parallel. The inductance L consists primarily of N SQUIDs, where N varies across different
TKC designs. The waffling pattern covering the niobium ground plane pins magnetic flux trappedin the superconductor in place. A closeup view of part of the SQUID array shows four Josephson
junctions, located at the ends of the bright ‘H’ shape. With the base niobium electrode (dark gray),they form two SQUID loops, as illustrated in Fig. A.1e. Figured reproduced from Ref. [204].

Eqs. (3.73), (3.76), and (3.58), targeting Qe = 50 and ZLC = 50 Ω.
Electromagnetic simulations of the resonators shown in Fig. 3.18 are shown in Fig. 3.19.79
Figure 3.19a shows the behavior the phase of S11 , the reflection coefficient off of the JPA’s only
port, with frequency. The resonance frequency lies at the center of the 2π phase drop. It and the
79

These simulations treat the SQUID arrays as linear inductors.

77

Figure 3.18: Layouts and design parameters for a suite of eight current-pumped TKCs. The
frequencies listed are the targeted values, logarithmically spaced between 4 and 12 GHz. The
number N of SQUIDs, and the design values for resonator and coupling capacitances C and Cc arelisted for each device.

external coupling γe (Fig. 3.19c) were fitted for each device using the equationarg(S11 ) = atan2γe (ω − ωTKC )(ω − ωTKC ) − γe2 /4

,(3.199)where atan2 is the two-argument arctangent function. Equation (3.77) then gives the external
quality factor Qe (Fig. 3.19b). The resonator impedance ZLC was determined from the individual
resonator components and Eq. (3.58).

3.6.3

Characterization measurements

A basic set of measurements is performed to confirm proper JPA functionality and characterize the device properties, including the bare resonance frequency, tunability, gain, and bandwidth.
The basic configuration of the TKC together with other microwave elements that realizes thecurrent-pumped JPA is shown in Fig. 3.20. Here, a directional coupler [187] is added in order toapply a pump tone.
These measurements, as well as those in Sec. 3.7.2, require combinations of three key instruments: a microwave generator for producing pump tones, a vector network analyzer (VNA)
for sending and receiving probe tones, and a DC current source for sourcing a DC magnetic flux
through a superconducting coil. Most commonly, we used a E8257D PSG Analog Signal Generator

78

Figure 3.19: (a) Simulated responses for the current-pumped JPAs of Fig. 3.18 at their maximum
resonance frequencies. The resonance frequencies νTKC = ωTKC /2π fitted from reflected scatteringparameter phase arg(S11 ) versus probe frequency ν are indicated, along with the number N of
SQUIDs for each device. (b) Simulated external quality factors Qe for each device. (c) Simulated
external coupling rates γe . Coupling rates increase with νTKC as a consequence of holding Qe fixed.
(d) Resonator impedances ZLC obtained from simulated circuit element values. Simulations were
performed using the Sonnet Software’s electromagnetic simulator.

and a E5071C ENA Vector Network Analyzer, both from Keysight Technologies, for the microwave
generator and VNA, respectively, and a Yokogawa GS200 DC Voltage/Current Source to provide
our DC current.

79

Figure 3.20: JPA configuration for current pumping. The TKC (Fig. 3.6b) is shown within the JPA(large triangle) as in Fig. 3.7. Here, details specific to biasing the JPA for current pumping are
included. A directional coupler (rectangle) couples a large pump tone into the signal path. The
cylinders represented segments of transmission line. Both pump and signal emerge from the output
port of the current-pumped JPA.

Figure 3.21 shows measurements of the resonant frequencies for the eight different JPAs.
In this measurement, only a weak signal tone is sent into the JPA to probe its linear properties(Fig. 3.21a). The results, Fig. 3.21b, are seen to match up reasonably well to the simulated values
from Fig. 3.19a.
Using the same setup, we look in in Fig. 3.22 at the flux tunability of one of the JPAs. Here, acurrent is varied through a superconducting magnetic coil placed a few cm from the SQUID loops.
The mutual inductance between the SQUID loops and the coil allows an applied DC current tosource an external flux Φext which together with the loop inductance gives rise to the total flux
Φtot penetrating the loop, Eq. (3.38). The total flux tunes the SQUID inductance up according
to Eq. (3.28), and hence the resonance down from its peak value according to Eq. (3.73). Ideally,
the peak value should occur at zero DC bias current, but it is often offset, as shown in the figure.
The offset indicates the presence of some trapped flux80 that is reasonably homogeneous over the
spatial extent of the array, and is therefore not a large problem, as it can be compensated for with
80

This is distinct from entering the flux-trapping regime of Sec. 3.2.2.

80

Figure 3.21: (a) State of pump and probe for measurement of resonance frequency. No pump isapplied, and a probe tone is applied in the low-power limit to measure the resonance frequency of thecurrent-pumped JPAs. (b) Reflection measurements of phase versus frequency alongside simulations
for eight different current-pumped JPAs with nominal resonances spaced logarithmically between
4 and 12 GHz.

the coil. Other nonidealities include the presence of standing waves (faint horizontal streaks), andsmall structure81 off of the main lobes. But overall, this is a healthy JPA, and these effects did notinterfere significantly with operation.
In Fig. 3.23, a current-pumped JPA is operated for two-quadrature gain. This time, we
do apply the pump tone (Fig. 3.23a). Figure 3.23b shows three gain curves, with higher gains
being achieved closer to the critical point, beyond which bifurcation sets in. Here we observe thepreservation of the GBP, Eq. (3.133).
This concludes our introduction to the specific current-pumped TKC/JPA devices used inthis thesis. We will return to them in Ch. 4.
81
An earlier fabrication produced JPAs with much more structure off of the main lobes. This turned out to be aconsequence of placing the SQUID array too close to the ground plane of the TKC, causing an inhomogeneous flux
environment across the SQUIDs. These JPAs were more difficult to tune, but still operable. We fixed this error ona subsequent fabrication by simply adding 50 µm of line on either side of the array. The JPA in Fig. 3.17 is a device
from the problem batch: observe that it’s SQUID array runs right up to the ground plane on either side. This issue
was distinct from the flux-tuning problem experienced by our flux-pumped JPAs, Fig. 3.32 and Appx. B.

81

Figure 3.22: Flux tuning of a nominal-7.5 GHz current-pumped JPA resonance with a DC current.
No pump tone is applied, as in Fig. 3.21a. When overcoupled, resonance occurs at the probe
frequency ν where the reflection coefficient phase arg(S11 ) drops 2π, and tunes with a DC current
applied to a superconducting coil which threads a magnetic flux through the SQUID loops. The
resonance is ideally periodic in current, and the two lobes closest to 0 current are displayed here,
The faint horizontal streaks are standing waves in the experimental setup. The small features off tothe side of the main lobes reveal slight nonidealities either in the array fabrication or the uniformityof the magnetic flux penetrating the SQUID loops.

3.7

Flux-pumped JPA implementation
Recall from Sec. 3.4.7 that flux pumping works, in an idealized picture, by creating a time-varying inductance (and thus resonance) at pump frequency ωp in order to amplify (or deamplify)
tones about ωp /2 [141, 182]. With flux pumping, this is achieved without a strong current tone
near the amplification band. The flux-pump tone modulates the resonance frequency along one of
the lobes in Fig. 3.22. In order to vary the resonance at ωp , the variation, whose scale is set by the
linewidth of the resonator according to Eq. (3.170), should not cross the peak, otherwise the resonance will go up then down and then up and down again over one period of pump-tone oscillation.
Even adjacent to the peak, the lobe’s slope is often too shallow to achieve good performance. In
practice, our flux-pumped JPAs perform best when operated at least ∼ 100 MHz below the top.

82

Figure 3.23: (a) State of pump and probe for measurement gain and bandwidth. A large pump tone
is applied in order to amplify a small signal. (b) Gain and bandwidth measurements performed
on a nominal-8.8 GHz JPA. Three different Lorentzian gain profiles show the two-quadrature JPA
power gain Gs for different operating points on the same device. The bandwidth B is the full-widthat half-max of the Lorentzian, and in the inset its product with the amplitude gain is seen to be a
fixed quantity.

The flux- and current-pumped JPA exist to perform similar measurements, but each havedistinct advantages and disadvantages. The current-pumped JPAs of the previous section are
simpler, smaller82 devices, and easier to fabricate. They have a slightly broader tunability, as fluxpumped JPAs cannot operate all the way up to the top of the flux-tuning curve. They also do
not require operation in a second frequency band a factor of two away from the band of interest.
For the most part, these advantages concern the design of and hardware used to operate the
JPAs. If one is willing to pay the overhead cost, flux-pumped JPAs confer significant operational
advantages [141, 183–186]. First, though they require a second port on the TKC for pumping, theyno longer require the lossy directional coupler of Fig. 3.20 in the signal path.83 Second, the pumptone is not routed to the JPA output, and is furthermore not within the band of interest. Third, an
equivalent flux-pumped JPA will have higher power handling than its current-pumped counterpart,82

Smaller here refers to the TKC footprints on a given chip. Designs used in this thesis include four (two)current-pumped (flux-pumped) devices per chip.
83
Current-pumped JPAs can also avoid this, but doing so requires adding on a second, weakly coupled port to thedevice [140] not so different than the one used for flux pumping.

83
and will not have the sizable Kerr-induced frequency shift of the resonance.84 And, as a bonus, aflux-pumped JPA, if desired, can even be current pumped (see e.g. Fig. 3.35).

3.7.1

Flux-pumped JPA design

We optimized several constraints in the design of our flux-pumped JPAs.85 First, we tookcare to ensure uniform coupling between the pump line and each of the SQUIDs that collectively
constitute the tunable inductor. Uniform coupling guarantees that the JPA’s dynamics are spatiallyhomogeneous, resulting in improved power handling. Second, we placed the pump line close to the
SQUID array, in order to minimize the pump power required to drive the LC resonator. At thesame time, the coupling of the LC circuit to the pump line is kept much lower than the coupling
to the line carrying the signal near the amplification band, in order to avoid losing part of thesignal through the pump port: the transmission between the two ports is kept to nearly -30 dB.
Finally, we shaped the flux line as a ‘U’ around the SQUID array. With this configuration, thepump couples to the differential-mode current circulating inside each SQUID loop and is isolated
from the common-mode current, unidirectional across the SQUID array.
We considered and simulated many designs, including those shown in Fig. 3.24, for fluxpumped JPAs, before settling on the relatively simple layout of Fig. 3.25a. The TKC, shown up
close in Fig. 3.25b, is laid out the same as in our current-pumped JPAs of Sec. 3.6. New to theflux-pumped designs, an AC bias line runs across the length of the chip. The line tapers andgoes through a small interdigitated finger capacitor Cf  Cc acting as a choke filter to suppress
low-frequency noise before wrapping around the SQUID array, crossing over the line containing the
SQUIDs as it reverses direction. As shown in Fig. 3.25c, the current flowing both left and rightcontributes to an AC flux within the SQUID region. The bias line ends back at the ground planeto the right of the array.
84

It is not that the Kerr nonlinearity is gone; it is just that there is no large current tone that causes the resonanceto shift downward because of it. Small Kerr-induced shifts can still occur because of signals and nonideal flux-pumptones.
85
For flux-pumped JPAs, we will frequently refer to the chip containing the TKC and flux-bias line, which is intruth neither the TKC or nor the entire JPA, as simply the “JPA.”

84

Figure 3.24: Discarded designs for TKCs with flux-bias lines. Six different concepts for laying out
the SQUID array and and flux-bias line for our flux-pumped JPAs are shown here. The shaded
region indicates superconductor, and the white region dielectric.

Figure 3.26 shows simulations performed in AWR Software’s Microwave Office of importantcircuit behaviors which were used to guide the design of the flux-pumped JPAs. The layout andport numberings used throughout the rest of the figure are shown in Fig. 3.26a. Ports 3–5 are used
to sense current along the flux bias line. Ports 6–11 are used to detect current within the SQUIDs.
As discussed in Sec. 3.2, the current within the two SQUID arms can be considered in the basisof total and circulating currents. As illustrated in Fig. 3.26b, we here refer to I from Eq. (3.15) as
the common-mode current, Icom , and recast Icirc as the differential current, Idiff = 2Icirc . It is thedifferential current that is essential to flux pumping, while any common current is an undesired
side-effect. The AC pump current Ip flows along the bias line as in Fig. 3.25c.
The first design consideration is the uniformity of current on the bias line. Figure 3.26cshows the amplitude of the scattering parameter from port 2 to each of three internal ports spaced
along the bias line.86 The positive and negative numberings on the ports act as the positive andnegative nodes of a port for the purpose of simulating the scattering matrix. The response to the86

For a discussion of internal ports in AWR, see Ref. [205].

85

Figure 3.25: (a) Design for the full TKC connected to the flux-bias line, which extends across theentire chip (length 6.3 mm). Bridges of superconductor cross the flux-bias line to ensure that the
ground plane is not interrupted. (b) Closeup on the TKC, with capacitor C (blue) and coupling
capacitor Cc (green) indicated. The capacitively coupled flux-bias line enters from the right. (c)Further closeup on the flux-bias line and SQUID array. The flux-bias line meanders around the
SQUID array such that the contributions from the left- and right-moving current Ip cos(ωp t) add
according to the right-hand rule within the SQUID loops, producing an AC flux ΦAC at the same
frequency.

stimulus from port 2 is shown to differ from one extreme end of the array to the other by about 5
dB (difference of green and black lines; less than a factor of two in current amplitude) independentof frequency. Since the device in Fig. 3.26 is meant to operate around ωTKC ≈ 6 GHz, we careabout the behavior of currents on the flux-pump line at 2ωTKC ≈ 12 GHz. Here, we see that theeven for port 3 (black curve), farthest from the flux-pump port (port 2), the transmission is greaterthan -10 dB. Whether this is sufficient to bias the SQUIDs for flux pumping depends on the mutual
inductance between the bias line and the SQUID loops.
The second design consideration concerns this mutual inductance: in particular, we need a86
sufficiently large differential current to modulate the SQUIDs with ∼ 1/Qe flux quanta in order
to flux pump. For Qe ≈ 50, Eq. (3.37) implies that we need Idiff ∼ 0.6 µA. However, the factthat we wish to operate these devices in a cryogenic environment around Tbase <∼ 30 mK limits
how much current Ip we can supply along the pump line. In order to reduce Johnson-Nyquist
noise [206,207] along the flux-pump port, we will couple the pump tone in with a 10 dB directional
coupler (Fig. 3.29).87 We measured the cooling power of our Oxford Triton 200 dilution refrigerator
to be roughly 10 µW at 30 mK. This is the maximum power we can send to the mixing chamber
plate of the dilution refrigerator, and the 10 dB coupler implies that only 10% of the power, 1 µW,
can make it through to port 2 of the flux-pumped JPA. For a 50 Ω characteristic impedance, 1 µW
corresponds to ≈ 140 µA. We therefore conclude that we require enough inductance coupling toachieveIdiff> 0.004,
Ip(3.200)

the ratio between 0.6 and 140 µA.
Figure 3.26d shows that our flux-pumped JPA meets the criterion of Eq. (3.200). By simulating the admittance matrix Yij [187] of the system, we identify
Idiff
Y63 − Y73=
Ip
Y33

(3.201)for port 3 and the adjacent SQUID. The result of Eq. (3.201) (black curve), along with analogous
curves for the ports 4 (red) and 5 (green) and their neighboring SQUID loops, indicates that
Idiff /Ip ∼ 0.025, independent of frequency. As a confirmation on our understanding that the powerthat makes it to each port 3–5 is sourcing the differential current in the neighboring SQUIDs, weobserve that the curves maintain the same hierarchy in Fig. 3.26d as in Fig. 3.26c.
The third and final design consideration, plotted in Fig. 3.26e, is that the common-mode
current be much smaller than the differential-mode current,
Icom
 1,
Idiff
87

(3.202)

Unlike the directional couplers used for current pumping, this one does not go in the signal path, and will
therefore not be harmful to squeezing.

87

Figure 3.26: (a) Design layout for the SQUID array region of the flux-pumped JPA shown in
Fig. 3.25c. Green (red) shaded region indicates the location of base electrode (wiring) superconductor. Josephson junctions are modeled with lumped element nonlinear inductances. Numberedinternal ports are used to detect the current flowing at different physical locations. Ports 1 and 2(not shown) are the input ports for signal and flux pump, respectively. Ports 3–5 measure current
along the bias line. Ports 6–11 measure currents on individual arms of SQUIDs. Not all of thecurrent sent in the 50 Ω coplanar waveguide (CPW) pump line from port 2 makes it to the biasline because of an impedance mismatch at the entry to the SQUID region. (b) Definitions of thecommon (Icom )- and differential (Idiff )-mode currents within a SQUID, as well as the pump current
Ip along the AC bias line. (c) Probe frequency ν-dependent scattering parameter magnitude fromport 2 to ports 3, 4, and 5, corresponding respectively to locations on the bias line near the left,
center, and right of the SQUID array. (d) Ratio of differential-to-pump currents for ports 3–5
and the adjacent SQUIDs versus ν. (e) Ratio of common-to-differential currents for the SQUIDs
adjacent to ports 3–5 versus ν. For (c)–(e), we are chiefly interested in the behavior around 12
GHz for this flux-pumped JPA, designed to work near 6 GHz.

88
when port 2 is excited. Failing to meet Eq. (3.202) would mean a substantial fraction of our pumptone would be wasted creating a common-mode current, useless for flux pumping and able topartially escape out the signal port of the TKC. For port 3,
Icom
Y63 + Y73.
=
Idiff
Y63 − Y73

(3.203)Equation (3.203) (black curve) and its analogues for ports 4 (red) and 5 (green) with neighboring
SQUIDs show that the differential current for ωp = 12 GHz is at least threefold larger than thecommon current across the array.88

Figure 3.27: (a) Simulated reflection coefficient phase arg(S11 ) versus probe frequency ν for thesignal port (port 1) of the flux-pumped JPA of Fig. 3.26. (b) Simulated transmission coefficientmagnitude |S21 | = |S12 | versus ν between the signal and pump port (port 2) of the same JPA.

Satisfied that the flux-bias line behaves as desired, we now turn our attention to the behaviorof the entire, two-port TKC. Figure 3.27a shows the reflection coefficient phase arg(S11 ) off of thesignal port, indicating Qe ≈ 50, as desired. Figure 3.27b shows transmission between the signal
and pump ports. We are concerned both with the behavior at the signal ∼ 6 GHz and pump ∼ 12
GHz frequency. Near the pump frequency, transmission is below -40 dB, which is practically ideal
behavior. On resonance, transmission reaches -28 dB, indicating that only 0.15% of a tone sent
into the signal port will be lost out of the flux-pump port.
Finally, using the harmonic balance functionality in Microwave Office, we verify in Fig. 3.28
that the device can achieve gain if the SQUIDs are properly biased with both DC and AC currents.
88

Since Microwave Office does not account for the Meissner effect, we might expect the precise values in theseplots to change slightly when the environment seen by the magnetic flux is partially superconducting.

89

Figure 3.28: Simulated gain Gs versus probe frequency ν curves for different AC bias strengths
using harmonic balance. Pump powers used range from -55 to -48 dBm input to the flux-pump
port. Probe power is set to -120 dBm.

Figure 3.29: JPA configuration for flux pumping. This figure is the specification of Fig. 3.7 andanalogue of Fig. 3.20 for flux pumping. The directional coupler is now outside of the signal path(and is not considered part of the JPA), and the pump tone no longer emerges with the signal.

We do not apply the DC currents through the port 2 bias line (which would not work in any casedue to its capacitive coupling), but rather manually by equipping each of the SQUIDs with a DC
current bias that creates a circulating current.
We designed six flux-pumped JPA chips with bare resonances logarithmically spaced between

90
(a)

100 µm(b)
I(ωp)(c) port 1:
signalport 2:pump

Figure 3.30: (a) False-color scanning electron microscope image showing the main elements in oneof our flux-pumped JPAs. Elements include a 550 fF interdigitated finger capacitor (purple), an
inductance comprising a SQUID array of 6 µA Josephson junctions (red), and a flux line, shorted
to ground (blue). (b) A closeup view shows several SQUID loops (red), surrounded by the flux line
(blue). The pump current I(ωp ) sources a differential-mode current around the SQUID loops. (c)The JPA equivalent circuit is shown with the same color-coding used in the images of (a) and (b).
Figure reproduced from Ref. [208].

4.6 and 10.3 GHz.89 Next, we will characterize their performance with some basic tests, as we didfor the current-pumped JPAs in Sec. 3.6.3.

3.7.2

Characterization measurements

The configuration of the TKC with other microwave components to realize a flux-pumped
JPA is shown in Fig. 3.29. Two of the advantages of flux-pumped JPAs — the removal of thedirectional coupler from the signal path and the absence of the flux-pump tone at the output — arevisible here. False-color, SEM images are shown in Fig. 3.30 together with the equivalent circuit forcomparison. The JPA is pictured mounted and wire-bonded to a printed circuit board in Fig. 3.31.
89

This is essentially the same set of frequencies as for the current-pumped JPAs of Sec. 3.6.2, but leaving off thelowest (4 GHz) and highest (12 GHz) frequency chips.

91

Figure 3.31: Photo of a mounted flux-pumped JPA. The JPA is wire-bonded to a printed circuitboard and attached with Elmer’s rubber cement to a gold-plated copper mount. The assembly isshown next to a penny for scale. It is later in a light-tight box, the bottom of which is formed by
the mount, and interfaced with via SMP connectors.

Figure 3.32 shows a two-dimensional (bias current, frequency) sweep of the bare JPA phaseresponse with a weak probe tone and no pump. This measurement is identical to that of Fig. 3.22
for a current-pumped JPA. The immediate difference between the two is obvious: the flux-pumptuning curve is substantially more structured. This structure is undesirable, and should not bepresent in flux- or current-pumped designs. Its source remained a mystery over the course of theexperiments carried out with flux-pumped JPAs in Chs. 5 and 8, but was ultimately discovered
to be trapped flux in the “dummy SQUID” loops placed at the ends of the SQUID array. The
problem, and its resolution, are discussed in Appx. B. While the additional structure made tuning
somewhat more difficult, it generally did not limit performance once a satisfactory tuning location
was found.
In Fig. 3.33, we apply a flux-pump tone and successfully realize gain. We can modestly (∼ 50
MHz) tune the device’s amplification band at a given DC flux bias by changing the half-pumpfrequency ωp /2. To achieve more dramatic tuning, the coil bias current is tuned.
The broader tuning landscape in the space of half-pump frequency and pump power is shownin Fig. 3.34a for a given flux-bias. For linear amplification, the JPA should be operated below the92

Figure 3.32: Flux tuning of a nominal-7.5 GHz flux-pumped JPA resonance. This figure shows thesame measurement of reflection phase arg(S11 ) versus probe frequency ν and DC bias current that
Fig. 3.22 shows for a current-pumped JPA. The tuning landscape is here significantly more messy,
because dummy SQUIDs placed on the end of the SQUID arrays to create a more homogeneous
superconducting environment around each SQUID had the unintended effect of trapping flux near
the arrays.

point where gain diverges (or would diverge, in the small-signal limit) at a given frequency — thatis, below the center of the bright red feature in Fig. 3.34a. Figure 3.34b–d show gain curves taken
at various powers along the three dashed lines in Fig. 3.34a. Better operation tends to occur closer
to the line of 3.34b, where the high-gain region bottoms out in pump power.
Finally, Fig. 3.35 shows the 1-dB compression point for the flux-pumped JPA as a characterization of the power handling, and compares it to that for current pumping on the same device. The
1-dB compression is defined as the signal power where the gain falls off by 1-dB from its low-power
value. For higher gains, we observe that it takes smaller signals to saturate the power handling of
the device. However, we see that flux pumping outperforms current pumping on the same device

93

Figure 3.33: Flux-pumped JPA gain Gs versus probe frequency ν curves showing tunability between
and within different coil tunings. Within two DC flux tunings ∼ 300 MHz apart in resonancefrequency, Gs curves with maximum gain over 30 dB are achievable across ∼ 50 MHz.

for all gains.
With a knowledge of both current- and flux-pumped JPAs in hand, we are now ready tomove onto several key experimental results of this thesis. We will begin in the next chapter with anexperimental technique to improve the power handling for current-pumped JPAs used for squeezing.

94

Figure 3.34: (a) Color map showing flux-pumped JPA two-quadrature gain Gs in the twodimensional parameter space of pump power and half-pump frequency νp /2 = 2πωp /2. Pump
powers are estimated at the input of the JPA flux-pump port from. (b)–(d) Gain versus probe
frequency ν curves taken along the linecuts indicated in (a). Some curves taken higher in power
than the main feature in (a) at a given frequency sometimes show a self-oscillation spike at thehalf-pump frequency.

Figure 3.35: Power-handling measurements for flux- and current-pumped JPA operation. The 1-dB
compression point input signal power decreases with gain Gs for both flux- and current-pumped
operation of the same nominal 7.5 GHz flux-pumped JPA. Flux-pumped operation typically achieves∼ 5 dB higher power handling for this device.

Chapter 4

Optimal current-pumped Josephson parametric amplifier operation for vacuum
squeezing

When operated in a phase-sensitive mode, the Josephson parametric amplifiers (JPAs) discussed in the previous chapter were seen to amplify one quadrature of the electromagnetic field while
deamplifying the other, enabling the squeezing of vacuum states below the standard quantum limit.
Such squeezed states can improve qubit readout fidelity in dispersive measurements [209–212], help
to better measure the motion of a mechanical oscillator [213], and enhance the signal-to-noise ratioof spin-echo detection in a magnetic resonance experiment [214]. Relevant to this thesis, squeezing
can also accelerate haloscope-based searches for axionic dark matter. Where we left off in Ch. 2,
JPAs had already been successfully deployed in haloscopes, albeit only in a phase-insensitive modeof operation [90, 91, 106, 107].
In the idealized picture of Sec. 3.4.6, a JPA squeezes one quadrature of vacuum while amplifying the other by the same amount [140, 215, 216]. Unfortunately, in practice vacuum squeezing
is limited by microwave losses and distortion due to amplifier saturation, or finite pump stiffness [179,217]. Minimizing loss through improved, low-loss elements or via fewer microwave connections remains an important challenge, and, to that end, on-chip circulators [218–222], flux-pumped
JPAs (Sec. 3.7), and directional amplifiers [149, 151, 218] are promising. In this chapter, we will
focus on the complimentary challenge challenge: distortion.
In Ch. 3, we saw that a JPA is essentially a nonlinear LC resonator whose nonlinearity
comes from the Josephson junctions comprising its inductance. Parametric oscillation of the circuit

96
converts a pair of pump photons at frequency ωp into a signal photon at ωs and a complementary
idler photon at ωi in a four-wave process: 2ωp = ωs + ωi . In the degenerate case, the pump iscentered about the amplification band and ωp ≈ ωs ≈ ωi .
Different combinations of pump power and frequency will produce the same gain, and one
commonly operates at points first order-insensitive to pump power fluctuations, i.e. on the line of
maximum gain (LMG) [181]. Here a typical JPA can achieve more than 20 dB of gain and maintain alinear relationship between input and output signal powers. Squeezing, however, is more susceptible
to nonlinearities that distort the output, arising when signal photons in the resonator greatly affectthe pump’s amplitude. These nonlinear effects impact the squeezed quadrature at gains much lower
than those where they affect the amplified quadrature. Pump stiffness can nonetheless be improvedby double pumping or flux pumping at the cost of increased design and control complexity.
In this chapter, which presents the work of Ref. [204] adapted to the language, conventions,and context of this thesis, we will see how a current-pumped JPA can be optimized for squeezing.
In particular, pumping it with a power greater than that which maximizes gain at a given pumpfrequency improves pump stiffness and minimizes signal distortion. We will first experimentally
study this effect on a coherent tone whose power is comparable to vacuum, and measure how it istransformed by a JPA operated at constant gain, for various pump powers and frequencies. Then,we will determine the optimal operating point for which the distortion is minimized. Finally, using
a second JPA as a phase-sensitive preamplifier to measure a vacuum squeezed state prepared by
the first JPA, we will see that squeezing follows the same dependence on operating point. With
optimized pump power and frequency, and while keeping microwave losses low, this technique willallow for the generation, transportation, and readout of a state 3.87 ± 0.03 dB below vacuum, overa bandwidth of 30 MHz, in an experimental setup designed to be equipped to an axion haloscope.
We will see that we are able to improve further on these numbers in the next chapter byswitching to flux-pumped JPAs. Nonetheless, the technique for optimizing current-pumped JPAs
for squeezing presented in this chapter may be of use to some axion haloscopes, as current-pumped
JPAs are often easier to design and/or acquire.

97

4.1

Signal transformation in phase space
In Sec. 3.4.4, we derived the input-output relation between small input signal and output

signals âin and âout reflecting off of a tunable Kerr circuit (TKC; Fig. 3.6). Here, we will consider
just the classical behavior of the system.
A classical limit of the input-output theory derivation in Sec. 3.4.4 gives the complex field
amplitude, or phasor, αout , at the TKC output from the incoming field amplitude αin asαout = αin 1 −γei(Knα − ∆) + γe /2


.

(4.1)

Equation (4.1) follows as an immediate consequence of Eq. (3.106) given the boundary condition
Eq. (3.100). The average photon number nα , detuning ∆, Kerr constant K, and power decay rate
γe are given in earlier sections by Eqs. (3.110), (3.108), (3.96), and (3.77), respectively. This semiclassical model does not properly treat the quantum noise at the JPA’s input, but it nonethelessmakes useful predictions of its behavior on a qualitative level.
In Sec. 3.4.4 we treated the JPA as losslessly amplifying an infinitesimally weak signal ain eiθ
in the presence of a large pump tone αin (here taken as phase reference), such that
Ain = αin + ain eiθ ,(4.2)

where we are treating both pump and signal classically. Equation (4.1) provides a mapping between
input and output within the bandwidth of the JPA, and remains valid even without the stiff pumpapproximation (Footnote 55). This transformation being unitary below bifurcation, the gain derives
from the sharp dependence of the output phase on the input amplitude. The photon number nα isdetermined by the cubic Eq. (3.107). Figure 4.1a shows how a family of real input pump tones withvarying amplitude αin maps onto complex outputs αout . They are calculated from Eqs. (4.1) and(3.107), with ∆ constant. A particular input phasor and its corresponding output are indicatedwith black arrows, illustrating the JPA’s behavior at the greatest phase sensitivity, i.e. at maximum
gain.

98

Figure 4.1: (a) Input-output mapping. A family of input pump phasors of different amplitudes
describes the horizontal, colored line. These inputs map onto output phasors distributed along thecorrespondinglyq √ colored curve in the complex plane. They are normalized by the critical amplitude
αin,c = γe / 3 3|K|. Two black arrows indicate the pair of input and output phasors at maximum
gain. At this operating point, a zoom-in shows how a circular input signal phasor distribution maps
onto a banana at the output. Two other operating points (red and blue) are indicated. (b) Signal
distortion in phase space. The input phasor, simulated with 360 different angles θ (dashed circle,
deformed due to the unequal axis scaling), is processed by the Josephson parametric amplifier(JPA) for three different pump powers corresponding to the operating points from (a). The lower
panel shows the distribution of points NpX of these phasors, when projected onto the deamplified
X-axis. (c) JPA direct gain and (d) deamplified quadrature variance plotted as a function of pump
amplitude. All calculations are for γe = 2π × 109 MHz, ∆/γe = −0.77, K/γe = −4.2 × 10−4 , anda quality factor Q = ω0 /γe = 65, typical values for our array-based JPAs. Figure adapted from
Ref. [204].

When adding a signal ain eiθ , the total input phasor Ain undergoes a θ-dependent transformation. One quadrature of the output signal aout is amplified while the orthogonal one is deamplified.

99
At low JPA gain or infinitesimal input signal, aout exhibits an elliptic shape. However, as the input
signal or gain increases, it distorts into a “banana,” indicating that the pump is no longer stiff.
Figure 4.1b represents such distortions, or deviations from the elliptical output of an ideal degenerate parametric amplifier [179, 214], at three different JPA gains obtained by varying the pump’s
amplitude, for a probe tone whose power is equivalent to half a photon over the JPA’s bandwidth.
They are rotated to share their amplified and deamplified quadratures, respectively aYout and aXout .
At the point of maximum gain, aout actually has a larger projection on the deamplified axis X thanat intermediate gain, where the projection onto this axis is minimal. When a signal is detuned fromthe pump, it creates output rotating phasors such as these, and the direct, or signal (see Footnote61), gain is expressed as [176]Gs =hGθ i + 1
,
2

(4.3)

with hGθ i the average ratio of output to input rotating phasor power. Figure 4.1c shows the direct
gain as a function of pump amplitude αin .
A vacuum state, considered semi-classically as a statistical ensemble of phasors, should undergo similar distortion when squeezed by a JPA. This effect degrades the maximal achievablesqueezing, because it increases the noise variance along the deamplified quadrature axis. To makea qualitative estimate of the attainable degree of squeezing, we first project the distributions of
input and output points onto the deamplified quadrature axis (Fig. 4.1b), and then compare the2 ) and (σ 2 ), of these projections. Figure 4.1d shows the ratio σ 2 /σ 2 , as a funcvariances, (σin
out
out
in

tion of pump power, for a given pump frequency. Deamplification is maximized when this ratio
is minimized. Strikingly, the pump amplitude that produces maximum gain fails to yield maximum deamplification due to phasor-distortion of the vacuum-sized input. Instead, deamplification
reaches a global maximum at an amplitude greater than that at maximum gain, i.e. above theLMG, where phasor distortion is better mitigated. We will examine this behavior experimentallyin Sec. 4.2, enabling the optimization of vacuum squeezing discussed in Sec. 4.3.

100

4.2

Coherent state deamplification
The first experiment discussed in this chapter investigates the distortion of a vacuum-sized

coherent tone when varying the JPA operating point by changing the pump power and frequency. A
single cryogenic experimental setup is used for studying distortion and vacuum squeezing, discussed
in Sec. 4.3. It comprises two identical JPAs (Sec. 4.4.1) connected to each other, as shown in
Fig. 4.2. The first we label the squeezer (SQ), and the second the amplifier (AMP). Each can be
independently tuned via its own superconducting coil, and a variable temperature stage (VTS)
is used to inject known thermal noise into the amplifier chain and measure the SQ added noise(Sec. 4.4.3). Microwave generators are used both to pump the two JPAs and to send coherent tones
to the input of the chain. An in-phase/quadrature (IQ) mixer demodulates the output quadratureswith respect to a local oscillator (LO). In this first experiment, the AMP is not used; tuned morethan 1 GHz below the SQ band, it acts as a passive element of unit gain.

pump
SQ

pump
AMP
LO

coherent
tone

20 mK

4K

VTS

SQ

AMP

I

L
R

Q

HEMT
ADC

coil

coil

Figure 4.2: Experimental setup for coherent tone distortion and vacuum squeezing. The squeezer(SQ) and (AMP) are mounted in series. When studying distortion, the AMP is detuned and not
pumped, while a microwave generator sends a known coherent tone at the chain’s input (dashedcircle), then processed by the SQ. Microwave losses between SQ and AMP are parametrized by η.
After subsequent amplification by a cryogenic high-electron-mobility transistor (HEMT) amplifier
the tone is mapped to low frequency using an in-phase/quadrature (IQ) mixer and a local oscillator(LO). An analog-to-digital converter (ADC) records the output signal’s I and Q quadratures. Figure
reproduced from Ref. [204].

In order to compare distortion and SQ direct gain GS at particular operating points, we find

101
GS (measured with a vector network analyzer, see Sec. 4.4.1) as a function of pump power Pp andfrequency νp . We identify the critical point, Eqs. (3.117)–(3.119), at which the gain diverges, anddescribe the pump power and frequency as fractions of these critical values (Fig. 4.3a). The dashedblack line shows the LMG. We compare deamplification when operating the SQ at points withequal gains below, above, and on this line. From the weaker dependence of gain on incident power
above relative to below the LMG, visible in Fig. 4.3a, we anticipate more linear behavior above theLMG.
To characterize the nonlinear behavior and visualize phasor distortion, we now input avacuum-sized coherent tone ain eiθ whose phase θ slowly rotates in the SQ pump’s frame (Sec. 4.4.1).
With LO and SQ pumps at the same frequency, the tone describes traces at the output in the IQplane such as those in Fig. 4.3b. They are obtained with the SQ operated above the LMG. Thus,
decreasing the pump power, GS increases from 6 to 13 dB, as the output transitions from ellipse
to banana.

Figure 4.3: (a) SQ gain map, with critical frequency νc = 7.0032 GHz and input critical power
Pc = −93.1 dBm. The different symbols N, ‚, and • correspond to those in Fig. 4.4. (b) Output
tone in the IQ-plane (input is a circle of unit radius, not shown). The SQ direct gain is varied
between 6 (blue) and 13 dB (green), and operated above the line of maximum gain (LMG) at
νp /νc = 1.0015, a frequency indicated by arrows in (a). The asymmetric axis scales mean all
output tones are longer along the Y dimension. Figure adapted from Ref. [204].

102
Upon repeating the measurement with the SQ off, we characterize deamplification as a func2 /σ 2 , as seen in Sec. 4.1. Figure 4.4a showstion of the SQ operating point by forming the ratio σoutinthese ratios, calculated from the points in Fig. 4.3b. When operating above (relative to on or below) the LMG, we observe the highest deamplification at fixed gain. Above GS = 9 dB, however,deamplification decreases due to phasor distortion.
Finally, we study deamplification along a contour of constant direct gain, encircling thecritical point, to look for an optimal operating point at which distortion is minimized. Figure
4.4b characterizes the tone’s deamplification along the GS = 8 dB contour, at operating points
2 /σ 2 isaway from bifurcation (i.e. for νp > νc ), indicated in black in Fig. 4.3a. Below the LMG σout
inabout -7 dB, whereas above it decreases to about -10 dB, as a result of reduced distortion. Thus,
operating the SQ above the LMG and with a gain of about 9 dB maximizes deamplification. We
will leverage these findings in Sec. 4.3 to optimally bias the SQ.

Figure 4.4: (a) Deamplification shown as a function of SQ direct gain. The SQ is operated below(N) and above (‚) the LMG (at νp /νc = 1.0015), as well as on (•) the LMG (where the pumpfrequency has been varied). (b) Deamplification shown for the black points in Fig. 4.3a, where
GS = 8 dB. Figure adapted from Ref. [204].

103

4.3

Optimizing vacuum squeezing with a current-pumped JPA
Most applications of microwave squeezed light require the ability to both generate and ef-

ficiently detect a squeezed state. These dual capabilities can be demonstrated with the AMP
cascaded after the SQ. Upon transport from SQ to AMP, the state suffers from microwave loss η(Fig. 4.2), which replaces part of the state with unsqueezed vacuum [223], diminishing the measurable degree of squeezing. These losses were reduced to η = 1.2 ± 0.2 dB (Sec. 4.4.2) using
superconducting cables and a narrowband triple-junction circulator with insertion loss < 0.5 dB.
To make a homodyne measurement of vacuum squeezing (see full setup in Fig. 4.7, Sec. 4.4.1),
one power generator provides the AMP pump and LO tones, while another, 20 kHz detuned,provides the SQ pump tone. These spectrally adjacent pump tones must be cancelled to avoid AMP
saturation and preserve the squeezing of the pure vacuum state generated by the VTS at TVTS =55 mK  ~ωS /kB . These cancellations are performed by combining π-phase-shifted pump tones atthe output of each JPA. The AMP acts as a nearly noiseless (Sec. 4.4.3) phase-sensitive amplifier
with roughly 30 dB of single-quadrature gain so as to overwhelm the rest of the amplification chain’s
added noise. After amplification, the squeezed vacuum is then digitized. A full 2π phase difference
θ between SQ and AMP pumps is swept every 1/20 kHz = 50 µs, and the phase for which thedigitized voltage signal is periodically minimized is taken to be the SQ squeezed quadrature. The
variance of the voltage fluctuations in this quadrature is thus computed with the SQ on, and thenagain with it off. The reduction in variance when the SQ is on is taken to be the squeezing,
S=

2
σon
2 .
σoff

(4.4)

Optimal squeezing occurs at νp /νc = 1.0015, with direct gain GS = 8 dB, and above theLMG, as expected from Sec. 4.2. Figure 4.5a shows a histogram of the voltage fluctuations Vθ
retrieved along the AMP amplified quadrature as a function of θ. Compared to panel (b), where
the SQ is off, vacuum has been squeezed for θ = π/2 and 3π/2. Figure 4.5c presents a cut in thehistograms at θ = π/2, demonstrating a clear reduction in the Gaussian’s standard deviation, andFig. 4.5d displays the squeezing S as a function of θ, with a minimum of -3.87 ± 0.03 dB. This

104
reduction in variance, which is measured over a full 30 MHz band centered on the AMP’s pump,is not a value inferred at the output of the squeezer, but rather one directly measured90 at theterminus of the measurement chain without any losses or added noises calibrated away.

Figure 4.5: (a) Histogram of output voltage fluctuations Vθ along the AMP amplified quadrature
with SQ on as a function of the phase difference θ between SQ and AMP pumps. (b) Histogramwith SQ off. (c) Profile of the fluctuations and Gaussian fits (solid lines) along the dashed lines
drawn in panels (a) and (b), at θ = π/2. (d) Squeezing, Eq. (4.4), as a function of θ. The SQ is
operated with direct gain GS = 8 dB, obtained with a pump frequency νp /νc = 1.0015 and pump
power Pp /Pc = 0.5045 dB. Figure adapted from Ref. [204].

Moving the SQ operating point, it was verified that squeezing decreases as we approach the90This directly measured value should not be compared straightforwardly to inferred values of squeezing such as
those cited in e.g. Refs. [224,225]. The directly measured reduction in variance suffers from loss and added noise thatcomes after the creation of the squeezed state. To see the consequences of our choice, consider Ref. [215]. There, thedirectly measured squeezing of -1.7 dB corresponds to -9.2 dB inferred squeezing upon correcting for microwave losses
and detection efficiencies. The directly measured value most closely tracks the realizable benefit of using squeezed
states in the presence of loss and added noise.

105

Figure 4.6: (a) Squeezing plotted as a function of pump power for νp /νc = 1.0015. The points inred, black, and blue are, respectively, below, on, and above the LMG. (b) Squeezing plotted as afunction of SQ direct gain. The red and blue points are those used in (a). For the black points, νp
has been varied to obtain gains between 6 and 14 dB. Figure adapted from Ref. [204].

LMG, due to increasing gain. Adjusting the pump’s power while keeping the frequency constant,
we can move the operating point from below to above the LMG, describing a line cut in the gain
map, indicated by arrows in Fig. 4.3a. Figure 4.6a shows the squeezing along this cut, in qualitative
agreement with the theoretical and experimental results on coherent tones seen in Secs. 4.1 and 4.2.
Furthermore, a slightly better squeezing is obtained above the LMG than below, as expected. Thisfeature is displayed in Fig. 4.6b, where squeezing is reported as a function of SQ gain.

4.4

Experimental details and calibration
This section describes additional experimental details and calibration measurements rele-vant to the experiments of Secs. 4.2 and 4.3. Section 4.4.1 overviews the experimental setup, andSecs. 4.4.2 and 4.4.3 respectively detail measurements to determine the loss in the lines between
the JPAs and the noise added by the amplification chain.

106
4.4.1

Experimental setup

The full experimental setup used throughout this chapter is represented in Fig. 4.7. SQ and
AMP TKCs like that of Fig. 3.17 are individually magnetically shielded. They can be tuned via
coils biased by current sources. The generator which provides the AMP pump and cancellation
tones also sources the LO. A switch may be used to turn off the AMP pump without interrupting
the LO. A third generator (far left in Fig. 4.7) is used to create the probe tone. Output signals
are demodulated with an IQ mixer, then digitized. A vector network analyzer (VNA) is used todetermine the direct gains and a spectrum analyzer (SA) monitors pump cancellation. Each cancelline consists of a variable attenuator and a phase shifter. Several isolators and 3 dB attenuators
are placed in the setup to minimize harmful reflections.
At base temperature, a triple-junction circulator links the SQ and AMP TKCs. It provides> 50 dB isolation between SQ and AMP, and has low insertion loss (< 0.5 dB). A VTS, weakly
coupled to the base temperature plate and composed of a 50 Ω load, a heater, and a thermometer,
generates the chain’s input noise and allows for measurement of SQ added noise. Superconducting
Nb/Ti lines are placed between TKCs and directional couplers to further reduce microwave loss,
and the directional couplers are directly connected to the circulator with swept elbows.
To measure the distortion of a coherent tone (Sec. 4.2), its power is adjusted to PPR = 12 ~ωB
in order to keep it comparable to vacuum, with B the SQ bandwidth. With the LO generator
at the same frequency as the SQ pump, the tone is detuned by νPR − νp = 20 kHz, so that its
phasor slowly rotates in the LO’s frame, enabling the measurement of 360 independent phasors
within the 15 MHz intermediate frequency (IF) bandwidth of the readout chain. Due to the feeble
input signal, the two output quadratures must be heavily averaged in order to describe traces in
the IQ-plane such as those in Fig. 4.3b.

E5701C

E4407BATS 9462

VNA

SA

ADC

1

2

107
3 20 attenuator
50 Ω termination
isolator
amplifierI

R
L

directional couplerQ

power divider
90° hybrid coupler
switch

E8257Dlow pass filter
θvariable attenuator
θ

3

10 dB

3

phase shifter
Nb/Ti coax.
normal coax.

3
10 dB

3

GS200

3

6 dB

3
θ

θ

300 K
4K

20

20

20 mK

HEMT

20

20

20

20

VTS

coil

20

SQ
cryoperm + Al

20

20 dB

QuinStar
QCE - 070100CM30

20 dB

coil

AMP
cryoperm + Al

Figure 4.7: Full experimental setup used for both coherent tone deamplification and squeezing.
The input and output coaxial cables are colored in black and purple, respectively. The green lines
are superconducting. Figure reproduced from Ref. [204].

108
4.4.2

Inter-JPA loss calibration

Calibrating the lines relies on measuring the overall input attenuation AI from the probe
tone’s microwave generator to the SQ input, and also the overall (single-quadrature) amplifications
O
I
GO
A,1Q and GS,1Q from the TKC outputs to the analog-to-digital converter (ADC). Knowing A

allows for calibration of the probe power incident on the SQ TKC, and the ratioOη = GO
A,1Q /GS,1Q(4.5)

provides, ceteris paribus, an estimate of the loss experienced by the squeezed state when travelling
between SQ and AMP.
O
Operating one JPA at a time (without pump cancellation), GO
A,1Q and GS,1Q are calculated byintegrating the power spectral density of vacuum fluctuations recorded by the ADC over a window
W = 500 kHz within the JPA’s bandwidth. A heterodyne measurement, with LO and JPA pump
detuned by 5 MHz, and W ending 400 kHz below the pump, is performed to avoid DC offsets.
Thus, when measuring GO
A,1Q , the generator used for pumping is the one connected to the SQ,
itself detuned. The SQ TKC then simply acts as a mirror, and the pump tone reaches the AMP
after reflection. Given a quantum at the SQ or AMP input, the integrated power for both cases is
O
PS/A= ~ωW GOS/A GS/A ,(4.6)

with GS/A the SQ or AMP gain, measured with the VNA. Operating the JPAs at various gains
between 20 and 30 dB in order to ensure that the output spectral density linearly tracked the JPA
gain yieldedGO
S,1Q = 75.3 ± 0.1 dB

(4.7)

GO
A,1Q = 76.5 ± 0.1 dB.

(4.8)

Therefore, we estimate that η = 1.2 ± 0.2 dB. In principle this value should enable us to observe
about 5.6 dB of squeezing, a bit more than what was measured. But in practice, we are limited
by a combination of squeezed state distortion and the contribution from the high-electron-mobility

109transistor (HEMT) amplifier added noise, which becomes harder to overwhelm when squeezing
because the HEMT added noise is larger relative to the squeezed state than to vacuum.
To estimate AI , we use a similar protocol. Turning off both JPAs, a probe tone with input
power PPI travelling in the lines creates an output power PPO on the ADC as
I
PPO = PPI GO
S,1Q A ,(4.9)

leading to an estimate of AI = −81.4 ± 0.2 dB.

4.4.3

Added noise calibration

Any resistor at temperature T generates noise of known variance, whose spectral density inunits of quanta at angular frequency ω is
Sin =

1
1
.
+
2 e~ω/kB T − 1(4.10)The VTS therefore allows us to generate a known thermal state at the chain’s input, which is thenamplified by the SQ, giving at its output:
SS,out = GS,1Q (Sin + NS ),(4.11)

where GS,1Q is the SQ gain and NS its added noise (in this experiment only the SQ is used; the
AMP is detuned). Since we are operating the SQ in a phase-sensitive mode, NS ≥ 0.
In practice there are inefficiencies along the thermal state’s path from VTS to SQ. We canmodel these with a simple beamsplitter picture, such that a fraction λ of Sin is replaced with noise
Sf at the fridge’s temperature. Considering the system added noise as
Nadd = NS +

NH
,
GS,1Q

(4.12)

where NH is the HEMT added noise, we haveSout = GO
S GS (λSin + (1 − λ)Sf + Nadd )in= GO
S GS Sout(4.13)

110

Figure 4.8: Noise spectral density as a function of variable temperature stage temperature. The
noise is referred to the SQ input. Points in black, blue, and red are for fridge’s temperatures
Tf = 50, 300, and 500 mK, respectively. The statistical error is smaller than the point size. The
solid lines are obtained from fitting the parameters Nadd , λ and the overall gain of the chain
GO
S,1Q GS . Figure reproduced from Ref. [204].

inat the output of the chain, where GO
S,1Q is the chain’s gain, and Sout is the output noise spectral

density referred to the SQ input. Varying Sin and Sf , we thus can deduce Nadd and λ.
With a homodyne configuration (LO and SQ pump at the same frequency), we integrate theoutput noise spectral density Sout over the SQ bandwidth B for various VTS temperatures. The
fridge’s base temperature Tf is fixed and GS,1Q ≥ 25 dB is tracked by amplifying and measuring
in as a function of the VTS temperature for three different
a small pilot tone. Figure 4.8 shows Sout

fridge temperatures. A fit gives Nadd = 0.045 ± 0.001 and λ = 0.79 ± 0.01, where the quoted
uncertainties correspond to statistical errors in the measurements.

4.5

Concluding remarks on current-pumped JPAs optimized for squeezing
Pump stiffness is crucial when generating a microwave squeezed state with a JPA, because

any deviation from the stiff-pump regime strongly affects the squeezed quadrature. A lack of

111
stiffness distorts output signals, thereby limiting the maximum degree of squeezing. We have seenthat operating above the LMG therefore constitutes a partial solution to the problem of finite
pump stiffness by means of choosing the operating point to lie above the LMG. The coherent tone
deamplification measurements of Sec. 4.2 confirm the efficacy of this approach, with less phasor
distortion above the LMG. When squeezing vacuum, we see the same trend. Maintaining low
microwave loss, it was thus possible to generate and deliver a 3.87 ± 0.03 dB vacuum squeezed statefrom one JPA to another over a band of 30 MHz.91
The benefits of optimizing JPA operation explored in this chapter will grow as losses shrink.
With relatively high losses, it does little good to create a minimally deformed squeezed state, only tohave most of it replaced by unsqueezed vacuum. However, as losses come down for experimentalists,
preventing squeezed state deformation will have impact beyond the few percent level achieved inthis chapter, as observed in Sec. 4.2 for the deamplification of a coherent tone.
For the purposes of axion detection with present-day technology, however, the gains achievedin this chapter are insufficient to recommend current-pumped JPAs in light of their two key disadvantages for axion detection. First, the losses introduced by the directional couplers in Fig. 4.7
degrade the itinerant squeezed state more than is absolutely necessary.92 Second, the pump cancellation of Sec. 4.3 poses a significant experimental challenge for an active haloscope which must
be automatically tuned through frequency space. This difficulty is further compounded by theKerr shift of current-pumped JPAs. Unlike with flux-pumped JPAs, changing the drive powerchanges the resonant frequency, making it harder still to stabilize both pump and cancellation
tones for two JPAs simultaneously. In light of this conclusion, subsequent work in this thesis uses
the flux-pumped JPAs of Sec. 3.7.

91
This 30 MHz is more than enough for axion haloscope purposes: as we will see in Chs. 5 and 8, the usablebandwidth for an axion haloscope at these frequencies is closer to 1–2 MHz.
92
While switching to flux-pumped JPAs will get rid of these directional couplers, the introduction of the second,
flux-pump port on the TKCs does allow for signal loss. In practice, however, the amount of loss out this port
(Fig. 3.27b) is far less than that from the reflection off and transmission through a directional coupler.

Chapter 5

Squeezed vacuum used to accelerate the search for a weak classical tone

In Ch. 2, we saw that the present generation of axion haloscope experiments [91, 107, 108] isor soon will be limited by vacuum fluctuations in the microwave field. The quadrature operators
X̂ and Ŷ of a resonant cavity’s electromagnetic (EM) field, Eqs. (3.137) and Eqs. (3.138), carry
the imprint of the dark matter signal as a slight excess in their power spectra. Problematically,
the quantum noise [128] intrinsic to a measurement of these observables overpowers the signal by
roughly three orders of magnitude,93 and the signal frequency is a priori unknown [226]. Amongthe most promising advances are those which allow access to a fundamentally distinct parameter
regime in which axion searches are no longer limited by quantum noise.
In detectors measuring the quadratures of a resonant mode, quantum noise can be circumvented by preparing the mode in a squeezed state. Squeezing unbalances the uncertainties in thetwo quadrature observables, thereby permitting precise knowledge of one at the expense of theother. Leveraging quantum squeezing to enhance measurement sensitivity has been a longstanding
goal [227–230] in the optical domain, now realized for the sensing of gravitational waves [231].
At microwave frequencies, squeezing had, prior to the work described in Ch. 8 of this thesis, been
demonstrated in principle [209,213,214], without yet aiding in a search for new physical phenomena.
Microwave squeezing can be used to aid in an axion dark matter search subject to current
experimental constraints. This chapter presents the work of Ref. [208] in order to substantiate that93

A three order-of-magnitude discrepancy between signal and noise spectral densities requires six orders of magnitude worth of independent realizations of the axion field merely to achieve an SNR of 1 [77], itself insufficient todetect or exclude the axion.

113
claim, first theoretically and then experimentally, before the conclusive experiment is performed in
Ch. 8. In particular, we will see in this chapter that squeezing increases the scan rate in a search
for a weak, axion-like signal of unknown frequency. This improvement exists in spite of the factthat squeezing does not improve the sensitivity of a haloscope to a tone of known frequency. This
speed-up is demonstrated in an apparatus designed to mimic the behavior of existing haloscopes.
With a maximum of 4.5±0.1 dB of squeezing delivered from one flux-pumped Josephson parametricamplifier (JPA; Sec. 3.7) to another in a squeezed state receiver (SSR) configuration, we will see
implemented a realistic acquisition and processing protocol both with and without squeezing enabled. The SSR presented in this chapter accelerates the scan rate to axion-like signals by a factorof 2.12 ± 0.08 over 200 independent experimental trials. In Ch. 8, we will equip the HAYSTAC
experiment with the SSR developed in this chapter to speed it up by very nearly that amount.

5.1

Theory of the squeezed state receiver
Figure 5.1 shows a representative experimental apparatus in which a resonant cavity is cou-

pled to an SSR comprising a pair of JPAs. We start by considering the cavity, whose internal mode
(frequency ωc ) has HamiltonianĤ = ~ωcX̂ 2 + Ŷ 2
,
2

(5.1)

equivalent to that of Eq. (3.62) subject to the definitions of Eqs. (3.137) and (3.138). The quadratures of the cavity field obey
[X̂, Ŷ ] = i,(5.2)

a direct consequence of their definitions and Eq. (3.59). This cavity is modeled as exchanging
energy with three ports. First, a measurement port couples the cavity mode to the propagating
modes of a transmission line with power decay rate κm . Along this line, a microwave circulator
spatially separates incoming and outgoing propagating modes. Second, a loss port, connected toa fictitious transmission line, models the cavity’s internal energy dissipation at rate κl . Third,
the cavity’s coupling to the signal of interest at rate κa is modeled as occurring through another114
fictitious transmission line; the signal itself is modeled as a microwave generator characterized by
its frequency ωa , spectral width94 ∆a , and amplitude Ea . We assume Ea  1, implying that thedisplacement of the cavity mode by the signal is classical (i.e., the contribution of the signal tothe cavity’s quantum fluctuations can be neglected in comparison to those coming from the loss
and measurement ports). We also assume a narrowband signal (∆a  {κl , κm }) so weakly coupled(Ea2 κa  κl ) that the time required to resolve the displacement is much longer than the signal’s
phase coherence time. Therefore, on average, this displacement yields a small excess power above
the vacuum fluctuations, isotropic in quadrature space (bottom panel of Fig. 5.1). These inequalities
are very well satisfied in the case of the axion field (Sec. 5.2).
The SSR itself comprises two JPAs coupled respectively to incoming and outgoing modes atthe cavity’s measurement port. This configuration exploits the fact that a portion of the vacuum
noise exiting the measurement port arises from vacuum noise incident on that same port. The first
JPA (called SQ, as in Ch. 4) squeezes these input fluctuations along the X̂ quadrature, reducing
the observable’s variance below vacuum levels:
1
2
σX
< .
2

(5.3)

To satisfy the uncertainty principle, the opposing quadrature’s fluctuations exceed vacuum
1
σY2 > ,
2

(5.4)

but are not measured. The squeezed input field subsequently enters the cavity, where a small displacement by the signal would yield a small excess power in both quadratures. At the measurement
port output, the second JPA (again called AMP) noiselessly amplifies only the X̂ quadrature with
sufficient gain to overwhelm the noise added by following amplifiers and mixers.
The benefit of squeezing can be understood by analyzing the microwave network formed by
the combination of SSR and cavity using input-output theory (Sec. 3.4.4). We will here qualitatively
overview the results of more detailed calculations that we will perform first in the lossless case in94

A closely related quantity is the standard deviation of the axion lineshape, σSHM in Fig. 2.5. The numbers
used the quote the axion spectral width traditionally encompass a bit more of the axion lineshape than the standard
deviation does.

115

𝜅a
𝜅l
ℰa

cavity

𝜅m

AMP

SQ
(b)

(c)

(d)
noise
signal + noise

(a)
(e)

(f)

(g)

Figure 5.1: Top panel: schematic of the SSR and cavity. Two Josephson parametric amplifiers,
squeezer (SQ) and amplifier (AMP), respectively squeeze and read out a microwave field interactingwith a cavity at rate κm . An axion-like field Ea , coupled to the cavity at rate κa , displaces the cavity
state. Energy leaves the cavity through internal absorption at a rate κl . Bottom panel: quadrature
representation of a vacuum state detuned from the cavity resonant frequency, travelling throughthe SSR. At the SQ’s input (a) it is Gaussian, azimuthally equiprobable in the (X̂, Ŷ ) plane (reddisk). The state is squeezed along X̂ by the SQ (b), displaced along a random phase within thecavity (c), and amplified along X̂ by the AMP (d). Comparing to what happens without squeezing(e, f and g), the size of the signal-plus-noise (green) relative to the noise (red) in this quadrature islarger with (d) than without (g) squeezing. Figure reproduced from Ref. [208].

Sec. 5.1.1, and then with experimental transmission losses modelled in Sec. 5.1.2. These calculations
show the benefit of squeezing against unsqueezed, single-quadrature axion detection. The picture
is completed in Sec. 5.1.3, where we will see that, in the absence of squeezing, there is neither abenefit nor a penalty associated with measuring only one quadrature compared to the usual twoquadrature case. Before describing experimental results in the rest of the chapter, in Sec. 5.2 we will
will map parameters of the model shown in Fig. 5.1 onto the physical parameters of e.g. Eq. (2.16),
which are conventionally quoted when searching for dark matter axions.

116
The behavior of the apparatus shown in Fig. 5.1 is characterized by its signal and noise
spectral densities. The signal spectral density at the measurement port output is equal to the signal
spectral density at the measurement port input, weighted by the susceptibility of the measurement
port output to the signal port input. Similarly, the noise density at the measurement port output
is a susceptibility-weighted sum of squeezed and unsqueezed noise from the measurement and loss
ports, respectively. In the absence of transmission losses between the JPAs and the cavity, we will
see that the ratio of output signal spectral density to total output noise spectral density (hereafter
called the signal visibility) isα(ω) ≈

nT + 12nA κa κm.
κl κm + Gβ(ω)
S,1Q

(5.5)

Here, ω is the frequency relative to cavity resonance,nT =1e~ωc /kB T − 1(5.6)

is the mean thermal photon number incident on the cavity from the measurement and loss ports,
nA = Ea2 is the mean photon number sourced by the fictitious generator,β(ω) =

κm − κl
2

2

+ ω2,

(5.7)

and GS,1Q is the single-quadrature power gain of the SQ, Eq. (3.139), along its amplified quadrature(ideally equal to the reduction of the squeezed state variance below the vacuum value).
When optimizing α(ω) in Eq. (5.5), we assume that κm and GS,1Q can be freely varied,
whereas nT , nA , κl , and κa are fixed by the physics of the signal source or technical constraints ofthe detector. On cavity resonance (ω = 0), α is maximized at critical coupling (κm = κl ). Because
β(0) = 0 at critical coupling, α(0) is independent of GS,1Q and there is no benefit from squeezing;
physically, the squeezed state injected into the cavity is completely absorbed in it, while all theunsqueezed noise from the loss port reaches the AMP. For ω 6= 0, squeezing increases α(ω) for any
value of κm , as GS,1Q reduces the amount of measurement port noise reaching the output. In thelimit where GS,1Q → ∞, the β(ω) term in the denominator can be neglected, and α(ω) approaches

117
the critically coupled resonant value α(ω = 0, κm = κl ) for any κm and all ω. This illustrates animportant point of principle: squeezing cannot improve the peak sensitivity of a haloscope, but
it can almost without limit95 enhance the detector bandwidth over which this peak sensitivity isachieved. When GS,1Q is finite, overcoupling (κm > κl ) increases the cavity bandwidth at the costof reducing α(0). This can be a favorable tradeoff (even in the absence of squeezing) because thesignal’s frequency ωa is a priori unknown [60, 95], and broader bandwidth enables larger cavity
tuning steps. Moreover, squeezing mitigates the reduction of α(0) from overcoupling, thus enabling
faster tuning without significant degradation of sensitivity.
To quantify this speed-up, we will also calculate the scan rate R, Eq. (2.16), at which we
can tune the cavity resonance through frequency space in search for a signal. This scan rate isinversely proportional to the measurement time at each tuning step, which in turn scales with α−2
as a consequence of Gaussian noise statistics (Sec. 5.1.1). HenceZ ∞R ∝ ∆aα2 (ω)dω;

(5.8)

−∞

carrying out the integral we obtainp∆a GS,1Q n2A κ2a κ2mR∝ h i3/2 .
κl −κm 2
1 2
1
nT + 2κl κm + GS,1Q
2

(5.9)

Without squeezing (GS,1Q = 1), R is maximized when the cavity is twice-overcoupled (κm = 2κl ),and at this optimal coupling the scan rate scales as Rumax ∝ κ2a /κl , a known result [90].96 When
GS,1Q  1, the optimal coupling is κm = 2GS,1Q κl and the scan rate scales as Rsmax ∝ GS,1Q κ2a /κl .
Comparing the two situations, the scan rate is improved by GS,1Q , which shows that an ideal SSR
greatly accelerates the search for a weak classical signal when squeezing and overcoupling.
In practice, however, we will see in Sec. 5.1.2 how losses in microwave components reduce
the SQ-to-AMP transmission efficiency η and hence the benefit of squeezing, because part of the95

Presumably, once the receiver bandwidth approaches the frequency of the cavity, it would become inviable tomeasure the cavity state faster than its own oscillations. However, this limit is over three orders of magnitude awayfrom the performance achieved here and in Ch. 8.
96
In particular, a twice-overcoupled haloscope (without squeezing) scans 32/27 ≈ 1.19× faster than the same
haloscope in its critically coupled configuration, despite losing a little sensitivity at the cavity resonance frequency.

118
squeezed state is replaced with unsqueezed vacuum. Figure 5.2 compares the theoretical scanrate enhancement Et = Rs /Rumax (a) when η = 1 (perfect transmission), and (b) when η = 0.69
(efficiency observed in practice), as a function of GS,1Q and κm /κl . In the first case, Et improves
arbitrarily as squeezing and coupling are together increased. In the second case, it plateaus at
Etmax ≈ 2.2 for GS,1Q > 20 when optimally coupled. In Secs. 5.3 and 5.4, we will see experimental
results for the scan rate enhancement consistent with this theory.

Figure 5.2: (a) Theoretical scan rate enhancement Et , calculated as a function of SQ singlequadrature power gain GS,1Q and coupling ratio κm /κl . With perfect efficiency, i.e. η = 1, scanrate improves steadily with GS,1Q and κm . (b) Same as (a) for η = 0.69. Scan rate plateaus at 2.2
for κm /κl = 5 and GS,1Q > 20. The color scales for (a) and (b) differ by close to a factor of 10.
Figure adapted from Ref. [208] to correct an error whereby the lowest value on the color axes werelabeled “1.” The other axis labels have been verified.

5.1.1

Lossless calculation

In this section, we track the propagating EM fields through the SSR and cavity, so as toderive the susceptibility matrix of the entire system. We then calculate the signal visibility α(ω)
and the scan rate R. We first consider the case where the propagating fields experience no loss
(the cavity mode still decays partially out its loss port), which nonetheless faithfully illustrates theutility of the SSR, then treat the full system in the presence of transmission losses in Sec. 5.1.2.
We model the energy exchange between the cavity’s ports in its own rotating frame. The

119time evolution of the cavity is governed by the Heisenberg-Langevin equation
X√
dÂ
κT
= − Â(t) +κj âin,j (t),dt
2

(5.10)jwhere Â is the cavity ladder operator,
κT = κm + κl + κa ,(5.11)

and âin,j (j = m, l, a) are the annihilation field operators of the input modes incident on ports
indexed by m, l, and a. At the measurement port, we physically separate input (âin,m ) and output(âout,m ) fields with a circulator. Given the input-output (or boundary conditions) relations√
κj Â(t) = âin,j (t) − âout,j (t),(5.12)

analogous to those of Eq. (3.100), the input and output field operators are related in the Fourier
domain byâout,j (ω) =Xχjk (ω)âin,k (ω),(5.13)kwhere√− κj κk + (κT /2 + iω)δjk
χjk (ω) =κT /2 + iω

(5.14)

are the elements of a 3 × 3 susceptibility matrix fully describing the behavior of the cavity [200].
By cascading the input-output relations for each element presented in Fig. 5.1, we can calculate the benefit from squeezing. We work in the quadrature basis and consider the vector of input
quadratures~xin = [X̂in,m , X̂in,l , X̂in,a ]T(5.15)aligned with our squeezing. We will calculate the SSR/cavity susceptibility matrix ΞX , in termsof which the vector of output quadratures is~xout = ΞX ~xin .

(5.16)

120
The first element in the system is the SQ, which performs a one-mode squeezing (OMS)
operation on the measurement port’s input quadrature:10 √GS,1Q
~xS = SX ~xin =  0
1


0
0


0


~xin ,0
1(5.17)

where the subscript S refers to the SQ output port and GS,1Q is the SQ single-quadrature power
gain. The OMS operation also amplifies the other quadrature of the measurement port mode, Ŷin,mp
by GS,1Q in order to preserve the Heisenberg uncertainty relation. We do not track the evolutionof the Ŷ quadrature here, as it is irrelevant for SSR performance.
Next, the cavity transforms the quadrature operators. In the cavity rotating frame, thevectors of quadrature operators are obtained from the vectors of ladder operators by
 



I3   ~a(ω) 
1  I3~x = √ 

,
2
†
~y
−iI3
iI3 ~a (−ω)(5.18)

or[~x, ~y ]T = P [~a(ω), ~a† (−ω)]T(5.19)

Here I3 is the 3 × 3 identity matrix, and the mode ordering in each vector is m, l, a: for example,~a(ω) = [âm (ω), âl (ω), âa (ω)]T .
Thus, the cavity susceptibility matrix in the quadrature basis χ̃ is:03  −1 χ(ω)
03 
χ(ω)
χ̃ = P 
P = ,03
χ∗ (−ω)03
χ(ω)(5.20)(5.21)

where 03 is the null matrix. We see that the cavity’s effect on field and quadrature operators is
identical, which leads to~xo = χ(ω)~xS ,where o refers to the cavity output port.

(5.22)

121
Finally the AMP performs a second OMS operation so as to amplify the quadrature that the
SQ originally squeezed:p GA,1Q~xout = AX ~xo = 
0


0


0

0


~xo ,
0


1

1
0

(5.23)

where GA,1Q is the AMP power gain. The SSR and cavity thus transform input to output quadratures according to~xout = AX χ(ω)SX ~xin = ΞX ~xin .

(5.24)

We can then calculate the single-quadrature output spectral density matrix
Σout,X =h[~xout ]† [~xout ]T i,
2π(5.25)

where the Hermitian conjugation here does not transpose the vector, in order to determine thetotal output signal and noise powers. Substituting for ~xout yields
Σout,X = Ξ∗X Σin,X ΞT
X,(5.26)

h[~xin ]† [~xin ]T i
2π(5.27)

where
Σin,X =

is the input noise spectral density matrix. Since the SSR is connected to three incoherent anduncorrelated modes (Fig. 5.1),
hX̂in,j X̂in,k i
1= δjk nin,j +,
2π2(5.28)

where nin,j is the input mean photon number per unit time per unit bandwidth (henceforth simply
mean photon number) of mode j. Thus,1nT + 200

nT + 12000nA + 21Σin,X = 



0



.



(5.29)

122
Taking the first entry in the matrix Σout,X we obtain the single-quadrature output spectral density
at the measurement port:Σout,X,m ≈





GA,1Q
1
β(ω)
nA κa κm + nT +κl κm +
,
B(ω)
2
GS,1Q

(5.30)where
B(ω) =

κm + κl
2

2

+ ω2,

(5.31)β(ω) is given by Eq. (5.7), and we used κa  {κl , κm } and nA  1/2.
The visibility α(ω) can be directly extracted from Σout,X,m , since the signal is the term
proportional to nA and the noise is Σout,X,m (nA = 0). The result is exactly that quoted in Eq. (5.5).
We now consider a search protocol comprising many measurements of Σout,X,m , as the cavity
frequency is tuned by discrete steps δc . The spectral scan rate R is obtained by taking the limit:δc,δc →0 τR = lim(5.32)

τ →0where τ is the duration of each measurement. In the rest of this section we will derive Eq. (5.9) for
the scan rate. We begin by defining the signal-to-noise ratio (SNR) at a single tuning step asα(ω) =

p
α(ω)
τ ∆a
,
2

(5.33)

which scales as the square root of the number of independent measurements of the power containedin the bandwidth ∆a , centered on a frequency detuned from cavity resonance by ω/2π [77, 106].
The factor of 2 is a consequence of our single-quadrature measurement scheme, which is discussed
in Sec. 5.1.3. The SNR is a useful quantity, as it relates the visibility to how much time is actuallyneeded to resolve an axion signal.
For δc <∼ κT /2π, multiple tuning steps will contribute to the SNR at each frequency. Without
loss of generality, we evaluate the net SNR for some putative signal frequency ωa which coincides
exactly with the cavity resonance at a particular step. We then consider the contributions to the
SNR of the surrounding 2n + 1 tuning steps, where
2πδc = 2κT /(2n + 1).

(5.34)

123
Contributions to the SNR add in quadrature, so the net squared SNR is
k=nα22n+1 =τ ∆a 1 X 2
α (k2πδc )δc .
4 δc(5.35)

k=−n

In the limit n → ∞, τ → 0, we obtain the integrated squared SNR
∆a 1
α2I =

4 R

Z ∞

α2 (ω)

−∞

dω
,
2π

(5.36)

where we have extended the limits of integration to ±∞: the contributions from tuning steps where
the cavity is detuned by more than κT are negligible.
Note that our arbitrary choice of ωa appears nowhere explicitly on the right-hand side ofEq. (5.36); thus the integrated SNR will be frequency-independent, insofar as the quantities contributing to α(ω) remain constant as we tune the cavity. In practice, we set a certain target valueof αI for which a real signal would appear as a sufficiently prominent peak in the grand spectrum(see e.g., Fig, 5.7b); this choice determines the required spectral scan rate R. Solving for R, we
obtain
∆a
R=
4α2I

Z ∞

dω
α2 (ω)
2π−∞p∆a GS,1Q n2A κ2a κ2m= h i3/2 ,κl −κm 2
1
1 2
2
κl κm + GS,1Q
16αI nT + 2
2(5.37)

in agreement with Eq. (5.9). In a comparison of squeezed and unsqueezed scan rates, the ∆a /4α2I
factor drops out; thus, we need only compare the integral in Eq. (5.9) for the two cases.

5.1.2

Calculation with transmission losses

Here we generalize the calculations of Sec. 5.1.1 to account for imperfect power transmission
efficiency η between SQ and AMP. Integrating the squared SNR over all frequencies, we obtain thescan rate as a function of η, used in Fig. 5.2. For simplicity, we treat the transmission efficiency λbetween the SQ and the cavity as identical to that between the cavity and the AMP, hence η = λ2 .
Note that in principle, loss between SQ and cavity is slightly less harmful than loss between cavity
and AMP, as loss that occurs after the cavity degrades the signal along with the squeezing.

124
As in Sec. 5.1.1, we track the vector of input quadratures ~xin throughout the SSR and cavity
system, neglecting the orthogonal quadratures. First, the SQ performs the OMS operation definedby Eq. (5.17): ~xS = SX ~xin . Next we must account for transmission losses between the SQ and thecavity, modeled as a beamsplitter interaction between the quadrature operators of the measurement
line at the SQ output {X̂s,m , Ŷs,m }, and those of another, uncontrolled mode {X̂s,λ , Ŷs,λ }. We could
define generalized quadrature vectors ~xS and ~yS to include X̂s,λ and Ŷs,λ respectively, but we do
not need to keep track of the EM field’s evolution in loss modes. Among the modes we do keep
track of, only the mode entering and exiting the measurement port experiences the loss:


√
0
0
 λ




~xi =  0
~x ,
1
0

 S


0
0
1

(5.38)

or ~xi = LX ~xS . Here i refers to the cavity’s input, and 0 ≤ λ ≤ 1 is the single-sided transmission
efficiency.
Since we are not keeping explicit track of the propagating loss mode between the SQ andcavity in this calculation, we account for the unsqueezed vacuum that it introduces as an added
noise term,1(nT + 2 )(1 − λ)

0

0

0

0

0



NX = 



0


.
0


0

(5.39)We thus have the single-quadrature noise spectral density at the cavity input:
Σi,X = [LX SX ]∗ Σin,X [LX SX ]T + NX ,where Σin,X is given by Eq. (5.29). This expression yields


 λ
1
n
+
+
1
−
λ
0
 T 2 GS,1Q

Σi,X = 0nT + 12
00(5.40)


00
nA + 12




,



(5.41)

125
where both the partially attenuated squeezed vacuum term (∝ λ/GS,1Q ), and the unsqueezed
contribution from the loss mode (∝ 1 − λ) appear clearly. Note that in the absence of loss,∗TΣi,X = SX
Σin,X SX,

(5.42)

in agreement with Eq. (5.26).
We can now calculate Σout,X , the output spectral density matrix along ~x. Given Eq. (5.21)
for the cavity’s susceptibility in the quadrature basis we may write
Σo,X = χ∗ (ω)Σi,X χ(ω)T .

(5.43)

Then losses between cavity and AMP are accounted for in the same manner as before, and finally
the AMP performs another OMS operation, amplifying the SQ squeezed quadrature. We thus
obtainT
Σout,X = A∗X [L∗X Σo,X LT
X + NX ]AX(5.44)T= A∗X [L∗X χ∗ (ω)Σi,X χ(ω)T LT
X + NX ]AX

as the analog of Eq. (5.26) in the presence of loss.
The first entry in the matrix Σout,X is the output spectral density at the measurement port
along X̂:GA,1Q λ11Σout,X,m = nT +GA,1Q (1 − λ) +
nA +
κa κm
2
B(ω)
2





1
λ
+ nT +κl κm + 1 − λ +β(ω) ,
2
GS,1Q(5.45)

where B(ω) and β(ω) are defined as in Eq. (5.30), and the same approximations have been made.
From Σout,X,m we can extract the signal visibility α(ω) in the presence of loss:α(ω) =

κa κm
λnA
h


1 ×
nT + 2
B(ω) (1 − λ) + λ κl κm + 1 − λ +

λ
GS,1Q



i.
β(ω)

(5.46)

When λ = 1, Eq. (5.46) reduces to Eq. (5.5).
Finally, the scan rate enhancement Et in the presence of loss, presented in Fig. 5.2b, isR∞

2−∞ α (ω)dω, normalized by the same integral with GS,1Q = 1 and κm = 2κl .

126
5.1.3

Equivalence between single- and double-quadrature amplification

Single-mode squeezing can only enhance sensitivity to displacements along a single quadratureof the cavity field. In a situation such as an axion search, the signal distributes its excess power
equally between the two quadratures. Thus switching to single-quadrature measurement fromdouble-quadrature measurement, currently the operational mode of choice of axion haloscopes[105, 106], seems detrimental. In this section, we will see that in the absence of squeezing, neither
measurement scheme has an advantage over the other.97
If we neglect amplifier added noise, the signal visibility α is independent of whether wemeasure one quadrature or both. Specifically, an axion signal characterized by its spectral density
Sa at the amplifier input divides itself equally as Sa /2 between the two quadratures. Similarly, thevacuum noise spectral density ~ωa /2 divides its power equally as ~ωa /4 between the quadratures.
However, we must account for the quantum limits on two-quadrature measurements. Anylinear amplifier that measures both quadratures adds at least a second half-quantum of inputreferred noise, evenly distributed between the two quadratures [128]. An ideal double-quadrature
measurement thus yields
α2Q =

Sa
~ωa

(5.47)

in each quadrature. By comparison, there is no quantum limit on single-quadrature amplification,so an ideal single-quadrature measurement yields
α1Q =

2Sa
= 2α2Q~ωa(5.48)

in the amplified quadrature.
To make a fair comparison between the single-quadrature and double-quadrature cases, wemust consider the improvement in the SNR (defined in Sec. 5.1.1). Because all pertinent differences
between the two cases enter when considering a single tuning step, we neglect tuning in the following
97
This will be true insofar as there is negligible added noise at all frequencies, and no bandwidth limitations in eithercase. Johnson-Nyquist noise from an identical source is assumed in order to make an apples-to-apple comparison.

127
discussion. The SNR α is given in terms of the spectral density ratio α by
r
α=

N ∆a
α,
2 ∆

(5.49)

where ∆a /∆ is the ratio of signal to noise bandwidths, and N is the number of independentmeasurements of the voltage contained in the noise bandwidth ∆. In considering the appropriate
values of N and ∆ for the two cases of interest, we will find two independent effects, each reducing√
α1Q by a factor of 2 relative to α2Q . Together, these effects cancel the apparent benefit stemmingfrom the absence of quantum noise limits in the single-quadrature case.
First, the Nyquist theorem guarantees that there are
N2Q = 2τ ∆2Q(5.50)

independent measurements of the noise voltage in a double-quadrature measurement of duration τand bandwidth ∆2Q , where the factor of 2 counts the two independent quadrature amplitudes foreach resolved Fourier component. Thus there are
N1Q = τ ∆1Q(5.51)

measurements of the noise voltage in a single-quadrature measurement of bandwidth ∆1Q .
Second, noiseless single-quadrature measurement with a parametric amplifier creates an irreversible ambiguity between the output signal and idler frequencies, equally spaced about theamplifier band-center. This ambiguity necessitates mapping amplifier outputs at a given detuningfrom band-center to the input signal and idler frequencies (Sec. 5.7). The consequent addition ofspectral densities, half of which are guaranteed not to have an axion-induced excess power, effectively increases the noise bandwidth in a single-quadrature measurement by a factor of 2 relative
to the signal bandwidth: ∆1Q = 2∆a . In the standard double-quadrature measurement scheme,
the signal and noise bandwidths are equal: ∆2Q = ∆a .

128
Putting this all together, the two measurement schemes are seen to be equivalent:rτ ∆1Q ∆a
α1Q
α1Q =
2 ∆1Q
r
2τ ∆2Q ∆a=
2α2Q
2 2∆2Q(5.52)= α2Q .
The first line agrees with Eq. (5.33) for the single-quadrature SNR. The advantage of squeezed overunsqueezed single-quadrature haloscope operation can thus be taken as the true advantage thatsqueezing confers over current haloscopes that amplify both quadratures.

5.2

Model for the axion field
Figure 5.1 models the axion field as a fictitious generator that drives the cavity througha weakly coupled port. In this section, we relate the fictitious port coupling κa and the power
spectral density nA at the generator output to physical parameters normally found in the haloscopeliterature, and show that for representative values, the axion field acts as a classical force.
The measurable axion-sourced power is obtained from the first term in the output spectral
density, Eq. (5.30). Referring to the cavity output and multiplying by the axion linewidth ∆a , theon-resonance (ωa = ωc ) signal power is
Psig = 4~ωa nA ∆aκa κm.
(κm + κl )2(5.53)The two model parameters κa and nA can readily be related to physical parameters by comparing
Eq. (5.53), to e.g. Eq. (1) of Ref. [90] evaluated on resonance. In our notation this expression takes
the form3ωc2 κm2
2 ~c ρaPsig = gaγγ
B0 V Cmn`,µ0 ωa2(κm + κl )2(5.54)

where µ0 is the vacuum permeability and other parameters are as defined in Eqs. (2.8) and (2.16).
Equating Eqs. (5.53) and (5.54), we obtainnA κa =

2 ρ c3gaγγ
a
B 2 Cmnl V.
4ωa µ0 ∆a 0

(5.55)

129
To derive a second expression relating κa to nA , we observe that the fictitious generator in
our haloscope model may equivalently be represented as a second harmonic oscillator mode withvery high occupancy but very weak coupling to the haloscope cavity. This equivalence is illustrated
in Fig. 5.3. The axion field in any laboratory-scale volume constitutes such an oscillator mode, withresonant frequency ωa . Specifically, we model the oscillations of the axion field as a fictitious cavity
occupying the same volume as the real haloscope cavity. The quanta of this fictitious cavity areaxion particles, so its total occupancy is
NA =

V ρa
;
~ωa

(5.56)

it is coupled to the haloscope cavity with interaction HamiltonianĤint = ~g(Â + Â† )(B̂ + B̂ † ),(5.57)

where g is the interaction strength and Â (B̂) is the annihilation operator of the haloscope (fictitious)
cavity. The Hamiltonian of the closed system isĤ = Ĥ0 + Ĥint ,(5.58)





1
1
†
†
Ĥ0 = ~ωc Â Â ++ ~ωa B̂ B̂ +.
2
2

(5.59)

where

Coupling the haloscope cavity to measurement and loss ports at rates κm and κl , respectively, we
write down the Heisenberg-Langevin equations of motion for the open system:√√dÂκm + κl= −iωc Â(t) − ig[B̂(t) + B̂ † (t)] −Â + κm âin,m (t) + κl âin,l (t)dt2dB̂= −iωa B̂(t) − ig[Â(t) + Â† (t)],dt(5.60)(5.61)

where we describe the input bath associated with the measurement (loss) port with the annihilation
operator âin,m (âin,l ).
We restrict ourselves to the classical limit of the system, with operators demoted to complex
amplitudes, and the case where the resonances coincide, ωa = ωc , with no power entering via the130

Figure 5.3: (a) Model for the haloscope cavity (with mode Â) corresponding to Fig. 5.1, wherein anaxion signal is modeled as a microwave generator sourcing mean photon number nA and coupled
to the haloscope cavity at power decay rate κa . (b) Equivalent representation of the axion signal
as a second, fictitious cavity (mode B̂) filled with NA axions coupled to the haloscope cavity withfield exchange rate g. The other couplings of the haloscope cavity remain unmodified.
quantity
ρa
B0
gaγγ
∆a
ωa /2π
V
Cmnl

value
0.45 GeV/cm3
9T-7.7 × 10−24 eV−1
5 kHz
5 GHz
1.5 L
0.5

Table 5.1: Representative physical values used to estimate axion port parameters.

measurement and loss ports,
ain,m = ain,l = 0.

(5.62)

Transforming into the rotating frame of the haloscope cavity,
A(t) → A(t)e−iωc t(5.63)

B(t) → B(t)e−iωc t(5.64)and making a rotating wave approximation, Eqs. (5.60) and (5.61) reduce todA
κm + κl= −igB(t) −
A(t)
dt
2
dB
= −igA(t),dt(5.65)(5.66)

131
where g plays the role of the field exchange rate corresponding to the power decay rate out of theaxion cavity,
g=

κa
.
2

(5.67)

These equations of motion describe an exchange of energy between the two cavities and a decay ofthat energy out from the haloscope cavity via the measurement and loss ports. We are interested
in the steady-state (dA/dt = 0) output field√
Aout,m = − κm A

(5.68)

when the occupancy of the axion cavity is|B|2 = NA .
We find a steady-state occupancy of the haloscope cavity

2
κa
2
|A| =
NA ,
κm + κl(5.69)(5.70)

implying an output signal power of
Psig = ~ωa |Aout,m |2 =~ωa NA κ2a κm.
(κm + κl )2(5.71)

Eqs. (5.53) and (5.71) for the output power must agree, implying
nA
NA
V ρa
=
=
.
κa
4∆a
4~ωa ∆aIn terms of physical parameters, Eqs. (5.55) and (5.72) yields|gaγγ |ρa B0 V Cmnl c3
nA =4ωa ∆a~µ0s
Cmnl ~c3κa = |gaγγ |B0.
µ0(5.72)(5.73)(5.74)

Using representative values for the HAYSTAC experiment [90] in the presence of a 5 GHz
KSVZ axion shown in Table 5.1, we obtain the values for our model parameters shown in Table
5.2. We see that the fictitious generator is well into the classical regime,
1nA  ,
2
while its extremely feeble coupling κa nonetheless makes its presence a challenge to detect.

(5.75)

132
parameterNAκa /2πnA

value
3.3 × 1016
2.3 µHz
2.4 × 107

Table 5.2: Model parameter values calculated using the physical values from Table 5.1.

5.3

Enhanced signal visibility with squeezing
In practice, the scan rate of Eq. (5.37) can be degraded not only by transmission losses,

but also by distortions in the squeezed state (Ch. 4) and added noise in the amplifier chain. In
order to investigate how accurately Eq. (5.37) predicts the performance of a squeezing-enhanced
haloscope, we work with an apparatus that mimics its microwave network, but without some of
the cumbersome features. Specifically, in an axion haloscope the mechanically tunable cavity must
reside in a large static magnetic field to enhance κa and nA . In our apparatus, the cavity has a fixed
frequency ωc = 2π × 7.146 GHz and there is no applied field. Consequently the cavity, constructedfrom superconducting aluminum and shown in Fig. 5.4, has much lower intrinsic loss than a copper
haloscope cavity. Additional loss is added through an explicit port that extracts energy from thecavity at a rate κl = 2π × 100 kHz, a typical value for a haloscope cavity. An axion-like signal iscoupled into the cavity through a microwave generator connected to a second port weakly coupled
at rate κa = 2π × 100 Hz. Finally, a third port couples the cavity mode to an SSR with a rate
chosen to be close to the optimum value for the case with (κm = 10κl ) or without (κm = 1.5κl )squeezing, creating a physical realization of the model in Fig 5.1.
For the largest increase in scan rate, the SSR should be attached to the cavity measurement
port with as little transmission loss as possible. To investigate the transmission loss independentof the cavity loss, we use the fact that the JPAs are narrowband (∼ 5 MHz), tunable amplifiersand detune both the SQ and AMP far off cavity resonance (∼ 10 MHz) so that the squeezed
state is promptly reflected from the cavity. Figure 5.5 illustrates the ability to efficiently generate,transport, and amplify a squeezed state in this off-resonance configuration. It shows histogramsof the measured voltage in the AMP’s amplified quadrature X̂out,m as a function of θ, the phase

133

Figure 5.4: (a) Photo of the aluminum cavity (closed up) used for the experiments in Secs. 5.3 and5.4. The three SMA ports are for measurement, loss, and signal injection. A penny is shown forscale. (b) Same cavity, opened up. The 7.146 GHz resonant mode lives in the slot visible in both
halves of the cavity.

between the amplified quadrature of the SQ and the amplified quadrature of the AMP. When
θ = π/2 or 3π/2, one quadrature is first squeezed and then amplified. At these points, comparing
2 and σ 2 measured with SQ on and off (not pumped), respectively,
the output noise variances σonoff

the squeezing S = −4.5 ± 0.1 dB using the definition of Eq. (4.4). As in the last chapter (Footnote90), this is not an inferred squeezing; rather, it is a direct measurement of an overall 4.5 dB
reduction in the noise floor of X̂out,m over the whole bandwidth of the quadrature measurement.
This amount of squeezing (consistent with the estimate of η = 0.69 ± 0.01, Sec. 5.5.3) required
particular care in reducing the transmission losses between the two JPAs, which was facilitated byusing the flux-pumped JPAs of Sec. 3.7.
This large amount of delivered squeezing implies that the SSR should improve our ability toresolve a weakly coupled signal detuned from cavity resonance. To demonstrate this improvement,
the SQ and AMP bands are centered on cavity resonance with the cavity’s measurement port
overcoupled (κm = 10κl ), and the phase θ set to π/2 or 3π/2 (Fig. 5.5). To perform the singlequadrature measurement, the output X̂out,m is mixed down with a local oscillator (LO) at thecavity resonant frequency its power spectral density is computed. As for any such measurement,
the frequency component at ω in the down-mixed output is a linear combination of two frequency
components at the mixer’s high frequency input, ±ω detuned from the LO frequency. Figure 5.6ashows the spectral density when the tone is 1 MHz detuned from the cavity’s resonance. Comparing

(a)counts
1000

134

500

(b)

0

Figure 5.5: (a) Histograms of voltage fluctuations VX measured along X̂out,m , as a function of the2 /σ 2 . Data taken off of
SQ-AMP relative phase θ. (b) Corresponding vacuum squeezing S = σonoffcavity resonance. Figure reproduced from Ref. [208].

two situations, SQ on and SQ off, the signal visibility improves by roughly 4 dB in the presence ofsqueezing, with 0.5 dB degradation from cavity loss.
To estimate the associated increase in scan rate from the SSR, the tone is stepped acrossthe cavity’s resonance and the visibility α(ω) measured at each detuning ω. In order to compare
the optimal squeezed and unsqueezed cases, Fig. 5.6b shows measured values of α(ω) for two cases:for κm = 10κl with squeezing, and for κm = 1.5κl without squeezing (see Sec. 5.6 for two othercomplementary cases). Without squeezing, α(0) is greater but the bandwidth is poor. WithR∞
squeezing, α remains relatively high as ω increases. Extracting R ∝ −∞ α(ω)2 dω for the twocases, the estimated scan rate enhancement is Ee = 2.05 ± 0.04. Independently measured valuesfor η, κm , and GS,1Q , yield expected values for α(ω), shown as solid lines, in excellent agreementwith the measured values in Fig. 5.6b. Finally, from the expected α(ω), Et = 2.11 ± 0.07, also inquantitative agreement with the data-based estimate Ee .

135

Figure 5.6: (a) Improvement in microwave tone visibility due to squeezing. Power spectra normalized to the unsqueezed vacuum power, with (red) and without (purple) squeezing are shownin for a tone 1 MHz detuned from the resonant frequency of the overcoupled cavity (κm = 10κl ).
The x-axis has been shifted between the two situations for visual clarity. (b) Visibility α(ω)
measured as a function of the tone’s detuning from cavity resonance (ω = 0) for two cases: nosqueezing while near critical coupling (κm = 1.5κl , blue circles), and squeezing while strongly overcoupled (κm = 10κl , red circles). In both cases α(ω) is normalized to the expected maximum value
αmax = α(0) evaluated at κm = κl . The theoretical expectation for each case is superimposed as a
black curve, calculated using κl = 2π × 100 kHz, η = 0.69 and GS,1Q = 13 dB. Figure reproduced
from Ref. [208].

5.4

Mock-haloscope experiment: accelerated search for a weak classical tone
In a real axion search, the aim is to detect a signal many orders of magnitude smaller than thatin Fig. 5.6. Inferring the presence or absence of an axion at each frequency requires combining themeasured powers from many adjacent cavity tunings [78,125]. Furthermore, over a long integration
time the benefit inferred from Fig. 5.6 is vulnerable to practical non-idealities. Drifts of either JPA’s
gain, drifts of the SQ-AMP relative phase θ, non-Gaussian noise processes, and interfering radio
frequency (RF) or intermediate frequency (IF) tones are of particular concern. In this section,
we will see that the SSR indeed matches the performance presented in the previous section whensearching for a feeble tone over a wide frequency range.98
In order to fully validate the enhancement gained from the SSR, a fake axion, or “faxion,”
98

This experiment was informally given the name “the Dummy Experiment for Axionic Dark Matter Active
Upwards of 5 GHz,” or DEADMAU5.

136
tone is sent through the cavity weakly coupled port in order to be detected. The tone is synthesizedfrom a randomly modulated microwave tone, whose power is adjusted such that the faxion spectral
density emerging from the cavity is roughly 1% of vacuum, and whose width is broadened toroughly ∆a ≈ 9 kHz, comparable to expectations for a realistic axion at frequency ωa ≈ 2π × 7
GHz (Sec. 2.4.3). Stepping the faxion tone frequency “backwards” past a stationary cavity simulatesa realistic axion search without the hardware demands imposed by a tunable cavity. The faxion’s
initial frequency is chosen randomly within a 2 MHz window around the cavity resonance, andis then tuned in discrete -10 kHz steps over a 4 MHz window. At each tuning step, an output
power spectral density is recorded as in Sec. 5.3. These spectra are mixed down and referred to themixer’s input by symmetrizing about ω = 0. Spectra are artificially shifted in steps of +10 kHz tosimulate cavity tuning. The spectra are then rescaled by α(ω) such that frequency bins with higher
sensitivity to the faxion are weighted more. These rescaled, shifted spectra are then added into a
grand spectrum (Sec. 5.7 and Ref. [78]). With this procedure, the faxion’s contributions in eachindividual spectrum add at its initial frequency as if the cavity had been tuned, creating a clear
excess of power in the grand spectrum. Figure 5.7a presents some symmetrized spectra, obtained
while squeezing with an overcoupled cavity (κm = 10κl ), in which the faxion excess power is too
small to rise above vacuum fluctuations. The spectra were obtained at different fictitious cavity
tunings, normalized to their measured standard deviations, and vertically offset from one anotherfor visual clarity. In the resulting grand spectrum (Fig. 5.7b), a prominent faxion peak emerges 6σ
above the noise.
Extracting a scan rate enhancement from such a realistic signal search requires two distributions of faxion powers: one when squeezing and one when not squeezing, with near-optimal κm foreach case. Each distribution is obtained by repeating the faxion injection and detection protocol
200 times. Over the course of the measurement, which takes roughly 9 hours per configuration,
the relative phase θ between SQ and AMP quadratures is stabilized at π/2 via a feedback loop
(Sec. 5.5.2). Figure 5.7c displays the two faxion power distributions, as well as the noise power

137

Figure 5.7: (a) Processed spectra. The acquired power spectra are symmetrized, rescaled and thenshifted in frequency to align all the frequency bins containing the faxion with each other (red dots),
thereby effectively tuning the cavity. The power excess of the processed spectra is plotted in unitsof their standard deviation σ γ . (b) Grand spectrum constructed from all processed spectra. The
grand spectrum excess power is plotted in units of its standard deviation σ η . A large power excess
is observed at the frequency of the faxion. (c) Distributions of faxion powers for many separate
experiments each culminating in a grand spectrum. When squeezing with κm = 10κl , the powers(red points) are drawn from a Gaussian distribution N (µs , (σ η )2 ) (solid line), with µs = 6.05±0.07.
When not squeezing with κm = 1.5κl , the powers (blue points) are drawn from N (µu , (σ η )2 ) (solidline), with µu = 4.15±0.07. The powers (black points) drawn from the faxion-less noise distribution
N (0, (σ η )2 ) (solid line) are also represented for comparison. Figure adapted from Ref. [208].

distribution. The no-faxion distribution of mean zero and variance (σ η )2 ,99 Eq. (6.12), is guaranteed by the central limit theorem to be Gaussian distributed, N (0, (σ η )2 ). The faxion adds a smallmean excess of power µ insufficient to enlarge the variance. The faxion distribution is therefore
N (µ, (σ η )2 ). The signal and noise distributions separate as the total measurement time squared,99

For consistency with future chapters, the superscript η borrow from the notation of Sec. 6.1.

138
and thus the speed-up due to squeezing is equal to

Em =

µs
µu

2
,

(5.76)

where µs (µu ) is the mean of the faxion power distribution obtained when squeezing (not squeezing).
The two distributions of faxion powers are characterized by µs = 6.05 ± 0.07 and µu = 4.15 ± 0.07,
leading to a measured scan rate enhancement of Em = 2.12 ± 0.08, in quantitative agreement withthe estimate Ee obtained from visibility measurements in Sec. 5.3.
This measurement provides the strongest proof out of all the arguments presented in thischapter that quantum squeezing not only can in principle be used to accelerate dark matter searches,
but that it can do so subject to realistic transmission losses in a haloscope-like system. The
following sections describe some of the auxiliary measurements and protocols necessary to performthe experiments in this chapter.

5.5

Experimental details and calibration
This section presents the experimental setup used for the experiments in Secs. 5.3 and 5.4,

describes the phase-locking protocol used to ensure θ = π/2 or 3π/2 between SQ and AMP pumptones, and discussed the calibration measurement used to infer the intra-SSR transmission losses.

5.5.1

Experimental setup

The full experimental setup is represented in Fig. 5.8. The SQ-cavity-AMP ensemble isattached to the bottom plate of a dilution refrigerator. For each JPA, the pump tone is routed
though a 10 dB directional coupler connected to the pump port. The placement and configurationof these directional couplers ensures that the large pump tone required for flux pumping primarily
heats up a 50 Ω termination whose Johnson-Nyquist noise [206, 207] propagates back up the pumpline, away from the JPA. A coil around each chip, connected to a DC current source at room
temperature, generates a DC magnetic field, and the chip-coil ensemble is magnetically shieldedwith aluminum and cryoperm. Each JPA is connected to a circulator through superconducting

139
Nb/Ti cables in order to minimize transmission losses.
An ensemble of four circulators route the squeezed state and provide microwave isolation
between the SSR elements. Two circulators between the SQ and the cavity protect the SQ frompower reflected back from the cavity. These circulators provide sufficient isolation when the SQ is
operated with 13 dB of signal gain. Similarly, three circulators separate the AMP and the cavity, as
the AMP is operated with higher signal gain, around 25 dB. Experimentally, it was observed thatwith only two circulators, undesirable feedback between the cavity and AMP perturbs the AMP’s
gain by effectively changing its pump’s power.
At the chain’s input, either vacuum noise or a probe tone can be injected via a 20 dB
directional coupler. This probe tone is useful when characterizing the JPA gain profiles or whencharacterizing the cavity. Finally, at the output, a double-junction isolator protects the AMP fromsignals reflected from the next amplifier, a high-electron-mobility transistor (HEMT) amplifier at
4 K.
At room temperature, a single microwave generator (Keysight E8257D) drives both JPAs andalso serves as LO for the in-phase/quadrature (IQ) mixer in the readout line. On the path leading
to the SQ pump’s input, a voltage-controlled variable attenuator and phase shifter provide control
of both the SQ pump amplitude and phase. On the path leading to the IQ mixer, a frequency
divider converts ωp to ω0 , the bare resonant frequency of both JPAs and the cavity. This divider(Pasternack PE88D2000) can input a wide range of powers (from -20 to 5 dBm) while always
outputting the same power (roughly -4 dBm). Thus, the AMP gain can be tuned freely with themicrowave generator’s output power. A second generator injects a tone through the weakly coupledport of the axion cavity. This tone can be shaped into a 9 kHz wide Lorentzian with an arbitrary
waveform generator (AWG).
At the output, a 6 dB directional coupler routes a fraction of the power to a vector network
analyzer (VNA), which monitors signals either from the SSR chain’s input or from the cavity’s
weakly coupled port. The VNA is used to measure in situ the couplings κm and κl + κa . The otherportion of the output power reaches the IQ mixer’s RF port. After being mixed down, the in-phase140
and quadrature signals are amplified and directed onto 1.9 MHz anti-aliasing low-pass filters, thenfinally digitized by an analog-to-digital converter (ADC).

5.5.2

Phase-locking of SQ and AMP pumps

Enhancing the scan rate with squeezing is only beneficial if the AMP amplifies the SQ
squeezed quadrature. To this end, as suggested by Fig. 5.5, the phase θ between SQ and AMP
pumps must be maintained at π/2 (or 3π/2) to a high precision, given the sharp dependence of thedegree of squeezing S on θ. When S > 0 dB, the output is noisier with the SQ on than off, andthe SSR is therefore detrimental to the axion search. Furthermore S also depends on the SQ gain
GS,1Q , i.e. on the SQ pump power, and there is an optimal GS,1Q for which S reaches its minimum
value Smin . In fact, as the gain increases, S first improves as the squeezed state elongates in phasespace, but then saturates due to distortion effects (Ch. 4). For this type of JPA, S = Smin at
GS,1Q ≈ 13 dB is determined experimentally.
2 as its control parameter on the voltageA feedback loop that uses the output variance σoncontrolled variable attenuator and phase shifter is used in order to achieve S = Smin throughout
a data run. When initializing a 9 hour acquisition, a two-dimensional sweep of the variable at2 . Then, periodicallytenuation As and phase shift θ is used to estimate the global minimum of σon
2 .
throughout the data run, a fast gradient descent-type algorithm corrects for small drifts of σon

Note that this approach is robust to possible phase shifts due to changes in the variable attenuationand vice versa. Empirically, S can be maintained at Smin ± 0.1 dB, over the course of the entire
experiment.
The AMP pump power remained stable around 25 dB without adjustment. However, when
implementing the SSR in HAYSTAC (Ch. 8), it has to be adjusted because the frequency of the
JPAs is also stepped in time. Note that in a practical haloscope run, sizable fluctuations of the net
receiver gain in a SSR-integrated setup on timescales shorter than the raw spectrum acquisition
time can be detrimental to axion detection. This is true even if all sources of added noise are
overwhelmed. For our raw spectra acquisition times of 0.32 s, the receiver gain fluctuations are

141
20

attenuator

power divider

θ phase shifter

dc lines

50 Ω termination

90° hybrid coupler

f/2 frequency divider

coil

isolatorswitchcryoperm + Al

amplifier

low pass filter

Nb/Ti coax.

directional coupler

variable attenuatornormal coax.

E8257DAWG

E5701CATS 9462

VNA
2
1

ADC

E8257D

26
f/2

I

L
R

Q

GS200
6

θ

GS200

15

300 K
20

20

20

20

4K

20

20

20 mK

10
SQ

10
AMP

20

cavityFigure 5.8: Full experimental setup used for the mock-haloscope experiment. The SSR is shown
together with its room temperature control electronics. The input (black) and output (red) coaxial
cables are in bold. At the 20 mK stage, Nb/Ti superconducting coaxial cables are shown in green,
and the three adjacent circulators form a triple-junction circulator fabricated by QuinStar (QCE070100CM30). The thinner (orange) lines represent DC cables. Figure reproduced from Ref. [208].

142
negligibly small.

5.5.3

Intra-SSR loss measurement

The presence of four circulators and several microwave connectors, including adapters from
SMP (used on the JPAs chips) to SMA standards, inevitably reduces the transmission efficiency
η between SQ and AMP. A triple-junction circulator100 and superconducting SMA cables between
the two JPAs and the circulators are used to minimize η. However, η still provides the primary
limitation on the efficacy of the SSR in accelerating axionic dark-matter searches.
O of vacuum fluctuations in a single quadraMeasuring the output power spectral density PS/A

ture amplified by the SQ with the AMP off then repeating this measurement with the role of
the JPAs interchanged yields an estimate of η. Performing the exact measurement described inOOSec. 4.4.2 gives η = GOSQ /GAMP = 0.69 ± 0.01. Figure 5.9 shows 4PS/A /(~ωW ), when varying eitherO
the SQ or the AMP gain. We extract η = GOSQ /GAMP = 0.69 ± 0.01.

The efficiency η observed in the mock-haloscope setup should not be significantly degraded by
the large magnetic fields required in a real haloscope experiment such as HAYSTAC or ADMX, as isborne out in Ch. 8. This is because the primary constraint imposed by the large field is an increased
spatial separation between the axion cavity and the circulators and amplifiers that process thesignal. This ∼ 1 m distance is bridged with superconducting coaxial cables, whose attenuation [232]is far subdominant to other sources of transmission loss, such as the microwave circulators in the
setup. At sufficiently high fields, the cables would eventually experience significantly increasedtransmission losses. However, empirically, at the fields we operate in, the performance remains
unaffected [90].
From η we can estimate the squeezing S that we should obtain far off cavity resonance (∼ 10
MHz detuned). Considering the ideal case where the reduction of the squeezed state variance is100
Perhaps the most obvious upgrade available to this setup would be an L-shaped, cryogenic, quadruple-junction
circulator to replace the combination of single- and triple-junction devices in Fig. 5.8. This option was not pursed
here or in Ch. 8 purely on grounds of financial cost.

143

Figure 5.9: Intra-SSR transmission loss calibration measurement. The normalized singleO
quadrature output power spectral density PS/A
is plotted as a function of the SQ or AMP gain
GA/S,1Q . Two configurations are represented: when operating only the SQ (red circles) and when
operating only the AMP (blue circles). A linear fit (solid lines) to these linear responses allows
yields η. Figure adapted from Ref. [208].

equal to the SQ single-quadrature power gain GS,1Q , we obtain:S=η+ 1 − η,GS,1Q

(5.77)

which, for GS,1Q = 13 dB leads to S = −4.6 dB, in quantitative agreement with what was measure
in practice.

5.6

Complementary measurements of the microwave tone’s improvement in
resolution
We saw in Fig. 5.6b the visibility α(ω) of a microwave tone as a function of ω, the detuning

between the tone’s frequency and the cavity bare resonance. We compared α(ω) between the two
relevant cases: squeezing with a strongly overcoupled cavity versus not squeezing with a nearcritically coupled cavity. Figure 5.10 presents these, along with measurements of α(ω) for the two

144

Figure 5.10: Complementary measurements of signal visibility in the SS. Signal visibility α(ω) is
plotted as a function of the signal’s detuning from cavity resonance. It is normalized by αmax ,obtained at zero detuning for κm = κl . Four cases have been measured, with the corresponding
theoretical predictions represented as solid lines: overcoupling (κm = 10κl ) with squeezing (redcircles), near-critically coupling (κm = 1.5κl ) without squeezing (blue circles), overcoupling withoutsqueezing (purple triangles), and near-critically coupling with squeezing (green triangles). The
theory lines have been calculated for κl = 100 kHz, η = 0.69, and GS,1Q = 13 dB. Figure reproduced
from Ref. [208].

complementary cases: strongly overcoupling without squeezing, and near-critically coupling withsqueezing. In all four cases, there is excellent agreement with predictions from the theory, developedin Sec. 5.1.2 and shown in solid lines. The theory curves are not fits; they use parameter valuesfor κl , η, and GS,1Q measured independently. Furthermore, the values are the same across all fourcases.
Squeezing is beneficial, even when not overcoupling the cavity’s measurement port, as itenhances α off cavity resonance while having no effect on resonance. Experimentally, the complementary estimate Eec1 = 1.72 ± 0.03 is obtained when squeezing and near-critically coupling,
compared to the same situation without squeezing, in agreement with the theoretical prediction,
Etc1 = 1.77±0.03. Conversely, the scan rate is worse when strongly overcoupling without squeezing,
compared to near-critically coupling with squeezing: from the data, Eec2 = 0.57 ± 0.01, not far from145
the theoretical value of Etc2 = 0.52.

5.7

Processing of the SSR spectra
In order to detect the small cavity displacement of Sec. 5.4, 401 “raw” spectra are acquired,processed, and combined into one “grand” spectrum. The injected faxion tone is sufficiently small
relative to the size of the vacuum fluctuations that in any one raw spectrum it typically does notstand out. However, after processing the spectra, the faxion emerges often well above the level of
vacuum noise, as in Fig. 5.7b. This section provides a brief overview of the steps in the processingof the spectra and provides the specific numerical values used throughout the processing in orderto produce Fig. 5.7.101 The processing procedure is based closely on the work of Ref. [78], which is
described quantitatively in Sec. 6.1. This section borrows the terminology used for the intermediate
processing stages set forth in Ref. [78], and also omits many details and rationales which are there
covered extensively.
Using the setup of Fig. 5.1, fluctuations emerging from the cavity, possibly in the presence ofsqueezed vacuum noise, are directed into a chain of amplifiers led by the AMP. The fluctuations
are mixed down to DC, further amplified, low-pass filtered up to 1.9 MHz, and sampled with an
Alazar ATS9462 digitizer at 6 MS/s. Each raw spectrum, distinguished as the least processeddata saved to a hard disk, itself actually comprises the frequency-averaged power spectral densitiesof 32 “subspectra,” each acquired over 10 ms and fast Fourier transformed to provide a spectral
resolution of ∆b = 100 Hz. Since 401 raw spectra are acquired for each of the 200 squeezed and
200 unsqueezed data runs, the live acquisition time of all the spectra totals just over 14 hours, notcounting the dead time. In practice it took roughly 18 hours.
Once the raw spectra are in hand, they are symmetrized. That is, the measured output at
each signal frequency is processed as if it came equally from the signal and idler inputs. Since theoutput of a JPA at a given detuning from the center of its amplification band is nearly identical tothe output at the opposite detuning from band center, Eqs. (3.128) and (3.129), the best one can101

Some of these values differ than those used to process the HAYSTAC data.

146
do is to infer the spectral density measured in the homodyne configuration as coming equally fromboth sides of the pump.
The symmetrized spectra are then averaged according to their real frequencies (i.e. not accounting for the fictitious stepping of the cavity used to simulate a haloscope search) in order todetect excess power in either the pertinent RF or IF band.102 This mean spectrum is then high-passfiltered by the profile of a Savitsky-Golay (SG) filter [233, 234] applied to the average spectrum.
A SG filter is simply a computationally quick means of applying the dth -degree polynomial generalization of a moving average within a window of 2W + 1 bins. If the width W is much larger
than the size of interesting spectral features, those features will be minimally attenuated when the
spectrum is divided by the filter response, whereas features extending over wide bandwidths will
be effectively removed. Following Ref. [78], d = 10, W = 500 for this first filtering.
The resulting real-frequency-averaged spectrum is used to detect and remove excesses ofpower which do not act like our faxion. Bins that exhibit power fluctuations more than 4 standard
deviations above the mean at their real frequencies are discarded along with their close neighbors
as contaminated for all spectra. In practice, over 99% of bins are preserved, with only extreme
outliers eliminated. Note that since in the real frequency space the faxion tone is being stepped,
its 400 tunings do not combine over any one narrow block of bins, and this bin rejection procedure
will therefore not eliminate a faxion signal. Once contaminated bins are removed, the SG filter isrecalculated, as it will be slightly different without the presence of outlier bins, and reapplied toeach spectrum individually.
This baseline removal leaves a set of nearly-flat “normalized” spectra, each centered near 1.
Residual structure left from fluctuations in the overall receiver gain or its profile is then removed
by dividing out the SG profile (d = 4, W = 500) of each individual spectrum. The mean value of 1is then subtracted from each spectrum to form the “processed” spectra, several of which are shownin Fig. 5.7a.103
102

In a real haloscope search, since the LO being used for homodyne measurement would be stepped along withthe cavity, non-axion-induced power excesses in the RF are trickier to reject than their IF counterparts [78].
103
For visual clarify, only a fraction of the bins are kept in Fig. 5.7a and b.

147
Next we must rescale the spectra to account for the varying sensitivity to a faxion tone asa function of its detuning from cavity resonance in any given spectrum. To form these “rescaled”
spectra, the processed spectra are divided by the relative visibility profile of the squeezed state
receiver: Eq. (5.46) with an additional term accounting for the small contribution of the HEMT’s
added noise referred to the input of the AMP, as the weak frequency dependence of AMP gain
over the bandwidth of the cavity in the presence of spectrally flat HEMT added noise contributes acorrespondingly weak frequency-dependence to the noise. The rescaled spectra have the important
property that a given power excess from the weakly coupled port of the cavity produces a constant
expectation value of rescaled spectrum power excess, regardless of detuning from the center of thecavity. As a consequence, the variance of the power distribution for each bin grows with detuningfrom cavity resonance.
Aligning bins along the fictitious frequencies for which the stepped faxion tone would appear
stationary, it is then possible to construct the maximum likelihood (ML) estimate of the powerexcess in each bin. Doing so yields the “combined” spectrum, a single spectrum whose bins contain
as few as one (at the extreme edges) and as many as hundreds of contributions from individual
rescaled spectra. The combined spectra bins each have a resolution of ∆b , far below the linewidth∆a of the faxion. The power within non-overlapping sets of nc = 10  ∆a /∆b bins are thus
averaged, to form the “rebinned” spectrum, which has nc ∆b = 1 kHz spectral resolution.
Finally, overlapping sets of nr = 41  ∆a /nc ∆b rebinned spectrum bins are combined,accounting for the independently experimentally determined lineshape of the faxion, in order toproduce the ML estimate of the faxion-shaped power centered on each bin. The resulting spectrumof estimated powers is the grand spectrum, shown renormalized to have mean power excess 0 andstandard deviation 1 in Fig. 5.7b. In the grand spectrum, the faxion typically stands out well above
the surrounding noise.
The acquisition and processing of sets of 401 raw spectra are repeated 200 times apiece forthe optimal squeezed and unsqueezed cases. The two data sets require slightly different processing,chiefly because the visibility profile of the unsqueezed case does not include the frequency-dependent

148
contribution from the squeezing. This results not from finite bandwidth effects of the SQ, but fromthe fact that squeezed noise is preferentially absorbed near cavity resonance by the cavity’s loss
port, wherefrom it is replaced with unsqueezed vacuum.
Each of the 400 total acquisitions provides one measured faxion power. The powers are
histogrammed for the optimal squeezed and unsqueezed cases, along with the far larger numberof grand spectrum powers of bins not containing a faxion, in Fig. 5.7c. While the absolute meanvalues of the squeezed and unsqueezed faxion distributions carry little meaning, as they scale withthe somewhat arbitrary choice of faxion power, their squared ratio, which would be preserved fora faxion of any power, gives the scan rate enhancement obtained from squeezing, as discussed inSec. 5.4.

5.8

Concluding remarks on the SSR for enhancing the scan rate of real
haloscope searches
In this chapter, we have seen both in theory and in practice that partially replacing the vac-uum noise in an axion haloscope-like experiment with squeezed vacuum circumvents the standard
quantum limit on the noise of the measurement apparatus, enabling high sensitivity over a broader
bandwidth and a more rapid search for a weak classical tone. Squeezing moreover promises to movehaloscopes into a qualitatively distinct design parameter space: whereas a quantum limited haloscope’s scan rate plateaus with improving microwave transmission efficiencies η, a sub-quantum
limited haloscope will benefit almost arbitrarily as microwave losses are further reduced. Thus,
haloscopes can reap larger benefits from increased efficiency (for example, a tenfold scan rate enhancement at η = 0.91) as low-loss quantum technologies such as on-chip circulators and directional
amplifiers mature [149, 218, 222].
In the following two chapters, we will turn our attention to the secondary focus of this thesis:
the statistics of axion searches. We will return in Ch. 8 to the technique of this chapter, and combine
it along with that of Ch. 7 to the HAYSTAC experiment.

Chapter 6

Axion haloscope processing and analysis

For this chapter and the next, we will turn our attention from quantum to classical information processing. This chapter details the processing protocol104 and statistical analysis105
framework used by HAYSTAC and other axion haloscopes prior to the work of this thesis. In
the following chapter, I will present a Bayesian analysis framework tailored to axion dark mattersearches. We will see that this framework improves significantly upon the standard analysis framework presented in Sec. 6.2 of this chapter, and validates the standard processing protocol presented
in Sec. 6.1 as already near-optimal. Both this chapter and the next present the work of Ref. [126].

6.1

Haloscope processing overview
This section provides a quantitative overview of the steps of the processing protocol describedin much greater detail in Ref. [78].106 In Sec. 7.2 of the next chapter, we will evaluate the protocol
for how well it preserves information content pertinent to the existence of the axion. This section
104

This processing protocol was partially previewed in Sec. 5.7 for the specifics of the mock-haloscope experiment.
It did not include an accounting for correlations necessary to accurately predict the power delivered by an axion, anddid not cast the steps in terms of their mathematical operations. The description in this section is intended to be
more general, more complete, more quantitative, and directly applicable to a full haloscope experiment.
105This thesis draws a distinction between what is meant by “processing” and “analysis.” “Processing” refers to
any manipulation of a measured quantity from the haloscope in order to obtain a final pair of numbers quantifying
the measured power excess and sensitivity at each frequency. The “analysis” comes after the processing, and refers
to steps that intake that pair of numbers in order to perform a statistical inference about the presence or absence of
the axion at each frequency.
106
The HAYSTAC processing protocol in Ref. [78] grew out of that developed by ADMX [125]. In practice, active
haloscopes all use similar data processing protocols, and the overview given in this section extends beyond what isdone by HAYSTAC, though for other experiments specifics such as the use of the Savitsky-Golay filter to remove
backgrounds may vary. At the time of writing this thesis, HAYSTAC is the only haloscope to have recently publisheda full description of its processing and analysis. The ABRACADABRA experiment [111], similar in some respects toa haloscope, has recently published its processing and analysis practices [75] as well.

150
focuses on the steps that transform each individual Fourier bin on its way to becoming part of the
final, or grand, spectrum. The processing of real haloscope data also involves steps for dealing withpractical nonidealities that are out of the scope of this section. We do not address issues such as
the identification and removal of persistent but nonaxionic spectral features, or the details of thedigital high-pass filtering of the data (see Sec. 5.7 or Ref. [78]).
The data from an axion haloscope arrives as a pair (one for each quadrature)107 of discretely
sampled, identically distributed time streams of measured voltages. The time streams are broken
up into separate pieces of duration 1/∆b and discretely Fourier transformed into so-called subspectra,108 with frequency bin spacing ∆b roughly two orders of magnitude narrower than the axion
linewidth ∆a . The subspectra are acquired in batches of Nb at each of Nt distinct haloscope tuning
steps. The Fourier frequencies in each subspectrum can be mapped faithfully back to the Nf radio
frequency (RF) frequencies109 of the fluctuations within the haloscope cavity. Ultimately, each RF
frequency is associated with a single bin in each of a large number of distinct subspectra. Therefore,α, where Q = X or Y denotes the quadrature,
the complex subspectrum voltages are denoted vQ,ijk

i = 1, . . . , Nf indexes the RF frequency, j = 1, . . . , Nt the haloscope tuning step (chronologicallyordered), and k = 1, . . . , Nb the subspectrum (also chronologically ordered) at a given tuning step.
The Greek superscript (here α) will advance through the processing steps detailed below, withletters farther down the alphabet being used to represent spectra obtained later at later stages ofdata handling.
At each tuning step, sensitivity data are acquired (or borrowed from a nearby tuning) as
well. The sensitivity data, in a simplified picture, fully specify the expected noise and possible
axion signal power at each frequency bin i. At each tuning step and for each frequency, theγsensitivity parameters ηij, defined as the ratio of signal power to noise power for an axion withgγ = 1 that delivers 100% of its power at the ith RF frequency, can then be calculated.
107

It is theoretically equally efficient to perform single-quadrature measurement (Sec. 5.1.3), in which case there is
only one quadrature time stream, but none of the conclusions of this section are altered.
108
Really, the subspectra come when we compute the power spectral density, but we will refer to the voltages before
that step as the subspectrum voltages.
109
The number of RF frequencies in earlier processing steps Nf is traditionally larger than N , the number of
frequencies in the grand spectrum, as a later data processing step will average sets of spectrally adjacent bins.

151
Using the terminology of Ref. [78], the subspectrum data are processed ultimately into grand
spectrum data through the following steps:(1) The real and imaginary parts of each subspectrum quadrature voltage are summed inquadrature so as to obtain the quadrature powers xαQ,ijk within each bin:110ααxαQ,ijk = (Im[vQ,ijk])2 + (Re[vQ,ijk
])2 .

(6.1)

(2) The quadrature powers are summed to obtain the total power xαijk in each subspectrum
bin:
xαijk = xαX,ijk + xαY,ijk .

(6.2)(3) All subspectrum powers at each tuning step and frequency are averaged to obtain the raw
spectrum powers:Nxβij =b
1 X
xαijk .
Nb

(6.3)

k=1

(4) The expected noise power hxβij i, obtained from the application of one or more SavitzkyGolay (SG) filter (or more generally a low-pass digital filter) at each frequency, is divided
from each raw spectrum to obtain the processed spectra as:111xγij =xβij
hxβij i

− 1.

(6.4)

The processed spectra (and subsequent spectra) are power excesses: that is, departures ofmeasured powers from the values expected absent an axion.
(5) The processed spectra are divided by the independently obtained sensitivities to obtain therescaled spectra:xδij =
110

xγij
γ
ηij

.

(6.5)

Equation (6.1) is for each quadrature individually. In the case of a heterodyne measurement such as that
performed in Phase I of HAYSTAC [90, 106], which uses an in-phase/quadrature (IQ) mixer to move tones to theintermediate frequency (IF) band, image rejection is first performed in software in order to separate components
coming from within the analysis band from those coming from outside of it. In practice, HAYSTAC Phase I achieved
better than 20 dB rejection of the unwanted sideband [42].
111
Dividing out a low-pass filter profile realizes a high-pass filter, which is what we ultimately want in order topreserve the (narrow) signature of the axion.

152
The rescaled spectra have the property that an axion with gγ = 1 unrealistically depositing
all of its power entirely within any one bin shifts the mean power excess for that bin up
from 0 to 1. Less axion-sensitive bins are therefore drawn from distributions with largerδ )2 .
variances var(xδij ) = (σij(6) In each RF frequency bin i, the combined spectrum power excess xi is taken as the maximum likelihood (ML)-weighted sum of all rescaled spectra power excesses:
Nt
X
xδij
xi
=

2 .
(σi )2
j=1 σ δ(6.6)

ij

The variance of the ith combined spectrum bin is obtained from the variance of all contributing rescaled spectrum bins as(σi )2 = 

−1

Nt
X
j=1

1


δ
σij


2 

,

(6.7)

δ )2 = ∞ at tuning steps that do not contribute to a given frequency bin i.
where (σij(7) Sets of nc  ∆a /∆b adjacent combined spectrum power excesses are scaled by nc nr , defined
below, and the rebinned spectrum power excesses xζi are formed from their ML-weighted
sums:
xζi
(σiζ )2=nc
xnc (i−1)+l
1 X

2 ,
nc nr

l=1 σn (i−1)+l
c

(6.8)

where only every nthc rebinned power excess is calculated and nr is the number of rebinned
bins that contribute to each grand spectrum bin (Step 8). The rebinned variances are
the analogously scaled, ML-estimated variances given by their combined spectrum counterparts:
−1
nc
X
1
1


(σiζ )2 =
2  ,
2(nc nr )
l=1 σnc (i−1)+l
(6.9)

The rebinned bin frequencies are taken as the average of the contributing combined bin
frequencies. For the rebinned spectrum and the subsequent grand spectrum, i now indexes
frequency in the more sparsely populated space.

153(8) In order to construct the grand spectrum, which contains both information about thehaloscope sensitivity (Step 5) and the axion velocity distribution, the axion lineshape,
Eq. (2.14), is calculated given an axion with rest frequency νa . The axion lineshape is
the probability density function (PDF) of axion particle energies given their Maxwellian
velocity distribution in the galactic rest frame, and our planet’s velocity through that rest
frame (Sec. 2.4.3). This PDF is then discretized into a probability mass function (PMF)112p̄a (νj |νa,i ) by integration over rebinned spectrum bins.113
The grand spectrum is then constructed. The grand spectrum has the same frequency
spacing as the rebinned spectrum, but each grand spectrum bin i includes contributions
from the
nr =

∆a
nc ∆b

(6.10)

rebinned spectrum bins beginning at i. Sets of 2nr + 1 grand spectrum bins hereby becomecorrelated.114 Specifically, the grand spectrum power excesses xηi are constructed from thesets of nr adjacent rebinned power excesses asnrXp̄a (νi+l−1 |νa,i )xζi+l−1
xηi
,
=

2
(σiη )2ζ
σi+l−1l=1(6.11)

where νi+l−1 denote the rebinned frequencies. Equation (6.11) provides the ML estimateof the power delivered by an axion to the ith grand spectrum bin, which has variance givenby"#2 −1nrXp̄a (νi+l−1 |νa,i ) (σiη )2 =l=1

ζ
σi+l−1.

(6.12)(9) If the combined spectrum bins did not contain correlations from the SG filtering (Step 4),
then the division by the independently obtained sensitivities (Step 5) would ensure thatxηi /σiη is drawn from a standard normal distribution (no axion) or a normal distribution with
112

This PMF is interpretable as the probability of an individual axion particle being contained in bin j given thatthe axion rest mass-converted frequency, Eq. (2.9) lies in bin i.
113
This conversion is made difficult by the fact that the axion frequency will in general not align perfectly with any
bin frequency, an effect that matters less the smaller the rebinned bin spacing nc ∆b is. Reference [78] accounts forthis uncertainty.
114
The implications of these correlations are addressed in Sec. 7.1.4.

154
unit mean and standard deviation (axion with gγ = 1). The negative correlations imprinted
by the SG filter (Sec. 7.1.4), however, reduce the grand spectrum standard deviations σiη bya spectrally uniform factor ξ, which can be extracted directly from the data and validated
through simulation.
Scaling the grand spectrum standard deviations down toσ̃iη = ξσiη(6.13)

makes the corrected grand spectrum excesses
(1)

xi

=

xηi
σ̃iη

(6.14)

standard normal random variables absent an axion. The superscript (1) denotes that Steps
1–9 have applied for the initial scan, and must be repeated for rescans. Equation (6.14)
represents the grand spectrum excesses referred to in the main text.
The SG filter has the additional effect of attenuating the visibility of a potentially present
axion. The frequency-independent magnitude of this attenuation can be simulated as well,and is captured along with other, frequency-dependent effects, in Eq. (6.16)’s sensitivity(1)parameters ηi(1)(for now, it suffices to say that ηi(1)quantifies the mean value of hxi i

imparted by a KSVZ axion).
(10) Steps 1–9 are repeated, possibly multiple times, at frequencies where rescans are deemed
appropriate. The corrected grand spectrum excesses and sensitivity parameters are labeled
(j)

xi

(j)

and ηi , respectively.

In Sec. 7.2, we will evaluate the consequences of each data processing step on our ability tosense an axionic signal in the data.

6.2

Axion search analysis considerations
The analysis starts where the data processing of Sec. 6.1 ends: with a grand spectrum of(1)

normalized power excesses xi

measured at frequencies νi . The superscript (1) denotes data taken

155
on an initial scan. Bins displaying large excesses on an initial scan are rescanned, potentially(j)multiple times, yielding rescan spectra xi

indexed as j = 2, · · · , M , where many frequency bins iwill not be measured in rescans.
For bins not containing an axion, acquisitions of order a millionfold times longer than the
inverse bin bandwidth yield, via the central limit theorem, normalized power excesses drawn fromthe standard Gaussian PDF, obtained by setting µ = 0, σ = 1 in
1(x − µ)2
fx (x; µ, σ) = √
exp −
.
2σ 2
2πσ

(6.15)

This distribution is shaded green in Fig. 6.1. For a discussion of the consequences of practical
departures from the Gaussian idealization, see Sec. 7.1.5.

Figure 6.1: Gaussian probability density functions fx (x) for a single-bin normalized power excess x
owing to noise alone (green) and noise plus-axion signal for a particular value of the axion-photon
coupling gγ and haloscope sensitivity (blue). In threshold-based inference frameworks (Secs. 6.3,
7.3.1, and 7.3.2), an excess power threshold (black dashed line) is predetermined, and the only
information analyzed is whether the measurement is above threshold. The light green (blue) region
denotes to a true negative (positive), while the dark green (blue) denotes a false positive (negative).
In the Bayesian power-measured (BPM) analysis of Sec. 7.1, the single-scan prior update u(x) (red)is simply the ratio of the two distributions. The exponential dependence of u upon x captures theinformation content of the measurement. Figure reproduced from Ref. [126].

A bin νi that falls at the axion frequency νa will have its power excess drawn from what weterm the axion distribution. There is in fact a family of such distributions parametrized by theaxion-photon coupling gγ , but it will suffice to consider just one. The axion distribution’s standard
deviation in the limit of weak axion signal power is approximately that of the no-axion distribution

156(i.e. 1) but its mean is offset to
(j)

(j)

2
µa,i = gγ,i
ηi .
(j)

The sensitivity parameter ηi(6.16)

is the independently calibrated SNR for an axion with couplingstrength gγ = 1 in the ith bin of the j th spectrum.115 The PDF of the axion distribution for the(j)ith bin (blue in Fig. 6.1) is thus Eq. (6.15) with µ = µa,i , σ = 1.
The purpose of the statistical analysis for an axion haloscope is to convert the corrected116(j)grand spectrum excesses xi(j)and sensitivity parameters ηiinto probabilistic statements distin-

guishing between the axion and no-axion hypotheses for a given bin or frequency window. The
particle physics community does not lack for statistical methods [235], but the challenge is not incoming up with a mathematically correct framework; rather, it is in finding one which deals wellwith the particulars of axion searches.

6.2.1

Distinguishing features of axion and haloscope searches

A given analysis framework can be evaluated in large part by how well it accommodates
each of the key, defining features of axion searches. Below is a list of some of the most prominent
distinguishing traits of axion — and axion-like particle (ALP; see Ch. 9) — dark matter searches.
This list is not meant to be comprehensive, nor to perfectly describe every such dark matter search.
Each item includes reference to the section of Ch. 7 which addresses it for the statistical frameworkpresented there.
(1) There is (by assumption) at most one frequency bin containing the axion rest mass among asea of many bins not containing it. Of interest to the experimentalist is both the probability
that the axion lies in any given bin, and in any range of bins. Evaluating the single- andmulti-bin hypotheses is covered in Secs. 7.1.1 and 7.1.2, respectively.
115

(j)

The parameters ηi account for the scaling of the haloscope sensitivity with integration time, experimental
parameters, and the dark matter density ρa . By normalizing the SNR to gγ = 1 and treating the multiplicative
coupling factor gγ2 as a parameter of the axion distribution, we are following the convention in the axion search
literature of assuming a fixed value of ρa and setting limits on gγ (Sec. 2.4.2). The i index on the coupling factor
indicates that we need not consider the same axion distribution in each bin.
116
Henceforth we will drop the word “corrected” for brevity.

157(2) Close to the bin containing the axion, there may be bins containing axion-induced powerexcesses. These and other correlations are handled in Sec. 7.1.4.
(3) The axion is a persistent signal117 with lineshape that is assumed to be known, at least
approximately (Sec. 2.4.3). The persistent availability of the signal contrasts sharply withsome other prominent searches for new physical phenomena. For example, in gravitational
wave searches, a given astrophysical event [236, 237] must be observed during its one andonly occurrence: both false negatives and positives carry important negative consequences.
The persistence of the signal is leveraged in all axion haloscope searches through the incorporation of rescan data. See Item 8.
(4) It is a priori unlikely that the axion will reside within the window (mass and coupling)in parameter space to which a given scan is sensitive. It is more unlikely still that theaxion will reside within any given bin. These small a priori probabilities are dealt with inSec. 7.1.3.
(5) Since the axion could (at least in principle) reside over many mass decades (Sec. 2.4.1),
linearly spaced Fourier domain bins do not on their own reflect the unstated belief whichclearly underlies the current fleet of axion direct detection experiments: that the axion
detection community’s belief in axions is nonuniform over frequency. Spectral weighting of
the performed hypothesis tests is covered in Sec. 7.1.3. Though the HAYSTAC results ofChs. 7 and 8 are too spectrally narrow for this effect to be of much consequence, the ALP
search in Ch. 9 relies heavily upon these considerations.
(6) The hypotheses for there not being an axion and for there being an axion of any given
coupling are generally both quantifiable as probability distributions in terms of the measured
data. The quantifiability of all hypotheses of interest recommends Bayesian statistics, or atleast a reformulation of the null hypothesis that takes into account both distributions.118
117

As mentioned in Footnote 19, sufficiently narrowband axions may actually move noticeably (in the context of a
haloscope search) in frequency due to the orbit and rotation of the Earth.

158
The axion and no-axion hypotheses are directly incorporated in the formulation of Sec. 7.1.1.
(7) To the extent that the previous item is not true, it is typically primarily because of excess
noise power coupled into the detector at discrete frequencies (or ranges thereof). Practical
departures from the idealized behavior of the axion and no-axion hypotheses are coveredin Sec. 7.1.5.
Additionally, some features more specific to haloscopes and other tuned, narrowband searches(these do not apply, for example, to the ALP search in Ch. 9) in particular are:(8) Frequency bins are scanned in groups that are small compared to the total number ofbins scanned during an entire data run.119 As such, targeted, narrowband rescans can be
performed with relative ease on power excesses that appear in an initial scan. Section 7.3
showcases key advantages of the framework in Ch. 7 pertaining to how rescan data may beincorporated.
(9) For a QCD axion (assuming it does exist, and has the characteristics put forth in Ch. 2)
singled out by a fully operational haloscope, nature has designed the expected signal behavior to be almost fool proof.120 Specifically, its particular scaling with applied magnetic
field and the cavity transmission profile, together with its lineshape and persistence, makes
it hard for spurious RF excesses to masquerade as axionic. This last point creates a fundamental asymmetry between exclusion and discovery. It has important implications for how
false positives and negatives should be guarded against in the analysis. Those implications
are discussed in the context of accounting for the look-elsewhere effect, Secs. 6.5 and 7.1.2.
118

Sharon Bertsch McGrayne traces the history of the development of Bayesian statistics in Ref. [238]. A quotefrom her work is pertinent to our discussion: “Egon Pearson’s idea was that the only correct reason for rejecting astatistical hypothesis was to accept a more probable one.”
119
While broadband searches are possible [110], it has been shown that a narrowband/resonant (quality factor Q)
approach is a Q-fold more effective strategy in terms of scan rate than using a broadband receiver [95].
120
However, as Edward Teller purportedly once said, “if you think you have made something fool proof, the fool isalways bigger than the proof” [239].

159

6.3

Frequentist threshold framework
The majority of existing axion exclusion has been obtained using either of two frequentist

methods. The haloscope exclusion literature [78, 89, 91, 93, 94, 97–109, 115] often does not do agood job distinguishing which framework is being used in which publication. As will be made clear
below, the two methods mean substantively different things by “exclusion,” so this ambiguity is arather unfortunate one. The first of the two, described in this section, uses a (typically) multi-step
threshold-based test to either exclude the axion or perform more advanced tests on a persistentsignal. The purpose of this section is twofold: first, along with the next section, it aims to lay
out for readers exactly how frequentist exclusion is and has been obtained for haloscopes. Second,
the framework in this section can be compared directly to one presented in the next chapter. The
remainder of this chapter, together with the next, constitute an argument that the frequentist
threshold framework considered in this section is on balance better suited to the task of axion
haloscope detection than that in Sec. 6.4, yet worse than that in Ch. 7.

6.3.1

Framework details

In order to perform an axion search subject to the FT framework, a power excess threshold(j)

xT is set for each scan j and this threshold is used to determine firstly whether or not to proceedwith rescans, and secondly whether or not to exclude the axion hypothesis at each frequency.
The logic of this exclusion is prescribed as follows. First, the experimentalist establishes a null
hypothesis. In this case, the null hypothesis for each bin is that there is an axion at the bin’s
frequency.121 Second, the experimentalist establishes and follows a procedure for acquiring data.
This procedure may include conditional steps: e.g. “acquire more data in a given bin via a rescan
if and only if the initial scan power exceeds the threshold set for that bin.” All possible paths ofaction eventually terminate either in a rejection of the null or a failure to reject the null, forming adecision tree. Figure 6.2 shows how a realistic grand spectrum of frequency bins might be scanned
121

The axion hypothesis is chosen as the null so that it might be rejected, or excluded, later. Because of theunconventional choice to make the interesting (i.e. axion) hypothesis the null, the language of false negatives andpositives will adopt the unconventional usage where a positive (negative) refers to the null being true (false).

160

Figure 6.2: Illustrative, fictitious grand spectrum data for a standard haloscope subject to a frequentist threshold (FT) analysis allowing M = 4 scans. The initial scan data from a typicalhaloscope [105, 107] consists of a normalized power excess in each of N ∼ 105 –106 frequency bins,
represented above by the boxes in the first row. At frequencies where the excess is above a predetermined threshold (red boxes), additional scans are conducted until a measurement below thresholdis recorded. Only if all M scans exceed threshold is failure to reject the axion hypothesis reported.
If any scan comes in below threshold (black boxes), the axion hypothesis is rejected for the bin in
accordance with Eq. (6.20) and further scanning need not be conducted (empty boxes). For the FT
framework of Sec. 6.3, both the threshold and the number of scans M must be predetermined, andthe exclusion therefore may take into account scans that were not performed. Here, Scan 4 was
not performed for any bins, but still impacts the final reported exclusion. Because the haloscope
bandwidth is typically larger than the expected axion linewidth, bins adjacent to those exceeding
threshold are automatically rescanned (gray boxes). However, the FT framework discards thesedata regardless of the power measured. Conversely, the BPM method discussed in Ch. 7 is able touse the information. Figure reproduced from Ref. [126].

according to the powers measured relative to threshold. Exclusion is set for all bins i where thenull is rejected at a confidence level of
E = 1 − Fn ,(6.17)

where Fn is the bin-i false-negative rate of the entire procedure: the total probability that the nullwould have been rejected, were it true.122 We refer to this class of methods as frequentist threshold
122

This approach is equivalent to defining a test statistic((j)1
x(j) ≥ xT ∀ jts =Fn (gγ ) otherwise(6.18)

from the total false-negative rate treated as an explicit function of gγ . Couplings where ts goes below some predetermined target rate Fn,target are rejected at 1 − Fn,target confidence level via Eq. (6.17). The piecewise definition of ts ,(j)Eq. (6.18), yields exclusion only when at least one scan’s power excess x(j) comes in below its threshold xT . When
all scans exceed threshold, ts = 1 indicates the presence of (possibly axionic) excess power.

161(FT) frameworks. FT frameworks answer the question, “assuming the axion exists, what is theprobability of failing to observe it?”An FT search procedure [107] is prepared to perform as many as M scans at each frequency
νi , reporting exclusion subject to its predetermined false-negative rate. At a given νi , no additional(j)scans are performed once any scan’s measured power excess xi

fails to exceed its predetermined(j)threshold xT , the vertical, dashed line in Fig. 6.1. For the general case where each scan may use
a different false-negative rate (dark blue region of Fig. 6.1), given from integrating Eq. (6.15) by
fn(j) =

Z x(j)T−∞
fx x; µ(j)
,
1
dx,
a

(6.19)

the total false-negative rate is
Fn =

M
X
j=1

fn(j)

j−1
Y


1 − fn(k) .

(6.20)

k=1

This total false-negative rate is what sets the confidence level of the reported exclusion E as in(j)

Eq. (6.17). For the special case where all M scans have equal false-negative rates, fn= fn0 ,

Eq. (6.20) reduces to
Fn0 = 1 − (1 − fn0 )M .

(6.21)(j)(j)

In practice, the false-negative rates fn and power thresholds xT are set to convenient values, re(j)moving the frequency-dependence of µa in Eq. (6.16) by forcing gγ to compensate for the frequency(1)dependence of the sensitivity. The initial scan sensitivities ηitherefore determine the minimum

coupling gγ,i that is excluded (or not) at each frequency. Subsequent (j > 1) scans, conversely,(j)integrate to a sensitivity ηidetermined by the coupling gγ,i reached on the initial scan. The

result, in the case where the axion hypothesis is rejected for all bins where it can be tested, is an(1)

exclusion plot [107] whose spectral structure reflects that of ηi .
The procedure for collecting data and reporting exclusion in these FT frameworks must be
rigidly defined and laid out in advance. If, instead, the experimentalist is allowed to alter thedecision tree that leads to negative or positive results while navigating that decision tree, the result
becomes susceptible to bias. To this point, the total false-negative rate Fn , Eq. (6.20), directly
responsible for setting the exclusion E, grows with the total number M of scans in the predetermined

162
experimental procedure. Suppose, then, that the experimentalist was willing to perform as manyas M scans at each frequency bin i before failing to reject the axion hypothesis, but that in practicesome number m < M was all that was required to get at least one negative result in every bin.
The correct exclusion to report, perhaps unintuitively, is that which takes into account all M
allowed scans, including even the M − m unperformed scans. The number of such unperformed(j)scans and their individual false-negative rates fn , Eq. (6.19), must be known to properly report
exclusion. If not written down in advance, these numbers are difficult to estimate without inserting
bias. In practice it is extremely difficult to rigorously adhere to FT logic. For example, one would
expect that some exclusion would be reported for a total number of scans M surpassing m, the exact
number performed, yet this never appears in the literature [89,94,97–104,106–108].123 Furthermore,exclusion has been reported for a number of scans less than the number performed [78]!
Though the preceding logic and inference is frequentist in nature, an axion haloscope analysis
can alternatively be carried out using the language of Bayesian statistics. In Sec. 7.3.2, I present a
threshold-based Bayesian analysis (denoted BT2) which is equivalent to the FT framework, given
simple assumptions applicable to all dark matter axion experiments. That is, the two frameworks
share operational procedures, and they ultimately output identical conditional probabilities. This
correspondence implies that we can quantitatively interpret FT axion exclusion as decreased belief
in, or probability of, the axion hypothesis.

6.3.2

Aggregate exclusion

The aggregate exclusion E(gγ ) will serve as our means of applying the logic of the FT framework to an entire scanned window (or portion thereof). We will see that E(gγ ) provides a moreconservative, yet more faithful assessment of the exclusion achieved by a haloscope than does thestandard practice of averaging the couplings at which a given exclusion E is achieved.
The aggregate exclusion E(gγ ) answers the same question that the exclusion E(gγ ) (written
123
The references cited here are those which either definitely use the FT framework or might plausibly do so. A
few publications which are not clear on framework usage have been determined to be using the framework of Sec. 6.4
with specific modifications discussed therein via private correspondences with the authors.

163
explicitly as a function of coupling gγ ) does, but for the entire frequency range. The exclusion
E(gγ ) answers the question: “supposing an axion of coupling gγ occupies a given bin, what is theprobability of having observed a predefined positive result (i.e. detection of the axion) there?” We
therefore define the aggregate exclusion E(gγ ) as the chance of having observed a predefined, global
positive result (since it is not known which bin the axion is in, the global positive result is defined
to mean at least one positive result across the N frequency bins.), assuming the axion rest mass
to occupy exactly one bin within the range. In keeping with the uniform-priors assumption ofSec. 7.1.3, the likelihood of an axion to be at any given bin is equal for all bins, and for simplicity
the powers in bins are assumed otherwise uncorrelated (see Sec. 7.1.4). In Sec. 6.3.1, the known(j)sensitivities ηi(j)are thought of as combining with predetermined power thresholds xT to preselectcouplings gγ,i to test at a uniform, desired false-negative rate Fn . This logic can be partially(j)inverted: assuming the same ηi(j)and xT , now use a frequency-independent coupling gγ to preselectfrequency-dependent total false-negative rates Fn,i (gγ ) at which to test. This framing allows forthe testing of a uniform coupling gγ across the entire range to a confidence level ofN

1 X
E(gγ ) ≈[1 − Fn,i (gγ )] ,N

(6.22)

i=1

where the approximation is valid in the low total false-positive rate limit of Sec. 7.3.2, Fp  1. The
confidence to test at will increase monotonically with the coupling gγ being tested. The coupling(j)coincident with a desired percentile (e.g. 90%) can be determined from the sensitivity profile ηi(j)alone, and a standard confidence test performed by comparing the measured power excesses xi(j)against the predetermined thresholds xT , with a positive result defined as a click on all M scansin at least one bin.124
The aggregate exclusion, by answering the question of threshold-based frequentist confidence
testing, proves a more natural figure of merit for FT exclusion over a given scan window than
the standard practice [107] of quoting the average coupling gγmean excluded to a given confidence
124

The aggregate exclusion E(gγ ) bears exactly the same relationship to the aggregate prior update U(gγ ) of
Eq. (7.7) that E(gγ ) bears to U (gγ ) of Eq. (7.6), making it central to comparing the Bayesian framework of Ch. 7 tothe frequentist framework of this section.

164
E. If the coupling gγagg is defined as that for which identical aggregate exclusion E(gγagg ) = E isachieved, then a comparison will reliably reveal gγagg > gγmean . The standard metric, gγmean , is not
somehow achieving deeper sensitivity, it is averaging in a space inconvenient to the natural logic
of frequentism, and biased relative to it. More directly, an experiment that achieves 90% exclusion
at a frequency-averaged coupling of gγKSVZ has not rejected the hypothesis that a KSVZ axion lies
within its scan window with 90% confidence.

6.4

p-value framework
The second commonly used framework for excluding axions is a standard frequentist methodin particle physics [235] that relies on p-values, which I will argue is ill-suited to the particularsof axion direct detection. To distinguish it from the frequentist framework of Sec. 6.3, I will refer
to it simply as the “p-value framework.” The p-value, or probability value, answers the question:“what is the probability that an event at least as extreme as the one observed would have occurred,subject to the null hypothesis?” The event in question is the measurement of a grand spectrum(j)power xi(or, more generally, a ML-weighted average of such powers over multiple scans). For thep-value method in its simplest form, exclusion at confidence level Ep is reported for the axion gγ
which satisfies(1) (1)Fx xi ; µa,i , 1 = 1 − Ep ,(6.23)whereZ x
Fx (x; µ, σ) =
fx x0 ; µ, σ dx0(6.24)−∞

is the cumulative distribution function (CDF) obtained from integrating Eq. (6.15), here evaluated(j)at the measured power excess xi

(1)

(1)

with µ = µa,i and σ = 1. The coupling gγ enters through µa,i

via Eq. (6.16).
The coupling excluded at each frequency is therefore the one for which the measured power
or a more extreme (that is, lower, for the inverted use of the axion hypothesis as the null hypothesis
for frequentist exclusion; Footnote 121) power would have been measured only 1 − Ep of the time,

165
were the axion really at that coupling. Intuitively, this sounds fine. However, p-values are famously
misleading quantities. The American Statistical Association in 2016 issued a statement cautioning
strongly against the widespread reliance on p-values found throughout the sciences [240]. They
are particularly dangerous when many independent hypotheses are being tested, giving rise to thephenomenon of p-hacking. If not used with care in the context of axion direct detection, we will
see in Sec. 6.5.1 that this danger applies here as well, in a modified form.
To be sure, there is nothing logically incoherent about a p-value. It is simply a mathematicalobject; any error is always on the user end. Let us consider two simple pitfalls associated with theuse of p-values in the axion exclusion context. We will consider a third pitfall associated with thetesting of many independent hypotheses in the next section.
First, a p-value essentially treats the measured power as itself-plus-a-probability-weightedsum-of-all-more-extreme-values, for the purpose of excluding. While there is nothing logicallywrong with this, an elementary, intuitive understanding of (say) 90% frequentist exclusion might
be that an axion at the excluded coupling would have been identified 90% of the time, were it
there. This is precisely the quantitative meaning of the FT exclusion of Sec. 6.3. However, bygrouping the measured power only with that subset of unmeasured powers which are more extreme(i.e. more favorable to more aggressive exclusion) than it, the p-value exclusion reliably rules out
axions which would not have set off even a single rescan with anything close to 90% probability,
were they present.
This pitfall is greatly exacerbated by a second feature of the p-value method: that it still
often uses a threshold to test for discovery, and that that threshold takes into account the lookelsewhere effect only with respect to discovery, yet not exclusion (Sec. 6.5). To give an example,(1)

a single-scan threshold may be set at 3.5σ (equivalently, xT = 3.5). A typical bin will come inaround 0σ, causing the axion at 1.28σ to be excluded at Ep = 90% confidence level via Eq. (6.23).
The probability that such an axion (mean at 1.28σ) would have actually set off the detector foreven a single rescan is, from Eq. (6.19), 1.3%. This can be made worse if, as is common, rescans are
performed. That is to say, when the FT framework reports 90% exclusion at a given gγ , it means

166
that an axion would have been identified 90% of the time. Indeed, we saw in Sec. 6.3.1 that theFT framework takes care to ensure that the exclusion is the complement of the false-negative rate
for an arbitrary M -scan protocol. But when the p-value framework reports 90% exclusion for anaxion a given gγ , the most typical result is that such an axion would have been identified 1.3% of
the time for the initial scan, and subsequent scans may add additional opportunities to miss thesignal.125 What would have in fact typically happened if an axion existed at 1.28σ is that an axion
at a typical value of 2.56σ, still well short of 3.5σ, would have been excluded. In a context whereof order 100,000 bins might be tested in a single scan, and there are not nearly that many pixels
displayed in a typical exclusion plot, this result could very well do nothing to modify even a single
pixel of the exclusion as displayed. In spite of all of this, there is nothing logically wrong with the
p-value framework: it is simply that most readers are not conditioned to look at an exclusion line
and understand that it might imply a typical false-negative rate of nearly 99%.
The second pitfall of the p-value framework is that it tends to exclude nonsensical couplings
if left unaltered. Simply put, any measured power below -1.28σ will require excluding an axion
with mean below 0σ at 90% confidence level. Of course, there is no axion with subzero mean, so
all axions are excluded. This is true independent of sensitivity. So in principle, an axion search
could be carried out not in the typical ∼ 8 T fields of modern haloscopes, but in the Earth’s field of∼ 50 µT, and 10% of the (say) 100,000 frequency bins would be excluded at all couplings down togγ = 0. Again, there is nothing logically wrong here. The 10% of bins that achieve this nonsensicalresult of perfect exclusion are what the quoted confidence level of 90% warns about, but the result
is highly inconvenient. One remedy is to artificially modify the axion distributions in a manner suchthat these results never happen, but it seems a strange price to pay when alternative frameworks
exist which allow for the use of the actual axion PDF.
The third pitfall, which we will see in the next section, is that the exclusion reported over the
125This story is complicated by the fact that adjacent bins tend to be scanned during rescans (Fig. 6.2). The
possibility that a bin with an axion might have been missed on the first scan, but showed up on the rescan ofa neighboring bin which did exceed threshold, can partially mitigate the extremely high effective false-negative
rate. Both the exacerbating and mitigating effects of rescans depends on the particulars of the rescan criteria andparameters, as well as the strength of correlations in nearby bins.

167
entire frequency range for any properly functioning framework ought to get reliably worse as more
independent frequencies are probed. The p-value method or close variants of it [75] as typically
implemented in axion searches [91, 93, 105, 109, 115] appear to not fully account for this effect,
whereas the other frameworks discussed in this thesis all do. For all of these reasons, the p-value
method should only be used with expert handling, or better yet not at all, given the availability ofmore intuitive alternatives (Secs. 6.3, 7.3.1. 7.3.2, and especially 7.1).

6.5

Look-elsewhere effect: implications for detection and exclusion
The look-elsewhere effect is the name for the phenomenon whereby rare events become com-monplace when many independent hypothesis tests are performed. It must commonly be dealt within particle physics searches [235] which entail testing many independent parameter values simultaneously. Typically, accounting for the look-elsewhere effect means shifting from a notion of how
locally unlikely an event is to how globally unlikely it is. Here, local refers to what occurs in any
given bin, and global refers to what happens in all the bins taken as a collective.126 For example, an
event that has a 0.01% chance of occurring locally, for example, will occur globally roughly 1% of
the time that 100 independent tests are performed.127 In a rare event search, therefore, a 5σ global
result is a typical scientific standard for discovery. However, in an axion haloscope context, accounting for the look-elsewhere effect is the conventional sense — that is, with respect to discovery(a modifier that will help us is distinguishing from its implications for exclusion, discussed below)— is somewhat trivial. In principle, a total unawareness that rare events occur more frequently overmany trials could lead a naı̈ve experimentalist to set a much-too-low rescan threshold and therefore
have to perform an over-abundance of rescans; in practice, that mistake does not appear to be
being made in narrowband, tuned experiments. Conversely, in broadband experiments without
targeted rescan capabilities, it is prudent to explicitly account for the look-elsewhere effect when126

The quantities in this thesis defined with “aggregate” in the name, Eqs. (6.22) and (7.7), answer questions about
global events.
127
A global occurrence is here taken to mean one or more local occurrence over a set of tests. This definition isharmonious with the hypothesis that an axion may lie in at most one bin, Item 1 in Sec. 6.2.1.

168
reporting discovery [75, 241]. Even in such contexts, however, the availability of simple and robust
manual tests can render the notion of reporting discovery based on a limited number of identicalscans overly simplistic. In searches for hidden photon dark matter [242] where the magnetic fieldplays no role [243, 244], accounting for the look-elsewhere effect with respect to discovery is also
important, and doubly so if those searches are also broadband.128 This might lead us to believethat accounting for the look-elsewhere effect is unimportant for axion haloscope searches. In fact,
the opposite is true.
In the context of an axion search, haloscopic or otherwise, the look-elsewhere effect worsens
the exclusion that is achievable across a scanned frequency window. The coupling excluded will
move upwards129 as more independent tests are conducted, reflective of the increasing difficulty inruling out a special frequency when it has more imposters to hide among.130 This effect is not fully
accounted for in the exclusion of some axion searches [75, 91, 105, 111]. If accounted for, it would
have the effect of scaling back existing exclusion results.131
We can be quantitative about the role the look-elsewhere effect plays in slowing down a scan.
Scan rate is traditionally calculated [42] as in Eq. (2.16), without the SNR2 → SNR02 correction to
the denominator of the first term. As promised in Sec. 2.5.2, we will here estimate that correction
term. This is not a rigorous derivation, but just a quantitative means of seeing what the lookelsewhere effect very roughly does to scan rate.
Without any involved calculations of the sort performed in Sec. 7.3.3, we can anticipate thata very rough criterion to be sensitive to an axion of some coupling gγ across an entire scan is that128

Another, less common example, but one relevant to this thesis (Sec. 7.4), is a haloscope that takes some dataand promptly stops working, because of, for example, a magnet quench [42, 90].
129
This effect can be observed in the numerical simulations of Sec. 7.3.3.
130This fact is analogous to flipping N coins M times apiece and looking to rule out the existence of one special,
anonymous coin with heads probability ph > 0.5 hidden among the whole set. The more coins you flip, the weaker
the overall ph you should be able to rule out. It is a question of semantics whether to call this the look-elsewhere
effect, which could be thought of as only applying to discovery by construction. This thesis takes the perspective that
because the effects on discovery and exclusion both originate directly from the tendency of rare events to occur morewhen there are multiple trials, it is appropriate to view them both as manifestations of the look-elsewhere effect.
131
More accurately, there is nothing wrong with the existing exclusion results from a purely logical perspective.
Rather, it is that, overlooked for sufficiently many independent tests, the logic itself ceases to imply that (frequentist)one would have detected the axion, were it in the scan window, or that (Bayesian) the probability of its being there
actually went down in light of the data. Subject to any reasonable definition of “exclusion,” this functionally meansthat the reported exclusion should be scaled back (i.e. upwards in coupling).

169
the expected number of background fluctuations hNB i that look like that axion obeys
hNB i  1.

(6.25)

This is because there can (by assumption) be at most one axion, and in any realistic scan of NI  1
independent bins,132 NB will be Poisson-distributed, with standard deviation equal to the squareroot of its meanσNB =phNB i.

(6.26)

Therefore, once hNB i gets close to unity, so does σNB , and it becomes impossible to reliably tell awhether a single additional axion-like excess is due to an axion, or due to the fluctuations expectedfrom the noise statistics.
To a rough approximation, what it means for a fluctuation to be axion-like (i.e. to counttowards NB ) is for it to achieve a grand spectrum power excess x of at least some significant
fraction of the SNR of the axion distribution.133 To put a number on it for the purposes of thiscalculation, we will say that a fluctuation in axion-like if it has134x ≥ SNR.

(6.27)

hNB i = NI [1 − Fx (SNR; 0, 1)],

(6.28)

Therefore,

where Fx (SNR; 0, 1) is the no-axion CDF, Eq. (6.24), here evaluated at the SNR per our criterion,
Eq. (6.27). To proceed further, we must understand the role of the SNR as it appears in the equation
for scan rate, Eq. (2.16).
When Eq. (2.16) is written in its uncorrected form (with SNR and not SNR0 ), a high SNR(e.g. of around 5, perhaps inspired by the common 5σ standard for discovery) is usually assumed.
However, as discussed above, in haloscope searches it is not necessary to build one’s fortifications
132
We will here think of the number of bins as some effective number NI of independent bins, to keep the mathsimple.
(j)133
in Sec. 6.3, we labeled this scan- and bin-dependent quantity µi , here we simply recognize it as the SNR, since
we are not working at a level of detail where we care about scan- or bin-dependencies.
134
One could argue for the use of a multiplicative constant in this equation, or perhaps even for a different criterion
entirely. The point here is largely just to show in what form NI enters the scan rate.

170
against accidental discovery into the statistical framework alone, when there are robust tests withvanishingly low error rates that can be performed on persistent signals. Rather, the high values of
the SNR usually chosen are, consciously or not, set to avoid paying the overhead of performing thesetime-expensive tests in a context where many independent tests are performed. In other words, thelook-elsewhere effect with respect to exclusion is already to zeroth order built into most estimatesof the scan rate by choosing a value for the SNR that only makes sense in a global context. But
either for a region of interest much smaller or (more practically, when considering the ambitions of
the field as a whole) larger than a typical scan, we can anticipate how the SNR term should change
to accommodate the different number of bins, while obeying Eq. (6.25).
Subject to the logic of the previous paragraph, a cleaner interpretation of SNR and SNR0 is
to temporarily think of SNR as the mean axion power one would desire if one were just scanning(unrealistically) a single bin. The desired value will depend on the overhead cost of performing
time-expensive tests off cavity resonance or with the magnet ramped down, or, if in a situation(perhaps an ALP search) where such tests are less fool proof, the (much scarier) risk of accidentally
reporting a false discovery. In any case, if we now go from 1 to NI independent bins, we may
ask how much higher the local SNR, which we now relabel SNR0 , must get in order to keep hNB iconstant: that is, how far we have to back off in coupling to maintain our global SNR given a fixedscan time.
To compare the two cases ({1 bin, SNR} versus {NI bins, SNR0 }), where
SNR0 = β(NI ) SNR(6.29)

is parametrized by a quantity β(NI ) ≥ 1 to be determined, we write out Eq. (6.28) for each:


1
SNR
hNB icase1 = erfc √
2
2


NI
β SNR
√
,
hNB icase2 =
erfc
2
2

(6.30)(6.31)

where erfc is the complimentary error function. Equating Eqs. (6.30) and (6.31) to preserve the171
expected number of background, axion-like fluctuations, we find
√




SNR
2
1
−1
β(NI ) =erfc √
,
erfc
SNR
NI
2
where erfc−1 is the inverse complimentary error function. According to Eq. (6.29),



√
1
SNR
0
−1
SNR ∼ 2 erfc
erfc √NI
2q≈ SNR2 + 2 log NI ,(6.32)(6.33)

where the last approximation, equivalent to Eq. (2.17), is seen in Fig. 6.3 to be valid at low NI andhigh SNR. The ∼ sign is a reminder that this is only a rough calculation of the effect of NI . When
NI → 1, SNR0 → SNR, as expected.

Figure 6.3: Effective correction to the required SNR from the look-elsewhere effect. Curves fordifferent global values of SNR showing how the local value SNR0 must increase to accommodate
higher numbers NI of independent bins and maintain sensitivity to a fixed axion. The solid curves
correspond to the exact form (first line) of Eq. (6.33) and the dashed curves to the approximate
form (second line).

The interpretation of Eq. (6.33) is that going to a higher number of independent frequency
bins NI entails searching for more strongly coupled (equivalently, higher local SNR0 ) axions. Figure

172
6.3 shows the effective look-elsewhere correction to the local SNR0 for several different global valuesof SNR. For example, for a scan like Phase I, Run I of HAYSTAC [78, 106] containing of order
105 independent bins and integrating to a local SNR0 of ≈ 5,135 we first determine that the global
SNR = 2 curve intersects SNR0 = 5 at NI ≈ 105 . The same curve passes through 106 around
SNR0 ≈ 5.5. Therefore, for a HAYSTAC Phase I, Run I-like scan to exclude the same couplings
over a scan of 10× as many bins, it would take an additional (5.5/5)2 − 1 ≈ 20% longer according
to Eq. (2.16) than naively expected from the tenfold increase in number of bins alone. This is asomewhat reassuring result: it means that for most scans of practical sizes, scaling up even by anorder of magnitude does not incur much further penalty. However, a ∼ 20% effect is not so small
as to be totally negligible. And, as we saw in Sec. 6.4 and will discuss further momentarily, whenframeworks do not account for the look-elsewhere effect with respect to exclusion at all, the effectis considerably more dramatic.

6.5.1

Frequentist frameworks and the look-elsewhere effectNow we can ask, do the frequentist frameworks discussed in this chapter account for the
look-elsewhere effect? The FT framework does, albeit in a somewhat clunky way. In order to avoid(j)

an over-abundance of false positives, the threshold xT must be moved up as more and more bins
are tested. Since the same axion that is tested for rescanning (in the case of a multi-scan protocol)
is also tested for exclusion, this necessarily implies testing more strongly coupled axions at fixed
confidence level. Hence, while the choice of exactly what (local) false-positive rate to set is left
somewhat subjectively up to the experimentalist, that rate must eventually get lower as more andmore bins are tested. The result is a worse exclusion line and worse aggregate exclusion, Eq. (6.22),
as required by the look-elsewhere effect.
The p-value framework in its simplest form (Sec. 6.4) accounts for the look-elsewhere effectonly with respect to discovery. The threshold used to test for rescans (in a multi-scan protocol)
has nothing directly to do with what coupling is excluded, as discussed in Sec. 6.4. While this135

The actual value used was 5.1.

173
threshold moves up for the same reason that the threshold in the FT framework does as morebins are scanned, the couplings excluded do not move up with it. The basic version of the p-value
framework therefore does not account for look-elsewhere with respect to exclusion, which explains
why the reported confidence level largely loses its intuitive meaning.136
The look-elsewhere effect can be accounted for in the p-value method numerically. Over agiven subset of Ns frequency bins, the coupling that would have produced the highest measured
power or more extreme 90% of the time (assuming all the other bins had no axion, and still generate
powers from the no-axion distribution), can be excluded over the subset of bins at 90% confidence.
If that subset is made identical with the entire set of bins for the data run,137 the resulting exclusion
properly accounts for the look-elsewhere effect with respect to exclusion: that is, it typically gets
worse as more bins are probed. In practice, this does not appear to be being done for exclusion
using the p-value framework. Though in at least some cases [91, 105] it is being done in some form
for subsets of bins at the pixel scale (i.e. over however many bins fit into a single frequency-axis
pixel in the exclusion plot as displayed in a publication). There, the look-elsewhere effect is being
accounted for properly at that spectral scale, but not beyond it. To the degree that the exclusion
plot consists of more than one independent pixel, the reported exclusion becomes progressively
more untethered from the notion that an axion, if present, would have been observed. In short, thephysics of the early universe did not know about the size of a printed page on which results about
it would one day be displayed.
In the next chapter, we will discuss a Bayesian analysis framework that is well suited todeal with all of the distinguishing features of axion searches listed in Sec. 6.2.1, including the lookelsewhere effect with respect to both discovery and exclusion.

136

This is closely related, but not identical, to the first p-value pitfall discussed in Sec. 6.4. In particular, the notionthat the exclusion should be getting worse is an alternate framing of the exacerbation of the first pitfall whereby, if it
doesn’t, the reported exclusion loses its intuitive meaning. However, the p-value’s effective grouping of the measured
power with only the subset of unmeasured powers which are more extreme than it is a distinct and separable facetof the problem. Both facets conspire to produce the particularly egregious 98.7% effective false-negative rate in theexample in Sec. 6.4.
137
In fact, there is no reason other than convenience to box in just the data run in this fashion. The look-elsewhere
effect with respect to exclusion should properly be accounted for between different data runs and experiments.

Chapter 7

A maximally informative axion dark matter analysis

In Ch. 5, we saw how quantum measurement may be used to speed up axion searches, albeit ata tangible hardware and complexity cost to the experiments. While the ultimate means of bringing
total haloscope scan times down from the formidable (millennia-plus) timescales of Sec. 2.5.3 will
have to involve hardware advances, the lowest hanging fruit for improving axion detection lies
within the data processing and analysis. Subject to the distinguishing features of axion searches(Sec. 6.2.1), a statistical framework can be more or less efficient with the information content of themeasurement, and allow for more or less operational freedom on the part of the experimentalist.
This chapter, together with the preceding one, presents the work of Ref. [126]. In it, I argue thatan optimization of the statistical analyses that haloscopes have historically used is readily available.
The standard threshold-based confidence tests used in the frequentist threshold (FT) analysis ofSec. 6.3 transform continuous measurements of power into binary outcomes, discarding knowledgeof what power was measured, and, with it, sensitivity to the axion. Because haloscopes are inpractice statistics-limited experiments, for which additional data continues to enhance sensitivity,
a more informative analysis will translate into tangible time savings during operation.
In Sec. 7.1, I will introduce the Bayesian power-measured (BPM) framework, a statisticalanalysis framework for axion searches that makes intuitive use of all of the information at theoutput of the data processing. We will see in Sec. 7.2 that the processing itself is nearly informationoptimal. Other sections exist to show how BPM accommodates the unique features of axion searches
put forth in Sec. 6.2.1 and to provide useful insight into BPM through comparison with competing

175
frameworks. Section 7.3 shows that BPM permits operational freedoms not available in its strongest
competitors. These freedoms can be leveraged in future work to determine more optimal scanstrategy [95], the other hardware-free advance which to date still has not been rigorously explored.
The BPM framework is put to the test in Sec. 7.4 by reanalyzing the HAYSTAC Phase I dataset[107]. The framework provides improved constraints on the axion-photon coupling gγ while also
identifying the most promising regions of parameter space within the 23.15–24.0 µeV mass range.
A comparison with the standard threshold analysis suggests a 36% improvement in scan rate fromthis analysis. In addition, it spotlights within the dataset the group of frequencies “least unlikely”
to contain the axion. The most prominent among these, though still unlikely to contain an axion,
nonetheless stands out relative to any other frequency within the scan range.

7.1

Bayesian power-measured framework
This section introduces the BPM analysis framework, which simplifies the operational con-straints placed on experimentalists while making use of all of the information content of coherent
axion detection data. This framework builds on the existing HAYSTAC processing procedure [78]and may be readily adapted to other experiments.
Adopting a Bayesian perspective, we can ask whether a more informative analysis than the
FT framework of Sec. 6.3 is possible, without the downsides of p-values (Sec. 6.4). We consider
frameworks which operate by applying Bayes’ theorem,
P (Y |Z) =P (Z|Y )P (Y )
,
P (Z)

(7.1)

to an axion search data set. Notationally, P (B) denotes the probability of event B being true,
while P (B|C) denotes the same, conditional upon event C being true. In Eq. (7.1), P (Y ) is theBayesian prior probability of event Y being true, and it is updated by the occurrence of event Z
to the posterior probability P (Y |Z). Section 7.3 describes two distinct Bayesian threshold (BT)frameworks in which Z is taken to be a binary outcome, or set thereof. We will see there that amore informative analysis is indeed possible, and that thresholding always imposes some operational

176
restrictions and needlessly discards valuable information. Nonetheless, the BT logic exemplifies
how Bayes’ theorem may be applied to axion detection. In doing so, it builds an important bridge
between the FT and BPM frameworks.

7.1.1

Updated belief

Bayes’ theorem, Eq. (7.1), may be applied to haloscope data so as to preserve the full information content of the measurement. The information at each frequency may then be aggregated(Sec. 7.1.2) into a statement about the change in probability of an axion of arbitrary couplingstrength gγ,i existing anywhere within the haloscope’s scan range.138 In the common case where
the data indicates the absence of an axion, the Bayesian language of updated belief maps onto thelanguage of frequentist exclusion (Ref. [235] and Secs. 7.3.2 and 6.3.2).
The BPM framework is readily motivated by the fact that both the FT (Sec. 6.3) and BT(Sec. 7.3) frameworks discard valuable information. Each power excess, measured on a continuum,
is reduced to a simple “click” or “no-click” response of an effective binary detector. A measurementthat comes in just below threshold on a higher-order rescan after already exceeding threshold on
previous scans is orders of magnitude more likely to indicate the presence of an axion than is atypical, 0σ power excess. Yet the threshold frameworks treat these events identically.
The BPM framework uses Bayes’ theorem to account for the precise effect of any initial(1)

measured power xi(0)on the prior probability Pa,i that an axion resides in bin i. After measurement,

the prior is updated to the posterior,
(1)

(1)

(0)

Pa,i = ui Pa,i ,(1)where ui

(7.2)
(j−1)

denotes the first scan’s prior update. If subsequent scans j > 1 are performed, Pa,iwill be further updated to
(j)

(j)

(j−1)

Pa,i = ui Pa,i

,

(7.3)

and so forth. In the appropriate limit of infinitesimal priors (Sec. 7.1.3), the single-scan prior update138

In the language of Bayesian statistics, a change in probability is synonymous with a change in “belief.”

177
for bin i and scan j follows from Eq. (7.1), independent of one’s choice of prior:
 




(j)
(j) 2
P xi |Aiµa,i(j)(j) (j)  = exp 
ui ≈ 
+ µa,i xi  .
−(j)2
P xi |Ni(7.4)

The second equality uses the Gaussian probability densities of Eq. (6.15), and Ai or Ni denotes
the event that an axion does or does not reside in bin i, respectively. The approximation in
Eq. (7.4) identifies the prior update with the Bayes factor, which compares the likelihood of two
hypotheses; it applies when the axion and other sources of excess power are sufficiently unlikely.139
The assumption underlying the validity of this approximation is made quantitative in Sec. 7.1.3, andin Sec. 7.1.5 we show that this assumption is conservative with respect to axion exclusion. The BPM
analysis can therefore alternatively be viewed as a Bayes factor analysis, but the interpretation of
the Bayes factor as a conservative approximation to the prior update makes for a more intuitiveand immediately useful end result. Equation (7.4) also indicates that the update to the probabilityof an axion’s existence, plotted in red in Fig. 6.1, scales exponentially with the measured power(j)excess xi . Rather than assume a particular axion distribution, we now consider the prior update
(j)

ui

(j)

(j)

(j)

as a function of µa,i . Formally, ui(j)where a positive excess xi(j)is maximized at µa,i = xi . Thus at each frequency> 0 is measured, we can single out a maximum likelihood (ML) axion
(j)

(j)

distribution parametrized by µa,i = xi .140 The update to the prior probability in the ML axion(j)scales sharply with µa,i ∝ gγ2 as"(j)ui,ML = exp(j)Since the updates ui(j)(µa,i )2

#

2

.

(7.5)
(j)

can be mapped back to their measured power excesses xi

and sensitivities(j)ηi , this use of Bayes’ theorem preserves the information content of each individual measurement.
(j)

Applying Eq. (7.4) to all bins xi

across all scans yields a spectrum of total updates Ui ,defined as
(M )

Ui =
139

Pa,i i

(0)
Pa,i

=

Mi
Y

(j)

ui ,

(7.6)

j=1

For a discussion of how Bayes factors relate to some other measures of evidence against a probabilistic hypothesis,
see Ref. [245]. For discussion of the properties of related quantities in rejecting hypotheses, see Refs. [246–248].
(j)140
Conversely, measurement of a negative excess xi < 0 reduces the probability of any axion relative to theno-axion hypothesis.

178
where Mi is the total number of scans performed on bin i. Unlike in the FT framework of Sec. 6.3,
scans that are not performed do not enter into the equation, and the number of scans need not
be specified in advance. Subject to its equivalence to the Bayes factor (Secs. 7.1.3 and 7.1.5), eachtotal update Ui is the change in probability that an axion of a specified coupling gγ resides in thebin. For example, Ui = 0.1 implies that it is 10% as probable that an axion resides in the bin after
all Mi scans were performed than it was prior to the experiment. A total prior update of Ui = 0.1
carries exactly the same meaning as 90% exclusion (E = 0.9) in the standard analysis framework,in the limit of vanishing false-positive rates (Sec. 7.3.2). Therefore the equivalent 90% exclusion
region consists of the (frequency-dependent) couplings above those where Ui = 0.1.141

7.1.2

Aggregation and the look-elsewhere effect(j)

The individual Pa,i , or Ui , regarded as functions of gγ , can be spectrally combined into an
aggregate prior update142 function:
PN
(Mi )
N
Pa0 (gγ )1 X
i=1 Pa,i (gγ )U(gγ ) =≈= P
Ui (gγ ),(0)N
Pa (gγ )NP (gγ )
i=1

a,i

(7.7)

i=1

where Pa0 (gγ ) and Pa (gγ ) are the aggregate posterior and prior probabilities, respectively, of thataxion existing. The aggregate update U(gγ ) does for the entire haloscope run what the total update
Ui (gγ ) does for each bin: it expresses the update to the probability that an axion of coupling gγ
resides anywhere within the scanned frequency window. The final equality of Eq. (7.7) holds in thelimit of approximately uniform priors,(0)Pa,i ≈
141

Pa
,
N

(7.8)

As discussed at length in Sec. 6.5, care should be taken in interpreting this region on an exclusion plot. A region
which accounts for the look-elsewhere effect with respect to exclusion can be generated above where the aggregate
prior update U of Sec. 7.1.2 is 0.1. However, even this region should, by the logic of the previous chapter, be aggregatedwith other such regions in order to properly gauge what parameter space has been excluded.
(j)142
In this text, we refer to four kinds of prior update. The single-scan updates ui of Eq. (7.4) are multiplicatively
combined into the total updates of Eq. (7.6) at each frequency. The total updates are subsequently aggregated via
Eq. (7.7) into the aggregate update. The subaggregated updates of Fig. 7.3a are likewise aggregated, but over 1% of the
HAYSTAC search window apiece. All updates are functions of axion-photon coupling gγ , and are written explicitly
as such where relevant. The quantities as calculated are Bayes factors, identifiable as updates subject to Secs. 7.1.3
and 7.1.5.

179
applicable when the total frequency scan range is small compared to the lowest scanned frequency,
νN − ν1  ν1 , as is the case for HAYSTAC or ADMX. For low-frequency experiments such as
ABRACADABRA [110] or DM Radio [244], or the axion-like particle (ALP) search of Ch. 9, logarithmically uniform priors, Eq. (7.14), should be used [95].
(j)Regardless of the measured excess powers xi(j)and sensitivities ηi , there will exist a couplinggγ,low below which virtually nothing is learned about the presence or absence of an axion,
U(gγ ≤ gγ,low ) ≈ 1.

(7.9)

Likewise, there will exist another coupling gγ,high above which the probability of an axion existing
vanishes
U(gγ ≥ gγ,high ) ≈ 0.

(7.10)

Between these two extremes, U(gγ ) will depend strongly on the measured power spectra. In thecontext of a realistic search, for which the experimental overhead is high, U > 1 suggests that atleast some bins within the range should be rescanned until either a cause of excess power is foundor U regresses to a lower value.
The aggregate prior update has several features that account for the look-elsewhere effect —the linear growth in expected number of locally significant excesses with the number of independent
hypotheses tested discussed in Sec. 6.5 (for dependent hypotheses, see Sec. 7.1.4). Because thedenominator of Eq. (7.7) also grows linearly with tests performed, these two linear factors cancel
out. Secondly, the typical coupling excluded will move upwards (Sec. 7.3.3) as more independent
tests are conducted, reflective of the increasing difficulty in ruling out a special frequency when it has
more imposters to hide among (Footnote 130). Finally, the ability to incorporate nonuniform priors
is a key feature for experiments sensitive over fractionally large spectral windows. Since 90% of all
independent axion tests in such an experiment will occur in the highest decade (assuming equal bin
widths), a standard frequentist trials factor approach [235] would set an artificially higher bar for
discovery in lower decades than it would have if the highest decade were not scanned; logarithmicpriors scale the height of that bar with the number of tests performed.

180
7.1.3

Choice of priors(0)

For any choice of prior belief Pa,i in axionic dark matter, Bayes’ theorem can be appliedand updates Ui calculated and reported. However, the utility of the BPM framework rests on thefact that its prior updates Ui can be treated as independent of the priors chosen for the axion andcompeting hypotheses, i.e. as reasonably approximated by Bayes factors. Additionally, as argued
in Sec. 7.3.2, it is highly convenient that in the appropriate limit the prior updates of the BT2framework, itself directly comparable to the BPM framework, become the conditional probabilities
quoted in frequentist exclusion. These conveniences hold only so long as one’s priors are infinitesimal
but nonzero.143 In this section, we will justify the assumption of infinitesimal priors, and consider atoy estimate of a prior for illustrative purposes. Additionally, we will discuss how the prior update
can be used to directly inform scan strategy in the presence of nonaxionic radio frequency (RF)power excesses.
The constraint that the prior be infinitesimal is only used in simplifying the denominator ofBayes’ theorem, Eq. (7.1), according to the law of total probability: Z ∞ 



(j)
(j)
(j)
P xi
=
P xi |Ai (gγ ) P (Ai (gγ )) dgγ + P xi |Ni P (Ni )0≈P(7.11)
(j)xi |Ni,

where the axion hypothesis Ai (gγ ) is here written explicitly as a family of hypotheses parametrizedby gγ > 0. The case of gγ = 0 is indistinguishable from the no-axion hypothesis Ni from theperspective of an axion haloscope.144 The approximation in Eq. (7.11) is justified so long as thetotal probability of there being an axion in any bin i,
Z ∞P (Ai,tot ) =P (Ai (gγ )) dgγ ,(7.12)

0is considerably smaller than any prior update that would be applied according to Eqs. (7.4) and(7.6).
143
The fact that the prior must be nonzero is justified by Cromwell’s rule, which states that priors of exactly 0 or
1 should be avoided generally. More directly, there would be no sense in performing a haloscope search if the chanceof success was truly zero.
144
The possibility that dark matter more generally might not couple to electromagnetism, or any of the nongravitational forces, is a distinct and disquieting one.

181
To make a coarse estimate of P (Ai,tot ), we can write down a Drake-like equation for axionicdark matter to which a typical haloscope searching for QCD axions such as HAYSTAC or ADMX
might be sensitive:
P (Ai,tot ) = P (PQ) × P (DM) × P (ma ≈ hνi /c2 ),(7.13)

where PQ denotes the event of the Peccei-Quinn hypothesis (Sec. 2.3.2) being correct and the QCD
axion field existing, DM denotes the event of that axion field actually accounting for an appreciablefraction of the galaxy’s dark matter (Sec. 2.4.2), ma ≈ hνi /c2 denotes the event that the axion iswithin roughly a linewidth of the bin of interest, i, and the probability of each event in parentheses
is implicitly conditional upon all leftward events. In practice several other propositions must betrue for an axion to be detectable. For instance, the axion must not be coincident with any otherRF spikes and its lineshape should roughly match that used in the data processing (Sec. 2.4.3).
It is beyond the scope of this thesis to attempt serious estimates of these probabilities, but
for the sake of justifying the infinitesimal prior approximation, we only need to consider the sample
optimistic set of probabilities shown in Table 7.1. For logarithmically uniform distributed priorbelief [95]Pa,i ∝

1
νi(7.14)

in a Qa ∼ 106 axion between mSN ∼ 10 µeV and mOC ∼ 1 meV, the approximate mass range
not disfavored by evidence from the SN1987A neutrino burst or overclosure arguments (Sec. 2.4.1)[52–54],
P (ma ≈ hνi /c2 ) =log(1 + Qa )= 2.1 × 10−7 .
log(mSN /mOC )(7.15)

If overclosure arguments do not apply [249], then a QCD axion could be far lighter than mOC .
Moreover, in practice, the prior probability for an axion to be in any given narrow coupling window
must be less than P (Ai,tot ). Hence, the rough estimate of
P (Ai,tot ) ∼ 9 × 10−8

(7.16)

should be considered a generous upper bound on the prior probability as the term is used in thisthesis.

182
event
PQ
DM
ma ≈ hνi /c2
Ai,totoptimistic probability
0.8
0.5
2.1 × 10−7
9 × 10−8

Table 7.1: Sample set of probabilities used to estimate prior axion hypothesis probability according
to Eq. (7.13). The probabilities P (PQ) and P (DM) are set to high (optimistic) values, where itis clear that raising them would not qualitatively change the conclusion that the priors are in
the appropriate infinitesimal limit. The probability of the axion mass coinciding with a given bin
assumes logarithmic priors [95], and is itself perhaps optimistic in assuming that the axion exists
within a three-mass decade window in which the haloscope operates.

In practice, one should not wait for a prior update of order 1/P (Ai,tot ) to manually interrogate
a bin;145 instead, an approximate prior estimate of the prevalence of spurious RF tones (Sec. 7.1.5)
should inform decision making. The prevalence of these RF spikes is a priori unknown, but thefact that in practice [78, 105, 107] the number of rescans required agrees with the predictions fromGaussian statistics (i.e. the measured distribution of excess powers looks Gaussian even in its righttail) indicates that only a few RF spikes large enough to push a bin above threshold are expected
in a scan. For an initial scan with NI independent bins, therefore, a prior update of U ∼ NI /NRF
should be sufficient cause to manually interrogate the bin. Here, NRF is the experimentalist’s
best guess for the expected number of real RF spikes in the scan window. Insofar as this guess isincorrect, the haloscope search will take longer than would be optimal. Rescanning and eliminating
spurious RF tones ensures that the bias in the updates remains small, as well as conservative.146
The prior probability P (Ai,tot ) estimated in this section makes our approximation of infinitesimal priors valid for any single-bin updates U  1/P (Ai,tot ). Since the prior update at
which manual interrogation ought to begin, NI /NRF , is generally orders of magnitude less than
1/P (Ai,tot ), the approximation of infinitesimal priors will always be valid.
145

Manual interrogation means performing tests that would discriminate between most spurious RF excesses andan axion: for example tuning the spike well off cavity resonance and/or ramping down the magnet and seeing if thesignal persists. See Item 9 in Sec. 6.2.1.
146
The conservative bias discussed in Sec. 7.1.5 results from interpreting the no-axion hypothesis, Ni in Eq. (7.11),(j)to incorporate spurious RF tones. This makes P (xi |Ni ) larger in reality than as calculated from Eq. (6.15). Usinga slightly undersized denominator in Eq. (7.4) results in a slight anti-exclusion bias.

183
7.1.4

Correlations between grand spectrum bins

In this section, we will see that correlations between grand spectrum powers147 at different
frequencies do not bias the BPM framework’s outputs, and are furthermore desirable. The majority
of Sec. 7.1 treats the grand spectrum power excesses as uncorrelated for simplicity. In practice, thisassumption is violated, as grand spectrum bin spacings are intentionally made spectrally smaller
than the axion linewidth and the frequency window of the digital Savitzky-Golay (SG) filter applied
to the data. The effect of estimating power excesses based on the axion lineshape (Step 8 in Sec. 6.1)
is to positively correlate nearby bins, whereas the effect of the SG filtering (Step 4) is to negatively
correlate bins over a somewhat longer frequency scale [78]. These are the only two processing steps
that will correlate adjacent bins.148 The data processing introduces no interscan correlations, as(j)

it handles separate scans independently. The single-scan prior updates ui (gγ ) can therefore be(j)treated as unconditionally unbiased (hui i = 1), scan-independent, frequency-dependent randomvariables.
(j)Given these properties of the single-scan prior updates ui (gγ ), the result that the aggregate
prior update U(gγ ) is unbiased then follows from the fact that aggregation entails multiplying
across scans and summing across bins. The total prior updates Ui (gγ ) are defined in Eq. (7.6) as
the product of grand spectrum prior updates from Mi scans. Using the fact that the expectation
value of the product of independent random variables is the product of the expectation values, here
*M
+
M D
E
Y (j)
Y
(j)
ui (gγ ) =hUi (gγ )i =ui (gγ ) = 1,
(7.17)
j=1

j=1

the total prior updates are seen to be unbiased, i.e. have unit expectation. Aggregation then occurs
through summation of the posteriors and priors via Eq. (7.7). The expectation value of the sum of
147

Recall from Sec. 6.1 that the grand spectrum is the final spectrum resulting from the data processing, accounting
for noise level, signal strength, and axion lineshape.
148
Systematic effects in the measurement will also correlate grand spectrum power excesses. The identification
and removal of nonaxionic power excesses and the SG filtering exist largely to undo such experimentally-induced
correlations. We do not consider the effect of such correlations on the grounds that these processing steps appear
largely successful [78].

184
dependent random variables is the sum of the expectation values, here
*N
+
N
X
X
hU(gγ )i =Ui (gγ ) =hUi (gγ )i = 1,i=1(7.18)

i=1

where the last equality uses the result of Eq. (7.17) to demonstrate that U(gγ ) is also unbiased.
As we relax assumptions of uniform or even low priors, the relevant updates, treated as randomvariables, can be shown to have unit expectation independent of grand spectrum correlations, but
the formulae used to compute and combine the updates become less elegant.
The primary effect of grand spectrum correlations on the prior updates is not to bias theanalysis, but only to alter the higher moments of the distribution that aggregate prior updates are
drawn from. In particular, the number NI of independent grand spectrum bins is smaller than thenumber N of grand spectrum excesses calculated. This increases the spread of the distributionthat aggregate prior updates will be drawn from for a given haloscope scan window. However,since the dominant correlations are produced by the axion lineshape (Step 8, Sec. 6.1), the numberof independent bins has as a rough lower bound the number of axion linewidths ∆a in the scan
window ∆W , and as an upper bound the number of bins used: ∆W /∆a <∼ NI ≤ N . The fact that
NI ≤ N broadens the probability distributions of the aggregate outcomes (Sec. 7.3.3), but does nototherwise impact the analysis.
To see that correlations are indeed desirable, note that their absence would imply a grand
spectrum bin spacing larger than the axion linewidth. As discussed in Ref. [78], such a coarse bin
spacing would effectively leave unprobed a large fraction of the scan window. Since the correlations
produce no deleterious effects upon the BPM framework, there is no reason not to oversample thespace of possible axion masses, and the correlations should be considered a feature, not a bug.

7.1.5

Practical departures from Gaussianity

The formula for updating priors using the BPM framework, Eq. (7.4), assumes that the axion
and no-axion distributions are Gaussian, Eq. (6.15). In particular, measured distributions often
rise in their extreme tails over what is ideally expected [250]. In the case of haloscopic axion

185
detection, the presence of spurious RF tones increases the probability density of the high-power
tail of the probability density functions (PDFs) of Fig. 6.1. In this section, we will see that theinterpretation of the exclusion achieved with the BPM framework is robust to the resultant bias
towards discovery. Conversely, there is little to be gained from Bayesian inference in the context ofmanually interrogating a persistent signal with the goal of reporting discovery. Efficiency is a far
less important consideration in this context, and test statistics such as those of Refs. [75, 241], in
conjunction with manual interrogation (Item 9 of Sec. 6.2.1 and Footnote 145) are well suited tothe task of claiming discovery, once all manual interrogation tests are passed.
Sources of RF noise in the haloscope’s physical environment can couple into the receiverchain. Real haloscopes go to significant effort to discriminate such artificial excesses from realaxion signals, but excesses sufficiently similar in spectral profile to an axion necessarily contributeto the measured power distributions, leading to an effective increase in the high power probability
density of both the axion and no-axion distribution PDFs.
The effect of increased probability density in the high-power tail in either distribution in
Fig. 6.1 is to bias the BPM analysis towards discovery of an axion. In any one frequency bin, thiseffect is very unlikely because it only affects the distribution tails. Indeed, for existing haloscopes,
the measured power excesses seem very well approximated out to several standard deviations by
Gaussian distributions [78, 125], indicating that the consequences of non-Gaussian features will bemodest. However, with tens or hundreds of thousands of independent frequency bins measured,
there may be a small augmentation of the BPM framework’s aggregate prior update, Eq. (7.7).
In the context of reporting exclusion over axion parameter space — the only context in whichdata from haloscope searches has appeared to date — a bias towards discovery is conservative, andhence acceptable. In the context of an FT framework (Sec. 6.3), the effect of a discovery bias is not
to change the reported exclusion, but rather to increase expected rescan time and the chance of a
false positive. Since, as we will see in Fig. 7.3b, the BPM framework still achieves a 36% scan rateenhancement relative to thresholding for the HAYSTAC Phase I dataset, the conservative bias is
manifestly not large enough (at least for this dataset, and also the Phase II dataset of Ch. 8) to186
overwhelm the improved exclusion over the whole scan range.
In the context of reporting the discovery of axionic dark matter, conversely, a bias towards
discovery would appear to be harmful. However, the nature of axion detection guards against
adverse outcomes in practice. A power excess presenting as an axion, once identified, is fairly
straightforwardly shown to persist or not by manual interrogation. The role of the statisticalanalysis frameworks discussed in this thesis is not to firmly identify the axion as such once a
persistent signal is found, but rather to indicate with the highest fidelity where such a signal islikely to lie. While a BPM analysis may bias the experimentalist towards believing that a few
nonaxionic excesses might be axionic, the real certification of an axion signal would look like thatfor other discoveries in fundamental physics: a series of careful measurements precluding beyond
doubt that no other known physical phenomena could account for the signal. In other words, theasymmetry between the criteria for exclusion and discovery ensure that BPM’s discovery-bias would
almost never149 lead to accidental discovery.

7.2

Effects of data processing on information content
This chapter takes as its motivation the unacceptably long scan times that will likely berequired to verify or falsify the axionic dark matter hypothesis over any meaningful fraction of theplausible parameter space. However, the BPM framework optimizes only the statistical analysisof the already-processed data of a specific measurement, raising two complementary questions:is the measurement optimal, and is the data processing optimal? The first question, of whether
the measurement itself is optimal — i.e. is the haloscope as typically operated the most efficient
tool for evaluating the axionic dark matter hypothesis? — is beyond the scope of this thesis,
and the reader is referred to Ref. [95]. Subject to a haloscope or haloscope-like search platform,
however, this section addresses the effect of the data processing — the intermediate set of steps
149

It is possible to construct scenarios where nature is sufficiently cruel as to source a tone with the distinguishingproperties of the axion. For example, imagine a microwave tone generated by the magnetic field itself which manages
to vibrate a piezoelectric object attached to the outside of the cavity, which in turn creates eddy currents inside it.
The result is a B02 -proportional microwave tone inside the cavity. Even still, however, it is unlikely (thought not
impossible) that such a tone would mimic the expected axion lineshape, Eq. (2.14).

187
between measurement and statistical analysis detailed in Sec. 6.1 — on the information content of
the haloscope data. We will see that the current data processing protocol of the leading haloscopeexperiments [91, 107] are already highly optimized, with very little room for further improvements,
though this has not historically been the case.
This section evaluates the consequences of each data processing step enumerated in Sec. 6.1
on the axion-pertinent (AP) information content of the data, where AP is understood to mean“potentially bearing on the probability of the axion hypothesis being true.” Information pertinent
to the existence of the axion can only be generated during the measurement, and may be degradedby the subsequent data processing. Our aim is to determine where in Steps 1–10, if at all, AP
information could be lost. A guaranteed way to make use of all of the AP information content
would therefore be to express the predictions of the axion and no-axion hypotheses, AQ,ijk andαand apply Bayes’ theorem as
NQ,ijk , respectively, in the space of the quadrature voltages vQ,ijk
α]|AQ,ijk )P (Re[vQ,ijk
α,ReuQ,ijk =,α]|NQ,ijk )P (Re[vQ,ijk(7.19)

where the standard infinitesimal prior limit has been assumed and the same update applies for theα].150 While this approach is impractical, it nonetheless serves asimaginary components Im[vQ,ijk

an excellent baseline to compare the effects of the data processing protocol against.
Step 1 includes two operations: squaring the real and imaginary parts of each quadrature
voltage, and then adding them. The real and imaginary parts of each quadrature voltage can beapproximated as the same independent, identically distributed Gaussian random variables with
2 . Anmean zero [75, 200]. Absent an axion, we label the variances of these random variables σv,ij
2 ,axion, assumed to be of a given coupling and at a given frequency, increases the variance to λij σv,ij

where λij is extremely close to but greater than unity. The prior update, Eq. (7.19), is the ratio of
Gaussian PDFs given by Eq. (6.15):
1uα,ReQ,ijk = p

λij

150

"
exp

1 − λ−1ij
22σv,ij#2αRe[vQ,ijk]

.

(7.20)

While in principle it is possible to evaluate the axionic dark matter hypothesis in the time domain, we do not
expect that this would provide any benefit, given the reversibility of the Fourier transform and the fact that the phaseof the axion field at any moment is unknown.

188
Since Re[vQ,ijk ] only occurs squared in this prior update (likewise Im[vQ,ijk ] in its identical equation), no AP information is lost by squaring the real and imaginary quadrature voltage components.
Furthermore, since Bayesian updates are by their nature combined multiplicatively, the update accounting for both components of the quadrature voltage is"#
1 − λ−1


1
2
2
ij
α,v
α
α
uQ,ijk =exp
Re[vQ,ijk] + Im[vQ,ijk
]
.
2
λij
2σv,ij(7.21)

Because the voltage quadrature components appear in the prior update only in their Eq. (6.1)
combination, Step 1 preserves AP information content.
The same argument applies to Steps 2 and 3. Multiplicative prior updates have their powers
add in the exponent, hence Steps 2 and 3 do not degrade AP information content. For Step 2,
this argument assumes that the axion and the thermal noise fluctuations distribute power evenly
between the X and Y quadratures. For a two-quadrature measurement where the phase of theaxion is unknown, that assumption is valid.
Step 4 is among the more vulnerable steps to meaningful information loss. If hxβij i in Eq. (6.4)
is treated as a number, then Eq. (6.4) amounts merely to the division and subtraction of constantsfrom the raw spectrum random variables xβij , which will not degrade the information content.
In practice, however, hxβij i must itself be estimated, making it in effect a random variable. The
estimation used by HAYSTAC and ADMX is performed via SG filters, which estimate hxβij i at eachfrequency bin i via the polynomial generalization of a moving average. If a wide spectral window
for the generalized moving average is used, then hxβij i is estimated with low variance, and acts
like a constant. However, the SG filtering creates undesired correlations between bins up to two
window-lengths apart, while also slightly attenuating an axion’s visibility, an effect accounted forin Step 9. The trade-off between the desired filtering effects and the undesired correlations andattenuation is beyond the scope of this thesis, but is discussed at length in Ref. [78]. So far, no
proof exists that SG is the optimal choice of digital filter151 for the data, so Step 4 may admit of151

Recall that the SG is itself a low pass filter, which the spectra are divided by so as to preserve high frequency
structure. See Ref. [78] for numerical tests performed to quantify the effects on synthetic data with real haloscope
spectral structure.

189
meaningful optimization.
γ
Calculating the processed spectrum sensitivity parameters ηij
of Step 5 relies on independentmeasurements of the thermal and added noise, magnetic field, and cavity quality factor and modestructure properties of the haloscope, as well as calculations of the power that a dark matteraxion would deliver. Subject to the accuracy of these measurements and calculations, Step 5
simply divides the processed spectra power excess random variables xγij by constants, preserving
information content.
The ML-weighting of Step 6 is the information-preserving generalization of the straightforward addition of Steps 1–3, for the case where the random variables being added have different
variances. By the central limit theorem, the rescaled power excesses xδij are Gaussian-distributed
δ well approximated as independent of the presence of any axion whichwith known variances σij

delivers power far less than vacuum. The pertinent effect of an axion is to shift the mean of therescaled power excess from 0 to µδ , which has no bin- or tuning step-dependence by construction.
The appropriate prior update delivered by each rescaled power excess xδij is thus"#−(µδ )2 /2 + µδ xδijδuij = exp
.
δ )2
(σij
When the updates of all Nt tuning steps are multiplied, the result is

!
Nt
Nt
δ )2 /2 + µδ xδ
Y
X
−(µ
ij 
uδij = exp 
δ )2
(σ
ij
j=1
j=1


−(µδ )2 /2 + µδ xi= exp.
(σi )2(7.22)(7.23)δ )2 can thus
Because the updates from all bin-i rescaled spectra power excesses xδij and variances (σij

be obtained using the single bin-i combined spectrum power excess xi and variance (σi )2 , given by
Eqs. (6.6) and (6.7), respectively, the ML estimation of Step 6 preserves AP information content.
The same argument about ML estimation guarantees that Steps 7 and 8 preserve AP information as well, in the limit where the axion lineshape is truly approximately constant over spectral
scales of nc ∆b . Step 9 is in effect properly accounting for the information loss suffered during Step4, but is itself simply multiplication by a scalar, and causes no additional information loss.

190

7.3

Comparison of analysis frameworks
One of the central claims of this chapter is that in a haloscope analysis an apples-to-apples

comparison may be made between the widely quoted frequentist exclusion E of Sec. 6.3 and theBayesian prior update Ui of Sec. 7.1.1. In general, these two are in fact not equivalent. The equivalence holds only between FT analyses with low false-positive rates and Bayesian analyses using
low priors. In this section, I introduce two Bayesian threshold analyses. The first framework, BT1(Sec. 7.3.1), is more informative than the second, BT2 (Sec. 7.3.2). In the dual limit of infinitesimalpriors/low false-positive rates, BT2 reveals the Bayesian prior update to be completely equivalent
to the frequentist exclusion. In Sec. 7.3.3, we will quantitatively examine the outcomes of all three
Bayesian frameworks (BT1, BT2, and BPM) discussed in this thesis. Together, BT1 and BT2 motivate the BPM framework as the maximally informative, minimally constrained limit of haloscope
analyses.

7.3.1

Bayesian threshold 1 framework

The first of the threshold-based Bayesian analyses we consider, BT1, updates its priors after
each successive scan. We are interested in the conditional probability that there is an axion in(j)

the ith bin, given that the measured power in the j th scan either did (xi
(j)

(xi

(j)

≥ xT ) or did not(j)< xT ) exceed threshold. For simplicity of notation, we will drop all frequency indices i forthe remainder of this and other sections where not necessary, with it understood that results canultimately be aggregated via Eq. (7.7). We denote the event of a measured power exceeding (notexceeding) threshold as a binary detector going “click” (“no-click”), and we denote the event thatthe measured power excess came from the axion distribution (no-axion distribution) as A (N ). The
posterior probability of an axion given a click is
P (A|click) =P (click|A)P (A),P (click|A)P (A) + P (click|N )P (N )(7.24)

where the denominator is equivalent to P (click). Identifying P (click|A) as the single-scan true(j)(j)

positive rate 1 − fn (Fig. 6.1, light blue) and P (click|N ) as the single-scan false-positive rate fp

191(Fig. 6.1, dark green), abbreviating the prior (posterior) probability that there is an axion in the(0)

bin as Pa

(1)

(0)

= P (A) [Pa,cl = P (A|click)], and noting that P (N ) = 1 − Pa , Eq. (7.24) simplifies to
(1)

(1)

ucl =

(1)

Pa,cl

1 − fn

Pa

(1)
fp

≈
(0)

≈

1
(1)

.

(7.25)

fp(0)

The first approximation holds in the appropriate limit of infinitesimal priors, here Pa(1)

 fp , and(1)

the second holds in the limit of low single-scan false-negative rates, fn  1.
(1)In the event of a no-click, the posterior probability Pa,no = P (A|no-click) is given by an(j)

expression analogous to Eq. (7.24) containing the true- (1 − fp ; Fig. 6.1, light green) and false(j)(fn ; Fig. 6.1, dark blue) negative rates. The expression simplifies as
(1)

u(1)
no =

(1)

Pa,no

fn

Pa

(1)
1 − fp≈(0)(0)

where the first approximation holds for low priors Pa(1)scan false-positive rates, fp≈ fn(1) ,(7.26)

 1, and the second holds for low single-

 1. Whereas single-scan false-negative rates in haloscope searches

using an FT framework are typically close to 5%, false-positive rates are often kept much lower, tominimize the need for time-expensive rescans.
Applied to a sequence of M scans, the BT1 analysis provides a crucial operational freedom,shared by the BPM framework, to the experimentalist. Since the ordering of multiple updates is
(j)

(j)

inconsequential, a series of M identical (fn = fn0 , fp

= fp0 ) scans has M + 1 possible outcomes,

corresponding to observing c ∈ {0, 1, . . . , M } clicks and M − c no-clicks. Using the simplest formsof Eqs. (7.25) and (7.26), the final prior update, obtained from Eq. (7.6), is
UcBT1 ≈

(fn0 )M −c.
(fp0 )c(7.27)The prior is upgraded by 1/fp0 for every positive result, and downgraded by fn0 for every negative
result.
In the BT1 framework, the threshold and false-negative rate for each scan must be set inadvance, but the number of scans M need not be. This operational freedom is a consequence ofthe fact that the expected value of the prior update before performing a given scan is unity, and so

192
the decision to perform another scan or not can be made on the fly, without biasing the outcome.
In contrast, in the FT framework of Sec. 6.3, adding another scan midway through a multiscan
protocol and recalculating the false-negative rate for the entire protocol, the experimentalist canmanipulate the expected value of the exclusion. This freedom has nothing to do with the useof frequentist versus Bayesian inference. We will see that Bayesian frameworks can be equally
restrictive.

Figure 7.1: The four possible outcomes of an M = 2-identical scan threshold protocol. The conditional probabilities of each outcome (1)–(4) subject to the axion (A) and no-axion (N ) hypotheses
are given in terms of true- and false-positive and -negative rates (Fig. 6.1). In both the frequentist
threshold (FT) framework of Sec. 6.3 and the Bayesian threshold 2 (BT2) framework of Sec. 7.3.2,
outcomes (2)–(4) (gray) are all treated identically as the negative result: in the FT (BT2) framework, the null is rejected (the prior is reduced). In this case, the exclusion E, Eq. (6.17), is trivially(BT2)
equivalent to the prior update Uneg ≈ Fn in the dual limit of low false-positive rates and priors.
Only region (1) (red) fails to reject the null (FT), or increases the prior (BT2). Figure reproduced
from Ref. [126].

7.3.2

Bayesian threshold 2 framework

Whereas the BT1 framework offers M + 1 possible outcomes for M scans and provides
interscan operational freedom to the experimentalist, the second BT framework (BT2) that we
consider is designed to be uninformative and restrictive by comparison; its outcomes, tellingly, will

193
map precisely onto those of the FT framework of Sec. 6.3. The standard FT analysis permits two
possible results at each frequency, illustrated for a simple case in Fig. 7.1: a rejection of the null(negative result) or a failure to reject it (positive result). For an M -scan protocol, the negative
result is defined as the case where at least one scan no-clicks; the positive result occurs only when
all M scans click. The multiscan false-negative rate, Fn from Eq. (6.20), sets the exclusion via
Eq. (6.17). False-positive rates do not enter directly into the FT framework’s reported exclusion.152

The BT2 framework applies Bayes’ theorem, Eq. (7.1), to this binary result landscape, ob(1)(1)

taining a posterior either for a positive result of Pa,pos or for a negative result of Pa,neg . For the
positive case,
P (A|pos) =P (pos|A)P (A)P (pos|A)P (A) + P (pos|N )P (N )(7.28)

yields(1)
Pa,pos
1 − Fn1
BT2
Upos = (0) ≈
≈
,
F
F
p
p
Pa

(7.29)

where the total false-negative rate Fn is that of Eq. (6.20) and
Fp =

M
Y

fp(j)

(7.30)

j=1

is the total false-positive rate for the M scans. The first (second) approximation in Eq. (7.29) is(0)

valid for infinitesimal priors, Pa Fp (low total false-negative rate, Fn  1). Note that BT2’s

positive prior update agrees in the case of identical scans (the general cases agree as well) precisely
with the c = M case of Eq. (7.27) for BT1. The agreement is due to the fact that they derive fromBT2 is a very
the same event — a series of M clicks. Since false-positive rates are very small, Upos

large number, corresponding to just how unlikely a positive result would be to observe, without anaxion present.
The negative result yields an update that will agree precisely with the standard FT exclusion
152
More completely, false-positive rates only enter indirectly, as the global false-positive rate — the chance of atleast one bin achieving a positive result (i.e. a click on all M scans) — must be set sufficiently low to account for the
look-elsewhere effect, as discussed in Sec. 6.5.1. Choosing a higher false-positive rate will achieve a deeper exclusion
if no positive results are recorded, but will increase the probability of such an event.

194
E. From the equation for P (A|neg) analogous to Eq. (7.28), we obtain:(1)
Pa,negFn
BT2
Uneg = (0) ≈≈ Fn ,
1 − FpPa(0)where the two approximations assume low priors Pa(7.31)

 1 and low M -scan false-positive rates

Fp  1. With these conditions met, the BT2 negative prior update is the complement of theBT2 ≈ 1 − E. In the limit relevant for haloscope searches, the frequentist
frequentist exclusion, Uneg

exclusion is therefore equivalent to the Bayesian prior update, an equivalence used in the context
of other searches for new physics [235]. Whether this update is obtained through one of the BT
frameworks or the BPM framework is immaterial.
Finally, a pair of differences between the two BT frameworks discussed in this section hints
at two important operational advantages of the BPM framework. The BT1 framework, by allowing
M + 1 possible results with differing outcomes (or up to 2M for scans with nonidentical falsenegative and -positive rates), has an informational advantage over the BT2 framework. The BT2framework deliberately blinds the experimentalist to which scan(s) no-clicked, combining M of theM + 1 possible outcomes into a single negative result. For an experimentalist intent on discovering
or excluding the axion, there can be no benefit to discarding this information. Taking the next
logical step, the power-measured information from each scan need not be discarded. The BPM
framework, unlike the BT frameworks, uses this information.

possible outcomes per bin
interscan freedom
intrascan freedom

p-value∞yes
yes

BT2/FT
2
no
no

BT1
2M
yes
no

BPM
∞
yes
yes

Table 7.2: Comparison of the Bayesian threshold frameworks (BT1 and BT2) of Sec. 7.3 with
the Bayesian power-measured (BPM) framework of Sec. 7.1 and the p-value framework of Sec. 6.4
for a protocol with M possibly nonidentical scans performed. More informative frameworks map
the continuum of possible measurements onto a larger number of reported outcomes. The more
informative frameworks also permit the experimentalist greater freedom to alter the scan protocol
without biasing the outcome. Interscan changes to upcoming scans are allowed within all but
the BT2 framework (equivalently, the FT framework of Sec. 6.3), while the BPM framework evenpermits intrascan adjustments as information compiles. The p-value framework has the same
properties as the BPM framework, but comes with a number of downsides discussed in Sec. 6.4.

195
The BT1 framework also proves operationally superior to BT2. Whereas the BT2 framework
locks in a commitment to M scans, if necessary, BT1 allows the experimentalist to reinsert himself
into the decision-making between scans without biasing the outcome. This interscan freedom is
improved to intrascan freedom for BPM, as indicated in Table 7.2, which summarizes the informativeness and operational constraints imposed by each framework. Under BPM, the expected priorupdate for the next iota of power measured always being unity protects the experimentalist from
inserting bias. To the degree that the data processing allows it, probability can be tracked in realtime and used to inform scan-protocol decisions on the fly. This capability provides an opportunity
for optimizing a BPM haloscope search algorithm, a promising direction for future analysis beyond
the scope of this thesis.

7.3.3

Outcome distributions for Bayesian analyses

The reanalysis of the HAYSTAC Phase I dataset that we will consider in Sec. 7.4 indicatesthat the BPM framework achieves superior exclusion to thresholding in the case of one real dataset(with a second added in Ch. 8). In this section, we demonstrate that this result is typical bytreating the prior updates in the BPM framework and the BT1 and BT2 frameworks discussed
in Sec. 7.3 as random variables. We make an apples-to-apples comparison between the aggregate
prior update probability distributions obtained from an M = 2-identical scan protocol in thesethree frameworks. In particular, while the mean aggregate prior update hUi = 1 for any unbiased
analysis, we will see that the median aggregate prior update in the BPM framework is a factorof two smaller than in either threshold framework. From this reduction in the median aggregate
prior update we predict a typical scan rate enhancement of 30% for BPM relative to thresholding,consistent with the enhancement observed in the HAYSTAC Phase I dataset. En route to this final
result, we obtain analytic expressions for the total prior update probability distributions in eachframework. Comparing the three frameworks at this level likewise elucidates the difference between
the BPM and threshold exclusion lines plotted in Fig. 7.3a.
Throughout this section, we assume an ideal, axionless haloscope dataset: the grand spectrum

196
(j)

excesses are independent, identically distributed random variables xi∼ N (0, 1). We compare priorupdate probability distributions for an M = 2-identical scan protocol in which an initial scan is
performed, followed by a rescan in each bin whose measured power excess exceeds a predeterminedthreshold.153 For definiteness, we assume scan parameters similar to those used in HAYSTAC
Phase I. For the BT1 and BT2 frameworks, we will first derive analytic expressions for the priorupdates in a more general M -identical scan protocol, and then specify to M = 2.
Starting with the least informative BT2 framework, the total prior update in a single bin has
probability mass function (PMF):
BT2
P (U BT2 = Upos
) = (fp0 )M

P (U

BT2

(7.32)

BT2= Uneg) = 1 − (fp0 )M .

In any scan of N independent frequency bins, k = 0, . . . , N of those will realize a positive result,where k is binomial distributed k ∼ B(N, (fp0 )M ). The PMF for the aggregate prior update is then!  
BT2 − U BT2
k 
N −k
U
N 
pos
neg
BT2
P U BT2 =k + Uneg=(fp0 )M
1 − (fp0 )M.
(7.33)Nk
Next we consider the BT1 framework, whose total prior update UcBT1 , Eq. (7.25), is written
in terms of the single-scan updates ucl and uno for a click, Eq. (7.25), and for a no-click, Eq. (7.26),
respectively. Each click occurs with probability fp0 , hence the probability of obtaining exactlyc = {0, 1, . . . , M } clicks is
Pc = (fp0 )c (1 − fp0 )1−δM c ,(7.34)

where δab is the Kronecker delta function. The total prior update PDF is therefore
P U BT1 = [ucl ]c [uno ]1−δM c = Pc .

(7.35)For the M -identical scan protocol over N independent bins, the aggregate prior update, Eq. (7.7),
is parametrized by the numbers nc of bins that click c times, which are multinomial distributed
153
As discussed in Sec. 7.3, BPM provides freedoms to deviate from such a rigid protocol, and can therefore
outperform the projections here. In Ch. 8, we take advantage of the enhanced operational freedoms of the BPM
framework.

197
with probabilities Pc given by Eq. (7.34). The aggregate prior update is
!
M
M
c
X
Y
1
N!
(u
)
cl
BT1
P U
=
= QM
nc
(Pc00 )nc00
N(uno )δM c −1
0 ! 00
n
0
c
c =0
c=0

(7.36)

c =0

if
M
X

nc = N,(7.37)

c=0

and 0 otherwise.

Figure 7.2: (a) Histograms of simulated total prior updates U (dots), Eq. (7.6), for the BPM (blue),
BT1 (green), and BT2 (red) frameworks. 225 independent bins were simulated for an axion thatwould produce a mean excess µa = 5.1, assuming there is in reality no axion present and using
a standard M = 2-identical scan protocol with scan parameters similar to those of HAYSTAC:
threshold xT = 3.455 (fn0 = 5%, fp0 = 0.03%). The error bars were estimated from binomial
statistics and are smaller than the data points. The simulated probability distributions are inexcellent agreement with the analytic predictions of Eqs. (7.44), (7.35), and (7.32) (blue, green,
and red lines). The total updates described in (a) are combined via Eq. (7.7) to form the aggregate
prior updates U described in (b). The aggregate prior updates are numerically generated over 221
independent trials of N = 215 independent bins for each of the three analysis frameworks. For
the frameworks for which we derive analytic predictions (green and red lines for BT1, Eq. (7.36),
and BT2, Eq. (7.33), respectively), the agreement with the simulations within binomial error bars
confirms our expectations. For all three frameworks, the (relatively few) bins which fall outside of
the plotted windows do indeed balance the scales so as to produce the unbiased expectation values
hU i = hUi = 1 (Sec. 7.1.4). The BPM framework displays by far the lowest median outcomes in allcases, indicating that it will typically achieve superior exclusion when no axion is present. Figure
reproduced from Ref. [126].

For the BPM framework, we restrict ourselves to M = 2 identical scans, where the second
occurs conditionally on the first exceeding threshold, and derive only the expression for the total

198
prior update. To derive the PDF for the two-scan BPM prior update, we consider two mutually
exclusive cases.
First, with probability 1 − fp0 , the initial scan does not exceed threshold (x < xT ; a true
negative). In that case, the prior update PDF is −1 (1 − fp0 )−1 fu (u)dufu,tn (utn ) = fx (x)
=

dx


0
where
fu (u) =0 ≤ utn < uT
(7.38)
otherwise,





2 /8

exp[−(log u)2 /2µ2a ]] exp[−µ
√a

0≤u
u3/2
µ 2π
a




0

(7.39)

u<0

and
 2

µa
uT = exp −+ µa x T
2(7.40)

is the single-scan prior update achieved at threshold (x = xT ). The second line of Eq. (7.38) follows
from Eqs. (6.15) and (7.4). The normalization factor (1 − fp0 ) accounts for the nonunit probabilityof drawing from this distribution in the initial scan.
Next we consider the case where the first scan exceeds threshold, (x ≥ xT ; a false positive),
which occurs with probability fp0 . The total prior update for the initial false-positive is the productof two single-scan prior updates Ufp = ufp u. The PDF for the first scan prior update ufp is(f )−1 f (u ) u ≤ u

p0
u fp
T
fp
fu,fp (ufp ) =(7.41)0
u <u ,
fp

T

and that for the second scan prior update u, which is not restricted to being above or below
threshold, is given by Eq. (7.39).

199
The PDF for Ufp is thenZ ∞fu (u)fu,fp

fU,fp (Ufp ) =

=



Ufp
u



du
|u|

−∞ h
i


2

exp −(µa /2)2 −(log[Ufp /2µa ])log[u2T /Ufp ]



erfc
3/2 √
2µa

0≤U




0

U < 0.

fp0 µa Ufp 4 π(7.42)The total prior update U for the procedure is a probability-weighted sum of PDFs:
fU (U ) = (1 − fp0 )fu,tn (U ) + fp0 fU,fp (U ).
Written out fully,log[u2T /U ]exp[−µ2a /8−(log U )2 /2µ2a ]exp[−(µa /2)2 −(log[U/2µa ])2 ]


√
√
erfc
+

3/2
2µ
a
µa U
4 π

µa U 3/2 2π





2
−(log[U/2µa ])2 ]log[u2T /U ]fU (U ) = exp[−(µa /2) 3/2
√
erfc
2µa

µa U
4 π






0

(7.43)

0 ≤ U < uT
uT ≤ U(7.44)

U <0

is the two-identical scan BPM total prior update PDF: that is, the probability density for obtaining aprior update U in a single axionless bin using the standard initial and conditional-rescan procedure.
The total updates for larger numbers of scans and the aggregate updates for multiple bins are too
cumbersome to write out in closed form, but may readily be simulated.
Equations (7.44), (7.35), and (7.32) are respectively the total prior update probability distributions for the BPM framework and the two BT frameworks discussed in Sec. 7.3. These threeequations are plotted as blue, green, and red lines, respectively, alongside simulated total priorupdates for the three frameworks (blue, green, and red dots) for a two-scan protocol without anaxion present in Fig. 7.2a.
The simulations are performed by drawing Gaussian random variables according to Eq. (6.15),
with µ = 0, σ = 1, and performing a second draw as well if x ≥ xT . Prior updates are thenapplied according to Eqs. (7.4), (7.6), (7.27), (7.29), and (7.31). The simulations match the analytic
expressions, and together reveal that the vast majority of bins will receive a downward update under
all three frameworks. In particular, the total prior updates for BT1 and BT2 almost always take

200
the precise values fn and 1 − (1 − fn )2 , respectively. The price we pay for these reliably low typical
updates are rare false positives with U  1 (far off the right side of the plot). The common U < 1and rare U  1 outcomes together ensure hU i = 1, the condition for an unbiased analysis. In
contrast, moderate positive and negative prior updates occur much more frequently in the BPM
framework, and hU i = 1 is enforced by a long tail of positive prior updates. The fact that themedian of the BPM distribution is lower than that of either threshold distribution is noteworthy,
as we will see presently in our discussion of the aggregate prior update.
Figure 7.2b shows the aggregate prior updates U for the identical two-scan protocol when
there are 215 independent bins, not far from the actual number in HAYSTAC [78]. In all cases,
the total updates from all bins are averaged according to Eq. (7.7) to obtain the aggregate priorupdate. All three methods deliver an unbiased aggregate prior update with mean hUi = 1, yet the
BPM framework typically delivers a stronger exclusion in the absence of an axion, i.e. its medianaggregate prior update is smaller than in either threshold framework by a factor of two. The
improvement is a consequence of the informational advantage of the BPM framework.
The factor-of-two typical improvement in aggregate exclusion illustrated for typical HAYSTAC
parameters in Fig. 7.2b yields a scan rate enhancement consistent with the 36% speedup inferredfrom real data in Fig. 7.3b, absent an axion.154 If the time for the BT2 framework to achieve thetargeted 90% exclusion across the scan window (factor-of-ten reduction in aggregate probability) is
T90 , the additional factor of two should take approximately T95 = T90 log10 (2), roughly 30% longer.

7.4

Test using HAYSTAC Phase I dataset
In order to test the BPM framework, it was used to reanalyze the full dataset from Phase

I of the HAYSTAC experiment, which includes at several frequencies as many as m = 3 scans.
Features specific to this dataset are discussed in Ref. [107]. Notably, several frequency bins in theoriginal analysis were discarded for having excess RF power that could with certainty be identified
154

With an axion present, additional simulations (not shown) reveal its presence in a manner consistent with theinterpretation of the global false-positive rate used in frequentist hypothesis testing.

201
as at least partially nonaxionic in nature. In the BPM reanalysis, we set Ui = 1 for these bins.
Otherwise, straightforward application of Eqs. (7.6) and (7.7) yield the results in Figs. 7.3a and b,
respectively.

Figure 7.3: (a) Reanalysis of HAYSTAC Phase I data as a test of the BPM framework. The
standard 90% exclusion line (red) achieved with the FT framework [107] is equivalent to the 10%prior update contour of the BT2 framework discussed in Sec. 7.3.2. The 10% prior update contour(blue) achieved with the BPM analysis constrains more aggressive couplings at nearly all frequencies. The logarithmic color scale indicates the subaggregated prior updates Us throughout the full
two-dimensional parameter space. Darker (lighter) shading corresponds to increased (decreased)probability of an axion of given frequency ν and axion-photon coupling gγ existing. Frequencies
where rescans were performed are marked with a black dot on the threshold exclusion line. The
five least unlikely (LU) axion candidates — those whose priors increased most — are marked attheir maximum likelihood couplings with orange circles. In the lower panel, corresponding orange(1)bars indicate on the orange logarithmic axis the total posterior probability PLU within the BPM
framework that an axion exists at each of the LU points normalized to the aggregate prior probability Pa that an axion resides anywhere in the window at that coupling. The LU candidate at? = 5.66417255 GHz, marked in pink, is more than two times higher than the remainder of theνLU
top five combined. (b) Aggregate prior update U (equivalently, exclusion E) taken as a function of
coupling gγ over the entire HAYSTAC Phase I frequency window for the FT and BT2 frameworks(solid red line), BPM framework (solid blue line), and BPM framework with several adjustments
made for a clean comparison with the performance of the thresholding (dashed blue line; see maintext). The dashed, gray line, zoomed in upon in the inset, indicates relative U = 10% (E = 90%)
scan time normalized to that achieved with the FT and BT2 frameworks [107]. Under typical experimental conditions, the exclusion achievable with the BPM framework would take an estimated
36% longer to attain using thresholding. Figure reproduced from Ref. [126].

202
7.4.1

Main results

The heat-map of Fig. 7.3a shows the updates to the probability of the axion existing throughout the two-dimensional parameter space. At high couplings (gγ ≥ gγ,high ≈ 3gγKSVZ ), the probability of an axion is greatly diminished from what it was prior to collecting data. At low couplings(gγ ≤ gγ,low ≈ gγKSVZ ), virtually no information can been gleaned from the data about the presenceor absence of an axion. At intermediate couplings, gγ,low < gγ < gγ,high , updates to the probability(j)of an axion being present depend sharply on the specific measured powers xi , and also on the(j)somewhat smoother haloscope sensitivity profile ηi .
The FT framework has its 90% exclusion line plotted in red in Fig. 7.3a. This corresponds
directly155 to the exclusion plotted in Ref. [107], and equivalently to the 10% prior update contourof the BT2 framework of Sec. 7.3.2. The solid, blue line is the BPM framework’s equivalent 10%prior update contour. At almost all frequencies, it constrains more pessimistic couplings than do
the threshold frameworks.
The frequency-dependent update (or exclusion) data of Fig. 7.3a has been aggregated in accordance with Eq. (7.7) and Eq. (6.22) for the BPM and FT frameworks, respectively, in Fig. 7.3b.
The solid, red line indicates aggregate updated probability (left axis) or exclusion of (right axis)
an axion anywhere within the HAYSTAC Phase I frequency window as a function of coupling.
The solid, blue line is the updated probability for the BPM framework. Its upward bulge around2.5gγKSVZ is largely due to the least unlikely (LU) candidate discussed in Sec. 7.4.2. Had the knowledge from the BPM framework been in hand before the HAYSTAC Phase I decommissioning [42],little additional scan time would have been required to either identify or promptly rule out an axion
signal. Therefore, an exclusion curve more representative of the BPM framework for comparison
purposes is given by the dashed, blue curve, where the prior update of the foremost LU candidate
is reset to unity. Other more minor differences between the dashed and solid blue curves due toparticular features of the HAYSTAC Phase I dataset not representative of typical haloscope data
155
The correspondence comes with the caveat that the data here has been subaggregated into bins of width ≈ 2
MHz, narrower than would be of practical concern to any experimentalist considering a haloscope run at nearby
frequencies.

203
are discussed in Sec. 7.4.3.
The figure of merit of an axion haloscope is the rate R at which it can scan a given frequency
window to a specified sensitivity (Sec. 2.5.2). The appropriate comparison to make between two
analysis frameworks is therefore the relative amount of time τrel taken to perform a given scan.
The dashed, gray line in Fig. 7.3b represents the time that would have been required to scan any
coupling gγ to the standard 90% exclusion (10% prior update) level using the FT (BT2) framework.
The line derives from theτrel ∝ gγ−4(7.45)

relation from Eq. (2.16) of scan times to couplings, and is normalized to the run time of HAYSTAC
Phase I. The inset of Fig. 7.3b shows all three curves in the vicinity of this line. Had the BPM? , its U = 10% aggregate priorframework been in place to recommend further scanning at νLU

update would have taken ≈ 36% longer to achieve with a threshold analysis. In the context ofdetection efforts that take years to cover unsatisfactorily little parameter space, this is a significant
enhancement available at no hardware or operational expense across a broad array of platforms.

7.4.2

Least-unlikely candidates

At the few frequencies where the BPM framework indicates weaker exclusion than the threshold frameworks, it does so with good reason. Most of these LU axion candidate frequencies had
excess power above threshold on an initial scan and not far below threshold on a rescan. This canbe seen from the spectral coincidence of several of these LU candidates, marked with orange dots
at their ML couplings, with the black dots indicating the performance of at least one rescan. The
LU candidates that were not rescanned corresponded to initial scan powers coming in just below
threshold.
In the bottom panel of Fig. 7.3a, the prior updates for the five LU candidates are plotted
normalized to the aggregate prior probability that an axion resides in the HAYSTAC Phase I
window. As discussed in Ref. [95], it is not obvious how to distribute one’s prior probabilityof an axion’s existence with respect to coupling gγ . However all five LU candidates have ML

204coupling strengths within a factor-of-1.5 range, wherein priors should not differ greatly. Therefore,?
the relative significance of the highest of these LU candidate prior updates (frequency νLU=

5.66417255 GHz) is noteworthy: it is more than two times higher than the remaining LU candidates
combined.
It should be emphasized that despite its exceptionally high prior update relative to therest of the dataset, it is overwhelmingly unlikely that this LU candidate is an axion. A roughattempt at estimating priors (Sec. 7.1.3) puts an optimistic probability of an axion in this bin
still below part-in-1,000, with more realistic estimates well below that. Secondly, as discussed inSec. 7.1.5, the BPM framework is inherently conservative with respect to exclusion in a real dataset,and consequently liberal in its identification of candidates. In particular, nonaxionic RF powerexcesses caused other spikes in the HAYSTAC Phase I power spectra. Many of these were manually
removed from the dataset on the grounds of supplementary evidence (e.g. failure to persist when
the auxiliary, weakly-coupled antenna was pulled out of the cavity) [107]. Since this LU candidatecame in under threshold on a third and final scan, it was rejected subject to the FT framework.
However, the more informative BPM framework suggests that significantly more than any of theother HAYSTAC Phase I frequencies, it merits further interrogation. Generously precluding thepossibility of nonaxionic excess power contamination, its probability of containing an axion at
its ML coupling is roughly the probability of 40 MHz of nearby, unscanned parameter space, or(1)

PLU /Pa = 20% of the 200 MHz scan window, containing an axion at that coupling.
7.4.3

Additional features of the Phase I dataset? discussed in Sec. 7.4, the dashed, blue curve of Fig. 7.3b
In addition to the LU candidate at νLU

also neglects two other features of the HAYSTAC Phase I dataset that appear in Fig. 7.3a. First,
the BPM reanalysis includes data from rescans over the upper 100 MHz of the scan window thatwere performed during Phase I, Run I of HAYSTAC [90, 106]. These rescans were discarded, andnew ones performed over some of the same frequencies, because an error in estimating the expectedaxion lineshape lowered their sensitivities below the levels required by the predetermined confidence205levels and initial scan sensitivities [78]. The extra rescan data itself is valid, however, and is thus
included in Fig. 7.3a and b, showcasing the flexibility of BPM to integrate all available information.
This extra data is not inlcuded in the comparison (dashed blue) curve, as the existence of extraneous
rescan data is not ordinary in haloscope analyses.
Secondly, a 3.94σ grand spectrum excess at grand spectrum bin 550 (ν550 = 5.59886155)
GHz was recorded in the initial scan. This excess occurs 0.3% of the way into the scan window,
where the number of contributing raw spectra (Sec. 6.1) was 14, considerably less than the typical
40. The lower number of spectra implies at once an increased susceptibility to systematic error(1)

and a reduced axion sensitivity η550 = 0.23. The reduced sensitivity implies that an axion here
has its prior update, Eq. (7.4), maximized at gγ ≈ 4.1. This excess ought to have been rescanned
during Phase I of HAYSTAC, but was not. As such, its positive prior update is included in the
BPM reanalysis. Its presence is indicated by the dark column at the left of Fig. 7.3a, and is solely
responsible for the failure of the aggregate prior update (solid blue line, Fig. 7.3) to vanish at high
KSVZ . Had the required rescans been performed, a source of excess would likely
couplings >∼ 3.2gγ? . As such,
have been discovered or rejected in short order, as in the case of the LU candidate at νLU

it is removed from the comparison (dashed blue) aggregate update curve.

7.5

Concluding remarks on statistical analysis frameworks for axion searches
Haloscopes belong to a class of experiments that are limited by the amount of time practicallyavailable to acquire data. As such, optimizing the extraction of the information content of their data
is a high priority. Analyzing haloscope data with threshold frameworks (Secs. 6.3, 7.3.1, and 7.3.2),
however, discards information pertinent to the presence or absence of an axion, and the alternative
frequentist approach of using p-values (Sec. 6.4) is even more problematic. The BPM analysis
framework straightforwardly applies Bayes’ theorem to incorporate the relevant information content
of haloscope power spectra into a posterior probability of an axion existing. Taken in ratio with theprior probability of that axion, the updated probability is seen to be completely equivalent to thestandard exclusion quoted in the literature for FT frameworks. Upon applying the BPM framework

206
to the Phase I data of the HAYSTAC experiment, we observe exclusion improved commensuratewith what under typical experimental conditions would translate to a 36% improvement in scantime — within the sensitivity error bounds for the HAYSTAC Phase I dataset, though independentof the contributing sources of error [106]. Additionally, the BPM framework better spotlights thoseregions of parameter space where the probability of an axion increases. For HAYSTAC’s data, asingle such frequency worthy of (but not demanding) further attention stands out.
The BPM framework may be straightforwardly applied to data from ADMX [91], HAYSTAC[107], and other haloscopes and similar dark matter searches [75, 94, 95, 108–111, 113, 116, 244, 251].
BPM will typically probe deeper couplings at no experimental cost while permitting experimentalists to operate nimbly, without the stricter constraints of preordained scan protocols. This latteradvantage is especially important for narrowband, tunable searches, in which the question of how to
optimize a scan protocol by responding in real-time to acquired data is of paramount importance.
Enhancements gained through operational efficiency will readily compound with those intrinsicallyavailable from BPM’s informational advantage over threshold-based analyses.
In the following chapter, the BPM framework will be used not in the context of a reanalysis,but on new data from the HAYSTAC experiment. Not only will it increase scan efficiency over FT
as predicted in this chapter, but its enhanced operational freedoms (Sec. 7.3), not available in thecontext of a reanalysis of already existing data, will compound that increase. In Ch. 9, we will see
the versatility of BPM as it is applied to a broadband search for ALPs extending as much as 17
orders of magnitude lower in mass than the QCD axions of this chapter and the next.

Chapter 8

A quantum-enhanced search for dark matter axions

In Sec. 2.5.3, we saw the strong need for improving haloscope scan rates in order to sweepout meaningful portions of parameter space on acceptable timescales. I have argued that the mostpromising advances are those that either (1) can be iteratively improved upon in order to continue
the decades-long trend of steadily increasing scan rates or (2) are easily and broadly applicable atlittle or no cost to the rest of the experiment.
In Ch. 5, we saw a direct demonstration of the squeezed state receiver (SSR) concept [208],
which achieved a factor-of-two scan rate enhancement in a realistic mock-haloscope experiment. Inline with objective (1) above, the SSR and quantum technologies for axion searches more generally
also promise scalability as transmission losses continue to come down [149, 218, 222] or reciprocal
elements are eliminated [252]. The history of improvements to haloscopic scan rate, wherein thevast majority of improvement over the past three decades has come from reducing noise as opposed
to increasing signal, further recommend the SSR approach.
In Ch. 7, we discussed an entirely separate and complimentary advance in the spirit of objective (2). There, the Bayesian power measured (BPM) analysis framework [126] improved the scan
efficiency of the HAYSTAC Phase I analysis (Sec. 7.4) by what would have been, under normal
operating conditions, ≈ 36%. While improved data handling and related scan strategy considerations are of secondary importance to fundamentally altering the noise regime in which modern
haloscopes operate, the advantage of improved data analysis is that it comes at zero hardware andnegative operational cost (Sec. 7.3).

208This chapter, which presents the work of [253], combines the advances of Chs. 5 and 7 topresent the most important result of this thesis: a live haloscope run directly improved by the SSR
apparatus and the BPM framework. We begin in Sec. 8.1 with the experimental setup and operationof the haloscope. Section 8.2 takes a detailed look into the data processing for the experiment,
and Sec. 8.3 presents analysis and results. Finally, Sec. 8.4 covers calibration measurements taken
periodically throughout the run to determine the noise level of the experiment. The combined
benefit of SSR and BPM amounts to a tripling of the scan rate for dark matter axions in Phase II156
of the HAYSTAC experiment over a mass range favored by recent theoretical projections [69, 70].
The data run reveals no signature of dark matter axions in the combined 16.96–17.12 and 17.14–
17.28 µeV/c2 mass window for axion-photon couplings above gγ = 1.38gγKSVZ , reporting exclusion
at the 90% level. The use of quantum measurement in particular in an axion search marks the firsttime quantum squeezing has been harnessed to improve the sensitivity of a search for fundamental
particles, inviting a new era of quantum-enhanced searches for physical phenomena.

8.1

Setup and operation
Phase II of the HAYSTAC experiment is shown schematically in Fig. 8.1. The axion cavity

shown in Fig. 8.2 resides in an LD250 BlueFors dilution refrigerator operated at a base temperatureof 61 mK and shown in Fig. 8.3. A solenoidal, superconducting magnet sources a spatially homogeneous B0 = 8 T field. The cavity sits at the center of the high magnetic field region, Fig. 8.4,
and has volume VC = 1.545 L (not counting space occupied by the tuning rod)157 and typicalQ0 = 47000 ± 5000 [96].
Two Josephson parametric amplifiers (JPAs; SQ and AMP, as in Chs. 4 and 5) hang from alower plate thermalized to the mixing chamber plate (Fig. 8.5a) approximately 1 m above the cavity
in a “field-free” region engineered by a counterwound bucking coil integrated into the magnet. The
156

The HAYSTAC data run discussed in this thesis is more properly Run I of Phase II. For brevity, I will leave off
the run designation herein, though at least one future Phase II run is planned.
157
Alternatively, the cavity could be treated as a 2 L volume with a commensurate decrease to the form factor,
Eq. (2.15).

209

Figure 8.1: Simplified HAYSTAC Phase II experimental schematic. A single signal generator
provides the local oscillator (LO) tone as well as the tones for pumping both JPAs. Each Josephsonparametric amplifier (JPA) has two ports: one for the input of probe tones and one where the signal
is input/output. The LO is set at half the pump frequencies via a frequency divider, and the relative
phase and amplitude of the pump tones are set using a variable phase shifter and attenuator on
the squeezer (SQ) pump line. Switches in the SQ and amplifier (AMP) pump lines (not shown) areused to toggle the JPAs on and off. Microwave circulators route signals nonreciprocally in orderto realize the time sequence of operations used to benefit from squeezing (Sec. 5.4). Circulatorswith a 50 Ω termination on one port act as isolators, shielding upstream circuit elements fromunwanted noise coming from further down the measurement chain. During data acquisition andcalibration measurements, signal and noise emitted from and reflected off the cavity are amplified
by a high-electron-mobility transistor (HEMT) amplifier at 4 K, fed into the radio frequency (RF)port of an in-phase/quadrature (IQ) mixer and mixed down to an intermediate frequency (IF),digitized (analog-to-digital converter; ADC), and recorded (personal computer; PC) where thepower spectral density is calculated. The cavity’s Lorentzian profile is monitored with reflection
and transmission measurements taken using a vector network analyzer (VNA), for which a portionof the output is split off before the IQ mixer. A switch that toggles between a hot (333 mK)
and cold (61 mK) 50 Ω loads is used for calibration measurements described in the text. Figure
reproduced from Ref. [253].

JPAs are further shielded by a 4-layer shielding can (niobium, Amumetal 4K, aluminum, Amumetal
4K), surrounded by three superconducting bucking coils (Fig. 8.5b). As demonstrated in Fig. 8.6,
this shielding is sufficient to reduce the solenoid-sourced flux through the JPAs’ SQUID loops tomuch less than one magnetic flux quantum, a roughly millionfold decrease in field strength fromthe nearby high-field region. The cavity is tuned by rotating the copper-plated tuning rod off-axis(Figs. 2.6b and 8.2).
At each operating point, The TM010 -like mode is stepped using an ANR240 piezoelectric

210

Figure 8.2: HAYSTAC cavity photo. The copper-coated stainless steel cylindrical cavity [96] hasheight 10 in and radius 4 in. The axion-sensitive TM010 -like mode is tuned by rotating the 1 intuning rod off-axis. The empty cavity has volume 2 L, which reduces to V = 1.545 L with the rod
present.

Figure 8.3: (a) Photo of the LD250 BlueFors dilution refrigerator housed at Yale before cabling andhaloscope components were installed. (b), (c) Side- and angled-view photos where key elements
labeled in (b) have been installed.

motor to rotate the tuning rod, whereupon a vector network analyzer is used to extract the cavity
resonance νc , loaded cavity quality factor158
QL =
158

The coupling rates κ are as defined in Ch. 5.

2πνc,κm + κl

(8.1)

211

Figure 8.4: (a)–(d) Radial (Br ), axial (Bz ), and total (Btot ) magnetic field specified values at
maximum field strength and R = 0 (a), 2 (b), 4 (c), and 6 (d) cm distances from cavity axis as afunction of distance along the cavity axis for the solenoid used in HAYSTAC Phases I and II. A
change in field component sign indicates a reversal of direction. The dominant contribution overthe entire region of the cavity of height 10 in and radius 4 in comes from Bz , the component usefulfor axion detection. The maximum field had to be lowered from 9 to 8 T for Phase II due to damage
sustained in magnetic quenches during Phase I operation [42, 90].

and coupling factor
β=

κm
κl

(8.2)

via pairs of transmission and reflection measurements. In order to center the amplification andanalysis bands on cavity resonance, a single microwave generator sets the two JPA pump frequencies to νp = 2νc and the same tone is further split off and frequency-divided to become the local
oscillator (LO) used for homodyne measurement about νc . The relative pump phase θ and ampli-

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Figure 8.5: (a) Mixing chamber stage layout shown in SolidWorks. The gold-plated mixing chamber
plate is thermalized to a gold-plated lower plate from which hangs the JPA housing and below that(not shown) the cavity. The JPAs are placed in the nominally “field-free” region, and the threegray rings show the locations of static superconducting bucking coils which are used to cancel theremaining field through Lenz’s law. The four circulators required to implement the SSR in practice(Fig. 5.8) are housed in magnetic shielding above the lower plate. (b) Photo of the JPA magnetic
shielding. The static bucking coils sit outside a four-layer magnetic shield; from inside-out: niobium,
Amumetal 4K, aluminum, and Amumetal 4K (visible layer). The Amumetal 4K, from Amuneal
Manufacturing Corp., is a high permeability material around the temperatures where the aluminumand niobium undergo their superconducting phase transitions, so as to minimize flux vorticies inthe superconducting layers when at base temperature. The whole configuration reduces the 8 T
magnetic field 1 m below to less than one part in 106 of its peak value (Fig. 8.6).

tude are optimized to maximize squeezing using an electronically actuated variable phase shifterand attenuator. Axion-sensitive voltage fluctuations VX (t) are collected for τ = 3600 s before being
Fourier transformed and having their power spectral densities (PSDs) computed.
As in Ch. 5, the SQ and AMP are coupled to the axion cavity in an SSR configuration via
a microwave circulator. The SQ prepares a squeezed vacuum state, which is coupled into theaxion cavity and subsequently measured noiselessly using the AMP. The setup delivers 4.0 dB of
off-resonant vacuum squeezing after the state is degraded by transmission losses and added noise(Footnote 90), yielding a 1.9-fold159 scan rate enhancement beyond what could be achieved at the159

This improvement comes relative to the optimal unsqueezed case for the system accounting for improper tuning
rod thermalization (Appx. C) in both cases. It also accounts for our measured added noise and the fact that thehomodyne measurement leaves a small range of frequencies near DC too noisy to use in both cases. In theory, theselost frequencies imply a higher overcoupling for optimal performance both with and without squeezing, though the213

Figure 8.6: Measurement of JPA tunability with magnetic field on and off. One of the flux-pumped
JPAs is tuned by a DC current source with the large magnetic field used to search for axions at 0
and 8.4 T. Tuning the magnetic field on causes only a very minimal shift in the tuning structureof the JPA, confirming the proper functioning of the magnetic shielding.

Figure 8.7: (a)–(b) Voltage fluctuation (VX ) histograms as a function of the phase θ between the
SQ and AMP pumps first with the SQ off (a) so that it acts like a mirror and there is no preferred
phase. With SQ on (b), the variance is minimized for θ = π/2 and 3π/2. Figure reproduced from
Ref. [253].

effect is slight, and neglected here for both cases. The calculation assumes an analysis band set to a fixed number of(overcoupling-dependent) cavity linewidths.

214
quantum limit. This measured squeezing,160 shown in Fig. 8.7, is consistent with the loss and added
noise measurements discussed in Sec. 8.4.

Figure 8.8: Illustration of the squeezed state receiver-equipped haloscope showing the transformation of the vacuum state in quadrature space. A vacuum state, whose Wigner function (color
surface) [254] is symmetric in quadratures X̂ and Ŷ , is sourced as Johnson-Nyquist noise from a
50 Ω microwave termination (black box) at 61 mK. It is routed by a nonreciprocal element (Circ)to the SQ JPA which squeezes the X̂ quadrature. The squeezed state may then be displaced by ahypothetical axion field a (red lines) in the axion cavity (Cav). It is subsequently unsqueezed by the
AMP JPA, in the process amplifying the axion-induced displacement. The resulting state is readout by a conventional readout chain led by a HEMT amplifier and digitized (spectrum analyzer;
SA). Figure reproduced from Ref. [253].

The data acquisition procedure at each tuning step in the initial scan161 is essentially thatof the mock-haloscope experiment discussed in Sec. 5.4, except here the cavity really is tuned insearch of the hypothesized axion signal, and the live acquisition time is brought up from 0.32 s to
τ = 3600 s as in Sec. 8.1. Figure 8.8 illustrates the journey of the squeezed state in quadrature space
as it propagates through the SSR-equipped haloscope. The JPAs in this picture transform phasespace, turning the vacuum state into a squeezed state and then back into a vacuum state.162 An
160

The measurement protocol is the same as that for Fig. 5.5, here showing SQ off in addition to SQ on data.
Because squeezing does not improve the maximum single-frequency sensitivity of the haloscope, it is not usedduring rescans. In theory, a marginal benefit could have been achieved for adjacent bins by squeezing even whilecritically coupled (Sec. 5.6).
162
In practice, the gain of the AMP is turned up higher than that of the SQ, so the final state is really ellipticalalong the opposing axis to the initial squeezed state. For illustrative purposes, however, it is helpful to think of the
AMP as simply undoing the transformation of the SQ.
161

215
axion, if present, imparts a displacement, and the portion of that displacement along the squeezed
quadrature stands out more strongly from the noise than in the case without squeezing.163

Figure 8.9: Advantage conferred by squeezing calculated with parameters used in HAYSTAC. Blue
curves show Johnson-Nyquist noise Sr reflected off of the cavity as a function of detuning δν fromcavity resonance. Red curves show cavity Johnson-Nyquist noise Sc which originates in the cavity.
Solid (dashed) lines indicate HAYSTAC Phase II (Phase I) operation. The combination of squeezing
and higher cavity coupling increases the bandwidth over which the haloscope is sensitive to axions.
Figure reproduced from Ref. [253].

As we saw in Ch. 5, squeezing improves the bandwidth over which the apparatus is sensitive
to an axion, rather than the peak sensitivity obtained at cavity resonance. In the setup of Fig. 8.1,
the reflected noise is sourced from a 50 Ω termination held at base temperature 61 mK, which
dominates away from cavity resonance. The ratio of the signal power that would be delivered byan axion at any given frequency to cavity Johnson-Nyquist noise Sc at that frequency is spectrally
constant. Because squeezing selectively removes reflected noise Sr , it is beneficial to overcouple thecavity’s measurement port by setting β, Eq. (8.2), to twice the deliverable squeezing S, Eq. (4.4), as
in Ch. 5. In Fig. 8.9, the dashed lines show cavity thermal Johnson-Nyquist in red and reflected noisein blue with β = 2, the ideal value for a quantum-limited haloscope. Cavity noise, and thereforea hypothetical axion signal, is only dominant over the narrow band shown by the upper black
163
This coherent picture of the axion displacement does a good job showing what each individual realization of
the axion field does. Figure 5.1 presents a complimentary picture: there, many coherent displacements of the sortshown in Fig. 8.8 average together to add variance to the squeezed or vacuum state over the course of a measurement.
In both pictures, the essential conclusion is that the axion is disproportionately more visible along the squeezed
quadrature.

216
arrow. The solid lines show the operating conditions for this work, where squeezing is employed
and the cavity is overcoupled at β = 7.1.164 Squeezing reduces the reflected noise by 4.0 dB, andincreasing β to 7.1 increases the cavity bandwidth. Without squeezing, β = 7.1 would make cavity
reflected noise dominant at all frequencies. Squeezing this noise below the quantum limit yields
near-maximal visibility extended far off resonance.

8.2

Data processing and analysis
The HAYSTAC Phase II data run used the SSR apparatus of Sec. 8.1 to probe over 70

MHz of well-motivated parameter space [69, 70] in half the time that would have been required forunsqueezed operation, saving approximately 100 days of scanning. Initial data acquisition occurredfrom September 3 to December 17, 2019, covering 4.100–4.178 GHz and skipping a crossing witha transverse electric (TE) mode at 4.140–4.145 GHz which does not couple to the axion [96].
A total of 861 spectra were collected as part of the initial scan, of which 33 were cut due tocavity frequency drift, poor JPA performance, or an anomalous power measurement in a probe
tone injected near cavity resonance. The cuts made on the initial spectra are shown in Fig. 8.10.
Analysis of the initial scan data yielded 32 power excesses that merited further scanning, consistent
with statistical expectations. 508 rescan spectra were collected from February 25 to April 11, 2020,
with 3 cut. None of the power excesses from the initial scan persisted in the analysis of rescan
data. Calibration measurements taken at each tuning step are described in Sec. 8.4.
The data processing for an axion haloscope, described in Sec. 6.1 and based upon Ref. [78], is acomplicated procedure prone to human error and confirmation bias. Furthermore, it requires modifications (Sec. 5.7) before being applied to a system performing single quadrature measurement andusing squeezing. To mitigate human error, HAYSTAC used two separate and semi-independent165
processing and analysis teams, led by Kelly Backes at Yale and myself at JILA, respectively. The
164

The overcoupling here is slightly higher than twice the delivered squeezing S because the cavity noise is higher
than the temperature of the fridge (Sec. 8.4). When not squeezing, the optimal overcoupling is actually β ≈ 2.8,
slightly higher than the value β = 2 plotted in Fig. 8.9. Fully accounted for, the cavity noise has negligible affect on
the scan rate enhancement achieved via squeezing. For a quantitative discussion, see Appx. C.
165
Each team was never allowed to look at the other’s processing and analysis code, but other communication was

217

Figure 8.10: Raw spectrum cuts represented by black, vertical lines for the 861 raw spectra (horizontal axis, in chronological order) in the HAYSTAC Phase II initial scan. Spectrum cuts wereapplied for spectra with unsuccessful squeezing (“noSQ”), cavity resonance drift as measured inreflection (“∆νrfl ”) or transmission (“∆νtx ”), anomalous average subspectrum power (“Pspec ”), or
anomalous average (“Ptone ”) or standard deviation (“σPtone ”) of heights of a probe tone situatedalongside the spectra and measured throughout their acquisition. Any one cut condition being met(“any”) was deemed sufficient to cut a spectrum. In total, 33 cuts were made on the initial spectra,
and 3 on the rescan spectra (not shown).

two teams ultimately agreed in the final prior updates Ui (gγ ) (Sec. 7.1.1) to better than 1%, withthe official results reported in Sec. 8.2 being the update-space average of the analyses.
This section follows the JILA processing and analysis from data acquisition to exclusion,presenting a comparison to the frequentist threshold (FT) analysis used in Phase I, Run I [106].
The improved informational efficiency and greater operational freedom of the BPM framework
together improve scan rate by 52%. Unlike for the Phase I data of Sec. 7.4, the final results are
remarkably clean, with no prominent least unlikely (LU) candidates of note.
permitted.

218
8.2.1

Subspectra and raw spectra

The 861 initial scan plus 508 rescan raw spectra166 are each acquired at a sample rate of 10
MS/s in 5 s acquisitions. These 5 s acquisitions are divided in code into 500 separate 10 ms chunks,
which are Fourier transformed at ∆b = 1/(10 ms) = 100 Hz Fourier bin resolution and rotated into
the AMP’s amplified quadrature prior to having their PSDs computed per Step 3 of Sec. 6.1.167 A
1 hr raw spectrum acquisition therefore collects and averages 360,000 subspectra.168 To save disk
space, only the raw spectra themselves are recorded.169
Figure 8.11 shows raw spectra acquired in the initial scan. The representative spectrum
in 8.11a was randomly chosen as spectrum no. 702. In order to follow the processing of a single
spectrum, the same spectrum is in several subsequent plots throughout this section. The irreversible
ambiguity between the AMP’s signal and idler inputs means that this and subsequent individual
spectra are mapped symmetrically about the LO (Sec. 5.7). The gap in the middle of this andall other spectra comes from low-frequency noise present in our homodyne measurement.170 The
predominant feature in the raw spectra is the large-scale structure owing primarily to frequencydependent gains in the amplification chain and the low-pass filtering of the data before digitization.
This broad spectral structure must be filtered out per Step 4 of Sec. 6.1.
In order to filter out large-scale structure, the raw spectra are grouped into three batches.
This is necessary because, over the course of the months-long acquisition, room temperature electronics were reconfigured to improve performance. These reconfigurations in turn changed the166
Some of these occurred to fill in small gaps in the initial scan, and therefore could alternatively be thought ofas belonging instead to that group. Since they were acquired with the rescan data, we tally them there.
167
For phase-sensitive measurement, Step 2 of Sec. 6.1 is replaced with the rotation of the measured quadratures
into the amplified and deamplified quadratures, the latter of which is discarded. Step 1 is discussed in Footnote 168.
168
The Fourier domain quadrature voltages are complex, so in practice the real and imaginary parts have their
PSDs separately computed and summed per Step 1 of Sec. 6.1. There are thus 720,000 independently measured
powers being averaged at each 100 Hz Fourier bin.
169
If the full timestream dataset for the above procedure were saved, it would occupy roughly 800 TB: not an
impossible number, but an inconvenient one, especially given that the raw spectra in principle contain the fullaxion-pertinent information content of the measurement per Sec. 7.2.
170
The lost low-frequency data is a practical disadvantage of the phase-sensitive measurement relative to the phaseinsensitive one. It is counterbalanced by the fact that phase-insensitive measurement operates off resonance, where
lower JPA gain gives rise to higher added noise. Unlike for the phase-sensitive versus -insensitive effects considered inSec. 5.1.3, there is no fundamental reason√ for these two effects to cancel out. In practice, it is reassuring that neither
rises to the level of even the smaller ( 2) impacts on scan rate discussed in Sec. 5.1.3, and that they work in opposite
directions.

219

Figure 8.11: (a) Individual raw spectrum no. 702 plotted in the RF band along with the SavitzkyGolay (SG) filter for its batch of spectra (batch 3). The SG filter shown is for removing structurecommon to all spectra within the batch. A subsequent filtering step, shown in Fig. 8.12a for thisspectrum, will remove the remaining structure. The spectrum is symmetric about its center as aconsequence of the phase-sensitive amplification. (b) All raw spectra shown together in the IF band.
Batches are determined by changes made to the measurement chain throughout the months-longinitial scan data acquisition. IF/RF subscripts are applied to frequency axes in these and otherplots for emphasis.

large-scale spectral structure (or “spectral baselines”) of the raw spectra.171 All 861 raw spectra
are shown together in Fig. 8.11, where the differences between batch 1 and batches 2 and 3, whichhave similar baselines, are evident. Each baseline is divided by a Savitzky-Golay (SG) filter (d = 10,
W = 500) applied to its entire batch in the intermediate frequency (IF) band. After the filter isapplied, bins which average in the IF (where an axion signal will not line up between different
spectra) to a power excess above 4.5σ are cut along with their close neighbors for all spectra withina batch.

8.2.2

Processed spectra

A second SG filter (d = 4, W = 500)172 is applied to each spectrum individually in theradio frequency (RF) band to remove the remaining large-scale structure which remains dominant
171
The three batches are each chronologically contiguous. The first contains 201 spectra, the second 74, and thethird 586.).
172
For rescans, the second SG filter has d = 6, following Ref. [78].

220

Figure 8.12: (a)–(d) Individualized SG filtering. The raw spectra PRF , normalized by the collective
filt and with 1 subtracted out, are plotted for four different
IF SG filters for their respective batches P̄IF
raw spectra. Also plotted are their their individualized SG filters. The structure of the individual
spectra all look different from one another because common structure was removed in the first SG
filtering stage. The individualized SG filters will be divided out to obtain the processed spectra ofSec. 8.2.2.

over the statistical fluctuations (plus possible axion-induced structure). These SG filters, shown
alongside four IF SG-filtered raw spectra in Fig. 8.12, look very different for each individual spectrum even within a batch (compare Fig. 8.12a and Fig. 8.12d, both in batch 3). The dissimilarities
among individual spectra indicate that the IF SG filtering of Sec. 8.2.1 successfully removed shared
structure.
Upon dividing out the individualized SG filters, we are left with the processed spectra. Figure
8.13a shows processed spectrum no. 702, which we have been following through the data processing.

221

Figure 8.13: (a) Individual processed spectrum no. 702 together with histogram of normalized
power excesses. The frequency-averaged inter-processed spectrum√ standard deviation hσmeas iν
(pink line) agrees well with the expected standard deviation 1/ τ ∆b (black, dashed line), andthe histogrammed power excess counts for spectrum no. 702 are seen to follow the corresponding
expected normal (N ) distribution. (b) Histogram of all renormalized processed spectra obtained by
dividing the normalized power excesses of (a) by hσmeas iν . The distribution of all processed spectra
is well approximated by the expected Gaussian distribution, and has mean 0, standard deviation 1
as expected. (c) Waterfall plot of many processed spectra near spectrum no. 702 (red) lined up in
the RF band. The black spectrum was one of the 33 spectra cut from the initial scan (Fig. 8.10).

√
It agrees very well with the expected mean 0, standard deviation 1/ τ ∆b = 0.0017 Gaussian,173
as does the collection of all processed spectra excesses shown in Fig. 8.13b. If noise were not averaging down properly in the experiment, the Gaussians would have larger-than-expected standard
deviations. In Fig. 8.13c, many processed spectra are shown together, with no. 702 highlighted in√
There are several √subtle factorsof two that go into Fig. 8.13.√ First, the standard deviation of 1/ τ ∆b could√ √really be written as (1/ τ ∆b ) × 2/ 2, where the multiplicative 2 comes from the standard deviation of the firstχ2 distribution and the divisive factor of two comes from the hidden factor of two in the number of independent
contributions
to each raw spectrum(Footnote 168). Separately, the error bars in the histograms in Fig. 8.13a and b√√
are 2 × counts, where the 2 multiplies the Poisson standard deviation because of the duplication of all measured
powers about the LO by the AMP (Sec. 8.2.1).
173

222
red and a cut spectrum in black. Observe that the spectral step size between processed spectra
is substantially smaller than the spectra themselves are, which ensures smooth coverage of thescanned frequency window (Sec. 7.1.4). Bins exceeding 6σ are cut from the processed spectra per
the procedure in Ref. [78].

8.2.3

Rescaled spectra

Figure 8.14: (a) Rescaling visibility (red line) for spectrum no. 702. Also shown are what therescaling would be if the AMP gain GA (ν) maintained its maximum value of GA (νp /2) across theentire analysis band (black, dashed line) and if in addition the cavity did not replace any reflected,
squeezed noise with internal, unsqueezed noise (|χml (ν)|2 = 0, black, dotted line). (b) Rescaled
spectrum no. 702, obtained from dividing processed spectrum no. 702, Fig. 8.13a, by the rescaling
visibility in (a). The rescaled spectra have the special property that bin containing 100% of thepower of a KSVZ axion would have mean 1.

The processed spectra are next each rescaled to account for their sensitivity to the axion.
KSVZ , shown in red in Fig. 8.14a for spectrum
Specifically, the single-bin KSVZ axion visibilities α1-bin

no. 702, are divided out from the processed spectra to produced the rescaled spectra. The rescalingthat would have been applied if the AMP maintained its maximum gain across the entire analysis
band is shown as a black, dashed line. The fact that the black, dashed line and the red line
nearly coincide indicates that JPA bandwidth is not at present a limiting factor, even with thecavity being strongly overcoupled to improve bandwidth upon squeezing. The black, dotted line,

223
shows the rescaling that would be applied if the cavity still sourced the same axion signal, but
without dissipating any of the external noise and replacing it with unsqueezed vacuum. As such,its deviation from the red line, largest towards the center, indicates how much of the squeezed stateis being replaced with cavity fluctuations. It shows that we are accounting for the fact that not allof the squeezed state (independent of transmission losses) makes it to the AMP.
Rescaled spectrum no. 702 is shown in Fig. 8.14b. Recall from Step 5 of Sec. 6.1 that therescaled spectra have the special property that a KSVZ axion whose power falls entirely into agiven 100 Hz bin would shift the bin’s expectation value from 0 to 1. Thus, bins closer to the centerof each raw spectrum, where axion transmission through the cavity is strongest, should be the mostsensitive (lowest rescaled standard deviation), as is the case in Fig. 8.14b. The reason for rescaling
the spectra to have uniform mean in the presence of an axion, with the sensitivity of each bin now
encoded in its standard deviation, is to facilitate the maximum likelihood (ML) estimation used to
combine individual spectra.

8.2.4

Combined spectrum

The rescaled spectra are summed with ML weights in order to form the combined spectrum
as in Step 6 of Sec. 6.1. The combined spectrum, shown in Fig. 8.15a, has the same property as therescaled spectrum: a single-bin, KSVZ axion has mean 1. Near the edges of the scan, the expected
standard deviation obtained from Eq. (6.7) grows very large, indicating insensitivity to the axion.
This is also true around the TE mode crossing near the center of the scan window. Mainly, thesensitivity profile is determined by the number of contributing raw spectra (shown in gray on theright axis) to each combined spectrum bin.
The combined spectrum can alternatively be viewed as in Fig. 8.15b, such that each bin has
standard deviation 1 regardless of the axion-sensitivity (and whether or not there is an axion).
The sensitivity to the axion is then encoded in the mean of the axion distribution for each bin(not shown). The representation of Fig. 8.15b is better for examining the statistics of the combined
spectrum. We find that its measured standard deviation matches the expected value, and it is well224
described by the expected Gaussian distribution plotted in the right panel.

Figure 8.15: (a) HAYSTAC Phase II initial scan combined spectrum, obtained by summing the
rescaled spectra with maximum likelihood (ML) weights such that a single-bin KSVZ axion hasmean 1. The standard deviation σ  (orange line) therefore encodes sensitivity to the axion. Grayline/right axis shows the number of raw spectra that contribute to each combined spectrum bin, withhigher axion sensitivity typically occurring where more spectra contribute. Small gaps in the initialscan combined spectrum were due to spurious, nonaxionic excesses which did not appear during
rescans. The large gap beginning at 4.14 GHz is due to a transverse electric mode. (b) Alternate
representation of the combined spectrum together with histogram of powers. The expected standard
deviation σ  has been normalized out to give all bins unit standard deviation. The measured
standard deviation σmeas for the combined spectrum (pink line) matches this expectation (black,dashed line), and the data agrees well with the expected normal distribution.

225
8.2.5

Grand spectrum

To account for the axion lineshape, we apply ML weights to overlapping sets of combined
spectrum bins as specified by Step 8 of Sec. 6.1.174 The resulting (normalized, as in Fig. 8.15b forthe combined spectrum) grand spectrum is shown in Fig. 8.16, where filter-induced correlations arecorrected for according to Step 9 of Sec. 6.1. As in Fig. 8.15b, the axion sensitivity is encoded in
the mean value µKSVZ that a KSVZ axion would impart to each bin, if its rest mass resided there.
Here, we use the standard halo model lineshape of Eq. (2.14).

Figure 8.16: HAYSTAC Phase II initial scan grand spectrum, which accounts for the axion lineshape. The sensitivity to the axion is encoded by the mean µKSVZ (brown line) that a KSVZ
axion with rest mass in each grand spectrum bin would have. Of particular initial interest was the
spike far above µKSVZ at 4.11190395 GHz, which did not persist during rescanning. The standard
deviation of the grand spectrum (pink line) matches the expected value (black, dashed line).

174
The BPM framework makes it more natural to skip Step 7 of Sec. 6.1, and simply perform Step 8 on the combined
spectrum. This is equivalent to setting nc → 1 and nr → nc nr in Sec. 6.1, and enhances sensitivity to the axion veryslightly while reducing complexity.

226

8.3

Results
The grand spectrum of Fig. 8.16 is combined with the rescan data (not shown) to report a

constraint on the existence of axions within the scanned window. We will first discuss in Sec. 8.3.1
the results of the JILA analysis, using the BPM framework to examine the prior updates in thefull two-dimensional space of frequency and coupling, the LU candidates, and the comparativeadvantage over a frequentist analysis. The official results, combining the JILA and Yale analyses,
are presented and discussed in Sec. 8.3.2.

8.3.1

JILA analysis results

Figure 8.17: (a) Two-dimensional (axion frequency νa , axion-photon coupling gγ ) map of subaggregate prior updates Us from the JILA analysis, with the 90% exclusion line (Us = 10% contour)
shown as the solid, blue line. The solid, red line shows the exclusion obtained using an frequentist
threshold (FT) framework, with rescans marked as black dots. The bottom inset shows the top five
least unlikely (LU) candidates with the ML couplings denoted by the corresponding orange circles(1)
in the main plot. None of the LU candidates posterior probabilities PLU occupy even 1% of thetotal prior probability Pa of the window at their ML couplings. (b) The blue (red) solid line shows
the aggregate prior update U (exclusion E) for the > 70 MHz region included within the two sets ofdashed vertical lines spanned by dashed horizontal lines in (a). The dashed, blue horizontal lines
in (a) indicate 90% exclusion for the Bayesian power-measured (BPM) framework, and the dashed,
red line does so for the FT framework. The τrel axis (dashed, gray line at 90% exclusion) indicatesthe relative amount of scan time (+52%) it would have taken to achieve the BPM exclusion with
the FT framework. The inset shows a closeup on the two frameworks’ aggregate exclusions near
90% exclusion.

227
Applying the BPM framework of Ch. 7 to the HAYSTAC Phase II dataset produces the twodimensional prior update map of Fig. 8.17a. Comparing to the Phase I reanalysis of Fig. 7.3 reveals
a much cleaner result: there are no regions of very high subaggregated updates Us , and the LU(1)candidates, shown in the inset, never reach even PLU /Pa = 1% (compare to 20% in Fig. 7.3). The
window quoted for aggregate exclusion in Fig. 8.17b is that spanned by the dashed horizontal lines
denoting 90% aggregate exclusion in Fig. 7.3a. In both figures, blue lines indicate BPM results andred lines FT175 results.
The aggregate exclusion properly accounts for the look-elsewhere effect with respect to exclusion for both Bayesian (Sec. 7.1.2) and frequentist (Sec. 6.3.2) analyses, and reveals a 52% more
efficient scan using BPM. In particular, the frameworks are compared at the conventionally used
90% mark, where the scaling of relative scan time with coupling, Eq. (7.45), converts the couplings
excluded at the 90% level over the whole range to relative scan times. The BPM framework’s advantage was aided by its operational flexibility (Sec. 7.3), which was used to rescan the 4.11190395
GHz initial scan excess (Fig. 8.16) multiple times.
The JILA results are not the official results of the Phase II HAYSTAC data run. Rather,
they were averaged with the prior updates of the semi-independent Yale analysis. The two analyses
agreed at better than the 1% level before averaging.

8.3.2

Official HAYSTAC Phase II results and conclusions

Averaging the JILA (Fig. 8.17) and Yale (not shown) analysis results, the final HAYSTAC
Phase II results constrain the existence of axions with ma between 16.96–17.12 and 17.14–17.28µeV/c2 . The final results exclude axions with gγ ≥ 1.38gγKSVZ . Figure 8.18a and b show the mostimportant quantities plotted in Fig. 8.17. The results from the HAYSTAC quantum-enhanced data
run are shown alongside other axion haloscope exclusion curves in Fig. 8.17c.
Quantum squeezing has allowed the HAYSTAC experiment to achieve a breakthrough insensitivity by conducting the first sub-quantum-limited search for new fundamental particles. The
175

The FT framework results shown here for comparison used the parameters of Ref. [106].

228

Figure 8.18: (a) Subaggregated prior updates Us in the full two-dimensional parameter space ofaxion frequency νa and coupling gγ achieved with the BPM analysis and shown in grayscale. The
10% prior update contour is shown in solid blue. The corresponding 90% aggregate exclusionlevel of 1.38gγKSVZ is shown as a dashed blue line. (b) Aggregate prior update U, Eq. (7.7), as afunction of coupling over the entire window covered by the dashed blue line in (a). (c) Resultsof this work shown in blue alongside previous exclusion results from other haloscopes shown in
red [89, 91, 93, 99, 102–105], orange [98, 101], purple [97, 100], and green [108]. Previous HAYSTAC
results [106, 107] are shown in faded blue. The QCD axion model band [59] is shown in yellow,
with the specific KSVZ and DFSZ model lines shown as black dashed lines. Figure adapted from
Ref. [253].

SSR, capable of delivering 4.0 dB of noise variance reduction relative to vacuum with the large
magnetic field turned on, has helped achieve world-leading sensitivity to axion dark matter inthe 10 µeV/c2 mass decade. As in Ch. 5, the current achievable squeezing is limited primarily
by microwave transmission losses between the two JPAs. As the development of superconducting
quantum technology [218, 222] continues to improve microwave transmission efficiency, the scan

229
rate enhancement from squeezing will improve even without new experimental techniques [208],
and advances in measurement techniques that remove nonreciprocal elements [252] offer a meansof eliminating the majority of the transmission loss in the current SSR configuration.

8.4

Calibration measurements

Figure 8.19: Model used for HAYSTAC Phase II noise calibration measurements. The system ismodeled for the purpose of calibration measurements with a cascaded series of circuit elementsthat perform discrete operations. A variable temperature stage (VTS) sources spectral density Sin .
Boxes with Greek letters model loss, amplifiers (triangles) have the gains listed within them, andall added noise NA is referred to the AMP. The cavity has resonance νc and coupling rates κm andκl , together giving rise to the susceptibility matrix elements χij (ν). The measured output spectral
density is Sout .

Characterization of the sensitivity to the axion requires in-depth measurements of the system
noise. To do this, we perform calibrations once every nine tuning steps. We use the model inFig. 8.19 to calculate all quantities of interest. The model is chosen to balance level of detailagainst the number of independent measurements and reasonable assumption that can be made on
the setup of Fig. 8.1. All spectral densities and gains in this section are single-quadrature unlessotherwise specified.176 The model consists of a variable temperature stage (VTS) modeled afterthat in Ch. 4 which sources Johnson-Nyquist noise with temperature-dependent spectral density Sin .
176
The “1Q” subscripts of previous chapters where there was cause for ambiguity have here accordingly been
dropped.

230
This noise is routed through the SQ (inferred squeezing GS ≤ 1)177 to the cavity with susceptibility
matrix elements (Sec. 5.1.1) χij determined from its resonance νc , and measurement and (fictitious)
loss port coupling rates κm and κl , respectively. From the cavity, the state goes through the AMP(gain GA ) and commercial amplifier chain (gain GH 0 ), led by a high-electron-mobility transistor(HEMT) amplifier. The system added noise spectral density NA is referred to the input of the AMP,
and any transmission loss that the state passes through between circuit elements is modelled withpower transmissivities λ, ρ, and α as shown in the figure. The fluctuation-dissipation theorem [255]
ensures that whatever fraction of the state is dissipated in these partially transmissive elements is
replaced with Johnson-Nyquist noise at the fridge spectral density Sf . Single-quadrature spectral
densities S (or N , which is used to represent added noise) have dimensions of photon number per
unit bandwidth per unit time (i.e. dimensionless energy), corresponding to physical temperatures
T at frequency ν via
1
S=
2



1
1
+ hν/k T
B
2 e
−1



1
= coth
4



hν
2kB T


,

(8.3)

half the two-quadrature value. The Johnson-Nyquist noise spectral density of the cavity is Sc ≥ Sf ,to account for nonideal thermalization of the tuning rod.178 Writing down the output spectral
density Sout is therefore a simple matter of cascading the multiplicative and additive effects of all
the elements:Sout =GH 0 GA NA + (1 − α)Sf + α 1 − |χmm |2 Sc(8.4)
2+ |χmm | [(1 − ρ)Sf + ρGS (Sin λ + (1 − λ)Sf )].

This equation will enable the calibration calculations in Secs. 8.4.1 and 8.4.2.
177

In this chapter, GS is the “gain” along the squeezed quadrature, which is less than 1. It is ideally the inverse of
GS,1Q ≥ 1 of previous chapters, which represented the single quadrature gain in the amplified quadrature of the SQ,
but may be increased (i.e. worsened) due to distortion or added noise of the squeezer.
178
The cavity spectral density cannot be determined from simply knowing the temperature of the tuning rod, as
the tuning rod and cavity walls both participate in sourcing the “cavity” spectral density Sc . Therefore, though wemeasure the tuning rod to be at Trod = 225 mK by slowly ramping up the fridge temperature and seeing where any
excess PSD due to the rod vanishes, we must still independently measure Sc .

231
8.4.1

Squeezing and cavity spectral density measurements

The first step in the calibration protocol is a set of three power spectrum measurements(Fig. 8.20) to determine two experimental parameters: the spectral density Sc emerging from thecavity and the IF frequency (νIF )-dependent squeezing GS (νIF ) inferred over the analysis band.
Measurement 1 is taken off cavity resonance with the squeezer turned off. Measurements 2 and 3 aretaken on cavity resonance with the squeezer turned on and off, respectively. All three measurements
are taken with Sin = Sf . From measurements 1 and 3 we calculate
( "
#
!
)
(3)

Sout
(1)
(3)
2
Sc =
GA (NA + Sf ) (1) /GA − NA − (1 − α)Sf /α − |χmm | Sf / 1 − |χmm |2 , (8.5)
Soutfrom Eq. (8.4) with GS = 1, where numerical superscripts denote quantities that are expected tovary between measurements 1–3. Other quantities are modeled as stable between measurements
and the AMP gain is measured independently. Averaged over all operating points,
Sc = 0.41 ± 0.02.

(8.6)

For a cavity at the base temperature of 61 mK, we would expect Sc = Sf = 0.27 from Eq. (8.3),
close to the zero-temperature value of 1/4. For a quantitative treatment of how the increased cavity
spectral density affects the scan, see Appx. C. From measurements 2 and 3, with Sc determined,
we calculate

(" "
GS =

(2)
Sout
(1)
GA (NA + Sf ) (1)
Sout

#

!
(2)
/GA − NA − (1 − α)Sf#)
− 1 − |χmm |2 Sc /|χmm |2 − (1 − ρ)Sf /(ρSf )

/α
(8.7)

from Eq. (8.4). Using Eq. (8.7), we infer a typical179
GS ≈ 0.1.
179

(8.8)

In practice, states squeezed beyond the ideal deliverable squeezing set by the transmission loss do not differ
greatly at the output. This is essentially why the scan rate enhancement in Fig. 5.2b rapidly levels out with respect
to GS,1Q . Fluctuations in the calibrated quantities, Eqs. (8.5), (8.7), and (8.9), tend to correlate or anti-correlate insuch a way as to roughly preserve the total noise level.

232

Figure 8.20: (a) Sample calibration measurements taken at a representative operating point with
(1, red) SQ off/off resonance, (2, green) SQ on/on resonance, and (3, blue) SQ on/off resonance.
Sets of 300 adjacent ∆b = 100 Hz frequency bins are averaged together. Some of the large-scale
structure owes to the spectral of the commercial amplification chain, and cancels out when ratioing
these measurements to obtain parameters of interest. (b) Calculated values of the cavity spectra
density Sc (in single-quadrature quanta) and squeezer gain GS ≤ 1 from the measurements in (a).
The squeezer gain is treated as frequency-dependent. The low-frequency rise in squeezer gain may
be due to an imperfect accounting of added noise by the added noise measurement, causing a slightincrease in inferred squeezing.

8.4.2

Added noise measurement

The second calibration measurement is a hot-cold load measurement with SQ off (GS = 1)
similar to the protocol used in previous HAYSTAC results [90],180 where the cold load is at thedilution refrigerator base temperature and the hot load is maintained at 333 mK and monitored bya Magnicon SQUID-based temperature sensor. Equation (8.4) gives us the IF frequency-dependent,
single-quadrature added noise NA (νIF ) referred to the input of the AMP asn hh
i


 i
h
NA =
λSin
+ (1 − λ)Sf ρ + (1 − ρ)Sf |χ̃mm |2 + 1 − |χ̃mm |2 Sc α + (1 − α)Sf Gha


h
h

 
 co
Sout
2
2
c South− c
Sf |χ̃mm | + 1 − |χ̃mm | Sc α + (1 − α)Sf GA / GA c − GA ,
Sout
Sout
(8.9)
180
h
c
The Y-factor measurement of Ref. [90], in which Y = Sout
/Sout
, is a simplified version of the VTS noise
calibration of Sec. 4.4.3, where the fridge temperature is not varied, and only two (one hot, one cold) VTS temperatures
are used.

233
where λ = −1.04 dB is measured with a separate thermal sweep in which the fridge temperature isvaried and α = ρ is assumed at half the independently measured value for the total intra-SSR lossof -2.0 dB.181 In Eq. (8.9), h and c superscripts denote quantities measured in the hot and cold loadconfigurations, and the cold-load VTS input is simply Sf . The tilde on |χ̃mm |2 denotes that wecorrect for the fact that, though this is an off-resonant measurement, the cavity’s higher spectral
density Sc is still close enough to be felt slightly, but only on one side of the LO.182 In Phase I
of HAYSTAC [90], phase-insensitive JPA operation required the addition of at least a quarter ofa quantum of noise per quadrature, NA ≥ 1/4. Now operating in phase-sensitive mode, the AMP
adds an average
NA = 0.03 ± 0.02

(8.10)

over the analysis band. This calibration measurement, together with those of Sec. 8.4.1, provides
an accurate measurement of the total noise against which an axion signal must be measured.
In practice, due to the intrinsic limitations of any model with just a few discrete, cascaded
elements, some of the calibration measurements for Sc , GS , and NA initially produce unphysicalvalues. In order to still use these calibration data, we adopt a procedure whereby the AMP gains
input to Eqs. (8.5), (8.7), and (8.9) are varied slightly relative to their measured values. Whenselecting a new value for the gains, we systematically bias towards values which produce lower
scan rates: that is, we systematically bias our measurements in the conservative direction. An
alternative approach to building in deliberate conservative bias in the face of imperfect calibration
measurements would have been to unbalance our assignment of the total measured intra-SSR loss
between ρ and α.183
181

The loss measurement protocol is exactly that of Sec. 5.5.3. Vacuum is amplified first by the SQ with AMP
off, and then by the AMP with SQ off at the same gain used for the SQ. The outputs for several different gains (toverify linearity) are compared at the output to determine how much of the state is lost between them. In Sec. 5.5.3,
the intra-SSR loss was -1.6 dB. Here, the benefit from going to lower frequencies is outweighed by the disadvantagesof a much less compact setup, constrained by the need for key circuit elements to be in the field-free region of themagnet, Figs. 8.4 and 8.5. Losses did not appear to increase when the magnet was ramped up.
182
Because Eqs. (8.5) and (8.9) are mutually dependent, the calibrations are in practice processed iteratively inorder to converge on values for Sc , GS , and NA . The contribution of Sc to NA is small, however, so a reasonablesimplifying assumption for future analyses would be to set Sc = Sf in Eq. (8.9), which would further eliminate thedistinction between |χ̃mm |2 and |χmm |2 .
183
Recall from Sec. 5.1.2 that, all else equal, loss in α is more harmful than loss in ρ, as in addition to degrading
the squeezed state, it also attenuates the axion signal.

Chapter 9

Search for axion-like dark matter with the JILA eEDM experimentThus far, this thesis has focused on the QCD axion as a well-motivated resolution to thedark matter problem; however, as we saw in Ch. 2, QCD axions are far from the only particulate
dark matter candidates, even among light particles. A popular and (as the name suggests) closely
related class of candidates are axion-like particles (ALPs) [256]. ALPs, like axions, are athermally
produced pseudo-goldstone bosons, but (by definition)184 do not resolve the strong CP problem
through the Peccei-Quinn mechanism of Sec. 2.3.2. For this reason, they are not as well motivated byparticle physics as QCD axions, but they nonetheless could address problems such as baryogenesis,
the cosmological constant, and small-scale structure formation [257]. Furthermore they are notunallowed by the laws of physics as we understand them, and the importance of solving the dark
matter problem itself serves as motivation to look for any such particles within experimental reach.
As with QCD axions, ALPs are expected to couple to normal matter, and may in fact do so morestrongly than their namesakes. However, different models predict starkly different mass windows,
with ma plausibly ranging from 10−24 –10−1 eV/c2 (10−10 –1015 Hz) [257–260]. ALPs on the lower
end of this mass range in particular could resolve problems raised by astrophysics pertaining tosmall scale structure [257,259,261,262]. Independent of mass, we will assume that ALPs, if present,
have the same standard halo model (SHM) lineshape as we have used for QCD axions, Eq. (2.14).
One way to detect ALPs is through their effects on measurements of the electric dipole
184

ALPs are somewhat loosely defined: they can alternatively be thought of as a super-class to QCD axions. This
thesis adopts the perspective that “axion-like particle” and “QCD axion” are nonoverlapping categories. In thischapter, unlike in previous ones, the QCD modifier will be prepended to “axion” when there might otherwise be
ambiguity.

235
moments (EDMs) of Standard Model particles.185 An ALP of mass ma may cause either a particle’s
EDM or its coupling to other particles to oscillate at frequency νa according to Eq. (2.9). The
practical case for such ALP searches is made especially salient by the fact that the precision
measurement of EDMs is a well-established field [263, 264], with an abundance of already existing,meticulously acquired, and potentially ALP-sensitive data [43, 265–270]. Inspired in large part by
ALPs, physicists have therefore recently begun to take seriously the possibility of time-varying
EDMs [271–275].
This chapter presents the work of Ref. [251], which uses the Bayesian power-measured (BPM)
framework of Ch. 7 to perform one such search for ALP dark matter by considering its effect on
the EDM of HfF+ ions. The EDM measurements were originally carried out to look for a static
electron EDM (eEDM) de in the JILA eEDM experiment [269, 276, 277]. A static eEDM above
the Standard Model prediction of |de | ≤ 10−38 e cm would constitute evidence of new physics
beyond the Standard Model [278–282], but as of yet none has been observed in either the JILA
eEDM experiment or the ACME experiment [270], currently the two most sensitive experimental
platforms dedicated to finding a nonzero eEDM. But the absence of a static signal does not preclude
the presence of an oscillating one, which could average to nearly zero over the long durations during
which data is collected.
The goal is to see if an oscillating signal fits to the existing, time-stamped JILA eEDM data;
the challenge is to properly identify and account for the myriad effects which shape an ALP signal
in that context. After discussing how the JILA eEDM experiment may be sensitive to ALPs in
Sec. 9.1, we will obtain the spectral content of its dataset in Sec. 9.2. Section 9.3 is the heart of thischapter: in it, we will fully quantify the no-ALP and ALP hypotheses, which entails accounting
for not only the behavior of dark matter in an entirely different regime than that of haloscopes,
but also for particulars of the experiment at hand. Finally, Sec. 9.4 presents the results of the fullsearch, which spans frequencies over seven orders of magnitude, from 27 nHz to 400 mHz (10−22 –185

In Sec. 2.3.1, we already saw the direct role of one EDM — that of the neutron — to motivating the QCD axion.
This chapter will look at measurements of a different EDM — that of the electron — to search for ALPs.

236
10−15 eV/c2 mass), and Sec. 9.5 discusses prospects for future searches. Observing no signatureof an oscillating EDM in this range allows us to place the first laboratory constraint on the ALPgluon coupling in the 10−17 –10−15 eV/c2 mass range, as part of a total exclusion result spanningseven mass decades.186 Moreover, the work presented here [251] is the first laboratory constraint
to account for the stochastic nature of the ALP field.

9.1

ALP sensitivity of the JILA eEDM dataset
The JILA eEDM experiment is a Ramsey spectroscopy measurement [283] of the stretched-state splitting in both levels of a Stark doublet of mF = ±3/2 states in 180 Hf19 F+ ions in an radio
frequency quadrupole ion trap (or Paul trap) [284]. The ions are polarized in rotating magnetic
and electric fields, and a spin precession measurement yields the Ramsey fringe frequency
fR = 3gF µB Brot /h + 3α0 frot R̃B̃ − 2WS CS D̃B̃ + 2Eeff de D̃B̃ + · · · ,(9.1)

which probes the mF = ±3/2 energy splitting directly. In Eq. (9.1), gF is the F = 3/2 state
Landé g-factor, µB is the Bohr Magneton, Brot and frot are the strength and frequency of therotating magnetic field bias, α0 is a parameter describing Berry’s phase [285], WS = 20.4 kHz is amolecule-specific structure constant, Eeff is the effective electric field seen by the valence electronin HfF+ , and CS is scalar-pseudoscalar nucleon-electron coupling. Each measurement is performed
in one of 23 = 8 “switch states” determined by {B̃ = ±1, D̃ = ±1, R̃ = ±1}, where B̃ is the signof the magnetic field, D̃ is the populated Stark doublet, and R̃ is the sense of the electric bias
field rotation. A complete description of the experiment can be found in Ref. [269] and references
therein.
Measurements of the eight switch states can be linearly combined in order to isolate different
components of the measured frequency. Of particular interest from Eq. (9.1) is
f BD = |2Eeff de − 2WS CS |,
186

(9.2)

For comparison, the Phase II HAYSTAC results of Ch. 8 spanned from 16.96–17.28 µeV/c2 , or 0.008 mass
decades.

237
used to probe the eEDM itself. While the first term in Eq. (9.2) may therefore seem conspicuous,
the known coupling mechanisms between de and ALPs are expected to be suppressed as ma [275,286, 287], deleterious to direct detection at the low masses considered in this chapter. Anothercandidate coupling is between an ALP field and the nuclear EDM, which would cause an oscillating
molecular EDM, but this is also ma -suppressed [288].
It is instead a putative ALP-gluon coupling interaction [289] that sources an ma -independent
effect on f BD . In particular, the interaction is hypothesized to cause an oscillating nucleon-electron
coupling CS . Therefore, for the rest of this chapter, we will assume
de = 0,(9.3)

consistent with the measurement of Ref. [269],187 and reducing Eq. (9.2) to
f CS = 2WS CS .

(9.4)

Figure 9.1a shows the full time series, CS -sensitive dataset of the JILA eEDM experiment,
acquired in a T ∼ 1 yr window centered in August 2016. The data is far from temporally uniform,
having been taken in several discrete data runs culminating with the 11 day run in 2017 shown as
the rightmost clump in Fig. 9.1a and in closeup in Fig. 9.1b. The data in this final window are thehighest quality measurements, with typical variances of order one hundredfold lower than those of
the other data points. For shorter timescale oscillations such as that illustrated in red in Fig. 9.1b,
this obviates the need for using the remainder of the data in Fig. 9.1a. However, the full dataset is
helpful in obtaining sensitivity to long timescale oscillators such as that shown in red in Fig. 9.1a.
Figure 9.1c shows a histogram of the normalized f BD measurements originally taken in Ref. [269].
They appear to follow a Gaussian distribution about the mean, which is itself consistent with zero
(Footnote 187).
The actual result reported in Ref. [269] is de = (0.9 ± 7.7stat ± 1.7syst ) × 10−29 e cm (90% confidence), where the“stat” and “syst” subscripts denote statistical and systematic errors, respectively.
187

238

Figure 9.1: (a) Time-series data from the JILA electron electric dipole moment (eEDM) experiment,
cast as measurements of 2WS CS versus time from August 16, 2016 (temporal center of the dataset).
An example sine wave (red) is shown overlain with the data, illustrative of how the full dataset
could in principle contain an oscillatory signal, even while averaging to zero. The full dataset
shown in (a) is used for the low frequency analysis. (b) Closeup on the last clump of data in (a),collected over 11 days in 2017 and used for the high frequency analysis. A higher frequency sine
wave (red) than in (a) is again imposed for illustrative purposes. (c) The full dataset in (a), cast as
measurements of f BD and histogrammed in normalized units relative to its mean. The data follows
a Gaussian distribution (black curve), and Ref. [269] shows it to be consistent with de = CS = 0.
Panels (a) and (b) adapted from Ref. [251]. Panel (c) adapted from Ref. [269].

9.2

Spectral content of the JILA eEDM dataset
The nonuniform time sampling in Fig. 9.1a and b necessitates a method other than the

Fourier transform for extracting the spectral content of the JILA eEDM dataset; least-squares
spectral analysis (LSSA) [290, 291] will serve us in this role. LSSA minimizes the least-squares fitof a sine function of frequency ν = ω/2π and unknown phase to a dataset. We use this to extract
A0 and B0 as the values of the quadrature variables188 Ã and B̃ that uniquely minimize
2

χ =

N
X
1 h
i=1

σi2

Ã cos(ωTi ) + B̃ sin(ωTi ) − f CS (Ti )

i2

.

(9.5)

In Eq. (9.5), Ti are the timestamps of the N data points f CS (Ti ) with variance σi2 shown in Fig. 9.1aand b.
188

This chapter will use several variants of the symbols A and B to represent closely related quantities: Ã and B̃
are the variables as used for fits in Eqs. (9.5) and (9.8), A and B are the results of simulations to determine LSSA
results, A0 and B 0 are the rotated results of those simulations (Sec. 9.3.1), A0 and B0 are the actual, measured values
for A and B, and A00 and B00 are the measured values for A0 and B 0 , obtained by rotating A0 and B0 .

239
The results of the LSSA performed on the JILA eEDM dataset are shown in Fig. 9.2. The
left portion of the plot comes from Eq. (9.5), and the right portion will be discussed below. The
even spacing of sample frequencies ν makes for a higher density of data points along the horizontalpaxis (within left and right portions of the plot), hence the higher amplitudes A20 + B02 on theright half of the plot are a simple manifestation of the look-elsewhere effect (Sec. 6.5). We canalready observe that no oscillatory component stands out prominently from its peers. This is notnecessarily an indication that there is no signal hidden within the noise, but it does mean that
we should not expect to see any extremely obvious ALPs upon performing our BPM statisticalanalysis. The BPM analysis will enable us to account conservatively (Sec. 7.1.5) for what spikes
are present, and put a constraint on ALPs over the seven frequency decades of Fig. 9.2.

Figure 9.2: Least-squares
spectral analysis of the JILA eEDM dataset. The total magnitudep2
2
of oscillation A0 + B0 corresponds to the full time-series dataset of Fig. 9.1a and b. The low
frequency analysis (< 126 µHz) is obtained from Eq. (9.5), and the high frequency analysis (>126 µHz) from Eq. (9.11). Figure reproduced from Ref. [251].

In order to examine those seven frequency decades, the JILA eEDM dataset is broken up into
two analysis regions: low frequency (27 nHz–126 µHz) and high frequency (126 µHz–400 mHz).189
For the low frequency analysis, LSSA is performed directly on the f CS data. The high frequency
analysis, conversely, involves breaking the individual data points in the 11 day 2017 data run into
smaller measurements that compose them, in order to be sensitive to short timescale oscillations.
189

The “high frequency” label is strictly relative to the lower frequencies considered within this chapter. It isroughly 10 orders of magnitude lower than the analyses of Chs. 7 and 8.

240
The Ramsey fringe frequency measurements fR of Eq. (9.1) are composed of a set of asymmetry measurements
A0i = ±

Ni − hHf + i,
hHf + i(9.6)

which are treated as individual data points in the high frequency analysis. In Eq. (9.6), Ni is acount of Hf+ ions generated by state-selective dissociation of the HfF+ ions in a single shot of
the experiment and hHf+ i is the mean Hf+ count for the set of shots containing the individual
shot in question. The +(−) sign corresponds to reading out the same (opposite) state as thatwhich was prepared. These asymmetry measurements, in the static case, already oscillate over thefree-evolution time tR (∼ 0.5–0.7 s).
The effect of an ALP is then to modulate that existing oscillation at νa . In practice, thismeans fitting the asymmetry measurements toAfit (tR ) = −Ce−γtR cos(2πfR tR + φ) + O,

(9.7)

where C is the fringe contrast, γ is the coherence decay rate, φ is the initial phase, and O is theoffset. We account for a potential ALP-induced time (t) dependence as a small perturbation term
δf (t) = f CS (t) = Ã cos(ωt) + B̃ sin(ωt)(9.8)

fR (t) = fR0 + M 0 f CS (t),

(9.9)

in

where fR0 is Eq. (9.1) with de = CS = 0 and M 0 are the matrix elements used to transform between
switch and parity bases [269]. Expanding Eq. (9.7) to second order in δf yields

Aosc (t) = −Ce−γtR cos 2π fR0 + M 0 f CS (t) tR + φ + O∼ −Ce−γtR cos [2πfR0 tR + φ] + O2+ Ce−γtR 2πtR M 0 f CS (t) sin (2πfR0 tR + φ) + Ce−γtR 2 πtR M 0 f CS (t) cos (2πfR0 tR + φ) .
(9.10)

241
For the high frequency analysis, we perform LSSA on the deviation of the N measured asymmetries,
Eq. (9.6), from the modulated fit asymmetries, Eq. (9.10), minimizingNX
1  osc
0 2
χ =
A
−
A
,
i
i
σ2
i=1 i
2

(9.11)

where σi2 are here the variances of A0i at acquisition times Ti . The results, cast as f CS quadratureamplitudes A0 and B0 through Eq. (9.8), are plotted summed in quadrature in the right portion ofFig. 9.2.

9.3

Quantifying the hypotheses
With the best-fit values from the LSSA in hand for a dense (relative to the scale on which

frequencies are independent, Sec. 7.1.4) set of frequencies, we now need to precisely quantify the
ALP and no-ALP hypotheses analogously to the axion and no-axion hypotheses shown in Fig. 6.1
in order to analyze the data with the BPM framework of Ch. 7. As in Sec. 6.2, it is helpful to thinkof the ALP (there, QCD axion) as modifying the noise-only, or no-ALP distribution. Therefore,
we will first determine the no-ALP distribution in Sec. 9.3.1, and then account for the presence ofa hypothetical ALP in Sec. 9.3.2. Key stochastic properties of the ALP distribution are discussed
in Secs. 9.3.3 and 9.3.4, and an additional correction specific to the dataset is accounted for inSec. 9.3.5.

9.3.1

No-ALP hypothesis

The no-ALP distribution is obtained for the low-frequency analysis through simulations based
off of the timestamps Ti and noise of the real data shown in Fig. 9.1a. Assuming each of the data
points has mean zero [269], standard deviation σi , a simulated time series of data is generatedand LSSA performed to determine values of A and B for frequencies evenly spaced up to 126 µHz
according to Eq. (9.5). 1,000 such simulations are performed in order to acquire reliable statistics.
The distributions of the A and B quadrature observables are assumed to be Gaussian (Fig. 9.1c),with mean zero and standard deviations σA and σB , which are determined separately for each242
frequency. The high frequency analysis is treated differently: a shuffling algorithm detailed in the
Supplemental Material to Ref. [251] is used, but the essential results are the same: distributions
for A and B characterized by σA and σB at each frequency.

Figure 9.3: Phase sensitivity of the JILA eEDM dataset. Simulated standard deviations σA0 andσB 0 for the rotated quadratures for all frequencies across both low and high frequency analyses.
The lower (higher) standard deviation, in pink (blue) belongs by construction to the more (less)sensitive rotated quadrature B 0 (A0 ). The difference between quadratures tends to diminish withfrequency, and small differences at high frequencies (where blue mostly resides behind pink) are
dominated by statistical precision, and not real phase sensitivity. Figure courtesy of Tanya Roussy.

The fact that (in the low frequency analysis) σA and σB are allowed to differ accounts forany phase sensitivity in the joint sampling profile of Ti and σi . Phase sensitivity matters because
we do not know the phase of the ALP signal, if one exists, and a fluctuation along a quadrature
to which we are sensitive should be taken more seriously than a fluctuation along a quadrature towhich we are not. As a simple example of how phase sensitivity could manifest, imagine a dataset
like that of Fig. 9.1a which only had measurements taken exactly one month apart. A sine wave
— the B̃ term in Eq. (9.5) — of frequency ν = 1/(2 months) starting at the first measurement
would be perfectly insensitive to the data, whereas a cosine wave (Ã term) would contain all of
the measurement’s sensitivity to any possible ALP signal. Typically, this effect diminishes withfrequency (though there is no reason to expect it to be perfectly monotonic for a real sampling
profile), where an oscillating wave (red lines in Fig. 9.1a and b) with many periods is unlikely to243behave in coordination with our very aperiodic Ti .
The simulated quadratures A and B are rotated into A0 and B 0 , the generalized quadratures with the highest and lowest standard deviations, corresponding to the lowest and highest
ALP-sensitivities, respectively. The insensitive (A0 ) and sensitive (B 0 ) quadrature standard deviations σA0 and σB 0 are plotted in Fig. 9.3. As expected, the phase sensitivity matters most at
low frequencies. At high frequencies, the apparent small discrepancy between quadratures appears(unsurprisingly) to be fully explained by the standard deviation owing to finite simulation statistics(limited by computational time constraints). Therefore, in the high frequency analysis, the quadrature standard deviations are averaged with one another and also smoothed in a 36-bin runningaverage. Aside from making evident the phase sensitivity of the data profile, rotation into A0 and
B 0 allows us to treat the measured values A00 and B00 at each frequency as statistically independent: as with phase-sensitive amplification (Sec. 3.4.6) of a vacuum state along the amplified versus
deamplified quadrature, knowing A00 is useless for predicting B00 , and vice versa. The same is not
generally true for A0 and B0 .
The no-ALP distribution is therefore the zero-mean two-dimensional Gaussian probability
function (PDF)



1
1 x2
y2
fx,y (x, y; σx , σy ) =
exp −
+
.
2πσx σy2 σx2 σy2

(9.12)

with {x, y, σx , σy } → {A0 , B 0 , σA0 , σB 0 }. The probability density will be evaluated for each measured
pair {A00 , B00 } and compared to those of the ALP distribution, determined in Sec. 9.3.2.
9.3.2

ALP hypothesis

To quantify the ALP distribution, we first determine the effect an ALP would have on anotherwise noiseless measurement, and then combine that effect with the no-ALP distribution ofSec. 9.3.1. We begin by anticipating that the only effect of an ALP will be to increase the variances

244
2 and σ 2 by some amount ζσ 2 as
σA
0
X
B0

2
2
2
σAX
= σA
0 + ζσX(9.13)

2
2
2
σBX
= σB
0 + ζσX ,(9.14)

where the ALP distribution will therefore be Eq. (9.12) with {x, y, σx , σy } → {A0 , B 0 , σAX , σBX }.
To justify this statement, first note that our lack of knowledge of the ALP phase implies that the
ALP cannot displace the means of the distributions of the quadrature observables. Put anotherway, the ALP will displace the mean for any one random realization of its phase, but averaged overall phases the effect is simply to broaden the two-dimensional no-ALP distribution. Furthermore,
we know the distribution will remain Gaussian because the effect of the ALP on the quadratures
is built of many small contributions from individual, interfering parts of the ALP field: a point
derived in Ref. [75] which we will return to in Sec. 9.3.3. Finally, the ALP and the noise in the JILA
eEDM experiment are uncorrelated190 so their variances add straightforwardly as in Eqs. (9.13) and2 must be the same for(9.14). Lastly, as a further consequence of not knowing the ALP phase, σX22
the two quadratures, even if the system noise, and thus σAXand σBX
are not.

The construction of the ALP distribution is represented in Fig. 9.4 for the procedure discussed
thus far: the unrotated quadratures A and B (Fig. 9.4a) are rotated to A0 and B 0 to obtain the
no-ALP distribution (Fig. 9.4b) before having their variances increased to account for the ALP(Fig. 9.4c). Having made and justified our ansatz for the form of the ALP distribution, we now set
2 , the variance of the ALP-only (i.e. noiseless) distribution, which we will arrive atout to derive σX

in Eq. (9.24). Subsequently, we account for the attenuation factor ζ from Eqs. (9.13) and (9.14) inSecs. 9.3.4 and 9.3.5.
190
Strictly speaking, while an ALP would not know about the noise in the JILA eEDM experiment, the reverse is
not true. An ALP, if present, could have had the effect of very modestly increasing the standard deviations of thedata points in Fig. 9.1. However, the effect is negligible compared to the signal such an ALP would have imparted.

245

Figure 9.4: (a) Two-dimensional Gaussian noise distribution at a frequency low enough to observe
phase-sensitive effects (ellipticity) along the unrotated quadratures A and B. (b) No-axion-likeparticle (no-ALP) distribution, obtained from rotating the distribution in (a) into the independent,
insensitive (A0 ) and sensitive (B 0 ) quadratures. (c) ALP distribution obtained by adding the same
variance to the no-ALP quadrature variances, reducing the ellipticity. The red dot marks a single
point in quadrature space as it is manipulated from (a)–(c). Figure courtesy of Tanya Roussy.

9.3.3

Coherent ALP

We first contextualize and consider the effect of a coherent, oscillating ALP signal. Many of
the ALP frequencies we wish to interrogate imply coherence times τc much longer than the durationof the measurement T (∼ 1 yr for the low frequency analysis, ∼ 11 days for high frequency): τc  T .
Recall from Sec. 2.4.3 that Qa ∼ 106 , which remains true for ALP dark matter. Therefore, we will
be in the regime of coherent ALPs so long an oscillating signal at the frequency of interest undergoes
much fewer than 106 oscillations during the measurement. The axion haloscopes discussed elsewherein this thesis are 10–17 orders of magnitude higher in frequency than the ALPs discussed in thischapter, with total measurement duration at each frequency typically lasting ∼ 30 hrs. They are
therefore in the opposite regime, τc  T . The results reported on in this chapter [251] represent
the first new laboratory constraint in the τc  T regime to properly account for ALP behavior.191
In Sec. 9.3.4, I will show results [251] that occupy the intermediate regime τc ∼ T , not directly
addressed elsewhere.
If the ALP is coherent, then the displacement amplitude it imparts along the quadratures is
191

Reference [292] offers a rough correction to previous constraints of other publications based on the authors’findings (Footnote 192), but not a new constraint.

246
fixed for a given amplitude of its field ; however, as already alluded to in Sec. 9.3.2, the amplitude of
the ALP field is itself a stochastic quantity [292].192 Reference [75] provides an elegant derivationof the ALP field amplitude, which can readily be translated into the quadrature picture.193 The
derivation, which I reproduce here, begins with an individual axion-like “particle”194
p




2ρA /Navi2
ai (vi , t) =cos ma 1 +t + φi ,ma
2(9.15)

where ρA is the local ALP dark matter density195 (Sec. 2.4.2), Na is the contributing number ofsuch particles (indexed by i), and vi is the speed of each one. The phases φi for each particle aretotally unknown: that is, distributed uniformly over [0, 2π). The particles are considered sufficiently
weakly interacting that we need not consider correlations between them.
In order to build up the local ALP dark matter field from the individual particles, Ref. [75]next partitions the particles into subsets Ωj with practically indistinguishable speeds between vj
and vj + ∆v. There are amply many ALPs that there can simultaneously be a very large numberof subsets, with each still treated classically. The expression for the coherent combination of all Naj
particles in subset Ωj is thenaj (vj , t) =

X
i∈Ωj

"
p
2ρA /Nacos ma
ma

vj2
1+
2

!

#
t + φi .

(9.16)The only i-dependent term in the sum is φi , so the sum can be performed by recasting cosine as
the real part of a complex exponential, and noting thatXeiφi = αj exp(iφj )(9.17)i∈Ωj

is the result of a two-dimensional random walk in the complex plane, with αj describing theRayleigh-distributed distance traversed in Naj steps. The phase φj ∈ [0, 2π) (as for φi ) as there is
192

This fact has until recently been underappreciated in the ALP-exclusion literature. As a result, Ref. [292] points
out that at least seven published constraints [273, 293–297] are overly aggressive by roughly an order of magnitude.
193
Reference [75] describes the analysis framework for ABRACADABRA [110–112], which occupies the same τc  Tregime as haloscopes, so the useful description of the coherent ALP therein is never directly used in the related work.
194“Particle” here means something more akin to “building block,” and quite distinct from a quantum of the
ALP field. The ALP is still very much a wave in this derivation. More precisely, the particle-as-building-block is acollection of (actual) particles of sufficiently high occupancy as to behave classically, but sufficiently low occupancy
as to make up an minute fraction of the total number of ALPs in a de Broglie volume.
195
In this chapter only, we treat the local dark matter density as 0.4 GeV/cm3 , in keeping with other published
ALP exclusion results (Footnote 15). We again assume the ALP dark matter density ρA saturates the local dark
matter density: ρA = ρDM .

247
qno preferred direction for the random walk. Redefining αj → αj Naj /2 yields

aj (vj , t) = αjq
ρA Naj /Nama"
cos ma

vj2
1+
2

#

!
t + φj(9.18)

where αj ≥ 0 now has PDF
2

fαj (αj ) = αj e−αj /2 .
The final expression for the ALP dark matter field is the sum of all subsets Ωj :
#
"
!
√ X q
vj2
ρA
a(t) =
t + φj ,αj fv (vj )∆v cos ma 1 +ma2(9.19)(9.20)

jand is of direct use to us in both the fully coherent (τc  T ) and intermediate (τc ∼ T , Sec. 9.3.4)
ALP regimes treated in this chapter. In Eq. (9.20), fv (vj ) is the ALP velocity distribution for the j thparticle subset. Equation (9.20) therefore has three random variables within the sum. Fortunately,
we know all of their probability distributions: αj is distributed according to Eq. (9.19), fv (vj )according to Eq. (2.13) for SHM ALPs, and φj is uniformly distributed as above.
For coherent ALPs (τc  T ), there is no sum in Eq. (9.20) (or rather j sums to 1), as there
are too few ALP oscillations throughout the measurement for it to be sensitive to the finite ALP
linewidth. This has two implications: first, we can safely set vj = 0 and fv (vj )∆v = 1 in Eq. (9.20).
Second, we can translate this expression into our two-dimensional Gaussian picture for the ALP
distribution (Sec. 9.3.2). This is because the two remaining random variables αj and φj describe aRayleigh distributed amplitude with random phase. One way to define a Rayleigh random variable√
is as X 2 + Y 2 , where X and Y are independent and identically distributed mean-zero, standard
deviation-σ Gaussian random variables. The Rayleigh PDF of Eq. (9.19) corresponds to σ = 1. We
can therefore directly relate σX of Eqs. (9.13) and (9.14) to the dark matter density.
2 cares both about the local ALP dark matter density ρ and the
The ALP-only variance σXA

ALP-gluon coupling CG /fa (where the symmetry-breaking energy scale fa is factored out). Writing
the oscillatory CS term in the Ramsey fringe as
f CS (t) = 2Ws η

CG
a(t),
fa

(9.21)

248
where η ≈ 0.11 is a coefficient relating CS induced in a Hf nucleus to the QCD θ angle196 [289,298]
yields√ρACG
A = 2WS ηαcos(φ)
fa ma
√
ρA
CG
0
B = −2WS ηαsin(φ)
fa ma
0(9.22)(9.23)

for the quadrature amplitudes of a pure (no noise), coherent (only one j term, with subscript leftoff) ALP signal. Equations (9.22) and (9.23) correspond to mean-zero Gaussians with
2
σX
=


√ 
CG ρA 2
2WS η
.
fa ma

(9.24)

We now have a simple expression relating physical parameters to the pure ALP distribution, absent
any attenuating effects captured by ζ in Eqs. (9.13) and (9.14).
The fact that we treat A0 and B 0 as normally distributed even in the presence of an ALP fully
accounts for the stochastic nature of the ALP. As pointed out in Ref. [292], “an unlucky experimentalist could even have near-zero field amplitudes during the course of their measurement.” This
fact is naturally handled by the Rayleigh amplitude σj which has standard deviation roughly equal
to its mean; however, it is better handled by the two independent Gaussian-distributed quadratureamplitudes (Sec. 9.3.1), as they can very easily accommodate asymmetric phase sensitivity throughtheir noise-only variances σA0 and σB 0 .197 Therefore, we choose to work in the quadrature picture.

9.3.4

Partially coherent ALP

2 added by the ALP
In the regime of intermediate ALP coherence (τc ∼ T ), the variance σXis attenuated by the interference of different velocity components vj in Eq. (9.20) over the du196

An ALP may couple to gluons through the QCD θ angle and yet still not solve the strong CP problem if thegluon coupling does not generate the largest contribution to the ALP mass.
197
It is worth noting that the statement about unlucky experimentalists is in a sense misleading: the coherent
interference of all the velocity eigenmodes has the effect of almost always giving the ALP a far greater amplitude
than it would have, all else equal, in the incoherent regime (τc  T ) occupied by haloscopes and similar platforms.
We will see this effect partially manifest in the intermediate regime (τc ∼ T ) of the next section. It is only bycomparing to the unphysical case where the ALP deterministically (as in the incoherent case) assumes an amplitude
that it could only achieve in the coherent case, that past publications have managed to report overly aggressive
exclusion results (Footnote 192). In other words, while the experimentalist could in principle be unlucky in the sense
described by Ref. [292], the wider ALP linewidths that could undo this effect, would, at constant dark matter density
and coupling to baryonic matter, only make things worse.

249
ration of the measurement. First we ask: will this affect our measurement at all? The answer
depends on both the ALP frequency (and by extension its coherence time, using Qa ∼ 106 ) andthe measurement duration T . In Table 9.1 I compute the phase slip
∆φ = 360◦ ×Tν
Qa(9.25)

between the frequency component at ν and that at ν(1+1/Qa ) over the duration of the measurement
for the low and high ends of the low and high frequency analyses. There are frequency components
beyond ν(1 + 1/Qa ), and the times used are approximate, so the values ∆φ just give a roughindication of whether we are in the coherent, intermediate, or incoherent regime. Evidently, thelow ends of both analyses are safely in the coherent regime of Sec. 9.3.3, whereas the high endof particularly the high frequency analysis should experience a significant interference effect198
between different velocity components of the ALP field. The high end of the low frequency analysis
undergoes a negligible phase slip for T ∼ 1 yr, but the effect may have to be accounted if, forexample, reanalyzing with another data run c. 2020 (and/or further in the future) incorporated.
The final row provides the approximate values for HAYSTAC Phase II (Ch. 8), which is emphaticallyin the incoherent regime, for comparison.
The quantitative effect of interfering velocity components was obtained through simulations.
Using Eq. (9.20), along with Eq. (2.13) for the SHM velocity distribution and Eq. (9.19) for Rayleighamplitudes, and independently generating each φj , the ALP field was numerically constructed at
frequencies spaced across the entire range. The field is “fit” with LSSA to get (upon performing
2
hundreds of such simulations) distributions of A0 and B 0 with (ALP-only) variances σX,A
0 and
2
σX,B
0 . We define the decoherence attenuations as

ζd,A =
ζd,B =
198

2
σX,A
0

(9.26)

2σX,A
0 ,narrow
2
σX,B
0
2
σX,B
0 ,narrow,(9.27)

We informally referred to this effect as “phase jarbling” while performing the analysis, and encourage future
analysis teams working in the intermediate coherence regime to do the same: we found reliably that while few people
had come across the word “jarble” (actual definition: “to wet; to bemire”; not listed in some dictionaries), mosteveryone knew what we meant by it.

low freq analysis, lowest freq
low freq analysis, highest freq
high freq analysis, lowest freq
high freq analysis, highest freq
HAYSTAC Phase IIν
27 nHz126 µHz126 µHz
400 mHz4.14 GHzT∼ 1 yr∼ 1 yr∼ 11 days∼ 11 days∼ 30 hrs typ.

∆φ (degrees)
3 × 10−4
1.4
0.04
140
1.6 × 1011

250
coherence regime
coherent
coherent*
coherent
intermediate
incoherent

Table 9.1: ALP coherence for high and low frequency ALP analyses and HAYSTAC Phase II. The
frequencies ν and approximate measurement durations T are listed for five different cases: thelowest and highest frequencies of the low and high frequency analysis, and the HAYSTAC Phase II
search of Ch. 8. The phase slips ∆φ between frequency elements roughly one ALP (or QCD axion)
linewidth (Qa ∼ 106 ) apart, calculated according to Eq. (9.25), provide a proxy for the amount ofinterference expected between different velocity components used to build up the dark matter field,
Eq. (9.20). The different cases are sorted into coherence regimes for organizational clarity. *Notethat the high end of the low frequency analysis is verging on the intermediate coherence regime.

Figure 9.5: Effect of ALP field decoherence on the ALP-only distribution variance for the highfrequency analysis. Dots with statistical error bars are decoherence-induced attenuations ζd fromsimulations of partially versus fully coherent ALP variances. The blue line is a fit to a second-orderpolynomial, whose values were used in performing the analysis. For the low frequency analysis (notshown) ζ = 1 for all ν. Figure reproduced from Ref. [251].

22
where σX,A0 ,narrow and σX,B 0 ,narrow are the resulting variances from identical simulations for an

infinitely narrowband ALP (Qa → ∞). In practice,199
ζd,A ≈ ζd,B

(9.28)

199
The symmetric attenuation is no surprise, but we considered the quadratures separately as a check on theintuition that there should be no meaningful difference.

251
so we defineζd =ζd,A + ζd,B.
2

(9.29)The simulations confirm that decoherence is negligible (ζd = 1) throughout the low frequency
analysis. The distributions for A0 and B 0 remain Gaussian, and the simulated attenuation of the
ALP-only distribution is plotted as a function of frequency in Fig. 9.5. The values used for the
analysis come from a second-order polynomial fit. The effect only becomes significant for ν >
∼ 100
mHz.

9.3.5

Finite sample time correction

Figure 9.6: Effect of finite sample time on the ALP-only distribution variance. Dots with statisticalerror bars are finite sample time-induced attenuations ζs from simulations of instantaneous samplingversus real sampling durations. Blue lines are separate fits to the low and high frequency data,
whose values were used in the analysis. Figure reproduced from Ref. [251].

The total attenuation ζ of the ALP variance in Eqs. (9.13) and (9.14) receives a second
contribution from sampling over time periods tR comparable to an ALP oscillation period. The
effect is an averaging over a fraction of a period of the oscillation, causing a slight attenuation insensitivity at the high end of both low and high frequency analyses. A procedure analogous to thatused in Sec. 9.3.4 is sufficient to quantify the effect. In particular, we compare LSSA results on
simulated sinusoidal data sampled much more finely than the timestamps in our real data set, and252on that same data averaged over the actual intervals of data collection (longer in the low frequency
analysis). This test, averaged over all signal phases, is performed at several frequencies with theattenuations quantified for each rotated quadrature as
ζs,A =
ζs,B =

2σX,A
0 ,avg(9.30)

2
σX,A
0
2
σX,B
0 ,avg2
σX,B0,(9.31)22
2
2
where σX,A
0 and σX,B 0 (σX,A0 ,avg and σX,B 0 ,avg ) are the variances obtained without (with) averaging.

Results indicate
ζs,A ≈ ζs,B ,(9.32)

promptingζs =ζs,A + ζs,B,2

(9.33)

analogous to Eq. (9.29). Figure 9.6 displays the results of these simulations, with separate fits used
for the low and high frequency analyses. The total attenuationζ = ζd ζs(9.34)

is given as the product of the individual attenuations from the two effects, Eqs. (9.29) and (9.33),
neglecting any small interaction the two might have with one another. This is the term that goes
into Eqs. (9.13) and (9.14) to fully specify the ALP distribution.

9.4

ALP exclusion results
To obtain final results from the BPM framework, we simply calculate the total prior up-

dates200 asCG
Ui νi ,
=
fa ma
f

x,y

h

fx,y [A00 (νi ), B00 (νi ); σA0 (νi ), σB 0 (νi )]



i ,
CG
G
A00 (νi ), B00 (νi ); σAX νi , faCm
,
σ
ν
,
i
BX
f
m
a
a a

(9.35)

200
As in Sec. 7.1.1, these reduce to the Bayes factors for our ALP dark matter. Unlike in Sec. 7.1.1, there is only
one scan, so we skip the individual scan updates ui and produce the total updates Ui .

253

Figure 9.7: (a) Logarithmically subaggregated Bayesian power-measured analysis results for the
JILA eEDM data. Grayscale shows subaggregated prior updates Us in the full two-dimensional
space of ALP frequency νa and ALP-gluon coupling CG /(fa ma c2 ), with darker (lighter) colors
indicating increased (decreased) belief in the ALP. Strings of blue dots show subaggregated 95%
exclusion for each decade, and the dashed, blue line does the same for the entire frequency window.
(b) Aggregate prior update U versus coupling for the entire seven-decade frequency window. A
small (U ≈ 2.5) peak likely due to technical noise is not enough to indicate the presence of an ALP.
The dashed, blue line corresponding to that in (a) marks U = 5% (95% aggregate exclusion). The
dashed, black line shows the coupling at U = 5% occurs. Logarithmically uniform priors are usedthroughout (a) and (b). Figure adapted from Ref. [251].

where dependencies on frequency and coupling have been made explicit in all terms, and the twodimensional Gaussian PDF is given in Eq. (9.12). Note that the coupling to ALPs only entersthrough the ALP distribution variances. The total updates are then aggregated, Eq. (7.7), here
using

 P
Ui /νi
CG
U νi ,= Pi,
fa mai /νi(9.36)

where /νi is our logarithmically uniform prior belief in the ALP, Eq. (7.14). The Ui are also
subaggregated (Us ) using Eq. (9.36) into the 100 logarithmically spaced frequency bins shown in
Fig. 9.7a. The corresponding aggregated results U of Fig. 9.7b actually go upwards to U ≈ 2.5 atintermediate couplings before plummeting downwards. As is apparent in Fig. 9.7a, the upward spikein U comes from contributions at many frequencies. Even if one of these happens to be an ALP(unlikely, given at most a 2.5-fold increase in our very low priors), the others are indicative of system
noise at sensitivities we would like to probe. Since the primary science goal of the JILA eEDM

254
experiment is (as the name suggests) to detect a nonzero eEDM, such features (and their successors
in future datasets) may have to be tolerated. It is therefore important to keep in mind, per the
discussion in Sec. 7.1.5, that the prior updates should really be viewed as conservative with respect
to exclusion. Therefore, the final result, 95% BPM exclusion for CG /(fa ma ) ≥ 4.79 × 1015 GeV−2
from 10−22 –10−15 eV, can be safely believed.
The exclusion of Fig. 9.7 is plotted alongside other laboratory and astrophysical constraints
in Fig. 9.8. The results of this chapter, shown in pink, include the first laboratory constraints on
the ALP-gluon coupling from 10−17 –10−15 eV/c2 .

Figure 9.8: Astrophysical and laboratory limits on the ALP-gluon interaction. The laboratory
constraints of this chapter (pink region) are shown alongside other constraints at 95% confidence
level. The purple region corresponds to ALPs with de Broglie wavelengths smaller than dwarf
galaxies [260]. The blue region is from laboratory results on neutron EDM (nEDM) measurements[273]. The green region represents constrains from big bang nucleosynthesis [299]. The yellow
region represents constraints from supernova energy loss bounds [249, 300]. Figure adapted from
Ref. [251].

9.5

Prospects for future JILA eEDM ALP searches
Future ALP searches will be straightforward to perform with data from the next generation

eEDM experiment, expected to be an order of magnitude more sensitive; however, two items areworth cataloguing at present for when those data arrive. First, the time duration T over which data
was taken will extend several years. Unlike for haloscope results, where prior updates should simply

255
be multiplied between separate data runs, update-multiplication will not be the information-optimal
way to integrate new data at low frequencies. Rather, where the ALP field exhibits coherent effects
over the duration of data acquisition, a “very low frequency” analysis should be performed on theentire combined dataset.201 At higher frequencies, updates may simply be multiplied, as there would
be no benefit, and significant computational cost, from a combined reanalysis. In the intermediate
case, as long as computational capabilities allow it, the coherent (“very low frequency”) approach issuperior. Second, the choice to stop at 27 nHz on the low end was not necessary. Though a coherent
ALP with oscillation period much longer than T could have its sinusoid be at a zero-crossing, theargument is analogous to that of the stochastic ALP field being at a null (Sec. 9.3.3): it does notprohibit us from accounting for the fact and conducting a search nonetheless. Acting on both ofthese items should, in conjunction with more sensitive data, produce more stringent constraints(or, preferably, a signal) at very low, low, and high frequencies. The techniques of this chapter may
also straightforwardly be applied to other existing, ultralight ALP-sensitive datasets.

201

In principle, the same could even be done between the datasets of multiple different ALP-sensitive experiments.
However, our experience with the JILA eEDM dataset recommends caution: often specific details of the measurement(e.g. Sec. 9.3.5) are required in order to produce an accurate accounting.

Chapter 10

Conclusion and outlook

Figure 10.1: Axion namesake-inspired representation of the two Josephson parametric amplifiers(JPAs; one performing the inverse gain operation of the other) and axion cavity comprising theSSR-equipped haloscope.

The mysterious nature of dark matter remains among the largest and most prominent problems in all of modern scientific inquiry. The most popular hypotheses now point to a dark sectorof abundant particles, which Occam’s razor suggests may in fact all be copies of the same particle.
Among these, the QCD axion (Ch. 2) is well motivated, detectable, and as-yet not meaningfully
disfavored by the results of direct detection efforts. The foremost goal of this thesis, and of the axion detection community more broadly, is to change that status quo: either by making a successful
detection or by sweeping out the QCD band [59] accurately and thoroughly, and discovering its
absence.

257

10.1

Conclusion
I have presented two separate but complimentary means for advancing towards that ambitious

but worthwhile goal. The first, demonstrated in Ch. 5, relies on the Josephson parametric amplifiers(JPAs) of Ch. 3 and builds from the experimental techniques of Ch. 4 [204]. With it, I showed how
two JPAs in a squeezed state receiver (SSR) configuration202 [208] can prepare, deliver, and readout a squeezed state of the microwave field in order to circumvent the standard quantum limiton haloscope noise performance. Presently, the primary limitation on this method comes fromexperimental losses in the system, 1.6 dB in the demonstration setup of Ch. 5. The second improves
upon the statistical analyses historically used for axion detection, which are discussed in Ch. 6. The
Bayesian power-measured (BPM) framework of Ch.. 7 makes full use of the information content of
the processed haloscope measurement to test the axion hypothesis. It provides a more intuitive andflexible means of analyzing axion-sensitive data that what has traditionally been done. In Ch. 9 we
saw its versatility in accommodating axion-like particle (ALP)-sensitive data as much as 17 ordersof magnitude in mass below its original region of intended use [251], where the ALPs behave very
differently than the QCD axions searched for by haloscopes.
The primary result of this thesis, that of Ch. 8, is the combined use of these two innovations to
speed up a real haloscope search roughly threefold [253]. The application of squeezing in particulardelivers on the long-held promise of quantum meteorology towards improving fundamental measurements [301], now in the microwave regime in addition to the optical [231]. The SSR-equipped
haloscope of Ch. 8 was limited by 2.0 dB of transmission losses, similar to in Ch. 5, allowing for anearly twofold speedup from squeezing alone.
The result of a quantum-enhanced axion dark matter search is of particular significance notsimply for the speedup itself, nor for the relatively small scientific contribution to our knowledgeof where in parameter space the axion is not (Fig. 8.18), but because it represents the first time asearch for new particles has benefited from a quantum-enhanced measurement. Axion haloscopes
202

For a representation of the SSR inspired by the dish detergent namesake of the axion [48], see Fig. 10.1.

258

Figure 10.2: (a) Approximate decrease of haloscope total noise levels as compared to the quantum
limit over time. The figure shows the noise levels quoted in axion haloscope publications whichmarked significant advances in the field: the first haloscope search (RBF 1987 [97]), the first
ADMX result (ADMX 1998 [89]), the first use of a SQUID-based amplifier in a haloscope (ADMX
2010 [104]), the first such use of a JPA (HAYSTAC 2017 [106]), and the work of Ch. 8 of this thesis(HAYSTAC 2020 [253]). We cannot reduce noise below zero (gray region), but there is a nearly
unlimited amount to be won in the region between zero noise and the quantum limit (thin yellow
region between the gray and white regions). (b) Approximate noise contribution to scan rate forthe same publications marked in (a) and plotted along a logarithmic axis. Scan rate, Eq. (2.16)
improves quadratically with noise relative to the quantum limit, making the linear noise decrease(1987–2017) of (a) a supralinear trend for the haloscope’s primary figure of merit. There is nobound on scan rate past the quantum limit (yellow region).

are among a small class of experimental platforms which had become limited primarily by quantum

259
noise fluctuations [91, 106, 229, 231]. Looking towards the future, it is unlikely that haloscopes(and long-baseline gravitational wave detectors along with them) will remain the exception andnot the rule. Figure 10.2a illustrates this point starkly within the historical context of haloscope
development. For the past three decades, axion noise levels (in quanta) have fallen almost perfectly
linearly to the point where they were near-quantum-limited [106] prior to the work of this thesis.
The corresponding effect on scan rate, shown in Fig. 10.2b, is by contrast quite nonlinear. If the
improvement of haloscope noise levels, nearly the sole driver of scan rate enhancement (Sec. 2.5.3)
since 1987,203 were to stop at the quantum limit, the time to sweep out the axion parameter space
might well be prohibitive. The blue dots in Fig. 10.2 correspond to the work of this thesis, which
enters into the yellow region,204 beyond the quantum limit. As indicated in Fig. 10.2b, there is nobound on scan rate in this new regime. Breaking through the quantum limit thus invites a newera of fundamental physics searches in which noise reduction techniques yield unbounded benefit
rather than the diminishing returns of approaching the quantum limit. In Sec. 10.2, I discuss how
axion physicists might most efficaciously accept that invitation.

10.2

Outlook
The two innovations in this thesis have distinct virtues. As the title of Ch. 7 suggests, theBPM framework is a terminal destination along the road to improving haloscope analyses, at least interms of the axion-pertinent information content they can extract from haloscope data. However, its
zero-cost applicability and tangible benefits recommend it strongly for rare event searches including
especially axion haloscope experiments, and its improved flexibility may facilitate advances in scanstrategy. Conversely, the SSR of Ch. 5 marks the beginning of an era of quantum enhanced searches
203

Notice that the left edge of Fig. 10.2 is 1983, the year the haloscope concept was first proposed by Pierre
Sikivie [54]. It took only four years to perform the first haloscope search [97]. The right edge of Fig. 10.2 is 2024,
four years after the writing of this thesis. In those four years, it is possible, but unlikely, that the axion may finallybe discovered. Perhaps a stronger possibility is that the technology which will one day lead to a detection will beinvented or advanced.
204
There are several ways one might perform the accounting in Fig. 10.2. I opt for a conservative one with respect
to the central result of this thesis, in which the contribution of poor tuning-rod thermalization (Appx. C) adds backmuch of the noise which squeezing removes. Even with this conservative accounting, the noise remains under thequantum limit.

260
for new particle phenomena. While its delivered twofold scan rate enhancement (Ch. 8) is on its
own an important step forward, as discussed above, its real advantage is in demonstrating theviability of sub-quantum-limited particle experiments.
Improving haloscope scan rates through improved scan strategy is a zero-cost advance enabled
by and related to the operationally flexible BPM framework. Scan strategy has received a small
amount of systematic treatment in the haloscope literature [95,126], but remains largely unexplored.
Meanwhile, the current protocol adopted by most haloscopes — perform an initial scan, rescan
peaks above a given threshold and/or below a certain sensitivity for roughly long enough to duplicate
the sensitivity of the initial scan, perform manual interrogation (Footnote 145) if some number ofrescans pass threshold — is not optimized. The discussion in Sec. 7.1.3 illuminates roughly how
one might use the prior update in a practical haloscope setting to decide when to transition fromrescans to manual interrogation. In addition, a more optimal scan strategy should account forlarger power excesses with (all else equal) longer rescans. A further consideration in determining
rescans could be the (cavity sensitivity profile-weighted) posterior belief that the axion is in any
frequency bin within a cavity linewidth. Real-time data processing and analysis could significantly
improve live times by informing the haloscope to simply wait longer near frequencies exhibiting
a power excess, rather than the time-expensive step of going back and rescanning them. Theseand related considerations recommend constructing a value function based on posterior belief todetermine initial scan, rescan, and manual interrogation strategy. A value function-based approachto scan strategy, enabled by the inter- and intrascan freedoms (Sec. 7.3) of the BPM framework,
would also need as inputs realistic time cost estimates for various haloscope operations, which would
differ between experiments. If implemented in software, it could passively improve scan rates forexisting haloscopes and other narrowband, tunable detectors.
The historical trend shown in Fig. 10.2 suggests that the strongest prospect for enhancing
haloscope scan rates is through continued improvement to noise performance. The SSR concept inthis paper suggests two avenues for doing this.205
205

These are not the only approaches available. At higher frequencies, for example, it would likely be advantageous

261
The first way to improve upon the SSR is by simply waiting. While such a suggestion may
sound frivolous, it is in fact one of the strongest arguments for using quantum measurement to reduce noise, rather than, for example, developing multiplexed cavity readout technologies to improvesignal [302]. Put simply, there are very few technologists outside of the axion haloscope community
working on coordinating the tuning and efficient readout of TM0n0 -like modes of multiple cryogenic
cavities immersed in large magnetic fields. Conversely, the technologies that are used to improvenoise and loss performance are the same ones that are constantly being developed and improved by
the superconducting qubit community in order to achieve ever more impressive results [303, 304].
As is shown in Sec. 5.5.3 and Fig. 5.2, the marginal benefit to scan rate from each 0.1 dB of reduced
transmission loss grows nonlinearly in the SSR setup. Specific technologies that could expedite
this progress include on-chip microwave circulators [222], wireless parametric amplifiers [305], anddirectional amplifiers [218, 306].
An alternative approach is to get rid of any nonreciprocal elements in the signal path altogether. Reference [252] explores one way of doing this which is the natural dual to the SSR
approach, and has one considerable advantage over it. Rather than deamplifying (or squeezing)
the noise reflected off of the cavity, which creates a fragile quantum state which must be transported around the haloscope setup, the alternative is to amplify the noise coming out of the cavity,
and with it an axion signal. The concept is illustrated in Fig. 10.3. Figure 10.3a shows the noise
power spectral densities (PSDs) along a single quadrature in a standard haloscope configuration.
The cavity and reflected noise PSDs complement one another, and, neglecting any added noiseor thermalization problems (Appx. C), the total noise spectral density is flat. The SSR reduces
the reflected noise, leaving the cavity noise untouched (Fig. 10.3b). In an idealized picture, it isequivalent to amplifying the cavity noise relative to the reflected noise (Fig. 10.3c). In practice, thelatter approach is far more robust to transmission losses, as the axion-imprinted state leaves theapparatus itself amplified, along with the noise fluctuations which also originated within the cavity.
In practice, this approach can be implemented with a setup like that of Fig. 10.3d. The axion
to measure in the photon basis [83, 84, 163].

262

Figure 10.3: (a)–(c) Manipulations of vacuum noise within a single quadrature. (a) shows the noise
profile of a standard haloscope. The reflected Johnson-Nyquist noise (red oscillation) is the same
size as the Johnson-Nyquist noise coming from within the cavity (blue oscillation). Since a feeble
axion signal (green oscillation) comes in fixed ratio with the cavity noise, the goal is to maximize
the ratio of cavity-to-reflected power spectral densities (PSDs), illustrated below with with thecavity’s Lorentzian transmission and reflection profiles about the resonance frequency ωA of its
mode (gray circle). The squeezed state receiver squeezed reflected noise without affecting cavity
noise as shown in (b) (compare to Fig. 8.9). An alternative, shown in (c), is to amplify cavity
noise relative to reflected noise, which has the advantage of making the resulting output state
more robust to transmission losses. (d) Schematic of a physical implementation of the concept in(c). The axion cavity with loss rate κl and axion-coupling rate κa is coupled through a nonlinear
element (Nonlin) to a readout cavity with resonance ωB 6= ωA . The axion is modeled as a fictitious
microwave generator as in Sec. 5.3. By modulating at the sum and difference frequencies ωΣ and ω∆ ,respectively, simultaneous two-mode entanglement (rate h) and state swapping (rate g) interactions
are generated between the cavity modes. The result is an amplification of the cavity noise emerging
from the readout cavity (external coupling rate κm ), before it encounters a nonreciprocal element
and subsequent amplification. Figure adapted from Ref. [252].

cavity at ωA is coupled to a second, readout cavity at ωB . The states of the two are entangled at
rate h by pumping a nonlinear element (such as SQUID array) between them at their sum frequencyωΣ = ωA + ωB ,(10.1)

ω∆ = |ωA − ωB |(10.2)

while a difference frequency

pump simultaneously swaps the cavity states at rate g [307]. When the rates are equal, the system

263
behaves as a phase-sensitive amplifier [308, 309]. Figure 10.3d models coupling rates κa and κl forthe fictitious axion and loss ports, and κm for the measurement port, analogous to Fig. 5.1. The
state decaying out of the measurement port encounters a circulator (followed by a JPA or otherleading amplifier) only after it has been amplified. This is essential, as the circulator has the effectof routing reflected signals from imperfections in the microwave elements before and after it away.
Conversely, an imperfection between the two cavities just changes their coupling slightly, which
can be compensated for via their antenna couplings. The reader is referred to Ref. [252] for a
full input-output theory treatment of the system, alongside practical considerations for near-term
implementation.206 As a final note, this approach can itself be complementary to the SSR: there is
no reason one cannot both reduce off-resonant noise and increase on-resonant noise (and with it,signal), all at the same time.
For now, the axion, if it exists, remains well cloaked in its vast parameter space. Quantum
measurement techniques, together with better classical information handling, has the potential tochange that.

206
We call this approach the Cavity Entanglement And Swapping Experiment For Improved Readout Efficiency,or CEASEFIRE.

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Appendix A

NIST trilayer fabrication process

The Josephson parametric amplifiers (JPAs) used for the work in this thesis were fabricatedat NIST Boulder by Leila Vale using the NIST Nb-AlOx -Nb trilayer process [202], modified byremoving the shunt resistor layer and minimizing deposited oxides [203]. JPAs, which reside on
individual chips, are produced in bulk on wafers which are then diced. The fabrication process isrepresented graphically in Fig. A.1.
Figure A.1a–d show the steps used to fabricate the Josephson junctions. First, Figure A.1ashows the three layers207 deposited uniformly across the entire wafer, which is made of Si withrelative dielectric constant r = 11.45 [310] plus a thin oxide layer. The layers are deposited withina dedicated load-locked vacuum system. A Nb layer on the bottom forms the “base electrode”(BE), which most of the superconducting circuit wiring will be made of. Atop this, a film of Al isdeposited and then oxidized. This AlO2 oxide is the insulating layer at the center of the Josephson
junction, Fig. 3.1. Finally, a second “counter electrode” (CE) layer of Nb sits atop the AlO2 . The
circuitry is then formed subtractively, by etching away the CE, Al–AlO2 , and finally BE layers.
What is left are just BE-connected pairs of Josephson junctions, Fig. A.1e, plus any BE wiring used
to from transmission lines, interdigitated finger capacitors, etc.
Fig. A.1f–h show the remaining steps performed to connect the junctions. An electron cyclotron resonance plasma enhanced chemical vapor deposition process adds an insulating layer of
207

The three layers in the “trilayer” name refer to the layers that comprise the superconductor, insulator, andsuperconductor of the Josephson junction, Fig. 3.1. These are the two layers of Nb in Fig A.1a, and the aluminumoxide between them. The Al exists here in order to accommodate its oxide; it is also a superconductor at thetemperatures at which we operate, so it can be thought of as part of the first superconductor.

291
SiO2 , which is selectively etched away to form vias above the junctions. Now, a wiring layer ofNb (called the wiring 1, or W1 layer) is added, forming the superconducting connections between
pairs of junctions shown in Fig. A.1g, before etching excess SiO2 away from the junctions in orderto minimize dielectric loss.
The result, Fig. A.1h, is a pair of SQUIDs. Current entering from the left can flow easily
through the superconductor. It splits into two parallel paths, tunneling up through the junctions,then left-to-right across the top layer of Nb, and finally down through the second pair of junctions.
Because our arrays always start and end with the BE layer in order to connect to the rest of thecircuitry, SQUIDs for us always come in pairs (junctions in fours): the cell shown in Fig. A.1h is
simply reproduced N/2 times to make an array of N SQUIDs. For a complete set of design rules
useful for laying out superconducting circuits to be fabricated with this process, see Ref. [310].

292

Figure A.1: (a) Trilayer growth. The layers of base electrode (BE) Nb, AlO2 grown from Al, andcounter electrode (CE) Nb, sit atop a Si wafer with a thin layer of oxide. Typical thicknesses areshown; for relevant lengths and widths, see Fig. 3.16. (b) CE etch. The CE is etched off away fromthe junctions. (c) Al–AlO2 etch. The same process is used to remove the Al and its oxide. (d) BE
etch. BE is removed anywhere it is not needed for junctions or other circuit wiring. The junction
is marked with an “X” symbol. (e) Fabricated junctions. A cell of four junctions, which are notyet connected, remains. (f) SiO2 deposition. A dielectric layer of SiO2 is deposited. (g) wiring
deposition. The SiO2 is etched away above the junctions, forming vias. A wiring layer of Nb isthen deposited on top, electrically connecting junctions to form pairs of SQUIDs. (h) SiO2 etch.
Undesired SiO2 is removed with a final etch. Figure not to scale. Graphics courtesy of Maxime
Malnou.

Appendix B

Resolution to the flux-tuning problem for flux-pumped JPAs

The flux-pumped Josephson parametric amplifiers (JPAs) used throughout this thesis worked
generally well, but for the problem with their flux tuning that is evident in Fig. 3.32, especially ascompared to the cleaner case of our current-pumped JPAs, Fig. 3.22. Every flux-pumped JPA wecooled down had some variant of this problem, though the specific structure varied from device todevice, and sometimes upon thermally cycling the dilution refrigerator.

Figure B.1: Location of cuts made on dummy SQUIDs. The dummy SQUIDs are the blue-hashedregion at the end of the SQUID array in this design layout. The pink hashed region above andbelow the SQUID array is the flux-bias line. In order to resolve the flux-tuning problem for ourflux-pumped Josephson parametric amplifiers (JPAs), we has the Nb base electrode layer cut witha focused ion beam (FIB) at NIST. Any two of the three cuts that were made (red, dashed lines)
on each pair of dummy SQUIDs is sufficient to break superconducting loops while maintaining
electrical continuity to the SQUIDs.

The culprit turned out to be pairs of “dummy SQUIDs” that were placed at each end of thearray. The purpose of the dummy SQUIDs, shown in Fig. B.1, was to mimic the superconducting
metalization pattern of the actual SQUIDs. The aim was for SQUIDs at the end of the arrays not
to see a different magnetic flux from those towards the center. Instead, the dummy SQUIDs, which

294
formed superconducting loops, readily trapped integer numbers of flux quanta when the devices
were cooled down.
The dummy SQUIDs were revealed to be the issue when cutting them solved the problem.
David Rudman at NIST Boulder performed the three cuts indicated in Fig. B.1 with a focusedion beam (FIB) for the dummy SQUIDs on both ends of the array, for two nominal-4.6 GHz
devices. The cuts were performed so as to remove the superconducting loops but preserve electricalcontinuity on both sides of the array.

Figure B.2: (a), (b) Bidirectional flux-tuning sweep of FIB-cut JPA showing fixed tuning problem.
Reflection coefficient phase arg(S11 ) is obtained via a flux-tuning sweep of DC bias current andprobe frequency ν such as that in Fig. 3.32, with the coil current being swept up (a) and down (b).
The flux-tuning curves no longer have the structure observed in Fig. 3.32. More than 1 GHz below
the bare resonance, the bidirectional sweep reveals some hysteresis.

Two flux-tuning curves taken on one of the FIB-cut flux-pumped JPAs are shown in Fig. B.2.
Figure B.2a shows the result of sweeping the coil current upward, and Fig. B.2b sweeping downward.
The flux tuning is much improved from that of the unmodified flux-pumped JPAs, Fig. 3.32, strongly
indicating that the superconducting loops formed by the dummy SQUIDs were causing the problem.
In particular, there is far less structure, the zero-current resonance sits closer to zero flux, and thebare resonance frequency rose 0.5–1 GHz above the nominal value for the two chips measured. All295
three effects can be explained by flux not being trapped near the SQUIDs.208
There is, however, a sharp feature that appears in Fig. B.2a at close to 3.5 mA of DC bias
current. The same feature occurs at the mirrored location in Fig. B.2b. Operation near this
hysteretic feature proved unreliable, but operation anywhere above it in resonance frequency was
excellent.

Figure B.3: (a) Two-quadrature gain map in the two-dimensional parameter space of half-pumpfrequency νp /2 and pump power estimated at JPA input. The white line shows the frequency of
maximum gain at each power up until the region where oscillation occurs. (b) Gain Gs versus
probe frequency ν traces taken at each point along the white line in (a), with colors matching thoseon the gain map.

Figure B.3 shows operation of the second FIB-cut flux-pumped JPA for gain at a bare resonance near 4.8 GHz. The gain map (Fig. B.3a) is clean, with traces shown up to 35 dB (Fig. B.3b),
before oscillation sets in.
The results shown in Fig. B.2 for the first FIB-cut JPA and Fig. B.3 for the second werevery similar across the two devices. The improved flux-tuning structure, zero-current behavior,and bare resonance, along with the robust amplifier operation, and the hysteresis, were visible inboth. The only noticeable difference was in the specific value of the bare resonant frequency, 0.5
GHz higher for the second device. Despite the hysteresis, whose cause is as yet unknown, these208
It is not surprising that the resonance actually moved above the nominal value absent trapped flux, as the criticalcurrents for most of our flux-pumped JPAs were observed to be ∼ 20% high.

296
devices represent an improvement over the flux-pumped JPAs used in this thesis. As such, future
fabrication runs will not include dummy SQUIDs which form superconducting loops.

Appendix C

Effect of improper tuning rod thermalization on scan rate

The HAYSTAC Phase II measurement presented in Ch. 8 of this thesis is adversely affectedby an improperly thermalized tuning rod. During Phase I operation of HAYSTAC, the issue wasnicknamed the “hot rod” problem, for the tuning rod’s tendency to maintain a higher physicaltemperature than the base temperature of the fridge [42]. Between Runs I [106] and II [107] of
Phase I, the hot rod problem was partially mitigated at some cost to cavity quality factor,209 but
did not go away entirely. In future runs of HAYSTAC, the thermalization problem will be addressed
further, but, in the context of the results presented in Sec. 8.3, it is important to understand theeffect of the hot rod on (1) the absolute scan rate and (2) the improvement available from squeezing.
We will find with respect to (1) that the hot rod reduces absolute scan rate by almost a factor of
two. With respect to (2), we will see that it has virtually no effect on the scan rate enhancement
available from squeezing. Indeed, since the reflected noise is still at the quantum limit, it is still
the quantum noise that is being squeezed.
To compare the effects of hot rod versus cold rod (i.e. rod at fridge temperature, Sc = Sf )when squeezing versus not squeezing, we first compute the noise and signal power spectral densities
for all four cases using the model represented in Fig. 8.19. Where applicable, we use Sc = 0.41 and
GS = 0.05, close to the values of Eqs. (8.6) and (8.8), respectively. For simplicity, we approximate
NA = 0 from (8.10), and note that the results do not depend strongly on any of these parametersin the vicinity of those measured in the HAYSTAC Phase II system (Sec. 8.4). The power spectral
209

The essential tension is that the most obvious ways of coupling heat out of the metal rod involve also coupling
microwaves out, as at low temperatures good thermal conductors tend to be good electrical conductors.

298

Figure C.1: (a)–(d) Signal and noise spectral densities S for the four different combinations ofcold/hot rod and squeezing/no squeezing as a function of detuning ω from cavity resonance. The
power spectral densitites are normalized to the respective signal and noise values for the criticallycoupled (β = 1), on-resonant, unsqueezed case. All plots are generated using the listed coupling
parameter β, obtained from the corresponding plots in Fig. C.3 so as to maximize scan rate. Allplots use ρ = α = −1.0 dB, Sf = 0.27, and NA = 0. When squeezing (not squeezing), GS = 0.05
(1). When the rod is hot (cold), Sc = 0.41 (Sf ). Note that κtot = κl (1 + β) differs for eachconfiguration.

densities (PSDs) are ratioed to obtain the visibilities, Fig. C.2, and the scan rate R is obtained
according to Eq. (5.8). By varying the coupling factor β, Eq. (8.2), we determine the scan rates for
all four cases relative to the optimal (β = 2) unsqueezed, cold rod case.
The relative scan rates for the four cases are plotted in Fig. C.3, from which, together with
Fig. C.1, we draw three important observations. First, a hot rod acts exactly like squeezing fromthe perspective of optimal coupling strategy: rather than lowering the off-resonant (reflected) noise(Fig. C.1c), it raises the on-resonant (transmitted) noise (Fig. C.1b). In both cases, overcouplingtakes advantage of the lower noise PSD off resonance. For HAYSTAC Phase II parameters, squeez-

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Figure C.2: (a)–(d) Normalized axion visibilities α obtained from the ratios of signal to noise
spectral density curves in Fig. C.1a–d. A value of 1 corresponds to the maximum achievable visibility(critically coupled and on resonance).

(a) cold rod, no squeeze(b) hot rod, no squeeze(c) cold rod, squeeze(d) hot rod, squeeze

optimal β
2
2.8
4.5
7.1

maximum R/R0
1
0.64
1.96
1.32

Table C.1: Theoretical parameters for the HAYSTAC Phase II haloscope with and without a hot
rod and/or squeezing. Values are calculated using the parameters quoted in Fig. C.1 and the modelof Fig. 8.19. (a)–(d) correspond to the panels of Figs. C.1, C.2, and C.3.

ing with a cold rod (Fig. C.3c) is optimized for scan rate at β = 4.5, while not squeezing with a hotrod (Fig. C.3b) is optimized at β = 2.8. When both squeezing and hot rod are present (Fig. C.3d),
an even higher β = 7.1 is required to optimize scan rate.210 Second, the hot rod has a little less than
210

In practice, the insertion of the antenna necessary to achieve high coupling seemed to slightly perturb theTM010 -like mode of the HAYSTAC cavity.

300

Figure C.3: (a)–(d) Scan rate R as a function of coupling parameter β normalized to R0 , the optimal
scan rate for a haloscope with neither squeezing nor hot rod, for each of the configurations shownin Fig. C.1. Higher levels of overcoupling partially compensate for a hot rod at fixed squeezing,
making the scan rate enhancement due to squeezing only weakly dependent on cavity spectral
density. Parameters are same as those in Figs. C.1 and C.2.

a factor-of-two impact on scan rate without squeezing present. Third, Squeezing improves both hot
and cold rod configurations roughly twofold. These results are summarized in Table C.1.211 We
conclude that HAYSTAC’s hot rod, while a problem nearly of the same size as the benefit conferredby squeezing, does not interfere with the ability of squeezing to improve scan rate, all else equal,
by reducing truly quantum fluctuations.

211
Note that the relative scan rates R/R0 predicted for squeezing are slightly inflated due to the NA = 0 simplifying
assumption used in this Appendix, which advantages squeezed over unsqueezed operation.

Appendix D

List of Abbreviations

This appendix provides an alphabetized list of acronyms and other abbreviations used in thisthesis.212 Most abbreviations that are used across multiple chapters are redefined the first timethey appear in the main text and figure captions of each chapter. Some exceptions are made where
it was judged to improve readability. For example, full names for the acronyms of some experiments
such as ABRACADABRA are listed only here.

ΛCDM

Λ cold dark matter (Λ is associated with dark energy)

ABRACADABRAA Broadband/Resonant Approach to Cosmic Axion Detection withan Amplifying B-field Ring Apparatus

212

AC

alternating current

ACME

Advanced Cold Molecule Electron electric dipole moment

ADC

analog-to-digital converter

ADMX

Axion Dark Matter Experiment

ALP

axion-like particle

AMP

amplifier Josephson parametric amplifierAPaxion-pertinent

AWG

arbitrary waveform generator

AWR

Applied Wave ResearchSome abbreviations such as those which appear only within Figs. 2.3 and 2.4 are excepted.

302
BE

base electrode

BPM

Bayesian power measured

BT

Bayesian threshold

BT1

Bayesian threshold 1

BT2

Bayesian threshold 2

C4C

Center for Community

CAPP

Center for Axion and Precision Physics Research

CAPP-8TB

Center for Axion and Precision Physics Research 8 tesla B-fieldCavcavityCDF

cumulative distribution f unction

CDM

cold dark matterCEcounter electrode

CEASEFIRE

Cavity Entanglement And Swapping Experiment For Improved
Readout Efficiency

Circcirculator

CMB

cosmic microwave background

CU

University of ColoradoDCdirect current

DEADMAU5

Dummy Experiment for Axionic Dark Matter Active Upwards of 5
GHz

DFSZ

Dine-Fischler-Srednicki-Zhitnitsky

DM

Dark Matter

EDM

electric dipole moment

eEDM

electron electric dipole moment

EMelectromagnetic

FIB

f ocused ion beamFT

f requentist threshold

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FWHM

f ull-width at half-maximum

GBP

gain-bandwidth product

HAYSTAC

Haloscope At Yale Sensitive To Axion Cold dark matter

HEMT

high-electron-mobility transistorIFintermediate f requency

IOT

input-output theory

IQ

in phase/quadrature

JILA(formerly) Joint Institute for Laboratory Astrophysics

JPA

Josephson parametric amplifier

KSVZ

Kim-Shifman-Vainshtein-Zakharov

LMGline of maximum gainLO

local oscillator

LSSA

least-squares spectral analysis

LSW

light shining through walls

LU

least unlikely

MACHO

massive astrophysical compact halo object

MB

Maxwell-BoltzmannMLmaximum likelihood

MOND

modified Newtonian dynamics

nEDM

neutron electric dipole moment

NIST

National Institute of Standards and Technology

OMS

one-mode squeezing

ORGAN

Oscillating Resonant Group Axion

PC

personal computer

PDF

probability density f unction

PMF

probability mass f unction

PNL

parallel nonlinear element

304
PQ

Peccei-Quinn

PSD

power spectral density

QCD

quantum chromodynamics

QUAX-aγ

Quaerere Axion (aγ refers to an axion and a photon)

RBF

Rochester-Brookhaven-Fermilab

RFradio f requency

RMS

root-mean-square

RWA

rotating wave approximationSA

spectrum analyzer

SEM

scanning electron microscope

SG

Savitzky-Golay

SHM

standard halo model

SHO

simple harmonic oscillator

SN1987A

Supernova 1987A

SI

spin-independent

SIS

superconductor-insulator-superconductor

SMA

sub-miniature version A

SMP

sub-miniature push-on

SNL

series nonlinear element

SQ

squeezer Josephson parametric amplifier

SQUID

superconducting quantum interference device

SSR

squeezed state receiver

SUSY

supersymmetryTEtransverse electric

TEM

transverse electromagnetic

TKC

tunable Kerr circuitTMtransverse magnetic

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UF

University of Florida

VNA

vector network analyzer

VTS

variable temperature stage

W1

wiring 1

WIMP

weakly interacting massive particle