the dark matter radio: a quantum-enhanced …...this thesis introduces dark matter (dm) radio, an...

THE DARK MATTER RADIO:

A QUANTUM-ENHANCED SEARCH FOR QCD AXION DARK MATTER

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Saptarshi Chaudhuri

August 2019

This dissertation is online at: http://purl.stanford.edu/qm978sp7183

© 2019 by Saptarshi Chaudhuri. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

ii

http://purl.stanford.edu/qm978sp7183

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Kent Irwin, Primary Adviser


Peter Graham


Giorgio Gratta

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Preface

This thesis introduces Dark Matter (DM) Radio, an electromagnetic search for wave-like axion

and hidden-photon dark-matter at rest-mass energies between 1 peV and 1 µeV, corresponding to

rest-mass frequencies between ∼240 Hz and ∼240 MHz.In Chapter 1, the problem of dark-matter detection is motivated, with a brief overview of the

QCD axion and hidden-photon dark-matter theories. Chapter 2 describes the basic operating prin-

ciple of DM Radio. DM Radio uses a tunable, superconducting lumped-LC resonator to probe

the small, oscillating electromagnetic signal that is generated by the conversion of dark-matter ax-

ions and hidden photons to Standard-Model photons. When the resonant frequency matches the

rest-mass frequency, the electromagnetic signal is resonantly enhanced; one can then conduct a sen-

sitive search for axion dark matter by tuning the resonant frequency across a wide frequency range,

much like tuning an AM radio to search for a particular radio station. The design of DM Radio is

motivated from fundamental statements regarding optimal electromagnetic searches for axion and

hidden-photon dark-matter subject to the Standard Quantum Limit (SQL) on phase-insensitive am-

plification. These statements, derived by the author in the course of this thesis work, are described

elsewhere.[1, 2] The optimization statements inform all aspects of the experiment, including the

method of coupling to dark matter, the choice of materials, the utilization of a single-pole resonator

rather than receiver structures with broader bandwidth, the coupling strength to the readout and

the choice of readout, and the tuning strategy.

Chapter 3 describes the DM Radio Pathfinder experiment, which serves as a small-scale test plat-

form for future, larger-scale versions of DM Radio. The Pathfinder experiment uses a 4.2 Kelvin,

liquid-helium-cooled detector with dc SQUID readout to search for hidden-photon dark matter be-

tween ∼0.4 peV and ∼40 neV. The detailed experimental design and first results, characterizingresonator performance, are included. Chapter 4 presents the sensitivity projections of DM Ra-

dio 50 Liter and DM Radio Cubic Meter, which are upgrades to the Pathfinder detector read out

by quantum-limited amplifiers.

As observed in Chapter 4, reduction of amplifier noise significantly enhances the sensitivity of

DM Radio. Additionally, a major implication of the fundamental optimization of electromagnetic

axion searches is that, even though the resonant detector possesses large thermal occupation, the

iv

scan rate of DM Radio can be enhanced by using readout evading the SQL of phase-insensitive

amplification.

As such, the second half of this thesis, introduced in Chapter 5, discusses plans for readout in

DM Radio. I derive four requirements for the optimal, phase-insensitive readout in DM Radio. In

Chapter 6, the Radio-Frequency Quantum Upconverter (RQU), a Josephson-junction-based read-

out device for signals in the DM Radio frequency range, is proposed. Basic device principles are

demonstrated, and a strawman design, which satisfies the requirements for optimal readout and is

appropriate for integration with DM Radio, is developed. In Chapter 7, I demonstrate that the

RQU can leverage techniques originally established for cavity optomechanics to evade the SQL on

amplification. In particular, a backaction evasion protocol is described which evades the SQL and

enables scan rate enhancement for DM Radio.

The thesis concludes with a summary in Chapter 8. The ultimate envisioned form of the DM

Radio campaign, DM Radio Ultimate, is presented. DM Radio Ultimateis a cubic-meter-volume

resonant experiment exploiting the full capabilities of RQUs and operating at 10 mK with a quality

factor of one million. The full-scale experiment may be able to probe well-motivated QCD axion

dark-matter parameter space down to ∼4 neV (1 MHz).

v

Acknowledgments

This thesis could not have happened without the support of the many outstanding people that I

have worked with over the past six years. I would like to start by thanking my advisor Kent Irwin. I

am awestruck by the depth of his scientific knowledge, and his approach to probing every aspect of a

problem has provided a model for me as a budding physicist. I will always appreciate his willingness

to give me the freedom to do whatever I wanted and to let me set my own research directions.

All members of the Irwin group deserve special thanks. The senior staff have provided valuable

advice during my time at Stanford. Dale Li has always been willing to sit down with me and

talk physics. His eye for trying to find the simplest explanation for a complex physical process

and drilling down to the essential details is inspiring. He produced many of the beautiful artistic

depictions of the DM Radio working principles that appear in this thesis. Sherry Cho is perhaps

most reponsible for keeping us grounded in practicality, which was important given the volume of

new ideas that the group produced. I also thank her for providing fine wine at all of the group

parties! Betty Young’s never-ending dedication to students is amazing. I will never forget the

evening during the first year of grad school when Betty spent until 10pm teaching me soldering and

wiring techniques. I have enjoyed the company of the many summer undergrads that Betty has

brought from Santa Clara. I would also like to thank Arran Phipps for providing leadership on the

DM Radio Pathfinder experiment. The Pathfinder would not have come together without him, and

he provided a reassuring voice during the countless times that some part of the apparatus failed to

work, an inevitable aspect of experimental physics.

The fellow inhabitants of the grad student “pen” have kept things lighthearted. Early on, I was

the only grad student student in Kent’s group, so it was great to share an office with Blas Cabrera’s

grad students, Ben Shank, Jeff Yen, Robert Moffatt, and Kristi Schneck, who always provided

interesting conversation. About two years after I came to Stanford, Jamie Titus joined the group.

I have learned much about x-ray spectroscopy from him, which has been refreshing, given that I

never come across the subject in my own work. He and Cyndia Yu have provided many delicious

pastries and baked goods for the office. Stephen Kuenstner has been a wonderful teammate on DM

Radio and quantum sensors, doing much of the work to get the Pathfinder instrument working. We

have spent innumerable hours talking shop on everything from the fundamentals of dark matter and

vi

measurement to the coolest quantum-sensing widgets. To Stephen, Drew Ames, and Carl Dawson:

there will have to be more “work-related beverage expenses” in celebration of hard work.

I would like to thank the front office staff, especially Maria and Violet, and the group adminis-

trators, Marcia Keating and Sha Zhang. They have gone above and beyond their duties in keeping

track of graduation milestones and paperwork and making sure that we met deadlines.

Finally, I would like to thank my mother Alpana, my father Murari, and my brother Debarshi

for their support. It will be fun spending six more months together before I ship out to Princeton.

One day, I hope to be able to explain to you all the cool things that I do on a day-to-day basis.

vii

Contents

Preface iv

Acknowledgments vi

1 Introduction 1

1.1 Dark Particles and Dark Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Probing Axions and Hidden-Photons Through Electromagnetism . . . . . . . . . . . 6

2 The Dark Matter Radio: Fundamental Principles 10

2.1 Basic Operating Principle of DM Radio . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Why Is Wave-Like Dark Matter Detection So Hard? . . . . . . . . . . . . . . . . . . 17

2.3 Optimization of DM Radio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Signal source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Matching network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.3 Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.4 Combining optimization steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 The DM Radio Pathfinder 29

3.1 Pathfinder Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Cryostat, shielding, and probe Infrastructure . . . . . . . . . . . . . . . . . . 30

3.1.2 Coupling inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.3 Capacitor and tuning strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.4 SQUID readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.5 Quality factor and materials considerations . . . . . . . . . . . . . . . . . . . 39

3.2 Pathfinder Performance: First Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Calibrating the resonator parameters . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.2 Noise performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Pathfinder Projected Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

viii

4 Improving the Sensitivity of DM Radio 48

4.1 DM Radio 50 Liter and DM Radio Cubic Meter . . . . . . . . . . . . . . . . . . . . . 48

5 Reading Out DM Radio 52

5.1 The Standard Quantum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Optimizing Imprecision and Backaction Noise in DM Radio . . . . . . . . . . . . . . 54

6 The Radio-Frequency Quantum Upconverter 60

6.1 Three-Junction Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.1.1 Critical current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1.2 Small-signal behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1.3 Interferometer as a variable inductor . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 RQU Microwave Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2.1 Circuit scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2.2 RQU responsitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2.3 Drive constraints from nonlinearity in the microwave resonator . . . . . . . . 71

6.3 Quantum Noise in an RQU Phase-Insensitive Amplifier . . . . . . . . . . . . . . . . 74

6.3.1 Imprecision noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.3.2 Backaction noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3.3 Reaching the quantum limit and tunable noise impedance . . . . . . . . . . . 79

6.4 RQU Amplifier Input Impedance and Dynamic Range . . . . . . . . . . . . . . . . . 80

6.4.1 Input impedance of an RQU amplifier . . . . . . . . . . . . . . . . . . . . . . 80

6.4.2 Dynamic range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.5 A Strawman Design for DM Radio Readout with the RQU . . . . . . . . . . . . . . 82

6.5.1 Interferometer design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.5.2 Microwave resonator design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.5.3 RQU drive power and follow-on microwave amplification . . . . . . . . . . . . 84

6.5.4 RQU noise performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.5.5 RQU optimized noise impedance . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.5.6 Damping from the RQU and dynamic range . . . . . . . . . . . . . . . . . . . 86

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7 Quantum Protocols for DM Radio Using the RQU 88

7.1 Mapping the RQU Onto An Optomechanical System . . . . . . . . . . . . . . . . . . 91

7.2 Quantum Metrology with the RQU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8 Conclusions 97

ix

A Impedance Matching to Dark Matter 98

A.1 The Dark Matter Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A.1.1 Hidden Photon Dark Matter: Visible and Dark Carriers . . . . . . . . . . . . 100

A.1.2 Lumped Circuit Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

A.1.3 Free-Space Propagation of Visible and Dark Modes . . . . . . . . . . . . . . . 105

A.1.4 Circuit Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.2 Impedance Matching to the Dark Matter Field . . . . . . . . . . . . . . . . . . . . . 118

A.2.1 Example Detection Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

A.2.2 Electromagnetic Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

A.2.3 Shielded Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

A.3 Mapping to the Axion Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

A.4 An Effective Current Representation for Axion and Hidden Photon Electromagnetism 157

A.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.6 The Electromagnetic Properties of Axion and Hidden Photon Dark Matter . . . . . 161

A.6.1 Modified Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.7 Hidden Photon Circuit Models–A Superposition Perspective . . . . . . . . . . . . . . 167

A.8 Axion Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Bibliography 176

x

List of Tables

3.1 Table describing resonator circuit parameters. The first seven parameters are mea-

sured, while the last four are derived. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.1 Table listing key design parameters for the strawman RQU amplifier. The imprecision

and backaction are evaluated at the reference drive strength α = α0, while the coupling

efficiency for optimal noise impedance is evaluated at Tph = 10 mK and drive strength

α = 3α0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.1 Table describing the mapping between electrical and mechanical oscillators. . . . . . 91

A.1 Table describing the mapping between the hidden photon and axion circuit models. . 155

xi

List of Figures

1.1 Example rotation curve from M33 galaxy showing the velocity of objects as a func-

tion of distance from the galactic center.[3, 4] Note the discrepancy between velocity

expected from luminous matter and the measured velocity. . . . . . . . . . . . . . . 2

1.2 Pie chart showing the composition of the universe. Figure from ESA/Planck.[5] . . 2

1.3 The two categories of dark-matter candidates, dark waves and dark particles. Dark-

wave dark matter cannot be arbitrarily light. At masses . 10−22 eV, the coherence

length (deBroglie wavelength) becomes larger than a dwarf galaxy, meaning quantum

pressure would prevent the formation of dwarf galaxies. Because dwarf galaxies are

indeed observed, dark matter lighter than . 10−22 eV cannot form an order-one

fraction of the abundance. The tick mark at 1019 GeV represents the Planck mass.

The Planck mass sets an upper limit on the mass of fundamental particles as dark

matter, but not on the mass of composite particles as dark matter. . . . . . . . . . 3

1.4 Spin-independent exclusion limits on WIMP-nucleon scattering cross section. Figure

from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Present exclusion limits for the axion-photon coupling. The frequency is related to

the axion mass by ν0DM = mac2/h. The blue regions correspond to bounds from the

Cern Axion Solar Telescope and SN1987a.[7] These bounds are independent of the

axion being dark-matter. The green regions are limits set by cavity haloscopes, most

notably ADMX[8, 9] and HAYSTAC[10]. The yellow band corresponds to models of

the QCD axion, in which the mass is proportional to the coupling strength (which is

in turn proportional to the inverse of the Peccei-Quinn scale). Two of the benchmark

QCD axion models are shown: the KSVZ and DFSZ models.[11, 12, 13, 14] . . . . . 8

xii

1.6 Present exclusion limits for the hidden-photon-photon mixing angle. Similar to axion,

the frequency is related to the hidden-photon mass by ν0DM = mHP c2/h. The light

blue region below .1 neV is excluded based on limits on CMB spectral distortions set

by the FIRAS instrument. The dark blue region are limits set by tests of Coulomb’s

law. These limits do not require the hidden photon to be dark matter. The green

regions are re-interpretations of axion haloscope experiments, such as ADMX. The

dashed tan line is a model-dependent bound on hidden-photon dark matter, based on

resonant conversion of hidden photons to CMB photons in the early universe.[15] . 9

2.1 Left: schematic of superconducting shield and sheath used in DM Radio architecture.

Right: Cross-section of sheath, cut top to bottom. Figure courtesy of Dr. Dale Li. . 11

2.2 Axion pickup schematic. a) Top-down cross section, showing the toroidal magnet

~Bb(~x) (orange) and resulting axion effective-current-density ~Ja(~x, t) (purple). b) The

oscillating magnetic fields ~Ba(~x, t) (green) produced by the current density, pene-

trating the sheath surface. c) The screening currents (red) generated in response to

the oscillating magnetic field. d) The slit cut allowing the screening currents to flow

along the outside of the sheath. The screening currents can be read out with a flux

amplifier, shown schematically as a dc SQUID. Figure courtesy of Dr. Dale Li. . . . 13

2.3 Hidden-photon pickup schematic. a) Side cross section, showing the hidden-photon

current density ~JHP (t) (purple). b) The produced oscillating magnetic field ~BHP (~x, t)

(green), threading flux circumferentially. c) The screening currents (yellow) generated

in response to the flux. d) The circular slit at the bottom face allowing the screening

currents to be read out with a flux amplifier. Figure courtesy of Dr. Dale Li. . . . . 14

2.4 DM Radio resonant detection schematics for a) axions and b) hidden photons. Figure

courtesy of Dr. Dale Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Circuit model for resonant detection scheme used in DM Radio. L represents the

resonator inductance, C represents the resonator capacitance, Lsheath represents the

sheath inductance, and Linput represents the input inductance. The resistance R

represents inevitable loss in the receiver, e.g. due to quasiparticles in superconductors,

eddy currents in normal metals, or dielectrics. Mt is the mutual inductance between

the resonator and slitted-sheath transformer, while Min is the mutual inductance

between the input coil and the flux amplifier. . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Schematic of a single-moded dark-matter receiver read out by a phase-insensitive am-

plifier. Double arrows indicate that signals can travel in both directions. For example,

the dark-matter signal propagates to the amplifier, but signal can be reflected off the

input of the amplifier and sent back toward the signal source. . . . . . . . . . . . . 20

xiii

2.7 Signal and noise in a dark-matter receiver, shown in a log-linear plot of current re-

sponse vs frequency. a) Rolloff of dark-matter signal (black vertical marks) following

the Lorentzian resonator line shape. b) Various noise sources in a dark-matter re-

ceiver, plotted alongside the resonator line shape. . . . . . . . . . . . . . . . . . . . 26

3.1 a) Cryostat (bottom) with dip probe mounting structure (top). b) Detail of supercon-

ducting shield. The body of the superconducting shield is made of a 0.08-inch-thick

niobium (Nb) cylinder, with a 5.7-inch diameter and 9.5-inch height. The flanges

have a half-inch radial thickness and #4-40 thru-holes for mounting the caps and for

mounting to the dip probe. Silicon diode thermometers are mounted at the top and

bottom of the shield to monitor temperature. . . . . . . . . . . . . . . . . . . . . . 31

3.2 Detector components within the shielding volume (green): slitted sheath (brown),

movable dielectrics (blue), capacitor (red), and annex for dc SQUID (magenta). . . 32

3.3 a) Slitted sheath, showing slit in one cap. b) Slitted sheath wrapped with toroidal

resonator coil. The wires are guided through slits in a teflon “bicycle wheel” structure.

The sheath metal thickness is 0.08 inches. The sheath possesses an inner radius of

1 inch, an outer radius of 3 inches, and a height of 7.5 inches. The slit, located

directly in between the inner and outer radius, is 0.1 inches wide radially. The sheath

is constructed in several steps, involving two cylinders, forming the center hole and

outer surface, and niobium plates. A niobium annulus is machined from a plate

and forms one cap of the sheath (not shown in figure). Two additional annuli are

machined and form the other cap with the slit (shown in figure). The annuli are

electron-beam-welded to the inner and outer cylinders, forming the sheath structure. 33

3.4 a) Hexagonal parallel-plate capacitor. b) Sapphire plates inserted into two sides of

parallel-plate capacitor, shown with inductor. . . . . . . . . . . . . . . . . . . . . . 35

3.5 Connection of SQUID to detector circuit. a) NbTi wires spot-welded to slitted sheath.

b) The wires are carried into the SQUID annex through a small hole. c) These

wires are attached to two Nb blocks, sitting on a printed circuit board, and the

blocks are attached to the on-chip SQUID input coil. Aluminum wirebonds carry

the SQUID output voltage signal onto copper traces, which are connected to wires

carrying the signal out of the annex and to room-temperature electronics. These wires

are displayed on the left side of b) and also carry the SQUID current bias and flux

bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.6 Assembled detector with support structures. Sapphire tuning plates not shown. The

readout and bias wires are shown at the top of the photograph. These are carried

through a hole in the top cap of the superconducting shield and into the center tube

of the dip probe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

xiv

3.7 Current injection scheme for measuring the sheath inductance. SQUID readout not

shown in figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.8 Input-coil-referred noise power spectrum. The 3dB resonator bandwidth is in purple,

while the sensitivity bandwidth is in blue. . . . . . . . . . . . . . . . . . . . . . . . 46

3.9 Sensitivity projection for the DM Radio Pathfinder. Experimental parameters for the

projection are described in the main text. See Fig. 1.6 for a description of the present

constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1 Sensitivity projections for the DM Radio hidden-photon dark-matter search. Experi-

mental parameters for the projection are described in the main text. See Fig. 1.6 for

a description of the present constraints. . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Sensitivity projections for the DM Radio axion dark-matter search. Experimental

parameters for the projection are described in the main text. See Fig. 1.5 for a

description of the present constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1 Schematic representation of noise associated with the Standard Quantum Limit. The

quantities ∆Xin, ∆Yin, ∆Xout, and ∆Yout are the total uncertainties in the quadra-

ture amplitudes in the input and output signals. The displacement of the circular

regions from the origin represents a putative dark matter (DM) signal, as signified by

the yellow arrow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Imprecision and backaction noise in a flux-to-voltage amplifier. The amplifier is dis-

played schematically on the right side as a flux loop with voltage taps. Flux trans-

formers, such as the sheath in the DM Radio detection scheme, are not shown. The

subscripts “S,” “n,” “I,” and “B” stand for source, noise, imprecision, and backaction,

respectively. Shown are the physical sources of signal in the circuit and noise in the

amplifier. The referred quantities Vout, InI , VnB , and InB are not shown in the figure. 55

5.3 Log-linear plots of noise response for two different couplings/noise impedances. Ther-

mal/ zero-point noise is in red, imprecision noise is in green, and backaction noise is

in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.1 The three junction interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.2 Critical current vs DC bias flux for n = 2. The point of interest is this paper is

θDC = π/2, corresponding to a differential flux bias of ΦDC = Φ0/4. At this bias, the

critical current is Ic(θDC = π/2) = 2I0. . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.3 Effective interferometer inductance in the low-power limit vs the differential applied

flux, represented by the angle θd/2π = Φd/Φ0. The inductance is given by the formula

eq. (6.19). It diverges at θd = π/2, corresponding to Φd = Φ0/4. A narrow region

around θd = 0 is the focus of our work. . . . . . . . . . . . . . . . . . . . . . . . . . 67

xv

6.4 Three junction interferometer embedded in an microwave resonator. ±M is the mu-tual inductance from the input coil to each loop, Lr is the resonator inductance, Cr

is the resonator capacitance, and Cc is the coupling capacitance to the feedline of

impedance Z0. The equivalent internal resistance Rr of the microwave resonator is

represented with a transmission line. The voltage drive tone on the left generates the

incoming wave a1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.5 Currents in the three-junction interferometer. Microwave drive and high-frequency

noise currents pass through the central branch (red), while direct current and low-

frequency noise currents circulate between the left and right junction branches (green). 78

7.1 Image of the LIGO Livingston site from [16]. The two arms of the interferometer,

oriented perpendicular to each other, can be seen at the top left and middle right. . 89

7.2 Analogy between LIGO and dark-matter detector readout by RQU. In LIGO, the

high-frequency optical cavity possesses resonance frequency dependent on mechanical

oscillator position x, ωa = ωa(x). In DM Radio read out by an RQU, the high-

frequency microwave resonator possesses resonance frequency dependent on the flux

input Φin to the junction interferometer, ωa = ωa(Φin). . . . . . . . . . . . . . . . . 90

7.3 Schematic representation of backaction evasion. Left: Backaction noise in a phase-

insensitive quantum-limited measurement, equally distributed between quadratures.

Right: Backaction noise in a backaction evasion scheme, in which the backaction noise

is reduced in the X quadrature at the expense of higher backaction noise in the Y

quadrature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.4 Log-linear plots of thermal+zero-point, imprecision, and backaction noise for DM

Radio readout, similar to Fig. 5.3. a) Reproduction of optimized coupling at SQL.

See Fig. 5.3. b) Reduction of backaction noise from BAE. c) Re-optimization of

coupling follow BAE, yielding a net gain in sensitivity bandwidth . . . . . . . . . . 96

A.1 Circuit models for visible and dark (hidden) electromagnetic wave propagation. a is

the unit cell size in the transmission line. We show the discretized form to emphasize

the “bottles” for electric and magnetic fields, which are the shunt capacitors and

series inductors, respectively. The continuous form can be recovered by taking the

limit a → 0. In both diagrams, the bottom trace represents the circuit ground. (a)Circuit model for visible-photon electromagnetic waves in vacuum. (b) Circuit model

for hidden-photon electromagnetic wave propagation. . . . . . . . . . . . . . . . . . 107

A.2 Circuit model in which not all variables can be solved. (a) Visible circuit. (b) Dark

circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xvi

A.3 Circuit model for the absorption process in Fig. 2a of ref. [1]. (a) Visible circuit

representing visible photon radiation and absorption. (b) Dark circuit representing

hidden photon radiation and absorption. . . . . . . . . . . . . . . . . . . . . . . . . . 120

A.4 Circuit model for a cavity detector. (a) Visible circuit. (b) Dark circuit. The length

of the transmission line in both circuits is set to L. . . . . . . . . . . . . . . . . . . . 123

A.5 Circuit model for maximum absorption of the dark matter wave. (a) Visible cir-

cuit. (b) Dark circuit. The vertical black lines represent perfect conductors. The

transmission line sections are labeled from above with their respective characteristic

impedances and are labeled from below with their geometrical lengths. . . . . . . . . 130

A.6 Circuit model for dark matter detection with an inductively-coupled lumped-element

circuit. (a) Visible circuit. (b) Dark circuit. The frequency-dependence of the ob-

servable impedance is omitted for brevity. . . . . . . . . . . . . . . . . . . . . . . . . 131

A.7 Strength of observable fields inside an electromagnetic shield, plotted as a function of

spatial coordinate. Three lengths are considered L/λ = 0.025, 0.1, 0, 2. Left: Electric

field. Right: Magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

A.8 Circuit model for the resistive sheet inside a shield. (a) Visible circuit. (b) Dark

circuit. The sheet, of impedance Zr, is positioned so that its distances to the left-

and right-side walls are L1 and L2, respectively. . . . . . . . . . . . . . . . . . . . . . 140

A.9 Circuit model for a lumped-element detector inside a shield. (a) Visible circuit. (b)

Dark circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.10 Circuit models for coupled dissipation processes at order ε2. (a) Visible-photon wave

incident on shield. Top: Visible circuit. Bottom: Dark circuit. (b) Hidden-photon

wave incident on shield. Top: Visible circuit. Bottom: Dark circuit. The perfectly

conducting shield wall is represented by the middle vertical line in all of these circuits. 146

A.11 (a) Purely radiative broadband search with incident hidden photon field. (b) Cavity

search with same incident hidden photon field. Each sheet is labeled with its sheet

impedance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

xvii

Chapter 1

Introduction

In 1933, Fritz Zwicky noted, upon applying the virial theorem to the Coma Cluster, that the galaxy

cluster must possess at least 400 times more mass than expected from luminosity measurements.[17,

18] He termed the mysterious excess mass Dunkle Materie, or “dark matter.” Work on dark matter

remained largely peripheral to the wider astronomical community until the 1970s, when Vera Rubin

performed revolutionary spectroscopic surveys illustrating that our understanding of the universe

was fundamentally and glaringly incomplete. By analyzing the velocity curves of a large number

of spiral galaxies, Rubin and colleagues showed that the amount of dark matter in most galaxies is

much larger than the luminous mass.[19] Specifically, Kepler’s Laws dictate that if all matter in a

galaxy is luminous, the velocity of stars and gases should decrease with increasing distance from the

galactic center. Rubin’s studies showed, in contrast, that the velocity stays roughly constant with

increasing distance from the galactic center (Fig. 1.1). Rubin’s work did much to propel the identity

of dark matter to the forefront of science as a major unsolved problem in fundamental physics and

astronomy. Over the last few decades, evidence for dark matter has been unearthed in a variety

of observational studies, including galaxy lensing surveys, Type 1a supernovae measurements, and

mapping of anisotropies in the Cosmic Microwave Background (CMB).[20] The case for the existence

of dark matter has also been strengthened by observational and numerical studies of structure

formation, which show that without dark matter, density perturbations in the early universe cannot

seed galaxies and the observed large-scale structure. It is now understood that dark matter accounts

for 26.8% of the energy density in the universe as well as 85% of all mass. (See Fig. 1.2.) In

accordance with the Lambda-CDM model of cosmology[20, 21], the dark matter is likely cold (non-

relativistic), as required for structure formation, and non-baryonic.

Despite the numerous astrophysical studies, the exact nature of dark matter remains unknown.

Because of its significance, the characterization of dark matter is a major area of research and the

direct identification of dark matter represents a holy grail in fundamental physics. While a number

of explanations have been developed for the form of dark matter, including modified gravity[22, 23]

1

CHAPTER 1. INTRODUCTION 2

Figure 1.1: Example rotation curve from M33 galaxy showing the velocity of objects as a functionof distance from the galactic center.[3, 4] Note the discrepancy between velocity expected fromluminous matter and the measured velocity.

Figure 1.2: Pie chart showing the composition of the universe. Figure from ESA/Planck.[5]


and Massive Compact Halo Objects (MACHOs)[24, 25], a favored explanation for the form of dark

matter is a new particle beyond the Standard Model. As such, direct identification of a dark-matter

particle would represent a major breakthrough not only in astronomy, but also in particle physics.

Figure 1.3: The two categories of dark-matter candidates, dark waves and dark particles. Dark-wavedark matter cannot be arbitrarily light. At masses . 10−22 eV, the coherence length (deBrogliewavelength) becomes larger than a dwarf galaxy, meaning quantum pressure would prevent theformation of dwarf galaxies. Because dwarf galaxies are indeed observed, dark matter lighter than. 10−22 eV cannot form an order-one fraction of the abundance. The tick mark at 1019 GeVrepresents the Planck mass. The Planck mass sets an upper limit on the mass of fundamentalparticles as dark matter, but not on the mass of composite particles as dark matter.

In this introduction, I give a brief overview of particle dark matter. In Section 1.1, I define

the two categories of particle dark-matter candidates, termed “dark particles” and “dark waves.”

Within each category, I discuss the favored candidate, the WIMP for dark particles and the QCD

axion for dark waves. I also discuss hidden photons, another well-motivated dark-wave candidate.

The primary focus thereafter, in Section 1.2, is on axions and hidden photons and their coupling

to the Standard Model via photons (electromagnetism). I introduce Dark Matter (DM) Radio, an

electromagnetic search for axions and hidden photons between ∼1 peV and ∼1 µeV.

1.1 Dark Particles and Dark Waves

A plethora of candidates for particle dark matter have been proposed[6], but the candidates can

generally be classified into two categories, displayed in Fig. 1.3. The categories, “dark waves” and

“dark particles,” are distinguished by the occupation number per mode.


Above several eV, the “dark particles” regime, the occupation per mode is below unity; equiva-

lently, the number of dark-matter particles in a box of sidelength equal to the deBroglie wavelength

∼ h/(mDMv) (where h is Planck’s constant, mDM is the dark matter mass, and v ∼ 10−3c is thevirial velocity in the Milky Way) is, on average, below one. Additionally, the deBroglie wavelength

is below a millimeter, so that, on the scale of a reasonably-sized detector (∼ 1 m3), the dark-matterdoes not possess coherence. In the context of the wave-particle duality of quantum mechanics,

the dark-matter is more appropriately described, for the purposes of detection, as particles rather

than waves. A search for dark particles thus aims to detect individual scattering events. For the

past thirty years, the favored dark particle dark-matter candidate has been the Weakly Interact-

ing Massive Particle, or WIMP, originally motivated by the so-called “WIMP miracle.” Under the

WIMP miracle, a thermal relic particle from the early universe with mass of ∼ 100 GeV whichinteracts through the weak force naturally produces the dark-matter abundance observed today.[26]

The WIMP is also motivated by supersymmetric extensions to the Standard Model, which predict a

particle with such mass and interactions. A number of experiments have been conducted to search

for WIMPs utilizing a wide variety of scattering targets.[6] A detection of the WIMP has yet to be

made; see Fig. 1.4 for recent exclusion limits on the parameter space.

In contrast, below several eV, the “dark-waves” regime, dark matter is more appropriately de-

scribed as a wave than a particle. The occupation number per mode exceeds one, so the Pauli

exclusion principle dictates that the particle must be a boson, rather than a fermion. The deBroglie

wavelength is, for a dark-matter candidate of mass mDM,

λcoh ∼h

mDMv∼ 100 km× 10

−8 eV

mDMc2. (1.1)

In other words, the dark-matter is coherent across any reasonably-sized detector. At sufficiently low

masses, it assumes a uniform field value not just over the detector, but over hundreds of kilometers.

A search in the dark-waves regime thus looks not for individual scattering events, but for collective

interactions with the detector.

The favored candidate in the dark-waves regime is the QCD axion, a spin-0 pseudoscalar field.

The QCD axion was originally motivated as a solution to the strong CP problem: why CP violation

is present in Standard-Model weak interactions, but is seemingly absent in strong interactions.

Perhaps the most notable consequence of the apparent lack of CP violation in the strong sector is

that the neutron electric dipole moment is much smaller than would otherwise be expected.[27] To

solve this mystery, in 1977, Roberto Peccei and Helen Quinn added a new global U(1) symmetry

to the Standard Model.[28, 29, 30, 31] The global symmetry is broken at some high energy scale

fa, known as the Peccei-Quinn scale, which dynamically nulls any CP-violating term in the QCD

Lagrangian. The Goldstone boson arising from the broken symmetry is the QCD axion. Soon after

the proposal of the QCD axion, it was realized that the QCD axion also makes a natural dark-matter

candidate, producing the observed abundance.[32] Unlike the WIMP, which is a thermally produced


Figure 1.4: Spin-independent exclusion limits on WIMP-nucleon scattering cross section. Figurefrom [6].


particle, the QCD axion is athermal. It is produced at zero temperature by the misalignment

mechanism, under which the initial value of the field can deviate from the minimum of the potential.

(A thermal particle with sub-eV mass would necessarily be relativistic, and thus, could not seed

structure formation. It would therefore not be an order-one fraction of the cold dark matter.)

The mass of the QCD axion ma is approximately related to the symmetry-breaking scale fa by

ma ≈ 6eV

c2

(106 GeV

fa

)(1.2)

QCD axions above ∼ 10s of meV are excluded as they would have transported substantial amountsof energy out of supernova SN1987a, shortening the neutrino burst duration observed on Earth.[33]

Originally, it was believed that QCD axions below ∼ 1 µeV would be produced in too high aquantity to be dark matter. However, recent theoretical work elucidating the behavior of the axion

field during cosmic inflation has shown this to be untrue.[34] As such, the QCD axion mass can be

anywhere between ∼ 1 peV, below which the Peccei-Quinn scale exceeds the Planck scale, and 10sof meV, above which the axion is excluded by SN1987a. (It should be noted that recent theoretical

work on black-hole superradiance [35] has disfavored QCD axions between 1 peV and 10s of peV.)

There is a broader class of particle candidates, which possess the same types of couplings to the

Standard Model as the QCD axion, but which do not possess mass linked to the Peccei-Quinn scale

(1.2) and which do not solve the strong CP problem. These are known as axion-like particles, or

ALPs. While they do not solve the strong CP problem, they are natural cold dark-matter candidates

which can produce the observed abundance. In this thesis, we refer to both ALPs and QCD axions

as “axions.”

Another well-motivated dark-wave candidate is the hidden photon, a spin-1 vector field. These

vectors emerge in many generic extensions of the Standard Model, in particular those with new

U(1) symmetries and light hidden sectors.[36] Like the axion, the hidden photon may be produced

athermally by the misalignment mechanism.[37, 15] They may also be produced by cosmic inflation.

As shown in [38], a hidden-photon vector in the 10 µeV-1 meV mass range produced by inflationary

fluctuations would naturally possess the abundance needed to be an order-one fraction of the dark

matter.

1.2 Probing Axions and Hidden-Photons Through Electro-

magnetism

Both axions and hidden photons may be probed by their very weak coupling to the photon. The

axion couples to the photon via the inverse Primakoff effect[39, 40], represented by the interaction

Lagrangian

Lint,a = gaγγaF F̃ , (1.3)


where a is the axion potential, F is the photon tensor-field-strength, F̃ is the dual tensor, and gaγγ is

the axion-photon coupling constant. The interaction dictates that, in the presence of a background

DC magnetic field, the axion converts to a photon. The hidden photon couples to the photon via a

kinetic mixing interaction[41]:

Lint,HP = εFF ′. (1.4)

where F ′ is the hidden-photon tensor-field-strength, and ε is the mixing angle describing the strength

of interaction. Because the mixing is a two-point interaction, rather than a three-point interaction, a

background field is not required for conversion to photons. The objective of axion and hidden-photon

dark-matter searches is to detect the tiny photon signal.

While WIMPs have been probed extensively over the last few decades, the axion and hidden-

photon parameter space remains relatively unexplored. For the axion, in particular, only a small

sliver of the QCD model band has been probed.[9] See Figs. 1.5 and 1.6 for present exclusion limits on

the axion-photon coupling and hidden-photon kinetic mixing angle. The wide-open parameter space

has recently motivated a number of experimental proposals to probe axions and hidden photons.

Refs. [7, 42] contain comprehensive reviews of these experiments. This thesis describes in detail the

Dark Matter (DM) Radio experiment, an electromagnetic search for axion and hidden-photon dark

matter between ∼1 peV and ∼1 µeV.


Figure 1.5: Present exclusion limits for the axion-photon coupling. The frequency is related to theaxion mass by ν0DM = mac

2/h. The blue regions correspond to bounds from the Cern Axion SolarTelescope and SN1987a.[7] These bounds are independent of the axion being dark-matter. Thegreen regions are limits set by cavity haloscopes, most notably ADMX[8, 9] and HAYSTAC[10].The yellow band corresponds to models of the QCD axion, in which the mass is proportional to thecoupling strength (which is in turn proportional to the inverse of the Peccei-Quinn scale). Two ofthe benchmark QCD axion models are shown: the KSVZ and DFSZ models.[11, 12, 13, 14]


Figure 1.6: Present exclusion limits for the hidden-photon-photon mixing angle. Similar to axion,the frequency is related to the hidden-photon mass by ν0DM = mHP c

2/h. The light blue region below.1 neV is excluded based on limits on CMB spectral distortions set by the FIRAS instrument. Thedark blue region are limits set by tests of Coulomb’s law. These limits do not require the hiddenphoton to be dark matter. The green regions are re-interpretations of axion haloscope experiments,such as ADMX. The dashed tan line is a model-dependent bound on hidden-photon dark matter,based on resonant conversion of hidden photons to CMB photons in the early universe.[15]

Chapter 2

The Dark Matter Radio:

Fundamental Principles

Section 2.1 describes the basic operating principle of the DM Radio search. The reader should

refer to ref. [41] for a more complete and more quantitative description. In Section 2.2, this

operating principle is contrasted with other detectors of electromagnetic signals in order to explain

the fundamental difficulties in dark matter detection. These difficulties motivate a first-principles

optimization of electromagnetic searches for axions and hidden photons, carried out in refs. [1, 2] and

summarized in Section 2.3. The optimization and the eight derived Design Conclusions in Section

2.3 are used to inform the design of DM Radio.

2.1 Basic Operating Principle of DM Radio

The description of the DM Radio detector begins with a superconducting shield. See Fig. 2.1. The

superconducting shield blocks external electromagnetic interference, critical to sensing tiny signals,

but allows the weakly-coupled dark matter through.

Solving the Euler-Lagrange equations inside the shield for the coupled dark matter-electromagnetic

system (1.3), (1.4) reveals that the effect of the axion and hidden-photon fields can be modeled as

an effective electromagnetic-current-density field modifying Maxwell’s equations. For the axion in a

background DC magnetic field, the current density is

~Ja(~x, t) = −gaγγ√~c�0

µ0c~B0(~x)∂ta(t). (2.1)

where a(t) is the axion pseudoscalar potential, and ~B0(~x) is the background DC magnetic field. ~ isPlanck’s reduced constant and µ0 and �0 are the vacuum permeability and permittivity, respectively.

10

CHAPTER 2. THE DARK MATTER RADIO: FUNDAMENTAL PRINCIPLES 11

Figure 2.1: Left: schematic of superconducting shield and sheath used in DM Radio architecture.Right: Cross-section of sheath, cut top to bottom. Figure courtesy of Dr. Dale Li.


For the hidden photon, the current density is

~JHP (t) = −ε�0(mHP c

2

~

)2~A′(t), (2.2)

where ~A′(t) is the hidden-photon vector potential and mHP is the hidden-photon mass. Both current

densities oscillate at frequencies near the rest-mass frequency, given by ν0DM = mDMc2/h.1

The volume of the DM Radio detector is limited to ∼1 cubic meter, as will be discussed inSection 2.3, so for the mass regime of interest, between 1 peV and 1 µeV (corresponding to rest-

mass frequencies between ∼240 Hz and ∼240 MHz), the magneto-quasi-static limit applies. Thecurrent densities then source predominantly magnetic fields. The aim of the DM Radio receiver is

to pick up these magnetic fields. This is accomplished with the superconducting sheath, shown in

Fig. 2.1 in the form of a long, hollow donut.

The pickup scheme for axions is illustrated in Fig. 2.2. A toroidal magnet embedded inside the

sheath generates a circumferential DC magnetic field ~B0(~x), shown in orange in Fig. 2.2a. As per

eq. (2.1), the axion-photon interaction produces a current density ~Ja(~x, t), shown in purple, which

follows the magnetic field. The current density sources a magnetic field ~Ba(~x, t) (green in Fig. 2.2b),

oscillating at the rest-mass frequency, whose flux lines penetrate the surface of the superconducting

sheath. The superconductor develops screening currents (red) to cancel flux from the bulk of the

sheath.[43] These screening currents flow along the inside surface of the sheath, as in Fig. 2.2c. By

cutting a slit along the side of the sheath, on both the inner and outer cylinders as well as on the

top and bottom annular faces, the current path is interrupted, and currents flow along the outside

surface of the sheath. This is illustrated in Fig. 2.2d. By connecting a coil across the slit, the

oscillating screening currents can be sampled and read out using a sensitive flux amplifier, shown

here schematically as a dc SQUID. One may note that the pickup mechanism is similar to that used

in a cryogenic current comparator [44], a device for measuring small currents.

The operating principle for hidden photons is similar and is illustrated in Fig. 2.3. The hidden-

photon effective-current-density ~JHP (t) (purple in Fig. 2.3a) points uniformly along the axis of the

sheath. From the right-hand rule, the current produces an oscillating, circumferential magnetic field

~BHP (t), in green in Fig. 2.3b. The circumferential flux produces screening currents (yellow) along

the inner surface of the sheath (Fig. 2.3c). By cutting a circular slit along one of the faces, as

opposed to the side-slit for axion detection, the screening currents are interrupted and flow along

the outer surface of the sheath. The screening currents can be sampled by connecting the input coil

of a flux amplifier across the slit.2

1I use mDM to generically denote the dark-matter mass. ma and mHP denote, more specifically, the axion massand hidden-photon mass, respectively. Additionally, since the detector dimensions are far less than a deBrogliewavelength, the axion and hidden-photon potentials can be treated as spatially uniform, i.e. no dependence on ~x.

2While the direction of the axion current density is set by the background DC magnetic field, the direction of thehidden-photon current density is a priori unknown. Thus, it is not guaranteed that the sheath is oriented to pick upthe hidden-photon current density. Mitigation strategies for misalignment with the vector field are described in [1].


Figure 2.2: Axion pickup schematic. a) Top-down cross section, showing the toroidal magnet ~Bb(~x)

(orange) and resulting axion effective-current-density ~Ja(~x, t) (purple). b) The oscillating magnetic

fields ~Ba(~x, t) (green) produced by the current density, penetrating the sheath surface. c) Thescreening currents (red) generated in response to the oscillating magnetic field. d) The slit cutallowing the screening currents to flow along the outside of the sheath. The screening currents canbe read out with a flux amplifier, shown schematically as a dc SQUID. Figure courtesy of Dr. DaleLi.


Figure 2.3: Hidden-photon pickup schematic. a) Side cross section, showing the hidden-photon

current density ~JHP (t) (purple). b) The produced oscillating magnetic field ~BHP (~x, t) (green),threading flux circumferentially. c) The screening currents (yellow) generated in response to theflux. d) The circular slit at the bottom face allowing the screening currents to be read out with aflux amplifier. Figure courtesy of Dr. Dale Li.


While the receiver could be run in a broadband mode, as shown in Figs. 2.2 and 2.3, such searches

require long averaging times to probe new parameter space. As we will show, broadband searches

are fundamentally suboptimal. Moreover, long averaging times leave the experiment vulnerable to

spurious interference, which complicate the data acquisition and analysis. DM Radio consequently

operates as a resonant experiment.

A schematic for the resonator in the axion detection configuration is given in Fig. 2.4a. The

axion-induced magnetic field of Fig. 2.2b produces a flux through the center hole of the sheath.

Thus, a wire-wound solenoidal inductor, placed in the center hole, couples to the axion signal. A

resonant LC circuit may be constructed by attaching a capacitor to the solenoid. A similar schematic

is given for hidden-photon detection in Fig. 2.4b. A toroidal inductor wrapped around the sheath

couples to the circumferential magnetic flux produced by the hidden-photon current density of Fig.

2.3. An LC circuit may be constructed by connecting the toroid to a capacitor. When the LC

resonance frequency matches the rest-mass frequency, the magnetic-field signal is enhanced by the

resonator quality factor. Consequently, less time is required to probe new parameter space at a

particular rest-mass frequency, relative to a broadband search. Since the dark-matter rest-mass is

unknown, the resonator is tuned via a variable capacitor to probe a wide range of masses. The

tuning may be performed with an insertable dielectric. In this manner, DM Radio operates much

like an AM radio, tuning into a radio station at particular frequency.

Figure 2.4: DM Radio resonant detection schematics for a) axions and b) hidden photons. Figurecourtesy of Dr. Dale Li.


Figure 2.5: Circuit model for resonant detection scheme used in DM Radio. L represents the res-onator inductance, C represents the resonator capacitance, Lsheath represents the sheath inductance,and Linput represents the input inductance. The resistance R represents inevitable loss in the re-ceiver, e.g. due to quasiparticles in superconductors, eddy currents in normal metals, or dielectrics.Mt is the mutual inductance between the resonator and slitted-sheath transformer, while Min is themutual inductance between the input coil and the flux amplifier.


An equivalent circuit model of the tunable, resonant detection scheme is shown in Fig. 2.5

and describes the signal flow to the readout. The dark-matter-induced magnetic field threads flux

through the resonator inductor, resulting, from Faraday’s Law, in a circuit voltage. When the circuit

resonance frequency, determined by the equivalent inductance and capacitance, matches the rest-

mass frequency, the dark-matter voltage signal drives enhanced currents in the receiver. The quality

factor of the receiver, i.e. the level of current enhancement, is governed by the resistance R. The

enhanced currents couple flux into the sheath, which acts as a one-turn pickup transformer to the

input coil of a flux amplifier. The signal is then read out through the amplifier.

2.2 Why Is Wave-Like Dark Matter Detection So Hard?

In Section 2.1, I discussed the basic detection mechanism for Dark Matter Radio. The detection

mechanism relies on a superconducting sheath coupled to a tunable, high-Q resonator, which en-

hances the dark-matter-induced magnetic-field signal when the rest-mass frequency matches the

resonance frequency. The signal is read out with a sensitive flux amplifier, such as a dc SQUID.

To shield the setup from external electromagnetic pickup, which can interfere with detection of the

dark-matter signal, a superconducting shield is used.

Considering how oscillating electromagnetic signals are detected more generally (not just for

dark-matter), this seems incredibly complicated! Standard receiving antennas, such as the whip

antenna used for car radios or the satellite dish used for televisions, do not require superconducting

or low-loss materials. Moreover, they operate unshielded. One might say that such components

are needed to detect a dark-matter signal which is much smaller than the signals routinely received

in antennas. The available power in the dark-matter field is governed by the flux of dark matter,

determined by the product of the local dark-matter density (0.45 GeV/cm3 in the Milky Way galaxy

[6]) and virial velocity and roughly equal to 10 Watts per square meter. In other words, in one square

meter, there is enough power in the dark-matter field to power a standard household LED lamp!

Furthermore, efficient broadband absorbers for electromagnetic fields are commonplace. Phased

array antennas, used routinely in the mapping of the Cosmic Microwave Background[45]—like dark

matter, a source of photons from the cosmos—as well as AM communication, can absorb 50% of

the power in an incident electromagnetic plane wave, independent of its frequency. If an analogous

receiver could be built for dark-matter detection, 5 Watts of power could be absorbed, independent of

the rest-mass frequency. This would result in an extremely efficient dark-matter search, without the

need for any high-Q, resonant enhancement! So why is the dark-matter signal necessarily “small”?

And why exactly is dark-matter detection so hard?

A traditional answer to this question is based on the concept of dispersion mismatch.[46, 47,

48, 49] Dark-matter waves are dispersive, owing to their mass, while free-space photons are disper-

sionless, with the phase and group velocity being equal to the speed of light. Momentum is then


not naturally conserved in the conversion of dark matter to photons, which limits the power in

photons. Resonators restore the momentum match by breaking translational invariance, allowing

greater photon production.

However, the question of detection is ultimately not one of how many photons are produced, but

rather, how much power is absorbed in the electron system of the receiver. It is thus a question of

impedance matching, rather than dispersion matching that fundamentally limits our scan sensitivity.

We must then consider the effective source impedance of dark matter and more directly the effect

of dark matter on electrons through the electromagnetic interaction. I have derived the source

impedance elsewhere[1]

ZDM ∼

(gaγγ

√~c�0cB0

mDMc/~

)2Zfs

cv , axions

ε2Zfscv , hidden photons.

(2.3)

See the appendix for an extended discussion of impedance-matching to dark-matter. B0 is the char-

acteristic strength of the background DC magnetic field in an axion experiment, and v ∼ 10−3cis the virial velocity. Zfs =

√µ0/�0 ≈ 377 Ohms is the free-space impedance. For instance,

for a KSVZ axion[11, 12] in a ∼10 T magnetic field (for which the prefactor gaγγ√~c�0cB0

mDMc/~ evalu-

ates to ∼ 3 × 10−16), the source impedance is ∼28 orders of magnitude lower than the free-spaceimpedance. As expected, axions and hidden-photons are “weakly coupled” to electrons through

electromagnetism, as is required to be dark matter. But more specifically, the dark-matter source

impedance is much smaller than the characteristic impedances associated with the photon-electron

interaction, i.e. the free-space impedance. Any impedance due to the photon-electron interaction ex-

erts a significant drag on electrons, limiting absorption of the relatively weakly coupled dark-matter.

In an efficient detector, the impedances due to the photon-electron coupling must be nulled. The

nulling can be achieved using a single-pole resonator, which is tuned on resonance to the dark-matter

frequency. However, because the source is weakly coupled, the quality factor of the resonator must

be extremely high, on the order of 1028, (low resistance in an equivalent series-RLC circuit to match

the low-source impedance) to impedance match to dark matter and absorb all ∼10 Watts per squaremeter of power. Practically, there exists no good impedance match to dark matter, so the power

absorbed in a receiver is very difficult to measure.

Because the dark-matter signal is so small and because the dark-matter mass/frequency is a priori

unknown, it is critical to develop optimal search strategies for the signal. While a resonator may

more efficiently absorb power from dark-matter at a particular rest mass than a broadband search,

that does not itself mean that a resonant dark-matter search is optimal. A receiver must search the

entire parameter space. The figure of merit for a dark-matter receiver is thus frequency-integrated

sensitivity, rather than sensitivity at a single frequency. In this respect, one must also consider


broadband search strategies, such as those proposed in the ABRACADABRA[50] and BRASS[51]

experiments. These strategies utilize information at the receiver output over the whole search

range, simultaneously probing a wide range of frequencies instead of a narrow band. Broadband

strategies require no tuning—in contrast to resonant experiments such as DM Radio—but also do

not benefit from resonant enhancement. In other words, broadband searches aim to compensate for

the lack of resonant enhancement with long integration times. The distinctions between resonant

and broadband search strategies beg the question of the fundamentally optimal search strategy for

axion and hidden-photon dark matter. To determine the fundamentally optimal search strategy, a

number of points must be considered:

• The set of all possible frequency-response functions of a dark-matter receiver (not just resonantand broadband). Because the receiver can be periodically varied, e.g. tuning in a resonator,

a determination of the optimal search must allow the frequency response to be periodically

varied as well.

• Signal-to-noise ratio. The question of optimal detection is not one of a signal, but one of signal-to-noise. Determination of the optimal search must thus consider irreducible noise sources in

a dark-matter receiver, such as thermal noise. If the search uses a phase-insensitive amplifier,

in which both cosine and sine components of the signal are amplified with equal gain, then

amplifier quantum noise must also be taken into account.[52, 53]

• Priors on the dark-matter properties. Because the dark-matter properties are a priori unknown,determination of the optimal search must consider priors on the mass and coupling of the axion

or hidden photon. For example, the prior probabilities may be weighted negligibly at regions

of excluded parameter space, but weighted favorably at regions of well-motivated parameter

space, e.g. the QCD-axion-model band in Fig. 1.5.

I carry out a fundamental optimization of electromagnetic axion and hidden-photon dark-matter

searches, investigating all of these points, in refs. [1, 2]. For the first time, these papers present

the optimal search strategy subject to the Standard Quantum Limit (SQL) on phase-insensitive

amplification.[52, 53] The amplifier quantum noise associated with the SQL is discussed in detail in

Chapter 5. For now, it is sufficient to simply define the SQL as the minimum measurement noise

that can be achieved with a phase-insensitive amplifier, as dictated by the Heisenberg uncertainty

principle. I summarize the results of the fundamental optimization in the next section and use them

to provide design rules for optimizing the DM Radio experiment.

2.3 Optimization of DM Radio

A receiver for axion or hidden-photon dark-matter consists of three components: an element coupling

to the dark-matter-induced electromagnetic fields (the “Signal Source”), elements transmitting the


Figure 2.6: Schematic of a single-moded dark-matter receiver read out by a phase-insensitive ampli-fier. Double arrows indicate that signals can travel in both directions. For example, the dark-mattersignal propagates to the amplifier, but signal can be reflected off the input of the amplifier and sentback toward the signal source.

dark-matter signal to the readout (the “Matching Network”), and the readout element (the “Read-

out”) measuring the signal. See Fig. 2.6. To determine the optimal search, each block must be

optimized in tandem with interactions across blocks. The optimization yields eight Design Conclu-

sions for DM Radio, listed below.

2.3.1 Signal source

The first block is the “Signal Source.” To optimize this block, it is important to quantitatively

treat dark-matter electrodynamics and consider the work done on the receiver currents by dark

matter. Consider a fixed rest-mass frequency. The dark-matter effective current densities produce

electromagnetic fields governed by Maxwell’s equations:

~∇ · ~DDM = ρDM, ~∇× ~EDM = −∂t ~BDM,~∇ · ~BDM = 0, ~∇× ~HDM = ~JDM + ∂t ~DDM (2.4)

~JDM(~x, t) denotes the axion ( ~Ja) or hidden-photon ( ~JHP ) current density. ρDM(~x, t) is the effective

charge density that accompanies the current density. It is given by the conservation relation

∂tρDM(~x, t) = −~∇ · ~JDM(~x, t) (2.5)

However, its effect is suppressed by the virial velocity v/c ∼ 10−3. The dark-matter-induced fieldsexcite currents in the receiver, described by free-electronic charge and current densities ρrec(~x, t) and


~Jrec(~x, t). These densities source response fields, also governed by Maxwell’s equations:

~∇ · ~Drec = ρrec, ~∇× ~Erec = −∂t ~Brec,~∇ · ~Brec = 0, ~∇× ~Hrec = ~Jrec + ∂t ~Drec (2.6)

We Fourier transform to the frequency-domain, writing, for example,

~JDM(~x, t) =√TrefRe

(∫dω

2π~JDM(~x, ω, ω

0DM) exp(+iωt)

). (2.7)

where Tref is a flexibly-chosen long reference time. The dark-matter signal is not monochromatic,

but possesses bandwidth determined by the dark-matter kinetic energy. Virialization and Earth’s

motion in the galactic rest frame (the frame in which the bulk motion of dark matter is zero) give

dark matter a velocity of v/c ∼ 10−3 in the receiver rest frame. The 10−3 velocity yields a 10−6

dispersion in kinetic energy and therefore, a bandwidth

∆ωDM(ω0DM) ∼ 10−6ω0DM. (2.8)

where ω0DM = 2πν0DM is the angular rest-mass frequency. The dark-matter current density and

sourced electromagnetic fields thus span the frequency range ω0DM ≤ ω . ω0DM + ∆ωDM(ω0DM).Consider a volume V surrounding the receiver. The rate of complex work[54] done on the receiver

currents by the dark-matter component at frequency ω is

1

2

∫V

d3~x ~J∗rec(~x, ω, ω0DM) · ~EDM(~x, ω, ω0DM) = −

iω

2

∫V

d3~x ~BDM(~x, ω, ω0DM) · ~H∗rec(~x, ω, ω0DM) (2.9)

+iω

2

∫V

d3~x ~EDM(~x, ω, ω0DM) · ~D∗rec(~x, ν, ν0DM)

− 12

∫∂V

( ~EDM(~x, ω, ω0DM)× ~H∗rec(~x, ω, ω0DM)) · ~da

The terms on the right describe three ways for dark-matter to couple power into the receiver.

The first term describes power coupled from the dark-matter-induced magnetic field into magnetic-

energy-storage elements in the receiver. We term this “inductive” coupling. This is the mode of

coupling used in DM Radio; see Chapter 2.1. The second term describes power coupled from the

dark-matter-induced electric field into electric-energy-storage elements in the receiver. We term this

“capacitive” coupling. This could constitute, for example, coupling to a parallel-plate capacitor, or

coupling to a mode in a free-space cavity. The third term, for sufficiently large volumes V , describes

power coupled through the free-space radiation pattern of the receiver, e.g. as in a radio antenna.

We term this “radiative” coupling.

In Sections II and III of ref. [2], we show that, on an apples-to-apples basis, all searches using a

radiative coupling are sub-optimal relative to a search using a high-Q cavity (inductive/capacitive)


coupling. The difference originates in the mismatch between the dark-matter source impedance and

the free-space impedance, i.e. the characteristic impedance of electromagnetic radiation. In the

limit that the experimental apparatus is small compared to the dark-matter Compton wavelength

λ0DM = c/ν0DM, the effective current density can be treated quasi-statically, so the dominant observ-

able is a dark-matter-induced magnetic field. One should then use inductive coupling, rather than

capacitive coupling. In the limit that the experimental apparatus is comparable to, or larger than,

the Compton wavelength, the dark-matter-induced magnetic and electric fields are generally compa-

rable. Inductive and capacitive couplings thus couple comparable amounts of energy. Because the

two forms of coupling also lead to comparable interactions across the blocks of Fig. 2.6, without loss

of generality, inductive coupling can be taken to be optimal. This yields the first design conclusion

for DM Radio:

• Design Conclusion 1: As DM Radio operates in the quasi-static limit, probing rest-massfrequencies between ∼240 Hz and ∼240 MHz (1 peV and µeV), it should use a lumped-elementinductor to couple to the dark-matter-induced magnetic field. (Inductive pickups for the axion

and hidden-photon detection schemes are illustrated in Figs. 2.2, 2.3, and 2.4).

To maximize excitation, the receiver should couple to as much of the dark-matter-induced

magnetic-field energy as possible. The available dark-matter-induced magnetic energy is

Umaxc (ω, ω0DM) =

1

2µ0

∫d3~x | ~BDM(~x, ω, ω0DM)|2, (2.10)

where the integral is performed over all space. Eq. (2.10) sets a maximum on the energy Uc(ω, ω0DM)

coupled to the readout circuit through the inductive pickup:

Uc(ω, ω0DM) ≤ Umaxc (ω, ω0DM). (2.11)

The larger the overlap of the pickup field pattern (the pattern being determined by inductor ge-

ometry) with the dark-matter-induced magnetic flux lines, the larger the coupled energy. For ax-

ion detection, the maximum Umaxc (ω, ω0DM) increases with background DC-magnetic-field strength.

(Larger DC field strength means a larger effective current density and larger sourced oscillating

fields.) We thus arrive at the second design conclusion.

• Design Conclusion 2: Pickup volume VPU3 and geometry should be co-optimized to max-imize coupled energy Uc and maximize receiver excitation. Due to practical constraints on

material fabrication, the volume is typically limited to ∼1 m3. For axion detection, magnetsproducing multi-Tesla fields may be constructed.

3VPU is a field volume, limited to the volume of the enclosing shield, rather than a physical volume. For wire-woundstructures, e.g. a single-turn pickup loop, the latter is often ill-defined.


Accompanying the pickup inductor is loss, i.e. resistance. This loss could be due to, for instance,

quasiparticles in superconductors, dielectric insulation on wires, or eddy currents in nearby normal-

conducting materials. The optimal signal source can thus be modeled as a series LR network. The

loss produces thermal noise, as well as effective zero-point-fluctuation noise. Thermal noise decreases

with decreasing temperature, so it is optimal to be as cold as possible.

• Design Conclusion 3: To minimize thermal noise, the temperature of the DM Radio receivershould be as low as possible. For cubic-meter-scale experiments, temperatures of 10 mK can

be achieved with a commercial dilution refrigerator.

2.3.2 Matching network

Having optimized the signal source and modeled it as a series LR network, I now turn attention to

the matching network, used to transmit the dark-matter signal to the amplifier. One example of

a matching network is a single-pole LC resonator, where a capacitance is attached to the pickup

inductance to provide resonant enhancement. Another example is a broadband inductive matching

network, in which the inductive pickup is attached directly to the input coil of a SQUID. See Figs.

2.2 and 2.3. One can also use a multi-pole resonator or arbitrarily-complex transformer structure.

In Sections IV and V of ref. [1], I determine the figure of merit for a matching network to be the

expectation value of the signal-to-noise-ratio-squared. The expectation value is taken with respect

to priors on the mass, coupling, and velocity distribution of dark matter. I define the log-uniform

search based on an uninformative prior for the dark-matter signal. In the log-uniform search, the

figure of merit reduces to the following measure of frequency-integrated sensitivity:

F [S21(ω)] =

∫ ωhωl

dω

2π

(|S21(ω)|2

|S21(ω)|2nth(ω) + 1

)2, (2.12)

where ωl and ωh are the lower and upper limits of the search range. nth(ω) = (exp(~ω/kBTph)−1)−1

is the thermal occupation number of the receiver, determined by the physical temperature Tph of the

resistance. S21(ω) is the scattering parameter describing transmission through the matching network,

from signal source to amplifier, at frequency ω. The aim of the matching network optimization is to

maximize the value of F [S21(ω)] over all posssible transmission functions.

The value functional is constrained by the Bode-Fano criterion[55, 56],∫ ωhωl

dω

2πln

(1

1− |S21(ω)|2

)≤ RL, (2.13)

which dictates the frequency-integrated impedance match between the LR signal source and the

amplifier. The Bode-Fano criterion enables the determination of an upper-bound on the value func-

tional, which is saturated by a multipole-LC Chebyshev filter. However, such filters are narrowband

matching networks and difficult to tune to cover a search range. An optimized single-pole resonator,


which is far easier to tune (it can, for example, be tuned using insertable dielectric in a capacitor),

possesses a value of F [S21(ω)] which is 75% of the maximum [2]. A single-pole resonator is thus not

only superior to broadband matching networks, but also fundamentally near-optimal.

• Design Conclusion 4: DM Radio should use a tunable, single-pole LC resonator as thematching network. Schematics of LC resonator schemes are shown in Fig. 2.4.

2.3.3 Readout

The final block in the receiver optimization is the readout. The readout optimization has far-reaching

consequences, and as such, the second half of this thesis, beginning in Section 5.1, is dedicated to

readout development for the DM Radio search. Here, I briefly summarize the optimization, assuming

the readout is performed with a phase-insensitive, quantum-limited amplifier, in which the sine

and cosine components of the signal are amplified with equal gain. The summary motivates the

subsequent Design Conclusions in this chapter, the Pathfinder design in Chapter 3, the experimental

upgrades in Chapter 4, the discussions on noise in Chapter 5, and the proposal and development of

readout technologies and protocols in Chapters 6 and 7.

As emphasized earlier, the figure of merit for a dark-matter search is frequency-integrated sensi-

tivity. As such, it is important to account for sensitivity not just within the resonator bandwidth,

but over the entire range of search frequencies. It is outside the resonator bandwidth where readout

noise plays a substantial role in the optimization.

To elucidate the role of readout noise, I return to the circuit model of the DM Radio resonator

presented in Fig. 2.5. The current response I(ω) to a voltage drive of complex amplitude V is given

by

I(ω) =Imax

1 + 2iQ(ω − ωr)/ωr(2.14)

where Q is the quality factor of the resonator, ωr = 1/√L′C is the resonance frequency, and

Imax = V/R is the maximum current in the resonator, achieved precisely on-resonance. Thus,

for fixed voltage drive, |I(ω)/Imax|2 follows a Lorentzian resonator line shape. The line shape isdisplayed in Fig. 2.7a. The dark-matter signal, resulting from dark-matter-induced magnetic flux

threading the pickup inductor, acts as a voltage in series with the RLC circuit (e.g. Faraday’s Law).

As such, the current response to dark-matter follows the same Lorentzian lineshape. The vertical

black marks above the line shape represent the resonator response to dark-matter signals at different

frequencies.

A dark-matter signal appears as excess power on top of the noise, i.e. a “bump” in the spectrum.

Thermal and zero-point-fluctuation noise also act as an equivalent voltage in the series RLC circuit,

described by spectral density

SFDV V (ω) = 4~ωR(nth(ω) + 1/2). (2.15)


“nth(ω)” represents the thermal noise, while the “+1/2” represents the zero-point-fluctuation noise.

The current response to the noise voltage is shown in red in Fig. 2.7b. Like the dark-matter response

current, the noise current is also Lorentzian with frequency. As long as the readout noise (in green

in Fig. 2.7b as a flat noise level) is subdominant to the thermal noise, the total noise is dictated

by the thermal noise. In this regime, because the dark-matter signal and thermal noise follow the

same Lorentzian lineshape, the signal-to- noise ratio stays approximately constant as a function of

detuning from resonance. Consequently, a resonator can possess sensitivity, undegraded by readout

noise, far outside the resonator bandwidth. The region of undegraded sensitivity is identified in pink

shade and labeled “Sensitivity Bandwidth” in the figure.

By reducing the readout noise, the sensitivity bandwidth may be increased, enabling faster

scanning of the parameter space. It is thus important to push the readout noise as low as possible.

The Heisenberg uncertainty principle dictates that a phase-insensitive amplifier readout must add a

minimum amount of noise to the measurement. This minimum is known as the Standard Quantum

Limit (SQL), discussed further in Section 5.1. As I demonstrate in refs. [1, 2], full utilization of the

sensitivity bandwidth with a quantum-limited amplifier can increase DM Radio scan rates by up to

five orders of magnitude, relative to using only resonator-bandwidth-only information. DM Radio

benefits enormously from the use of a quantum-limited amplifier!

• Design Conclusion 5: To maximize sensitivity bandwidth and scan rate, the amplifier noisein the DM Radio search should be as low as possible, approaching the Standard Quantum

Limit of phase-insensitive amplification. Near-quantum-limited amplification is possible with

Josephson-junction-based technologies, such as dc SQUIDs[57, 58] (which have demonstrated

near-quantum-limited performance in axion experiments[8]) and the RF Quantum Upconvert-

ers (RQUs) proposed in Chapter 6. Further scan rate enhancement is enabled with protocols

evading the SQL, such as those described in Chapter 7.

One should also note the practical advantages of larger sensitivity bandwidth. When using only

information within the resonator bandwidth, one needs to tune the resonator at a step size equal to

the resonator bandwidth (or finer) in order to ensure full coverage of the search band. In contrast,

when using the whole sensitivity bandwidth, one needs to tune the resonator at a step size equal to

the sensitivity bandwidth (or finer); this constitutes a considerably larger step size. The larger step

size significantly mitigates requirements on tuning resolution, and because fewer steps are needed,

reduces dead-time due to settling.

2.3.4 Combining optimization steps

Combining the optimization steps described in the previous three sections yields a fundamental,

standard quantum limit on performance of any single-moded dark-matter receiver. This quantum

limit is derived in Section VI of ref. [1]. For the hidden-photon search, the minimum kinetic mixing


Figure 2.7: Signal and noise in a dark-matter receiver, shown in a log-linear plot of current responsevs frequency. a) Rolloff of dark-matter signal (black vertical marks) following the Lorentzian res-onator line shape. b) Various noise sources in a dark-matter receiver, plotted alongside the resonatorline shape.


angle εmin(ω0DM) at rest-mass frequency ω

0DM to which the DM Radio receiver is sensitive scales with

experimental parameters as

ε−1min ∝U

1/2c Q1/4T

1/4int

T1/4ph η

1/4(2.16)

where Tint is the total integration time, and

η ≡ kBTn~ω0DM/2

(2.17)

is a normalized measure of amplifier noise temperature Tn. For a quantum-limited readout, η = 1. Q

is the resonator quality factor. We denote the factor on the right-hand side of (2.16) as the hidden-

photon performance figure of merit, PHPFOM . Similarly, for the axion, the minimum axion-photon

coupling gaγγ,min to which the receiver is sensitive scales as

gaγγ,min(ω0DM)

−1 ∝ B0U1/2c Q1/4T

1/4int

T1/4ph η

1/4(2.18)

where B0 is the root-mean-square DC magnetic field strength within the shielding volume. The

factor on the right-hand side of (2.18) is the axion performance figure of merit, P aFOM . Note that,

for both the axion and the hidden photon, for fixed inductor geometry (e.g. fixed aspect ratios), the

coupled energy scales with pickup volume as

Uc ∝ V 5/3PU . (2.19)

The performance figure of merit then scales with pickup volume as V5/6PU . Because the performance

figure of merit increases with quality factor, it is important to maximize resonator Q.

• Design Conclusion 6: DM Radio should aim to obtain as high a resonator quality factor aspossible. Thus, it should use only superconducting metals (no normal metals) for conduction

paths, including the wire-wound resonator inductance. Dielectric losses on these wires should

be minimized. Low-loss dielectrics, such as single-crystal sapphire, should be used for capacitor

tuning. Effort should be made to minimize loss in mechanical support structures. DM Radio

aims to achieve quality factors of one million.

Here, one should note a key benefit of using the toroidal geometry for axion detection, shown

in Figs. 2.2 and 2.4; the geometry enables a high-quality-factor superconducting resonator to be

obtained in the presence of a large DC magnetic field. Multi-Tesla magnetic fields usually preclude

the use of superconductors, which trap flux and become lossy if their critical field is exceeded.

Toroidal magnets, when designed properly, can possess sufficiently low fringing fields such as to not

exceed the critical field of the surrounding sheath (0.2 T if superconducting niobium is used[59]).

CHAPTER 2. THE DARK MATTER RADIO: FUNDAMENTAL PRINCIP

the dark matter radio: a quantum-enhanced …...this thesis introduces dark matter (dm) radio, an...

Documents