FERMILAB-PUB-22-739-T Searching For Dark Matter with Plasma Haloscopes Alexander J. Millar1 2 3 aSteven M. Anlage4Rustam Balafendiev5Pavel Belov6Karl van Bibber7Jan Conrad1

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FERMILAB-PUB-22-739-T

Searching For Dark Matter with Plasma Haloscopes

Alexander J. Millar,1, 2, 3, aSteven M. Anlage,4Rustam Balafendiev,5Pavel Belov,6Karl van Bibber,7Jan Conrad,1

Marcel Demarteau,8Alexander Droster,7Katherine Dunne,1Andrea Gallo Rosso,1Jon E. Gudmundsson,1, 5

Heather Jackson,7Gagandeep Kaur,9, 1 Tove Klaesson,1Nolan Kowitt,7Matthew Lawson,1, 2 Alexander

Leder,7Akira Miyazaki,10 Sid Morampudi,11 Hiranya V. Peiris,1, 12 Henrik S. Røising,13 Gaganpreet Singh,1

Dajie Sun,7Jacob H. Thomas,14 Frank Wilczek,1, 11, 15, 16 Stafford Withington,17 and Mackenzie Wooten7

(Endorsers)

Jens Dilling,8Michael Febbraro,8Stefan Knirck,3and Claire Marvinney8

1The Oskar Klein Centre, Department of Physics, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden

2Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden

3Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA

4Quantum Materials Center, Physics Department, University of Maryland, College Park, MD 20742-4111 USA

5Science Institute, University of Iceland, 107 Reykjavik, Iceland

6Narxoz University, Zhandossov street 55, 050035 Almaty, Kazakhstan

7Department of Nuclear Engineering, University of California Berkeley, Berkeley, CA 94720 USA

8Physics Division, Oak Ridge National Laboratory, TN 37831

9Centre for Lasers and Photonics, Indian Institute of Technology Kanpur; Kanpur, 208016, India

10Department of Physics and Astronomy, Uppsala University, Uppsala, 75237, Sweden

11Center for Theoretical Physics, MIT, Cambridge, MA 02139 USA

12University College London, Gower Street, London, WC1E 6BT, UK

13Niels Bohr Institute, University of Copenhagen, DK-2200 Copenhagen, Denmark

14X1, Department of Physics, Illinois Institute of Technology, Chicago, IL 60616 USA

15T.D. Lee Institute and Wilczek Quantum Center, Shanghai Jiao Tong University, Shanghai 200240, China

16Arizona State University, Tempe, AZ 85287, USA

17Department Physics, University of Oxford, Clarendon Laboratory Parks Road, Oxford, OX1 3PU

We summarise the recent progress of the Axion Longitudinal Plasma HAloscope (ALPHA) Con-

sortium, a new experimental collaboration to build a plasma haloscope to search for axions and

dark photons. The plasma haloscope is a novel method for the detection of the resonant conversion

of light dark matter to photons. ALPHA will be sensitive to QCD axions over almost a decade of

parameter space, potentially discovering dark matter and resolving the Strong CP problem. Unlike

traditional cavity haloscopes, which are generally limited in volume by the Compton wavelength

of the dark matter, plasma haloscopes use a wire metamaterial to create a tuneable artiﬁcial plasma

frequency, decoupling the wavelength of light from the Compton wavelength and allowing for much

stronger signals. We develop the theoretical foundations of plasma haloscopes and discuss recent

experimental progress. Finally, we outline a baseline design for ALPHA and show that a full-scale

experiment could discover QCD axions over almost a decade of parameter space.

CONTENTS

I. Introduction 2

II. Analytic Formulation 3

A. Solving the Axion-Maxwell Equations 3

B. Signal Readout 5

C. Eigenmode Calculation 6

D. Example: The Rectangular Resonator 8

TM Modes 8

TE Modes 9

E. Resistive Losses 9

III. Numerical Simulations 10

A. Mode Structure 10

aamillar@fnal.gov

B. Quality Factor 11

C. Tuning 11

IV. Experiment Prototypes 12

A. Measurements in Free Space 12

Experimental setup and measurements 12

B. Measurements in Brass Cavity 14

C. Measurements in Copper Cavity 15

V. Superconducting Wires 16

VI. Experiment Design 18

VII. Discovery Potential 18

VIII. Summary and Conclusion 21

Acknowledgments 22

arXiv:2210.00017v3 [hep-ph] 22 Mar 2023

References 22

I. INTRODUCTION

A wide variety of astronomical observations and the

modern theoretical understanding of structure forma-

tion in the Universe support the hypothesis that a new

form of matter, not accounted for in our standard model

of fundamental physics, supplies much of the mass of

the Universe. Indeed, what is often called the “standard

model of cosmology” and abbreviated ΛCDM incorpo-

rates both a cosmological term Λand Cold Dark Matter,

a hypothetical substance that interacts very feebly with

ordinary (baryonic) matter and with itself.

Because of the equivalence principle, the astronomi-

cal and cosmological evidence gives us only limited in-

formation about the nature of dark matter. The simplest

hypothesis, conceptually, is that it is a very long-lived

elementary particle that does not carry electromagnetic

or color charge. This particle must be abundantly pro-

duced in the early universe but then decoupled at a

time when its typical velocities are non-relativistic (i.e,

it is “cold”).

In recent years much attention has focused on axions

as a candidate to provide the dark matter in our Uni-

verse. Axions ﬁrst appeared superﬁcially in an unre-

lated context, namely the issue of why a parameter, θ,

that appears in the standard model is empirically con-

strained to be exceedingly small: |θ|.10−10. A “nat-

urally” large value of θ, i.e. θ∼1, would introduce T

violating effects - speciﬁcally, electric dipole moments

of protons and neutrons - at levels that far exceed ex-

perimental constraints. This difﬁculty led Peccei and

Quinn [1] to propose extending the standard model

to incorporate an additional (anomalous and sponta-

neously broken) U(1)symmetry, now known as Peccei-

Quinn or PQ symmetry. Weinberg and Wilczek [2,3]

independently observed that this proposal necessarily

leads to the existence of a new light spin 0 particle,

the axion, that is highly stable and interacts very feebly

with ordinary matter. Unfortunately, present-day the-

ory is unable to predict the precise value of the axion

mass ma, however the requirement to solve the Strong

CP problem tightly constrains the interactions of the

axion relative to that mass, leading to an almost one-

parameter theory.

Consideration of the production of axions in the early

Universe and their subsequent evolution [4–9] reveals

that their properties are consistent with the observed

properties of cold dark matter.

Big Bang cosmology incorporating axions follows

two qualitatively distinct scenarios, depending on

whether an inﬂationary epoch intervenes between ax-

ion decoupling and the present. If not, then it is pos-

sible – although challenging – to relate mato the den-

sity of axion matter present in the Universe today (post-

inﬂationary scenario). If axions dominate the observed

dark matter, the most recent calculations [10–17] favor

ma=40 −180 µeV (with ma=65 ±6µeV if the spec-

trum of axions radiated as during the decay of topolog-

ical defects is scale invariant).

On the other hand, reheating after inﬂation might not

restore PQ symmetry. In this case, inﬂation essentially

enforces a constant (but random) initial value θiwithin

the entire observable Universe, thus providing an un-

known initial condition. This opens the possibility of

axion models with signiﬁcantly smaller values of ma,

correlated with very small values of θi(pre-inﬂationary

scenario). But if we insist on avoiding ﬁne-tuning or an-

thropic reasoning, it is natural to estimate θi=O(1),

and then we ﬁnd 10−1µeV .ma.100 µeV.1

These considerations motivate experiments to test the

hypothesis that axions, with a mass in the range of

O(10)−O(100)µeV, exist and constitute the cosmolog-

ical dark matter. The traditional approach to axion (and

similar hidden photon [8,20–22]) dark matter searches

is based on use of a resonant cavity, designed so that

a resonant mode (typically the lowest transverse mag-

netic mode) matches the Compton wavelength of the

dark matter [23]. Whilst this is adequate for relatively

low frequencies (hundreds of MHz to a few GHz [23–

25]) the diminishing size of the cavity at high frequen-

cies leads to a rapid deterioration in signal power.

This problem, and the perceived promise of the axion

hypothesis, has inspired a torrent of new experimental

ideas. Some proposals relevant to the indicated mass

range involve using multiple or coupled cavities [26–

30]; others abandon the traditional cavity altogether,

and substitute either a mirror, as in the case of a dish

antenna [31–35] or an array of large dielectric disks

[36–39]. (Completely different ideas come into play for

ultra-small values of ma.) For recent comprehensive re-

views see [40–42].

These approaches, broadly speaking, attempt to over-

come the barrier to resonance that arises due to the

mismatch between the fundamentally massless pho-

ton and a massive, non-relativistic axion by breaking

translation invariance, thus providing momentum. The

leading idea of Ref. [43], pursued by ALPHA and dis-

cussed in detail here, is instead to provide the photon

with an effective mass corresponding to its plasma fre-

quency.2This has the fundamental advantage that it

allows wavelength matching up to the de Broglie rather

than the Compton wavelength of the dark-matter ax-

ions, and thus allows use of much larger resonant sys-

tems.

As is well known, photons acquire an effective mass

inside a plasma. No natural plasma has all the required

1Other options include modifying the cosmology so that θis not a

random choice between {0, 2π}, for example Refs. [18,19].

2In THz regime there have been some similar concepts to make an

effective massive photon quasiparticle (polariton) using condensed

matter axions [44,45] and optical phonons [46,47].

properties (viz., cryogenic operation, low loss, and tune-

ability) within the frequency range of interest, but an ar-

tiﬁcial plasma, consisting of an array of thin wires [48],

appears to be suitable. Notably, because the properties

of wire metamaterials depend primarily on the geom-

etry of the system, the plasma frequency can be tuned

through geometric changes [49,50]. These properties

make wire metamaterials excellent candidate materials

to implement the concept of a plasma haloscope.

The application of plasma haloscopes is not lim-

ited to the QCD axions from the spontaneously bro-

ken PQ symmetry. More generally, a plasma haloscope

can address any light particles interacting with high-

frequency microwaves. Physics beyond the Standard

Model, such as string theory, universally predicts new

scalar and pseudoscalar bosons or extra U(1)gauge

bosons. The former is often referred to as axion-like par-

ticles and is free from the theoretical constraint in mass

and coupling of QCD axions [8,51–56]. The latter are

hidden photons (HP) or dark photons, though parapho-

tons has also been used historically [57–60]. The tech-

nical requirements of searching for these particles are

very similar to that of axions but without needing a

static magnetic ﬁeld [22,61].

In this white paper we develop an analytic formalism

for plasma haloscopes, proving the formulas used in

Ref. [43] and developing an overlap integral formalism

applicable when the system exhibits a single mode. Us-

ing numerical simulations, we also explore the expected

quality factors of wire metamaterials at low tempera-

tures and practical tuning geometries. We then review

recent experimental work by Refs. [50,62] that validates

our theoretical analysis. We also introduce supercon-

ducting metamaterial as an alternative candidate for a

tuneable effective plasma with higher unloaded qual-

ity factor. Finally, we outline an overall design for the

ALPHA project and make projections for the expected

sensitivities of exploratory and full scale experiments.

II. ANALYTIC FORMULATION

Here we calculate the interactions of axions with our

detector, a tuneable plasma.

First, we will set up and formally solve the Maxwell

equations coupled with the Klein-Gordon equation of

axion (axion-Maxwell equation), estimating the mi-

crowave power produced in such a device. Then, we

further develop the single mode approximation, obtain-

ing a convenient overlap integral. Finally, we exploit

the analysis of Ref. [62] to ﬁnd explicit expressions for

rectangular plasma modes, enabling us to estimate the

unloaded quality factor for a given choice of wire ma-

terial.

A. Solving the Axion-Maxwell Equations

We are interested in calculating the conversion of ax-

ions to photons in a ﬁnite plasma. Throughout this

work unless otherwise speciﬁed we use natural units

with the Lorentz-Heaviside for electromagnetic units.

The axion-Maxwell equations in a medium are given

∇·D=−gaγB·∇a, (1a)

∇×H−˙

D=gaγ(B˙

a−E×∇a), (1b)

∇·B=0 , (1c)

∇×E+˙

B=0 , (1d)

a−∇2a+m2

aa=gaγE·B. (1e)

Here we have deﬁned Eas the electric ﬁeld, Bas the

magnetic ﬂux density with Dand Hthe displacement

and magnetic ﬁeld strengths, respectively. We have ne-

glected any free charges or currents as we are interested

in axion-sourced ﬁelds. Neglecting the small spatial

gradient of the axion, the axion ﬁeld is giving by the

real part of

a=a0e−imat, (2)

where a0is the axion ﬁeld strength coming from the

axion dark matter density

ρa=m2

aa2

2. (3)

The axion couples to the photon via a dimensional cou-

pling gaγ. Such a coupling is also often written as a di-

mensionless quantity Caγ, which makes use of the fact

that for the QCD axion the mass and coupling can be

calculated from the axion decay constant fa, numeri-

cally found to be [63]

ma=5.70(6)(4)µeV 1012 GeV

fa, (4a)

gaγ=−α

2πfa

Caγ, (4b)

Caγ=E

N−1.92(4), (4c)

where αis the ﬁne structure constant, Eis the elec-

tromagnetic anomaly and Nis the colour anomaly

(also known as the domain wall number) of the axion.

For the simplest standard benchmark KSVZ and DFSZ

models |Caγ|=1.92, and 0.746 respectively [64–67].

We will linearise the system around an external mag-

netic ﬁeld Bethat solves Maxwell’s equations indepen-

dently (its sources are left implicit). Thus, to ﬁrst order

in gaγwe have

∇·D=−gaγBe·∇a, (5a)

∇×H−˙

D=gaγBe˙

a, (5b)

∇·B=0 , (5c)

∇×E+˙

B=0 , (5d)

a−∇2a+m2

aa=gaγE·Be, (5e)

where now Bonly contains the axion-induced mag-

netic ﬁelds. We will now turn our attention to a plasma

that is inﬁnite in one direction (which we will take to

be the z direction), but bounded in the transverse direc-

tions. This deﬁnes a commonly encountered “plasma

waveguide” conﬁguration, whose analysis we will bor-

row [68].

As we anticipate a wire metamaterial (WM), we will

assume that the medium only has a plasma response in

one direction, aligned with the cylinder, acting as vac-

uum in the other two directions. This is summarised in

a an electric permittivity ez. For simplicity we will con-

sider µ=1. The symmetry of the system allows us to

break up the ﬁelds into the transverse and z directions,

writing

B=Bt+Bzˆz;E=Et+Ezˆz;Be=Beˆz, (6)

where the subscript tstands for transverse. We can an-

alyze the ﬁelds into harmonic components, assuming

that the ﬁelds oscillate with angular frequency ω, for

which we derive (5a) and (5d)

∇t+∂

∂zˆz×(Bt+Bzˆz)=−iω(Et+ezEzˆz)

−iωgaγaBeˆz, (7a)

∇t+∂

∂zˆz×(Et+Ezˆz)=iω(Bt+Bzˆz). (7b)

Note that the transverse curl of the transverse vectors

is necessarily in the z direction, so we can divide these

equations into

ˆz·∇t×Bt=−iωezEz−iωgaγaBe, (8a)

ˆz·∇t×Et=iωBz, (8b)

ˆz×∂Bt

∂z+∇tBz×ˆz=−iωEt, (8c)

ˆz×∂Et

∂z+∇tEz×ˆz=iωBt. (8d)

Taking the ﬁelds’ z dependence to be of the form eikzz,

we arrive at a closed form for Btand Et;

Et=1

ω2−k2

z∇∂Ez

∂z+iω∇tBz×ˆz, (9a)

Bt=1

ω2−k2

z∇∂Bz

∂z−iω∇tEz×ˆz. (9b)

Thus the transverse ﬁelds depend only on Ezand Bz.

This is a consequence of the fact that the axion driving

term arises only in the direction of the external B-ﬁeld.

Since the axion does not couple to Bzmode, for modes

that do bring it in geometrically we can neglect it, and

focus on Ez, obeying

ω2

ω2−k2

z∇2

tEz+ω2ezEz+m2

agaγBea=0 . (10)

For a cylindrical structure it is easiest to bring in cylin-

drical coordinates, yielding

ω2

ω2−k2r2∂2Ez

∂2r+r∂Ez

∂r+r2ω2ezEz+r2ω2gaγBea=0 ,

(11)

where we have dropped the subscript on k≡kz. This

is solved by

Ez=−agaγBe

+C1J0(ir√ezpk2−ω2)

+C2Y0(−ir√ezpk2−ω2), (12)

where J0,Y0are standard Bessel functions and C1,C2

are constants.

To incorporate a bounded medium we need to solve

the differential equations in both the external and

plasma region and to match the solutions by apply-

ing appropriate boundary conditions. For concreteness

we will consider the outside to be a conductor; other

boundary conditions can be handled similarly for ex-

ample including an air gap between the conductor and

plasma or absorbing material. Since Y0(0) = −∞, so for

the plasma region (centered, of course, on the z-axis) we

use C2=0. Thus the solution inside the plasma is

Ez=−agaγBe

+C1J0(ir√ezpk2−ω2). (13)

Here the ﬁrst term is the solution for a homogeneous

medium. It will be approached in the limit that the

plasma radius is inﬁnite.

To proceed further we assume that we are only look-

ing for modes which will maximially couple to the ax-

ion, and so will a uniform zstructure (i.e., k=0). The

transverse ﬁelds are given by

Et=0 , (14a)

Bt=i

∂Ez

∂rˆ

θ. (14b)

To ﬁnd the solution outside the cylinder, we must know

the relevant boundary conditions. We will assume that

only the plasma itself is magnetised (i.e., Be=0 out-

side the plasma) and that the plasma is surrounded

by a conductor, so that the tangential component of

the E-ﬁeld (i.e, Ez), vanishes. Specifying the plasma-

conductor boundary at a radius R, we see that

Ez=−agaγBe

+agaγBe

J0(√ezrω)

J0(√ezRω), (15a)

Bt=−agaγBe

√ez

J1(√ezrω)

J0(√ezRω)ˆ

θ. (15b)

We are primarily interested in resonant (longitudi-

nal) modes, where Re(ez) = 0. As J0(0) = 1, be-

haviour in the centre of the medium is determined by

J0(√ezRω). As R→∞,J0(√ezRω)→∞, so the second

term in the E-ﬁeld vanishes. Similarly, J1(0) = 0 implies

that the B-ﬁeld only exists in the outer portions of the

medium. Thus a sufﬁciency large medium has a bulk

that behaves exactly the same as in the inﬁnite medium

case. However, note that this behaviour depends on e00

smaller e00

zrequires a larger Rfor the medium to appear

homogeneous.

B. Signal Readout

Of course the goal is not just to create an E-ﬁeld, but

to measure it by coupling it to an ampliﬁer using an

antenna. To discuss the signal that could be read out

by an ampliﬁer, we must have some description of the

readout and losses. To look at the material losses, in

the Drude model, which is typically used to describe

metals, ecan be written as

e=1−ω2

ω2+iωΓp'1−ω2

ω2+iΓpω2

ω3, (16)

where ωpis the (angular) plasma frequency and Γp

is the inverse lifetime of the plasmon. To include an-

tenna losses, i.e., the power extracted by an antenna

which comprises the signal, we can add in an addi-

tional damping term, Γa(which may be frequency de-

pendent), giving a total dielectric constant

e=1−ω2

ω2+iωΓp

+iΓa

ω. (17)

(Such an expression implicitly assumes that the antenna

couples efﬁciently to all parts of the system. As we will

discuss below, for very large systems this can require

multiple antennas.) This allows for a total loss rate Γt

near the plasma frequency in the medium, where

Γt=Γp

ω2

ω2+Γa, (18)

We deﬁne the quality factor as

Q=ωU

P. (19)

Uis the stored energy and P=˙

Uthe power. The en-

ergy stored an isotropic medium with temporal disper-

sion is given by

U=1

4Z∂(eω)

ω|~

E|2+|~

B|2dV '1

2Z|~

E|2dV , (20)

where the latter holds exactly in the limit where the

axion velocity and higher order powers of gaγare ne-

glected. As our medium is simply aligned (i.e., the E-

ﬁeld excited by the axion is only in the direction aligned

with the wires), the isotropic formula gives the same

answer as an anisotropic one. We also neglect spatial

dispersion, as for uniaxial wires this exists only for mo-

mentum in the z-direction [69]. To ﬁnd the losses in the

medium we can use the imaginary component of the

effective dielectric constant e00

zusing [70]

P=ωe00

2Z|Ez|2dV . (21)

Putting Eqns. (20) and (21) together, we then ﬁnd

Q=1

e00

. (22)

Here we will treat the system as having a single loss fac-

tor, thus deﬁning the “loaded quality factor”, in other

words the quality factor of the system with an antenna

system included. Conversely, the quality factor which

only includes resistive losses is generally refered to as

the ”unloaded quality factor”. The signal power can

then be written as

Ps=κGVQ

ρag2

aγB2

e, (23)

where

G=1

0g2

aγB2

eVQ2Z|E|2dV . (24)

While we have deﬁned Gto serve a similar role to the

geometry factor in a cavity haloscope, the deﬁnition

and derivation of Gdiffers conceptually. It does not

contain the overlap of the axion and photon wavefunc-

tions directly, but is rather a normalisation of the stored

energy in the resonator. We have deﬁned κ=Γa/Γtto

take into account that only the power into the antenna

is read out.

Note that near the plasma frequency e'iΓt/ωp, so

Q=1/|e|, and after some rearranging we reproduce

the formula for the Gin Ref. [43] for ma=ωp,

G=|ez|2

0g2

aγB2

eVZ|E|2dV . (25)

So far, our derivation has been for axion dark mat-

ter. However, plasma haloscopes are also very sensi-

tive to hidden photons (HP) [49]. HPs are a massive

U(1) gauge boson which mixes with the visible photon.

This interaction is described to lowest order by the La-

grangian within the propagation basis (i.e. the mass

basis) as [60,61,71,72]

L=−1

4˜

Fµν ˜

Fµν −1

4˜

µν ˜

F0µν +χ

2˜

Fµν ˜

F0µν

+mX2

2˜

Xµ˜

Xµ+eJµ˜

Aµ,

(26)

where the dark vector ﬁeld is ˜

Xµ, the dark photon ten-

sor ˜

µν,χis the mixing coupling constant, mXis the

mass of the dark photon, eis the electric charge, Jµis

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FERMILAB-PUB-22-739-TSearchingForDarkMatterwithPlasmaHaloscopesAlexanderJ.Millar,1,2,3,aStevenM.Anlage,4RustamBalafendiev,5PavelBelov,6KarlvanBibber,7JanConrad,1MarcelDemarteau,8AlexanderDroster,7KatherineDunne,1AndreaGalloRosso,1JonE.Gudmundsson,1,5HeatherJackson,7GagandeepKaur,9,1ToveKlaesson,1Nol...

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