arXiv:2208.01640v2 [hep-ph] 6 Aug 2022

Twisted Anyon Cavity Resonators with Bulk Modes of Chiral Symmetry andSensitivity to Ultra-Light Axion Dark Matter

J. F. Bourhill, E. C. I. Paterson, M. Goryachev, and M. E. TobarARC Centre of Excellence for Engineered Quantum Systems and ARC Centre of Excellence for Dark matterParticle Physics, Department of Physics, University of Western Australia, 35 Stirling Hwy, 6009 Crawley,Western Australia.

(Dated: 9 August 2022)

In this work we invent the Anyon Cavity Resonator. The resonator is based on twisted hollow structures,which allow select resonant modes to exhibit non-zero helicity. Depending on the cross section the cavity,the modes have more general symmetry than that has been studied before. For example, with no twist themode is the form of a boson, while with a 180o twist the symmetry is in the form of a fermion. We show thatthe general twisted resonator is in the form of an anyon. The non-zero helicity couples the mode to axions,and we show in the upconversion limit the mode couples to ultra-light axions within the bandwidth of theresonator. The coupling adds amplitude modulated sidebands and allows a simple sensitive way to search forultra-light axions using only a single mode within the bandwidth of the resonator.

I. INTRODUCTION

Chirality is a fundamental property in many physicalsystems ranging from particle physics1–3, topological andquantum systems4–9, complex molecules and chiropticalphenomena10–15. Many of these phenomena occur as sur-face states, at high energy or frequency, or due to complexmeta-structures16,17 or plasmonic systems18,19 which in-evitably add loss. In this work we realise new class ofelectromagnetic resonator, the anyon cavity resonator,based on twisted and Mobius-like structures, which ex-hibit monochramatic bulk chiral modes at radio frequen-cies with near unity helicity. Such resonators will allowenhanced bulk chiral spectroscopy and sensing over largevolumes ranging from radio to millimeter wave frequen-cies. Furthermore, the modes naturally couple stronglyto ultra-light dark matter axions with near unity formfactors, equivalent to the mode helicity. Ultra-light ax-ions have been shown to solve the Standard Model strongCharge-Parity problem20–23 and could account for the en-tire dark matter density of the universe24–26, and are usu-ally searched for using putative axion interactions withgluons and neutrons27–30. In contrast, ultra-light darkmatter axions experiments proposed through the axion-photon chiral anomaly require two near degenerate pho-ton modes, and are limited by how close in frequency thetwo modes can be tuned31–33. We show, that modes withnon-zero helicity interact with ultra-light axions causingan amplitude modulation without the need for two sep-arate photon modes. This not only drastically reducesthe complexity, but also opens up the possibility of util-ising low loss superconducting resonators34–37, allowingsensitive searches in the ultra-light mass range of 10−22

to 10−14 eV.

II. THE ANYON CAVITY RESONATOR

A class of microwave resonators with anyon symmetry,which exhibit monochromatic modes with non-zero helic-

ity have been constructed and analysed (see Fig.1). Thecavity takes the form of a waveguide with a twisted crosssection with two possible configurations: 1) A twistedring resonator similar to a Mobius strip, with only dis-crete angles of twist allowed, depending on the shapeof the waveguide cross section: 2) A twisted waveg-uide with conducting boundary conditions at the startand end of the waveguide that allows all possible twistangles. To construct these resonators we have imple-mented 3D printing from aluminium, similar to what hasbeen achieved with superconducting cylindrical cavityresonators34. We show that the shape of the cross sectionhas significant impact on the mode helicity, with an equi-lateral triangle cross section exhibiting the strongest pos-sible mode helicity of order unity. To model the resonantstructures, we use finite element analysis to calculate theproperties, such as resonance frequency, geometry factor(and hence Q-factor) and helicity.

xy

v

xyz

a b

R

v1

2 3

✓

l

3l

xy z

c S1

S2

S3

FIG. 1. The geometries investigated in this work; a) thetwisted triangular waveguide resonator and b) the triangularMobius ring resonator. c) The dihedral group of polygons,Dn, which have 2n symmetries; n rotational and n reflection.

arX

iv:2

208.

0164

0v2

[he

p-ph

] 6

Aug

202

2

2

✓=

0�✓

=36

0�

� +

TE111TM110

120�

✓

0�

S(exp.)21

-150

-125

-100

-75

-50

-25

60�

240�

S(sim.)21 [dB]

17

17.5

18

Frequency[GHz]

-150-100

-50[dB]

-70-60-50[dB]

a

b

BzEz BzEz

~B? ~E?

-0.010.01

0[(J

m�

3)1/2]

FIG. 2. (a) ~E⊥ (magenta) and ~H⊥ (black) fields, and corre-sponding (normalised) axial fields, of TM110 and TE111 modeswith θ = 0◦ and ψ± modes with θ = 360◦ (b) ExperimentalS21 transmission measurements of the 60◦, 120◦ and 240◦

twisted triangular waveguide cavities (dark blue) overlayingcalculated frequency response (as the colour density plot).Light blue vertical lines indicate the corresponding locationsof the resonant transmission peaks of the experimental data.

The normalised mode mode helicity38–45 is given by,

Hp =2 Im[

∫Bp(~r) ·E∗p(~r) dτ ]√∫

Ep(~r) ·E∗p(~r) dτ∫Bp(~r) ·B∗p(~r) dτ

, (1)

Here Ep(~r) and Bp(~r) are the respective mode elec-tric and magnetic field vector phasor amplitudes. He-licity is tied to the conservation of Zilch46, or photonicchirality14. Usually such electromagnetic properties areconsidered when photons interact with chiral moleculesor materials11,13,14, analogously, the axion is calculatedto couple to photons through a chiral anomaly, whetherin condensed matter physics6,8 or in quantum chromody-namics (QCD)20,22,23,47, so one might consider that sucha mode with non-zero helicity would couple strongly tothe axion, which we show to be true, which means theintroduction of the axion as a modification to Maxwell’sequations introduces a non-zero Zilch to the equations ofmotion, which must respect the chiral continuity equa-tion.

First, we consider the triangular linear waveguide res-

onator as a function of twist angle θ, as shown in Fig.1(a).We can imagine this waveguide by taking the resonatorcross-section in the {x, y} plane (red in Fig.1a) and ex-truding it along the z-axis. As it is extruded, the cross-section is rotated such that the beginning and end facesof the volume are rotated by a total angle of θ. To con-struct the ring resonator depicted in Fig. 1(b), we canimagine taking the linear resonator and joining its twoend faces. In both cases, the conducting boundary con-ditions now form a curvilinear geometry, which now pos-sesses mirror-asymmetry and can impart some helicityon electromagnetic radiation inside it48.

It has been previously demonstrated how a Mobiusstrip cavity with rectangular cross-section is analogous toa fermion when compared to an “un-twisted” ring cav-ity, which is analogous to a boson49. The topologicaltransformation on the latter results in a sign reversal ofcurrent amplitude upon a 2π rotation of the solutions(In+1 = −In), and a 4π rotation is therefore required forinvariance of the eigenfunctions (In+2 = In), resultingin half-integer mode indices. These half-integral modeindices guarantee that the current amplitude eigenfunc-tions transform as spinors on the twisted structure.

In the case of a ring resonator whose cross-section is aregular polygons of the dihedral group (Fig. 19(c)), therotation angle θ is not fixed to be a multiple of π as it isfor the previously discussed rectangular Mobius ring cav-ities. Instead, we have the choice of θ = ±(2π/n)Z wheren is the number of vertices and the number of rotationalsymmetries of the cross-section. This means that upon a2π rotation of the solutions, the current amplitude willbe transformed according to In+1 = eiθIn. Thus, if theMobius ring resonator displays Fermion-Boson rotationalsymmetry, the triangular cavities discussed hereon mustdisplay Anyon rotational symmetry. For example, forresonators with θ = 2π/3 such as those in Fig. 1, threelengths of the linear resonator in Fig 1(a) are necessaryfor invariance of the eigenfunctions and a 6π rotation inthe ring case (as demonstrated by the arrowheads). Infact, in the case of the linear waveguide, there are norestrictions placed on the value of θ.

The explanation of the generated helicity in twistedcavities arises from two factors: (i) a magneto-electriccoupling resulting from the mirror-asymmetry (chirality)of the electromagnetic radiation inside such a twistedresonator50, and (ii) approximately equal in-plane prop-agation constants between orthogonal transverse electric(TE) and transverse magnetic (TM) modes. The com-bination of these two effects results in the emergenceof non-degenerate Kramers partners for the photonicstates17, which we label ψ±.

The emergence of photonic spinors is well documentedfor such a chiral geometry49,50. To understand theiremergence, consider that the TE and TM modes of alinear twisted waveguide (arising from either Neumannor Dirichlet boundary conditions, respectively) have non-

vanishing magnetic/electric field components ~Hz/ ~Ez and

in plane (transverse) components ~E⊥/ ~H⊥. If the in-

3

plane propagation constants ~qTE/TM = qTE/TM(ω)~e⊥(where ω is the frequency of the harmonic wave) for thesetwo polarisations become equal (condition (ii) above),the magneto-electric coupling (i) mixes the TE andTM polarisations creating new spin states ψ±(~x⊥; ~q) =

Ez(~x⊥; ~q)±Hz(~x⊥; ~q). Naturally, this results in ~E and ~Bfields no longer remaining perpendicular within the onephotonic mode and therefore a non-zero solution for thenumerator integral in (1).

The in-plane ~E⊥ and ~B⊥ fields of the mode solutionslabelled ψ± are plotted in Fig. 2(a), indicating thatthey appear as the mixing of TM110 and TE111 modes atθ = 0◦. Together with the Ez and Bz field magnitudesplotted over the cavity volume, these field diagrams in-dicate that for θ = 360◦, the TE and TM modes nowappear as ψ− = Ez − Bz, which results in a positive

H value given ~E and ~B are rotating in the same direc-tion, whilst the higher frequency mode ψ+ = Ez + Bz,with negative H . The separation in frequency of thenew orthogonality basis modes can be thought of as amagneto-electric coupling causing a splitting of the twodegenerate θ = 0◦ modes. There also exists modes thathave relatively low H values. These modes are of theTE10p family, where p is a mode index for the axial di-rection, which do not have a TM counterpart and there-fore cannot fulfil condition (ii) so do no mix into a neworthogonality spinor basis.

The corresponding ψ± modes in the same length lineartwisted cavities, with higher-order regular polygon cross-sections will still fulfil condition (ii) given they’re equian-gular and equilateral, but the maximum achieved H willbecome drastically reduced. This is because as the num-ber of vertices is increased, the cross-section closer ap-proximates a circle, which would form a resonator whoseconducting boundary conditions become invariant undertwist angle. Thus, the greatest H will be observed forthe lowest order regular polygon cross-section with whicha volume can be constructed; the triangle.

The predicted eigenmode frequencies of the twisted lin-ear waveguide resonator can be experimentally tested byconstructing them for different values of θ and measur-ing their resonant modes, the results of which are pre-sented in Fig. 2(b). Three different θ valued cavities are3D printed and their transmission spectra measured atroom temperature through co-axial probes placed at theend faces. There is excellent agreement between simu-lations and experiment, with fψ− = 17.380, 17.238 and16.990 GHz for θ = 60◦, 120◦ and 240◦, respectively,which gives a dfψ−/dθ = −2.15± 0.0742 MHz/deg. Themodes appear to preserve a bandwidth of κ ≈ 4.1 MHz,invariant of θ (or Q-factor of order 4.2× 103 at 17 GHz).

We can now consider the case of bending the waveg-uide resonator’s two end faces around and forming a ring.One immediately apparent advantage of this is that twometallic boundaries are removed from the resonator andhence we would anticipate modes with lower loss, thebenefit of which will become apparent below. Simula-tion results of such a ring resonator are presented in

|E|0 max.

ring

waveg

uide

FIG. 3. Simulated H values for eigenmodes in an equilateraltriangular cross-section ring resonator with R = 150/2π mm,v = 20 mm and θ = 120◦ (blue circles) compared to the samedimension equivalent waveguide resonator (yellow triangles).

| ~E| is plotted in the inset colour density plots.

Fig. 3, demonstrating close to unity |H | for a partic-ular mode family. It should come as no surprise thatthe |H | ≈ 1 mode family appears identical to the ψ±

modes in the linear resonator, with an apparent cut-offfrequency of ∼ 17.2 GHz. Once again it is the com-bination of magneto-electric coupling resulting from thechirality induced by the twist and the coincident frequen-cies of the untwisted orthogonal modes that generates thehelicity of the new modes.

Importantly, Fig. 3 highlights that |H | ≈ 1 modesexist in the ring cavity beyond the fundamental m = 0mode, which is not the case in the waveguide resonator.This is a result of the boundary conditions of the endfaces in the latter limiting the original TE/TM mode ba-sis so that the former has zero Hz field at the end faceswhile the latter has maximum amplitude Ez field. In thewaveguide case, as p becomes large, the mode overlapbetween the two field maxima becomes smaller resultingin a smaller |H |. With the end boundary conditions re-moved, both sin(mφ) and cos(mφ) solutions are permit-ted in the ring resonator, where the azimuthal coordinateφ now plays the role of the axial coordinate z and there-fore p → m. Now in the equilateral cross-section case,each mode has a double-degeneracy; a TE/TM and asin/cos orthogonality basis, meaning higher order TEm11

and TMm11 modes will maintain good mode overlaps,forbidden in the waveguide case. In addition, the doubledegeneracy results in each of the equivalent ψ±m modes inthe ring case existing as doublets, i.e. a ψ+ cos(mφ) anda ψ+ sin(mφ) doublet.

The net result is that the ring resonator presents with awider range of modes with |H | ≈ 1 to choose from, whichas will be demonstrated below leads to a sensitivity toultra-light axions. Furthermore, the ring cavity has lowerlosses due to having less surface area per unit volume,which will enhance its sensitivity to the axion-photon

4

coupling constant gαγγ .

III. SENSITIVITY OF ULTRA-LIGHT AXIONS TO THEANYON CAVITY RESONATOR

The QCD axion is a putative pseudoscalar particle,postulated to solve the strong Charge-Parity (CP) prob-lem in Quantum Chromodynamics (QCD)20,47. The ax-ion is predicted to couple very weakly to other knownparticles and has thus been postulated to be cold darkmatter22,23,51–60. Moreover, it is predicted to be gen-erated in the early Universe, either before after or dur-ing inflation26,61–71 and is thought to have a mass rang-ing from any where between 10−24 to an eV . The cou-pling to photons occurs due to the axion mixing withneutral pions through the chiral anomaly. This “axion-electromagnetic chiral anomaly” may be described by thefollowing interaction term, added to the Lagrangian ofthe photonic degrees of freedom by69,

Laγγ =gaγγ

4aFµν F

µν = θ(t) ~E · ~B, where θ = gaγγa(t).

(2)Here, the photonic degrees of freedom are represented bythe electromagnetic field tensor Fµν and its dual, Fµν ,while the axion modification to the equations of motionare given by the dynamic axion theta angle pseudoscalar,

θ(t) multiplied by ~E · ~B, where, a(t), is the dynamic axionfield, and gaγγ , the chiral photon-axion coupling term.

Ultra-Light Dark Matter axion experiments typicallysearch for masses between 10−24 to 10−8eV 72–75. Suchexperiments must be maintained for multiple years to beable to search for such low-masses, for example a particlemass of 10−22eV corresponds to a frequency of 24.2nHzwith a period of 1.3 years. Alternatively, upconversiontechniques, which utilises two real photons for the twophoton degrees of freedom can work in a similar way tothe ULDM frequency comparison experiments if they are

made near degenerate with non-zero∫~En · ~Bm dτ be-

tween the two photonic modes76–80. However non-lineareffects and injection locking are likely to limit how closetogether the two modes can get, so they are not likelyto be configurable for ultra-light axions. In contrast,

Axion Mass (eV)

(GeV

-1)

CASTSN1987Aɣ

CMB

µHz mHz Hz

10-22 10-21 10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-1210-1710-16

10-15

10-14

10-13

10-12

10-11

10-10

=107

=109

=1011

FIG. 4. Predicted exclusion regions of axion-photon couplingstrength gaγγ of a low-mass axion search conducted by anAnyon cavity with different Qp values, shown in green, aswell as limits set by astrophysical observations.

the anyon cavity resonators can have modes of effectively

unity helicity, and thus a non-zero self∫~Ep · ~Bpdτ . In

the appendix we show the square of the helicity is equiv-alent to the resonant axion form factor that couples tolow-mass axions as an amplitude modulation through aparametric interaction.

For the purposes of sensitivity estimation, the SNR toa dark matter axion is calculated to be (see appendix),

SNR =gaγγβp|Hp|√

2(1 + βp)

Qp√1 + 4Q2

p(ωa

ωp)2

(106tfa

) 14 √

ρac3

ωp√Sam

.

(3)Here Sam represents the spectral density of the pump os-cillator amplitude fluctuations, which excites the modefrequency and also assumes the measurement time, tis greater than the axion coherence time so that t >106

fa. For measurement times of t < 106

fawe substitute(

106tfa

) 14 → t

12 . Thus, a unique sensitive search for ultra-

light axions in the mass range 10−22 to 10−14 using onlya single mode without the need for dual-mode excitationor a large volume magnet. The coupling to low-mass ax-ions occurs through the axion modifications to Maxwell’sequations adding electromagnetic chirality, with an esti-mate of the possible sensitivity detailed in Fig 4. Thesesensitivities are attained by solving equation (3) for gaγγas a function of axion frequency fa = ωa/2π, settingSNR = 1, assuming critical mode coupling βp = 1,|Hp| = 1, ωp/2π = 1 GHz, the cold dark matter den-sity ρa = 8 × 10−22 kg/m3 (i.e. 0.45 GeV/cm3), c thespeed of light and the spectral density of the pump sig-nal based off the anyon cavity’s amplitude fluctuationsto be state of the art Sam ∼ −160 dBc/Hz. Sensitivitiesfor different values of cavity Q-factor Qp are shown. Wealso note that if we include the putative monopole cou-pling to the axion81, we can replace gaγγ → gaγγ + gaBB

as outlined in the methods section.

APPENDIX A: AXION ELECTRODYNAMICS ANDCALCULATION OF SIGNAL SIDEBANDS

The photon-axion coupling from Eq.(2) modifies partsof the electrodynamic equations of motion proportionallyto the dynamic theta parameter, Θ(t) = gaγγa(t). Con-sidering the action density for the electromagnetic andaxion fields, it has been shown that a set of modifiedMaxwell’s equations may be written as82,

∇ ·(ε0 ~E −Θε0c ~B

)= ρe,

∇×

(~B

µ0+ Θ

~E

µ0c

)− ∂t

(ε0 ~E −Θε0c ~B

)= ~Je,

∇ · ~B = 0 and ∇× ~E + ∂t ~B = 0.

(4)

5

Correspondingly, the axion equation of motion in thenon-relativistic limit leads to a solution of harmonic form,and in the quasi-static limit the local axion field has nospatial dependence, simply given by,

Θ(t) = θ0 cos(ωat) (5)

where ωa = mac2

~ , θ0 = gaγγa0 and a(t) = a0 cos(ωat).Here we consider cold dark matter in the non-relativisticlimit, where a(t) is a large classical non-relativisticpseudo scalar field and gaγγ is an extremely small cou-pling so θ0 << 1 and the axion is almost “invisible”.

It is widely considered that the best way to searchfor the axion is to apply a large background DC mag-

netic field ( ~B0), which will mix with the axion to gen-erate a second photonic degree of freedom that can bedetected. This works for specific cases, as long as the

integral over the volume of the dot product of ~B0 withthat of the electric field of the second photonic degree

of freedom ( ~E1) is non zero (∫~B0 · ~E1dτ 6= 0). Here,

~E1 usually takes the form of specific transverse magnetic

modes in a resonant cavity when ~B0 is orientated in the zdirection, as for most other resonant modes the integralis zero. This is the basis of the axion cavity resonatorhaloscope83,84, which now comes in a variety of resonantand non-resonant forms depending on the mass rangeof investigation85–129. Also, more recently the AC ax-ion haloscope has been considered to search for low-massaxions31–33,78–80,130,131, which excites two modes in a sin-gle resonator. This technique allows new ways to searchfor low-mass axions through upconversion, where the ax-ion Compton frequency is upconverted to the higher car-rier mode frequency, where the difference frequency of thetwo modes is approximately equal to the axion frequency.In contrast, in this work we investigate the possibility ofdetecting the axion with a single monochromatic moderesonator with anyon symmetry, without a backgroundDC field or a second monochtomatic mode frequency ex-cited within the resonator.

Upconversion of Weak Low-Mass Axion Modulations

Here we consider the case where the axion Comptonfrequency, ωa, is smaller than the pump frequency, whichis assumed to be equal to the cavity mode frequency ωp,so ωa << ωp. The axion converts to a very weak slowsingle tone modulation of the cavity mode resulting inaxion upconversion, with the creation of upper and lowerside band modulations at ωp − ωa and ωp + ωa.

For such harmonic monochromatic solutions, theaxion pseudo-scalar may be written as, Θ(t) =12

(θe−jωat + θ∗ejωat

)= Re

(θe−jωat

), and thus, in pha-

sor form as, Θ = θe−jωat and Θ∗ = θ∗ejωat. Giventhe phase of the axion is arbitrary we may set θ =θ∗ = θ0, related to the root mean square value by,〈θ0〉 = θ0√

2. In contrast, the electric and magnetic fields

FIG. 5. Phasor diagram for E(~r, t) with respect to the rotat-ing frame e−jωpt. The left diagram shows the amplitude mod-ulated carrier, while the right diagram shows the frequencymodulated carrier. The modulations are single tone and as-sumed small, so that mam, mam << 1. The lower sidebandis shown in red, while the upper sideband is shown in blue.

as well as the electric current are represented as vector-phasors with slowly time varying modulations at the

axion frequency, ωa. For example, we set ~E(~r, t) =12

(E(~r, t) + E∗(~r, t)

)= Re

[E(~r, t)

]. Thus, in the limit

that the modulation is weak compared to the carrierpower, we define the modulated electric field vector pha-sor by,

E(~r, t) = Ep(~r)e−jωpt

(1 +

mam

2(e−jωat + ejωat)

+mpm

2(ejωat − e−jωat)

)= Ep(~r)e

−jωpt(

1 +mam −mpm

2e−jωat +

mam +mpm

2ejωat

)(6)

and its complex conjugate by,

E∗(~r, t) =

E∗p(~r)ejωpt

(1 +

mam −mpm

2ejωat +

mam +mpm

2e−jωat

),

(7)where the AM and PM modulation indices, mam andmpm, are much less than unity, with the correspondingphasor diagram shown in Fig.5. Likewise, the magneticfield and electric current vector phasors will have similarform. Next we take the time derivative of eqn. (6) andobtain,

∂tE(~r, t) = −jωpEp(~r)e−jωpt(

1 +mam −mpm

2

(1 +

ωaωp

)e−jωat

+mam +mpm

2

(1− ωa

ωp

)ejωat

)≈ −jωpEp(~r)e−jωpt

(1 +

mam −mpm

2e−jωat

+mam +mpm

2ejωat

)≈ −jωpE(~r, t)

(8)

6

and likewise

∂tE∗(~r, t) ≈ jωpE∗(~r, t) (9)

in the limit where ωp >> ωa, which allows us to ignore

terms of order ∼ mam/pmωa

ωp.

Thus, from Eqn. (4), (8) and (9), the axion modifiedAmpere’s law in phasor form may be shown to be,

1

µ0∇× B = Je − jωpε0E + jωaε0ΘcB

1

µ0∇× B∗ = J∗e + jωpε0E

∗ − jωaε0Θ∗cB∗(10)

while, the phasor form of the Faraday’s law in (4) be-comes,

∇× E = jωpB

∇× E∗ = −jωpB∗,(11)

We now may apply complex Poynting theorem usingthe modified axion electrodynamics in phasor form tocalculate the sensitivity of the anyon cavity resonator toultra-light axions132. The complex Poynting vector andits complex conjugate are defined by,

S =1

2µ0E× B∗ and S∗ =

1

2µ0E∗ × B, (12)

respectively, where S is the complex power density, withthe real part equal to the time averaged power densityand the imaginary term equal to the reactive power,which will be zero at the frequency of a resonant mode asthe inductive and capacitive terms cancel. Thus, we onlyneed to consider the real part to calculate the sensitivityof a resonant detector. Unambiguously the real part ofthe Poynting vector may be written as,

Re (S) =1

2(S + S∗). (13)

Taking the divergence of S in eqn. (12) and keeping thefull vector-phasor form we find,

∇ · S =1

2µ0∇ · (E× B∗) =

1

2µ0B∗ · (∇× E)− 1

2µ0E · (∇× B∗)

(14)

Combining (14) with (10) and (11), we obtain,

∇ · S =jωp2

(1

µ0B∗ · B− ε0E · E∗)−

1

2E · J∗e+

jωaε0c

2E · Θ∗B∗

(15)

Then, by taking the divergence of S∗ in eqn.(12) and

FIG. 6. Equivalent parallel LCR circuit model of a resonantmode with a coupling of βp, when impedance matched βp = 1.

keeping the full vector-phasor form we find,

∇ · S∗ =1

2µ0∇ · (E∗ × B) =

1

2µ0B · (∇× E∗)− 1

2µ0E∗ · (∇× B)

(16)

Combining (16) with (10) and (11), we obtain,

∇ · S∗ =jωp2

(ε0E · E∗ −1

µ0B∗ · B)− 1

2E∗ · Je−

jωaε0c

2E∗ · ΘB

(17)

Finally, by using (13) then applying the divergence the-orem and substituting (15) and (17), we obtain,∮

Re (S) · nds =jωaε0c

4

∫(E · Θ∗B∗ − E∗ · ΘB dτ

− 1

4

∫(E · J∗e + E∗ · Je) dτ

(18)Equations (18) represents the Poynting theorem equa-tion, which we may use to calculate the sensitivity of thesingle mode haloscope.

First we recognise the integral on the left hand sideof (18) represents the incident carrier power entering thecavity equivalent circuit (see Fig.6) at the carrier fre-

quency, so∮

Re (S) · nds = − 4βpPinc

(1+βp), where Pinc is the

incident power and βp is the coupling, the negative signis because the power is entering the resonator from anexternal source, rather than leaving. Note when the cou-pling is unity the resonator is impedance matched and∮

Re (S) · nds = −Pinc. Thus, starting with eqn. (18) we

substitute the values of Je ≈ ωpε0Qp

E and J∗e ≈ωpε0Qp

E∗,

(where Qp is the mode Q-factor) to obtain,

−Pinc4βp

(1 + βp)=jωaε0c

4

∫(E · Θ∗B∗ − E∗ · ΘB dτ

− ωpQp

εp2

∫E · E∗ dτ.

(19)

Next we substitute the values of E(~r, t), B(~r, t) and Θ(t).

To first order in the modulation side bands, we find

E(~r, t) · E∗(~r, t) = Ep(~r) ·E∗p(~r)(1 +mam(e−jωat + ejωat)

),

(20)

7

which only depends on amplitude modulation, as thephase sidebands are second order. Next, to leading orderat the same modulation frequencies we find that,

E(~r, t) · Θ∗(t)B∗(~r, t)− E∗(~r, t) · Θ(t)B(~r, t) =

Ep(~r) · B∗(~r)θ∗ejωat −E∗p(~r) · B(~r)θe−jωat.(21)

Substituting (20) and (21) into (19) gives three equa-tions at the carrier frequency and the two modulationside bands. At the carrier frequency the stored electro-magnetic energy in the resonator mode is known to be,

Up =ε02

∫Ep(~r) ·E∗p(~r) dτ, (22)

so we obtain the following equation, which represents thedissipated power, Pd in the steady state,

Pd =ωpUpQp

=4βp

(1 + βp)Pinc, (23)

where Qp is the cavity loaded Q-factor. The fraction ofthe dissipated power in the coupling resistor Re is thengiven by,

Pp =βpPdβp + 1

=4β2

p

(1 + βp)2Pinc. (24)

Next, it turns out the modulation sidebands give re-dundant information, both lead to the following value ofthe AM modulation index,

mam =1

2√

2Qp

ωaωp〈θ0〉Hp, (25)

where

Hp =2 Im[

∫Bp(~r) ·E∗p(~r) dτ ]√∫

Ep(~r) ·E∗p(~r) dτ∫Bp(~r) ·B∗p(~r) dτ

, (26)

is the normalised mode helicity of a monochromaticfield38–45. Here we have set θ = θ∗ = θ0, the amplitude ofthe axion theta term, where the root mean square averageis given by 〈θ0〉 = θ0√

2. Thus, the fraction of the power in

the amplitude modulated sidebands with respect to theincident carrier power is given by,

PamPinc

=m2amPpPinc

= Q2p

4β2p

(1 + βp)2

(ωaωp

)2 〈θ0〉28

H 2p . (27)

Here, H 2p may be defined as the single mode resonant

axion detector form factor analogous to that defined fora DC haloscope69. The above calculation assumes thatthe axion frequency is within the bandwidth of the reso-nant mode such that ωa <

ωp

Qp, which will be true in the

ultra-light regime. However, in general we must substi-

tute Qp → Qp√1+4Q2

p(ωaωp

)2to take into account the filtering

nature of the cavity resonator, in a similar to parametrictransducers for displacement measurements, or gravita-tional wave detection133.

In actual fact the axion presents as a narrowband sig-nal oscillating at the axion frequency, fa = ωa

2π , viralizedas a Maxwell-Boltzmann distribution of about a part in106, equivalent to a narrowband noise source with a fre-quency spread of 10−6fa Hz. In this case, defining thespectral density of the axion field as, SA(f)[kg/s/Hz],from (27) the spectral density of amplitude modulationsis given by,

SAam(f) = g2aγγ

β2p

2(1 + βp)2Q2p

1 + 4Q2p(fafp

)2

(fafp

)2H 2p SA(f),

(28)where SAam

is the spectral density (per Hz) of amplitudemodulation of the anyon cavity mode.

To calculate the signal to noise ratio of the anyon cavityhaloscope to dark matter, we need to relate the axionangle, θ0, to the dark matter density at the earth, ρa,where the cold dark matter density is taken to be ρa =8 × 10−22 kg/m3 (i.e. 0.45 GeV/cm3) in this analysis.This may be done in the standard way, where 〈θ0〉2 =

g2aγγ〈a0〉2 = g2aγγρac

~2

m2a

= g2aγγρac3

ω2a

. Thus integrating

over the bandwidth of (28) and given the pump oscillatorhas an amplitude noise spectral density of Sam per Hz,the signal to noise ratio is given by,

SNR =gaγγβp|Hp|√

2(1 + βp)

Qp√1 + 4Q2

p(ωa

ωp)2

(106tfa

) 14 √

ρac3

ωp√Sam

.

(29)This assumes the measurement time, t is greater than the

axion coherence time so that t > 106

fa. For measurement

times of t < 106

fawe substitute

(106tfa

) 14 → t

12 .

Note, there will be a frequency modulation of the modedue to the axion interaction as well. However, this willbe proportional to the real part of the helicity given inEqn. (26), and the sensitivity will be reduced by a factoror Qp as it would have to include a complex dissipativepart of the field, and hence the SNR proportional to thesquare root of Qp, while the amplitude modulations aredirectly proportionay to Qp.

Axion Maxwell Equations with a Monopole Term

Recently Sokolov and Ringwald showed that therewould be a significant modification of conventional axionelectrodynamics through the existence of putative mag-netically charged matter81. This has lead to a more gen-eral modification of axion electrodynamics, where assum-ing ∇a = 0, the modified Ampere’s law may be written

8

in phasor amplitude form as,

1

µ0∇× B = Je − jωpε0E + jωaε0cgaγγ aB

− jωaε0gaAB aE1

µ0∇× B∗ = J∗e + jωpε0E

∗ − jωaε0cgaγγ a∗B∗

+ jωaε0gaAB a∗E∗,

(30)

while Faraday’s law may be written in phasor form as,

∇× E = jωpB + jωagaBB

caE− jωagaAB aB

∇× E∗ = −jωpB∗ − jωagaBB

ca∗E∗ + jωagaAB a

∗B∗

(31)Then from Poynting theorem and with some algebra itis straight forward to show that equation (18) is of thesame form, but with θ0 = (gaγγ + gaBB)a0. Thus, limitson gaγγ presented in this work is equivalent to limits ongaγγ + gaBB and in general includes the monopole term.

AcknowledgmentsThis work was funded by the Australian Research

Council Centre of Excellence for Engineered QuantumSystems, CE170100009 and Centre of Excellence for DarkMatter Particle Physics, CE200100008.

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