Sensitivity of Resonant Axion Haloscopes to Quantum Electromagnetodynamics

Recently interactions between putative axions and magnetic monopoles have been revisited by two of us [arXiv:2205.02605 [hep-ph]]. It has been shown that significant modifications to conventional axion electrodynamics arise due to these interactions, so that the axion-photon coupling parameter space is expanded from one parameter $g_{a\gamma\gamma}$ to three $(g_{a\gamma\gamma},g_{aAB},g_{aBB})$. We implement Poynting theorem to determine how to exhibit sensitivity to $g_{aAB}$ and $g_{aBB}$ using resonant haloscopes, allowing new techniques to search for axions and a possible indirect way to determine if magnetically charged matter exists.

The generalised axion electrodynamics, expanded to include all possible couplings between axion and electromagnetic field, was shown to be given by (in SI units), where g aBB and g aAB are additional axion couplings, which were shown to dominate the effective axion currents over the conventional term, g aγγ [1].Here, B 0 and E 0 are the electromagnetic background fields, which are generated from free charge and current densities defined by J e0 and ρ e0 , and to zeroth order in axion coupling satisfy the non-modified Maxwell's equations given by, while B 1 and E 1 are the axion generated fields of the first order in axion couplings to photons, which will also generate the associated free current and charge densities, J e1 and ρ e1 , respectively, within the haloscope detector.
For cold dark matter it is usual to assume ∇a = 0, and in this limit (1)-( 4) becomes, ∇ For this set of equations we notice the axion electric displacement current is generalised to, and an axion magnetic displacement current exists, Thus, in QEMD there are three extra axion current terms due to g aAB and g aBB compared to standard axion electrodynamics.
We may also use the method of writing the modifications as effective polarizations and magnetizations [1,[22][23][24][25], by implementing the following vector identities, ∇a 1)-( 4).In this case Gauss' and Ampere's laws become, ) where we define the effective polarization and magnetization as, Applying the same vector identities to the magnetic Gauss' and Faraday law we obtain, ) with the following definitions of effective polarization and magnetization, We have reversed the symbols and adopted an opposite sign convention for P m1 and M m1 compared to [1], this means we keep the vector P as an electrical polarization, and vector M as a magnetic polarization (or magnetization), then the subscript refers to whether it is induced from an effective axion like electric (e) or magnetic (m) charge.The opposite sign convention is used to keep them consistent with how the auxiliary fields ( D and H) are defined in matter, which then may be generalised to, Note in this representation of axion modified electrodynamics, both the electric field and magnetic flux densities may have both vector and scalar potentials as dictated by two potential theory [20,21,[26][27][28][29][30][31][32][33].One can also write these electrodynamic equations in terms of the auxiliary fields.Thus, assuming ∇a = 0 and combining ( 18) with ( 6)-( 9), we may write the axion modified electrodynamics as, another form of the modified axion electrodynamics.
To calculate the sensitivity of experiments to (g aγγ , g aAB , g aBB ) one can use Poynting theorem, with different choices of vectors considering the electric and magnetic fields and auxiliary fields [34].However, similar to what has been shown before [19], for resonant and radiative systems the choice of Poynting vector to calculate the sensitivity gives the same first order solution, so in the following analysis we use the simplest form given by equations ( 6)-( 9) (this is straightforward to show and not included here).

PHASOR FORM AND COMPLEX POYNTING THEOREM
For harmonic solutions, the axion pseudo-scalar a(t) may be written as, a(t) = 1 2 ãe −jωat + ã * e jωat = Re ãe −jωat , and thus, in phasor form, Ã = ãe −jωat and Ã * = ã * e jωat .In contrast, the electric and magnetic fields as well us the electric current are represented as vector-phasors.For example, we set E 1 ( r, t) = 1 2 E 1 ( r)e −jω1t + E * 1 ( r)e jω1t = Re E 1 ( r)e −jω1t , so we define the vector phasor (bold) and its complex conjugate by, Ẽ1 ( r, t) = E 1 ( r)e −jω1t and Ẽ * 1 ( r, t) = E * 1 ( r)e jω1t , respectively.Following these definitions, the axion modified Ampere's law in (2), in phasor form becomes, while Faraday's law in (4) becomes, To implement Poynting theorem to calculate the sensitivity of a resonant system, we need to calculate the real power flow at resonance as the reactive power is zero at the resonant frequency [19].The complex Poynting vector and its complex conjugate are defined by, respectively, where S 1 is the complex power density of the harmonic electromagnetic wave or oscillation, with the real part equal to the time averaged power density and the imaginary term equal to the reactive power, which may be inductive (magnetic energy dominates) or capacitive (electrical energy dominates).For resonant systems as analysed in this paper the inductive and capacitive imaginary terms cancel as verified in [19], so we only need to consider the real term, however for reactive systems it is the reactive power that dominates, but these systems are not considered in this paper.Unambiguously we may calculate the real part of the Poynting vector by, Taking the divergence of Eq. ( 26) we find The next step is to calculate and Substituting Eqs. ( 23) and ( 24) into Eqs.( 28) and ( 29) leads to, Now by substituting ( 30) and ( 31) into ( 27), we find, then applying the divergence theorem, we obtain Re (S 1 ) This equation may be used to calculate the expected power in the photon-axion energy conversion in a resonant system.

SENSITIVITY OF AXION RESONANT HALOSCOPES UNDER DC MAGNETIC FIELDS
In this section we calculate the sensitivity of conventional Sikivie-type resonant axion haloscope [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53].If we assume that the background degree of freedom is only excited by a DC magnetic field, then background field equations ( 5) become, Since the phase of the photon leaving the cavity is not an observable, we may arbitrarily set the axion phase, setting First thing we notice is that it is quite clear that a haloscope with a DC magnetic field is not sensitive to g aBB .
In the following, we assume a resonant cavity of volume, V 1 , with resonant modes of stored electromagnetic energy, U 1 , given by, where the first term is the electric stored energy and the second term is the magnetic.The electric and magnetic energy must be equal, thus we may also write . The first term on the right-hand side of (35) represents the power dissipated in the resonator.For example, the electric field on the surface of the cavity where the power is dissipated is given by E 1 = R s K 1 , where R s is the surface resistance and K 1 the surface current.Substituting these values into . From this relation the well-known geometry factor, which relates the cavity surface resistance to the cavity Q may be derived, In this case the dissipative components of E 1 and B 1 fields are attenuated in the cavity walls in a similar way and are effectively in phase, and we may write the fields as complex with respect to a loss tangent (tan , where the real terms relates to the oscillating fields over the volume and the imaginary terms relate to dissipative fields, and thus . Further-more, on resonance and in steady state we may assume Re (S 1 ) • nds = 0, meaning there is no external energy inputted at the resonant mode frequency, so the axion generated signal power, P s1 in the resonant mode is equal to the dissipated power, P d , so P s1 in equation ( 35) may be identified as, (38) Combining (36) with (38) we may obtain, Now defining the following form factors of the haloscope, and setting , where ρ a is the axion dark matter density, the signal gained at the output of the haloscope becomes, Considering the modes in a cylindrical cavity, with the z-axis aligned with a DC magnetic field, so B 0 = B z ẑ, then B 1 • ẑ dV = 0 for all modes if B z is constant, and the calculation is consistent with what has been derived previously [18,54,55], equivalent to the well-known sensitivity equation of a Sikivie-type axion haloscope, which to first order is not sensitive to g aAB or g aBB and is only sensitive to g aγγ .
To gain sensitivity to g aAB with a DC haloscope, one can apply a nonuniform DC magnetic field with other vector components or a DC electric field (see next section), in a similar way to scalar-field dark matter experiments recently proposed in [56], which set limits on the dilaton scalar coupling parameter, g φγγ .It turns out that the limits set on g φγγ in [56] are equivalent to limits on the axion-photon parameter, g aAB , so in effect this paper also sets limits on g aAB .

SENSITIVITY OF AXION RESONANT HALOSCOPES UNDER DC ELECTRIC FIELDS
If we assume that the background degree of freedom is only excited by a DC electric field, then background field equations ( 5) become, Since the phase of the photon leaving the cavity is not an observable, we may arbitrarily set the axion phase, setting ã = ã * = −a 0 = − √ 2 a 0 (33) becomes, Now defining the following form factors of the haloscope, the signal gained at the output of the haloscope becomes, Thus, by applying a DC electric field to a cavity resonator, the experiment becomes sensitive to g aBB and g aAB , where the latter is sensitive to T M 0,n,0 modes in a cylindrical cavity resonator if an electric field is applied along the cylinder axis.However, to attain sensitivity to g aBB a more complicated DC electric field is required.It is a much harder experiment to supply a large DC electric field across a high-Q tunable cavity, even though the QCD axion is supposed to have a larger coupling to g aBB than g aγγ [1].Thus, the resonant cavity technique might not be the optimum way to make use of this larger coupling, as it is much easier to configure an experiment with a large DC magnetic field in comparison to a large DC electric field.

SENSITIVITY OF UPCONVERSION RESONANT HALOSCOPES
The upconversion technique utilises two modes of a resonant cavity, a readout mode (subscript 1) and a back-ground mode (subscript 0), and will up converts a putative axion signal of mass equivalent to the difference frequency between the two modes, so ω a ∼ |ω 1 − ω 0 |, where ω a << ω 1 .This technique was first proposed in [57,58] and experimentally demonstrated in [59], and showed that a dark matter axion will perturb the frequency (or phase) and amplitude (or power) of the readout mode.The former we call the "frequency technique" and the later the "power technique".The power technique was also proposed in [60][61][62][63], and later performed in [64].In this work we calculate the sensitivity of this experiment to the three axion coupling parameters (g aγγ , g aAB , g aBB ).

Power Technique
Here we use the Poynting vector equation (33) to derive the sensitivity of the power technique, where the background field will mix with the axion to generate power at the readout mode frequency.Since the phase of the photon leaving the cavity is not our observable, we may arbitrarily set the axion phase, setting where we can identify the power generated by the axion as, Note, the sensitivity coefficients to g aAB drop out, as in the lossless limit they are identically zero.
Ignoring losses in the background fields we individually consider them real for both E 0 and B 0 .
Equating P s1 in (48) to P d = ω1U1 Q1 , we can write the stored energy as, Since Then defining the unit vectors, so cB 0 = E 00 b 0 and E 0 = E 00 e 0 , and cB 1 = E 01 b 1 and E 1 = E 01 e 1 then (50) becomes, where the overlap functions are defined by [57], where the square of the overlap functions are analagous to the form factors in the previous sections.
Here E 00 = 2Pc0 ω0ǫ0V where P c0 is the circulating power of the background mode over the cavity volume V ,which is related to the incident power, P 0inc by, where β 0 is the background mode coupling to the cavity and Q L0 the mode loaded quality factor.Now, we can determine the square root power in the coupling circuit of the readout mode to be, where we define δω a = ω 1 +ω a −ω 0 , so when δω a = 0 then ω a = ω 0 −ω 1 and the axion induced power is upconverted to the frequency, ω 1 .Thus, δω a defines the detuning of the induced power with respect to the readout mode frequency.Combining ( 51)-( 54), we obtain where which are the axion gain coefficients in units of square root power.Equation ( 55) can be used to calculate the sensitivity for the power technique, and is sensitive to the effective monopole coupling term g aBB unlike the Sikivietype detectors that utilises a DC B field.
To calculate the signal to noise ratio (SNR) for virialised axion dark matter from the galactic halo, we take into account that it presents as a narrow band noise source with a line width of a part in 10 6 .In SI units we may relate the axion amplitude to the background dark matter density in the galactic halo, ρ a , by a 0 = √ ρac 3 ωa .For i = aγγ or i = aBB, then limits on the axion couplings can be undertaken independently by calculating, where, P N (Watts/Hz) is the noise power competing with the axion signal and ∆f a is the axion bandwidth in Hz, where ∆f a = fa 10 6 for virialised dark matter.This assumes the measurement time, t is greater than the axion coherence time so that t > ∆f −1 a .For measurement times of t < 10 6  fa we substitute 10 6 t fa . The noise power in such experiments is dominated by thermal noise in the readout mode resonator of effective temperature, T 1 and the noise temperature of the first amplifier, T amp after the readout mode, and is given by [65,66], (58) In the case β 1 ∼ 1 and δω a ∼ 0 then P N ∼ kB (T1+Tamp) 2 , and assuming β 0 ∼ 1 the signal to noise ratios become,

Resonant Haloscope Frequency Shift from Perturbation Analysis
As pointed out previously, there is no first order frequency shift for DC Sikivie haloscopes as they are only second order sensitive to frequency shifts [57].However, when two AC modes are excited, the situation is different as the DC and AC haloscopes belong to different classes of detectors.Since virtual photons or static fields carry no phase, the DC haloscope belongs to the class of phase insensitive systems.In contrast, the AC scheme relies on a pump signal carrying relative phases to the readout signal and axion field, and since this occurs in a resonant cavity, phase shifts are converted to frequency shifts.This is analogous to existing amplifiers that can be grouped into DC (phase insensitive) amplifiers, where energy is drawn from a static power supply, and parametric (phase sensitive) amplifiers, where energy comes from oscillating fields.
To calculate frequency shifts here we adapt the perturbation theory technique to axion modified electrodynamics as opposed to the quantum optics technique used in the past [57,58], which gives the same result.We consider the perturbed readout resonator mode fields, (E ′ 1 , B ′ 1 ) and frequency, ω ′ 1 , due to the mixing of the axion and the background pump mode, where the unperturbed modes and frequency are given by Ampere's law and Faraday's law in standard electrodynamics, ) and the perturbed Ampere's law and Faraday's law, are derived from (23) and (24), and written as and Then we can implement the following integral, Substituting ( 60), ( 61) and ( 62) into (63), one obtains, where, δω 1 = ω ′ 1 − ω 1 .Now to apply perturbation theory we set all perturbed fields and currents to approximately their unperturbed values, and given that dV , where U 1 is the stored energy in the resonator, then from (64) we derive ) This equation may be used to calculate resonant cavity frequency shifts based on upconversion.

Frequency Technique
In this section we use perturbation analysis via equation (65) to derive the sensitivity of the frequency technique, where the background field mixes with the axion to perturb the frequency of the readout mode.Since the integrals must be real, then in terms of unit vectors, we obtain Now given that E 01 = 2Pc1 ω1ǫ0V and P c1 = 4β1QL1 (β1+1) 2 P 1inc , and by considering the sidebands transferred to the coupling circuit, then taking the root mean square average of both sides, then (66) may be written as, where and depend on the same normalised overlap functions as in the power technique.
To calculate the signal to noise ratio (SNR) for virialised axion dark matter from the galactic halo, for i = aγγ or i = aBB, then limits on the axion couplings can be undertaken independently by calculating, where, S y1 (1/Hz) is the fractional frequency fluctuations competing with the axion signal in the readout oscillator.
The lowest noise oscillators are frequency stabilised by a phase detection scheme, which in principle is limited by the effective readout system noise temperature T RS of the internal phase detector (and includes the resonator and amplifier noise temperature), which is close to ambient temperature for a well-designed system [67], and in such a case the oscillator noise will be, where P 1inc is power incident on the input port to the readout mode.In this configuration, usual operating conditions will require β 1 ∼ 1, β 0 ∼ 1 and δω a ∼ 0 then the signal to noise ratios become, and .
(72) Note, the power and frequency techniques derive the same signal to noise ratio, which depends on the system noise temperature.Thus, the inherent sensitivity of both techniques is the same, which one is better will depend on which can be configured better experimentally to be less influenced by the relevant noise sources and external systematics.

Sensitivity to ultra-light axions
Another upconversion technique worth mentioning is the use of the anyon cavity resonator, which uniquely allows the detection of ultra-light axion dark matter due to the non-zero normalised helicity of the cavity mode, given by, so the single mode may act as its own background field [68,69].This technique has been detailed in [68] and included the QEMD terms to show that this technique was sensitive to the sum of g aγγ and g aBB , where the helicity is equivalent to the overlap functions defined for the two-mode upconversion detectors.In this case both g aγγ and g aBB have the same overlap function.

DISCUSSION
In this paper we have applied Poynting theorem to axion modified electrodynamics.In QEMD there are three parameters to put limits on (g aγγ , g aAB , g aBB ), which in principle maybe related if the axion is a QCD axion [1] and is a generalisation of the two-photon chiral anomaly if magnetic charge exists.We have shown if the background electromagnetic field is a DC magnetic field then we can only simply configure the experiment to be directly sensitive to g aγγ .Nevertheless, with a more complicated DC background magnetic field, sensitivity to both g aAB and g aγγ may be obtained.Conversely, if the background electromagnetic field is a DC electric field then we can only simply configure the experiment to be directly sensitive to g aAB .Nevertheless, with a more complicated DC background electric field, sensitivity to both g aAB and g aBB may be obtained.However, if the background field is an oscillating electromagnetic field, the experiment can be sensitive to g aγγ and g aBB at the same time, as we have shown for the upconversion experiments.
Note, the upconversion experiments search mass ranges less than a µeV , which is a much lower mass range when compared to the DC Sikivie haloscope [59].In particular, the anyon cavity haloscope is uniquely sensitive to ultra-light axions [68].Other ways to search for axions in the low mass range include reactive axion haloscopes, such as those based on capacitors [19,56] or inductors [70], however they were not considered in this work.
The caveat is that we have treated all the axion photon couplings to the axion field, a, as independent, and then for each of the couplings the terms ∝ ∇a may be omitted.However, it was shown in [1] that ∇a multiplied by g aBB can be of the same order of magnitude as ∂ t a multiplied by g aAB , which could lead to extra sensitivity than to what is calculated in this paper.