Detecting high-frequency gravitational waves with microwave cavities

Abstract

We give a detailed treatment of electromagnetic signals generated by gravitational waves (GWs) in resonant cavity experiments. Our investigation corrects and builds upon previous studies by carefully accounting for the gauge dependence of relevant quantities. We work in a preferred frame for the laboratory, the proper detector frame, and show how to resum short-wavelength effects to provide analytic results that are exact for GWs of arbitrary wavelength. This formalism allows us to firmly establish that, contrary to previous claims, cavity experiments designed for the detection of axion dark matter only need to reanalyze existing data to search for high-frequency GWs with strains as small as h of order 10⁻²² to 10⁻²¹. We also argue that directional detection is possible in principle using readout of multiple cavity modes. Further improvements in sensitivity are expected with cutting-edge advances in superconducting cavity technology.

I. Introduction

The first direct observations of gravitational waves (GWs) by the ground-based interferometers LIGO and Virgo have ushered in the era of GW astronomy. While the central focus of such experiments has been on the hertz to kilohertz frequency range, an exploration across a much wider spectrum is warranted. The Universe is expected to be populated by GWs over many decades in frequency, analogous to electromagnetic (EM) radiation, carrying information that may revolutionize our understanding of nature. This fact has spurred the development of a large array of observational efforts with the aim of detecting much lower frequency signals compared to current interferometers. These include future ground-based and space-based laser interferometers, atom interferometers, pulsar timing arrays, and CMB observations, as well as new types of astrophysical signatures.

On the other hand, GWs of much higher frequency have more recently garnered renewed interest. A number of interesting proposals and operating detectors for high-frequency GW detection already exist, including interferometers, microwave and optical cavities, optically levitated sensors, mechanical resonators, superconducting rings, and detectors based on the inverse-Gertsenshtein effect and the excitation of collective magnon modes. However, there are still orders of magnitude in both GW frequency and amplitude, well-motivated by theory expectations, that we are currently unable to explore.

In this work, we focus on how GWs couple to electromagnetism, highlighting in particular the role of small-scale laboratory experiments for the detection of gigahertz-frequency signals. A GW propagating through a static background EM field sources a feeble EM field that oscillates at the frequency of the GW. Resonant detectors are well-suited to the detection of such oscillating fields provided that the GW is coherent over many oscillation cycles. In fact, similar signals arise from other new physics sources, most notably in the case of ultralight axion dark matter that couples to electromagnetism. Motivated by the tremendous progress in small-scale technology targeting dark matter detection, we focus on setups that are either identical or similar to existing experiments (such as ADMX, HAYSTAC, ORGAN, and CAPP), which feature a resonant conducting cavity of size of order a centimetre to a metre immersed in a strong static magnetic field. Since the resonant frequencies of conducting cavities are comparable to their inverse geometric size, such setups are naturally sensitive to GWs in the gigahertz regime. Our results also apply to other electromagnetic resonators, such as LC circuits.

From a more general perspective, a second goal of this work is to provide a description of how GWs couple to electromagnetism in a manner that is largely agnostic to the particular experimental setup. In performing such calculations, great care must be taken to preserve gauge invariance (equivalent to consistently incorporating the signal within a particular choice of frame). In particular, GW signals are often computed in the so-called transverse-traceless (TT) gauge, since the spacetime metric is especially simple in this case. However, in this frame, the background EM field and the cavity modes do not coincide with those in flat space. This has not always been taken into account in previous calculations, which has led several studies to conclude that no EM signal is generated when the background magnetic field is aligned with the GW’s direction of propagation. As we show in this paper, this statement is at odds with gauge invariance. Our treatment illustrates that existing experiments targeting axions, such as ADMX and HAYSTAC, already have sensitivity to high-frequency GWs and need only to reanalyze existing data with a different signal template.

For the detectors considered in this work, complications arising from gauge artifacts are avoided by noting that the laboratory defines a preferred frame, the so-called proper detector (PD) frame. For this reason, the majority of our calculations adopt the PD frame. However, in order to demonstrate gauge invariance, we also perform a simple toy example calculation in both the TT and PD frames to show that they yield identical results. More generally, the use of the PD frame has typically been restricted to situations where the GW wavelength is much larger than the size of the detector, such that it suffices to keep only the leading corrections to the flat spacetime metric. Here, we further improve upon such calculations by resumming the GW perturbation to all orders in the ratio of detector size to wavelength.

II. Gravitational-wave electrodynamics in the proper detector frame

In this section, we provide a detailed discussion of GW electrodynamics, paying particular attention to the role of gauge invariance. Before presenting the technical details, we give a conceptual overview of the signal strength and the process of graviton-photon conversion in the language of classical fields. As we show in the following sections, we find this formalism particularly convenient at the level of identifying optimal cavity modes and quantifying the dependence of the signal on the GW’s direction of propagation.

The GW-EM coupling is encapsulated in the Einstein-Maxwell action, Eq. (1). To isolate the effect of a GW, we first linearize the metric as the flat-space metric plus the dimensionless GW strain plus terms of second order in strain. In the presence of a static external B-field, the action contains terms of first order in strain schematically of the form strain times signal field times background field. This implies that a GW of frequency omega-g can generate an EM field of typical magnitude equal to the strain times the background field, at the same frequency. Inside an EM cavity, this signal will ring up coherently if the GW frequency matches the cavity’s resonant frequency. At the level of single quanta, this effect can be interpreted as graviton-photon mixing in a background magnetic field, known as the inverse-Gertsenshtein effect. We can also describe this effect in terms of a classical effective current, which as we show below is parametrically of size given by the GW frequency times the strain times the background field, when the cavity size is of order the inverse GW frequency. Because the graviton is described by a spin-2 tensor field, the direction of this effective current is nontrivially determined by the polarization of the GW.

As mentioned above, we make use of the PD frame throughout this work. This frame utilizes so-called Fermi-normal coordinates, which describe GWs according to a freely falling inertial observer and are written as an expansion in the proper distance from the detector’s center of mass. As we illustrate below, the EM signals generated by GWs in resonant cavities are most simply described using such coordinates. Regardless, this computation is nontrivial when the GW wavelength is comparable to the cavity size, in which case the expansion parameter is of order unity and the series expansion cannot be approximated by the first few terms. As far as we are aware, a closed-form expression for the metric, including terms to all orders in the ratio of detector size to wavelength, has not been presented previously. In particular, we show below that resumming the metric in the PD frame is possible for a monochromatic GW of any wavelength traveling along a fixed direction.

A. Analogies with axion dark matter detection

Though it is not strictly necessary for the logic of the paper, it is useful at this point to make an analogy with axion-photon conversion, since this will allow us to derive a quick back-of-the-envelope estimate for the sensitivity of existing axion experiments to GWs. Indeed, the similarity of the phenomenology of axions and gravitons interacting with EM fields has been noted since the seminal paper of Raffelt and Stodolsky, and the effective current formalism is often used when studying axion dark matter signals in the low-frequency (quasistatic) limit. The Lagrangian for an axion dark matter field interacting with EM fields is proportional to the axion field times the scalar product of the electric and magnetic fields, with the dimensionful axion-photon coupling as the prefactor. Taking the magnetic field to be a static external field, the Lagrangian now contains a bilinear term which allows an axion field at frequency omega-a to convert to an electric field that oscillates at the same frequency, with typical magnitude given by the coupling times the axion field times the background field. This is reflected in the equations of motion for the axion and EM fields, which can be written so that the time derivative of a nonrelativistic axion background field sources an effective current term on the right-hand side of Ampère’s law. Here, we defined the effective dimensionless field theta-a as the coupling times the axion field, which will allow for a useful comparison to the GW case discussed above. Since axion dark matter is described by a nonrelativistic spin-0 field, the direction of the effective current is determined straightforwardly by the external field, independent of the axion.

The effective current formalism helps elucidate the fact that the cavity modes that couple most strongly to GWs will in general be different from those excited by axions. Nonetheless, we will show below that for certain geometries, GWs do indeed have a nonzero coupling to the TM010 cavity mode currently employed in, e.g., the ADMX and HAYSTAC axion detectors, meaning that these experiments already have some sensitivity to GWs in their resonant frequency ranges. Momentarily ignoring very important differences in the spectral characteristics of the axion dark matter and GW fields, we can derive a conservative estimate for the sensitivity of axion dark matter experiments to coherent high-frequency GWs by comparing the respective forms of the effective currents. In particular, identifying theta-a with the strain h and noting that ADMX is currently sensitive to the QCD axion parameter space, corresponding to theta-a of several times 10⁻²², implies that such experiments are sensitive to similar values of the strain h (as well as smaller values for GW signals that are more coherent than axion dark matter). A more precise sensitivity estimate will be provided in Sec. V.

Aside from the difference in cavity mode selection rules, there is a second important conceptual difference between axions and gravitons related to the role of reference frames. The axion dark matter field is assumed to have a Maxwellian speed distribution in the galactic rest frame, and moving to the laboratory frame where the cavity fields are defined is a simple Galilean boost which does not parametrically affect the signal strength. On the other hand, the large gauge freedom of linearized general relativity allows the GW signal to be computed in different reference frames, but a gauge transformation will also transform the background EM fields at the same order as the signal strength. We explore these issues in detail below.

(Sections II B and II C — the general formalism for the effective four-current, the resummed proper-detector-frame metric, and the toy example computed in both the transverse-traceless and proper detector frames to demonstrate that they agree — are omitted for length; the complete text is at the source.)

III. Resonant excitation of cavities

In this section, we calculate the EM signal that arises in a resonant cavity immersed in a magnetic field. As emphasized above, in a realistic experimental setup, the background field is static and spatially uniform in the PD frame. Note that this is essentially the reverse of the toy example discussed above, where the background field was instead taken to be static in the TT frame. By contrast, our calculational setup corresponds to the physical situation of turning on a static and spatially uniform magnetic field in the PD frame.

A. General formalism

The components of the effective current enter as additional source terms in the inhomogeneous Maxwell equations. The piece of the current independent of the metric perturbation sources, for example, the external B-field. In the following we subtract all such zeroth-order pieces such that all fields are of first order in strain. On the other hand, the homogeneous Maxwell equations — Gauss’s law for magnetism and Faraday’s law — are not modified by the GW. Indeed, this is because these Maxwell equations come from a topological equation of motion which does not involve the metric. This will be of practical importance because the homogeneous Maxwell equations determine the resonant cavity modes, and thus we will see that the change in the modes induced by the tidal force of the GW affects our signal only at second order in strain.

Combining the inhomogeneous equations with Faraday’s law yields the standard form of the wave equation. The electric field in the cavity is expanded in terms of the resonant modes. In general, the sum over the modes includes both solenoidal and irrotational contributions. Since irrotational modes are not resonantly enhanced, they are omitted from our analysis below. Note also that we assume any degenerate modes have been diagonalized into orthogonal mode functions and indexed separately in the sum.

(The remainder of Sec. III A — the mode decomposition, the resulting signal power of Eq. (23), and the definition of the dimensionless cavity coupling coefficient of Eq. (22) — is omitted for length; the complete text is at the source.)

B. Selection rules for cylindrical cavities

The discussion in the previous section is valid for cavities of any shape. In the following, we focus specifically on cylindrical cavities in part because existing experiments use this geometry. For concreteness, we will consider such a cavity of equal radius and length. The solenoidal modes of a cylindrical cavity are classified into transverse magnetic (TM) and transverse electric (TE) modes. For cylindrical cavities, the generic mode number is represented by three integers standing for the azimuthal, radial, and longitudinal mode indices, together with an index labelling a pair of degenerate modes with distinct azimuthal dependence.

As illustrated by the coupling coefficient, it is advantageous to have a large overlap between the electric field mode and the GW-induced effective current. The geometrical coupling between the gravitational wave and the cavity is of order unity, as can be inferred from the similar spatial profiles of the effective current and the signal mode.

For a fixed GW polarization, the effective current is only invariant under rotations about the symmetry axis by integer multiples of pi, reflecting the spin-2 nature of the tensor field. The background field, being uniform and aligned with the cylindrical axis, is a scalar under the cylinder’s rotational symmetry. The EM fields of a cylindrical cavity can be decomposed into a sum over modes with definite transformations under that symmetry. Thus only modes with azimuthal index 2 couple to the effective current for this particular example. This matches the intuition garnered from the field maps, where the angular overlap of the effective current with an azimuthal-index-2 mode is visually apparent.

It is also instructive to apply this argument to the case where the GW remains coaxial with the cylinder, while the background field is not aligned with the cavity axis. In this example, the direction of the applied magnetic field does not preserve the symmetry of the cavity. Since the background field is a pseudovector under the cavity’s rotational symmetry, along with the spin-2 GW, this yields an effective current with a spin-1 component, which can couple to azimuthal-index-1 cavity modes. The coupling coefficient for such geometries is evaluated numerically in Sec. V B.

IV. Sources

Before moving on to calculating the sensitivity of an EM cavity to a generic GW in Sec. V, we first briefly survey the kinds of sources that could give rise to spatially localized (on the sky) high-frequency monochromatic GWs. Our main examples are mergers of subsolar mass objects — including primordial black holes (PBHs) and other exotic compact objects — and boson clouds from PBH superradiance. As we will see below, for binary mergers to generate inspiral signals in the gigahertz regime, the merging objects must be much lighter than a solar mass.

Exotic compact objects can be considerably lighter than a solar mass and therefore can emit GWs at high frequencies. Examples of such exotic compact objects are boson and fermion stars, gravitino stars, gravistars, and dark matter blobs. In general, the GW waveforms generated by merging exotic compact objects and PBHs are distinct, though this difference is small when the orbital radius is much larger than the spatial size of the merging objects; for simplicity, in our analysis we treat the merging objects as pointlike.

The frequency of the GW emitted by a merging binary increases during the inspiral phase as the two compact objects approach each other. There is an upper bound on the frequency of GWs emitted by a binary in a quasicircular orbit in the weak-field limit, corresponding to the GW frequency at the innermost stable circular orbit (ISCO). For a binary consisting of compact objects of equal mass, at sufficiently early times such that the orbital radius is greater than the ISCO radius, the GW frequency evolves as roughly 14 GHz times the ratio of a millionth of a solar mass to the binary mass, times the three-halves power of the ratio of the ISCO radius to the orbital radius. Thus, only very light binaries, such as sub-Earth mass PBHs, can generate GW signals in the gigahertz regime well before reaching the ISCO.

However, such GW signals are highly transient near the ISCO, as the frequency of the emitted GW evolves rapidly in time, which can limit the sensitivity of resonant experiments. In particular, GWs from the merger of light binaries can typically only resonantly excite a cavity for a short amount of time that decreases for heavier masses. This is quantified by the number of orbital cycles a binary spends emitting GWs within the resonator bandwidth. Requiring that a typical cavity is fully rung up by the GW signal requires that the binary mass be no greater than about 10⁻¹¹ solar masses, which, amusingly, is a scenario where PBHs could constitute an order-one fraction of the cosmological dark matter abundance.

The best case strain can be estimated under the assumption that the PBHs are 100% of the dark matter and all PBHs are paired in binaries. In this case we get a separation of 10⁻³ parsecs and hence a best case sensitivity of strain around 10⁻²⁶. A more realistic study finds a larger average distance at which one expects one merger event per year. Note that the merger rate is used as an approximation for the number of events where the gigahertz frequency band is crossed by a source nearby. Plugging the more realistic value for the distance into the strain formula yields a considerably smaller strain.

The experimental setup we discuss in Sec. V below has parametrically reduced sensitivity to sources that only spend a much smaller number of cycles exciting the cavity, which makes detecting heavier binaries (with a correspondingly larger strain) difficult. On the other hand, since transient mergers sweep through a large range of frequencies, the resultant GWs will hit multiple resonant frequencies of the cavity. Therefore, unlike searches for axion dark matter, one does not have to scan different cavity frequencies by tuning the cavity when looking for GWs from such signals; the frequency sweeping is done by the merger itself.

High-frequency GWs can also arise from the annihilation of boson clouds generated by black hole superradiance. For instance, bosons of mass of order a microelectronvolt times the ratio of 10⁻⁴ solar masses to the PBH mass accumulate in large numbers outside a PBH. When such bosons annihilate into gravitons in the background gravitational field, the frequency of the emitted GW is twice the boson mass, of order a gigahertz for a microelectronvolt boson. Thus, if such PBHs and sub-electronvolt bosons both exist, superradiant clouds emit very high-frequency GWs. From the point of view of the experimental signals discussed here, the advantage of this GW source is that the associated waveform is monochromatic and coherent over very long timescales. The coherence time is limited by the change in gravitational potential energy of the boson cloud as it annihilates, which leads to a small positive drift in frequency. This drift is small, such that the signal is effectively coherent for much longer than the cavity ring-up time. However, the masses of such hypothetical bosons and PBHs are unknown, so unlike the case of a transient inspiral signal, a scanning strategy must be implemented. The expected strain of such signals arising from a PBH a distance D from Earth is of order 10⁻²⁷ times the ratio of ten kiloparsecs to D, times the ratio of the PBH mass to 10⁻⁴ solar masses, where we have used that the fraction of PBH mass that the axion cloud carries is 10⁻³.

In agreement with all current constraints on PBH dark matter we can assume that PBHs constitute 1% of the dark matter density. This enables us to estimate the average distance to be about 1 parsec, from which we estimate a strain of about 10⁻²³. Note that this is a best case scenario and the actual strain might be orders of magnitude worse because we have not taken into account how the PBHs attained the spin that is necessary to generate the boson clouds via the superradiance mechanism.

While we do not focus heavily on noncoherent sources in this paper, there are also interesting sources of high-frequency stochastic GWs. The most prominent examples include GWs arising from first-order phase transitions, cosmic strings, inflation, preheating, and the thermal plasma (the so-called cosmic gravitational microwave background). We briefly discuss the detection of stochastic GWs with EM cavities below and will return to this issue in future work.

V. Sensitivity estimates and cavity coupling coefficients

In this section, we apply the formalism developed in Sec. III to a concrete experimental setup. We begin in Sec. V A with a general discussion of the sensitivity of a resonant cavity to coherent or stochastic GW sources, assuming an optimal cavity-GW coupling coefficient of order 0.1, and illustrate how this sensitivity scales with the assumed experimental parameters. In Sec. V B, we then examine the angular dependence of the coupling coefficient with respect to the direction of an incoming monochromatic GW for different cavity modes and GW polarizations. We find that for various cavity modes, the coupling coefficient is of order 0.1 over a large fraction of a solid angle, including the TM010 and TM020 modes, which are already employed by existing axion dark matter experiments, such as ADMX and ORGAN. The strong dependence of the coupling coefficient on the incident direction of the GW makes a plausible case for directional detection.

We note that our estimates are purely based on the signal induced by the direct coupling of the GW to photons that we described in the previous sections. In principle, the GW can also induce other effects. We have already discussed how deformations of the cavity walls are not observable in these setups, since they generate an EM signal at second order in strain. The GW can in principle also induce a relative motion between the laboratory apparatus generating the background B-field and the cavity itself. This relative motion can produce an additional oscillating EM field in the rest frame of the cavity. The resulting first-order oscillating component of the background electromagnetic field is shielded by the cavity itself (even in the absence of additional electromagnetic shielding). In practice only the static component penetrates the cavity walls efficiently and can produce a first-order signal within the cavity after interacting with the wave. Thus, any electromagnetic signal generated by deformations of the external B-field source generates a signal only at an order suppressed by the electromagnetic shielding of the cavity. An alternative justification for ignoring such effects is that the time-dependent electromagnetic fields generated by the GW interacting with the external electromagnetic source can only couple to the electromagnetic modes of the cavity by driving currents along the cavity walls. Such currents do not resonantly couple to the electromagnetic modes of the cavity since, by definition, the electric components of these modes vanish at the cavity wall.

A. Sensitivity estimate

The signal power due to a coherent GW on resonance with the cavity is given by Eq. (23). The signal-to-noise ratio is then given by the Dicke radiometer equation, Eq. (28), in terms of the effective noise temperature, the measurement integration time, and the signal frequency bandwidth. The sensitivity is estimated by taking a signal-to-noise ratio of at least one, which yields the reach quoted as Eq. (29): a strain of about 3 times 10⁻²², for a gravitational-wave frequency of 1 GHz, a coupling coefficient of 0.1, a background field of 8 T, a cavity volume of 0.1 cubic metres, a quality factor of 100000, a system temperature of 1 K, a signal bandwidth of 10 kHz and one minute of integration — parameters similar to those of ADMX. Recent advances in superconducting cavity technology suggest that achieving a quality factor of 10⁷ with a 6 T field may be possible in the near future, and, of course, a longer integration time is possible for a dedicated GW search.

In a realistic setup, the signal bandwidth will be determined by a combination of factors. For instance, it is bounded from below by, e.g., the intrinsic frequency spread of the GW source or the drift of the cavity resonant frequency, and it is bounded from above by the cavity bandwidth. Our conservative benchmark of 10 kHz corresponds to the cavity bandwidth for a quality factor of 100000 at 1 GHz, similar to that of the ADMX cavity. The fundamental lower bound on the bandwidth is given by the frequency resolution, which is saturated for a sufficiently monochromatic source with infinite coherence time in a phase-sensitive measurement scheme such as a lock-in amplifier. In this case, the signal-to-noise ratio scales linearly with integration time, resulting in the sensitivity quoted as Eq. (30): a strain of about 1 times 10⁻²³ for the same benchmark parameters. Indeed, this technique is already used in the GW detection community in the form of matched filtering, which ensures that the signal-to-noise ratio scales with integration time as long as the precise waveform of the signal is known.

The sensitivity estimate is illustrated in Fig. 4 for various existing axion experiments, as well as for superconducting cavities being developed at the SQMS Center at Fermilab which can support both large B-field and high quality factors. Note that the sensitivity is generally weaker at higher frequencies because the cavity volume typically scales with the inverse cube of the signal mode frequency. After taking the volume scaling into account, we find that the sensitivity to strain is proportional to the frequency, which degrades linearly at large frequencies. One might overcome this by using higher modes if the coupling coefficient for such modes does not decrease, though at very high mode numbers the quality factor will tend to decrease and mode crossings make isolating the signal mode difficult. Alternatively, one could consider a multiplexing strategy with N cavities each of a fixed volume, in which case the sensitivity would improve as the inverse square root of N. Indeed, this sort of strategy is being pursued by ADMX for their small sidecar cavities. A similar approach could be implemented in multicell cavities of the type typically used for radio frequency acceleration.

Although not the main focus of this study, we conclude this subsection with a brief discussion of stochastic GW signals and leave a more detailed investigation to future work. A stochastic GW background is described by the average strain power per unit frequency, i.e., the strain power spectral density. For a stochastic GW background of cosmological origin, this determines the energy density in GWs per logarithmic frequency, normalized by the critical energy density today. Such a noncoherent signal appears as an additional noise source in the detector, such that the signal-to-noise ratio is given by the ratio of the signal and noise power spectral densities, independent of the integration time. Although a stochastic signal does not resonantly excite the cavity, the signal-to-noise ratio scales linearly with the quality factor since larger values correspond to suppressed intrinsic thermal fluctuations. Experimental parameters similar to those adopted above lead to projected sensitivities that are cosmologically meaningless, although our setup may be applicable to stochastic sources of noncosmological origin, such as populations of merging PBHs.

(Sections V B and V C — the numerical evaluation of the coupling coefficient as a function of the incident direction and polarization for TM and TE modes, and its behaviour in the quasistatic regime — are omitted for length; the complete text is at the source.)

VI. Outlook and conclusions

In this work, we have analyzed the interaction between GWs and EM fields. By consistently working in the proper detector frame, which is the reference frame relevant for laboratory experiments, we have shown that axion dark matter haloscope experiments have sensitivity to gigahertz-scale GWs and only need to reanalyze existing data to set the current best bound on such signals. A generic feature of these setups is that the detailed structure of the signal is strongly tied to the incoming direction and polarization of the GW. Thus, the use of multiple cavities or multiple readout modes of a single cavity may enable the ability to localize the source and determine the polarization of a tentative signal. We have identified subsolar mass binary mergers and GW emission from superradiant boson clouds as two possible sources in the gigahertz range to which our setup could theoretically have some sensitivity, though the immediate prospects for detection are not strong.

Throughout, we have focused predominantly on EM conversion of GWs in a static background B-field using resonant cavity readout, since the signal is parametrically suppressed for experiments targeting much lower frequencies. Along these lines, we have noted that most axion experiments designed for lower frequency signals are not favorable for GW detection.

There is, however, at least one important exception to this parametric statement. For instance, using microwave cavities pumped with an oscillating B-field considerably improves the low-frequency scaling of the signal. This technique was proposed to detect the mechanical signal induced by GWs through the vibration of the cavity walls. A similar concept was subsequently applied to the EM signals generated by axion dark matter. In a future companion paper, we will extend previous studies to include the direct GW-EM coupling with an oscillating background field, which gives rise to a visible signal also in the idealized case where the cavity is completely isolated from external vibrations. It would also be interesting to compare the generic sensitivity of such a heterodyne detection scheme to the sensitivity of interferometers such as LIGO, extrapolated into the kilohertz to megahertz regime. More generally, we plan to explore the applicability of other axion experimental setups to GW detection, such as those planned for broadband readout at high frequencies.

A wealth of precious information on the fundamental laws of nature is encoded in GWs spanning orders of magnitude in frequency and strain. The high-frequency regime, well above known astrophysical sources, is particularly interesting to extract information on early times (and extremely high energies) in the history of the Universe. The interactions of GWs with electromagnetism have long been proposed as a possible avenue toward detection in this frequency regime. In this work, we have further fleshed out details of this approach, leveraging cutting-edge advances in high-quality-factor cavity technology, and we have set the foundations to study future experimental setups that can target this important new frontier in GW detection.

(The acknowledgments, the four appendices — cavity mode functions and their resonant excitation, the resummed proper-detector-frame metric, the transformation between frames, and the quasistatic limit — and the 137-item bibliography are omitted for length; the complete text is at the source.)