Axion and hidden photon dark matter detection with multilayer optical haloscopes
A well-motivated class of dark matter candidates, including axion-like particles and dark photons, takes the form of coherent oscillations of a light bosonic field. If the dark matter couples to Standard Model states, it may be possible to detect it via absorptions in a laboratory target. Current experiments of this kind include cavity-based resonators that convert bosonic dark matter to electromagnetic fields, operating at microwave frequencies. We propose a new class of detectors at higher frequencies, from the infrared through the ultraviolet, based on the dielectric haloscope concept. In periodic photonic materials, bosonic dark matter can efficiently convert to detectable single photons. With feasible experimental techniques, these detectors can probe significant new parameter space for axion and dark photon dark matter in the mass range.
There is overwhelming evidence that the majority of the matter density of the universe takes some beyond-Standard-Model form, referred to as dark matter (DM) Rubin and Ford (1970); Spergel et al. (2003). Despite this, the form of the dark matter remains almost entirely unknown. If, like Standard Model (SM) matter, it is a relic from the hot thermal plasma of the early universe, then the fact that it is ‘cold’ (low-velocity) today means that it cannot consist of particles lighter than YÃ¨che et al. (2017); IrÅ¡iÄ et al. (2017). However, there are a range of alternative, non-thermal production mechanisms which could generate a viable cold dark population of lighter particles. In many beyond-the-Standard-Model theories, these light new particles can naturally have very small couplings to SM states, allowing them to be stable and hard-to-detect. Thus, such particles can serve as attractive dark matter candidates. At masses , the dark matter must be bosonic, since the Pauli exclusion principle forbids fermionic DM from having the dense, low-velocity distributions observed in galaxies Boyarsky et al. (2009).
Unless there is a symmetry preventing it, the leading-order interaction between light bosonic dark matter and Standard Model matter will be absorption and emission of single DM particles. This is true for the simplest and most attractive light DM models, such as axions Weinberg (1978); Wilczek (1978); Peccei and Quinn (1977) or dark photons Holdom (1986). Accordingly, a range of existing and proposed experiments aims to detect the absorption of light DM through different mechanisms (see Irastorza and Redondo (2018); Graham et al. (2015); Jaeckel and Ringwald (2010) for reviews of axion and dark photon DM detection experiments). However, many of these are not sensitive to DM masses far above the microwave frequency range. In this paper, we discuss how to extend the search for light dark matter candidates to higher masses, from to .
For many kinds of dark matter couplings, DM to photon conversion is a promising experimental approach, transferring the entire rest mass energy of the dark matter to readily-detectable photons. At DM Compton wavelengths around or above meter scales, conversion experiments based on resonant receivers Sikivie (1983); Krauss et al. (1985); Sikivie (1985) are a practical solution, as illustrated by the ADMX experiment Asztalos et al. (2010); Stern (2016), and by a range of ongoing and proposed experiments at similar and lower frequencies Lamoreaux et al. (2013); Brubaker et al. (2017); Kahn et al. (2016); Chaudhuri et al. (2015). At higher DM masses, filling a large volume with resonant elements, such as cavities matching the DM Compton wavelength, becomes difficult. Consequently, other forms of target structure that can correct the mismatch between the DM and photon dispersion relations are more practical. ‘Dielectric haloscopes’ Caldwell et al. (2017); Phillips (2017); Sloan (2015) provide an example of this idea; a periodic structure of alternating dielectrics modifies photon propagation in the target volume, enabling DM-to-photon conversion for DM Compton wavelengths matching the target periodicity. The proposed MADMAX experiment Caldwell et al. (2017); Brun et al. (2017); Millar et al. (2017) aims to search for axion-like particle DM using this technique, over a mass range .
The main topic of this paper will be extending the dielectric haloscope concept to higher-than-microwave frequencies. At these shorter wavelengths, it becomes more difficult to construct and manipulate individual, wavelength-scale elements. On the other hand, it is possible to make bulk materials whose optical properties vary on the relevant scales, all the way down to ultraviolet wavelengths. These ‘photonic’ materials have been used to create many novel optical devices, such as very high quality cavities and filters Joannopoulos et al. (2008); Sekoguchi et al. (2014); Noda et al. (2007); Bushell et al. (2017). The simplest and most widely used examples are multilayer films, as employed in optical coatings.
There are a number of reasons why higher-mass bosonic DM is an attractive target for experimental searches. Practically speaking, single-photon detection becomes significantly easier at energies , corresponding to the energy resolution of superconducting detectors, as made use of in Arvanitaki et al. (2017). On the theoretical side, there are ranges of parameter space where simple early-universe production mechanisms can produce the correct DM abundance, with couplings below current constraints; in particular, purely gravitational production during inflation can result in a DM abundance of light bosons. Such cold bosonic dark matter acts as a classical field, oscillating at a frequency set by its mass and with an amplitude set by the mass and the dark matter density. It is coherent over times of order and lengths of order , where is the virial velocity in the galaxy.
In this work we outline an experimental proposal using multilayer films, combined with a sensitive photodetector, to search for bosonic dark matter. Alternating layers of commonly used dielectrics with different indices of refraction lead to coherent conversion of dark photon, and, in the presence of an applied magnetic field, axion, dark matter to photons. The resulting photons emerge in a direction perpendicular to the layers, and are focused onto a detector (Figure 1). These setups have close-to-optimal DM absorption rates (in a frequency-averaged sense). Small volumes () of layered material could achieve sensitivities several orders of magnitude better than existing constraints for these dark matter candidates.
Section II discusses the theory of DM absorption in layered materials; this can be skipped by readers more interested in experimental details. In Section III, we summarize these theoretical results, discuss concrete, illustrative examples of how such an experiment might be realized, and analyze the sensitivity of these setups. Section IV discusses sensitivity to other forms of DM. In Section V, we give a brief overview of DM production mechanisms, and conclude by discussing future extensions and comparing our proposal to other experiments in Section VI.
Ii Multilayer optical films
In this Section, we will start by giving a brief overview of the physics of DM to photon conversion, in the simplest photonic materials: multilayer films. By the scaling properties of Maxwell’s equations, this is exactly analogous to the physics of dielectric haloscopes at microwave frequencies, as derived in depth by various publications from the MADMAX collaboration Caldwell et al. (2017); Millar et al. (2017). One additional feature is that, at the shorter wavelengths we consider, the DM coherence length may be less than the scale of the experiment, requiring us to take the DM momentum into account (for example, in determining the direction of emitted photons).
ii.1 Axion conversion in layered materials
DM to photon conversion is an especially attractive experimental strategy for models in which the DM couples directly to the EM field — for example, an axion with Lagrangian
where we take the signature, and use the convention . Except where indicated, we use natural units with . An axion generally has a periodic potential, , where is the axion’s ‘decay constant’, and is consequently its mass. The ‘natural’ expectation for the coupling to photons is that , where is the fine structure constant Grilli di Cortona et al. (2016). As discussed in Section V, a dark matter abundance of these axions could be produced via a number of mechanisms. We will take such an axion as our prototypical example for the rest of this section, commenting later on sensitivity to other DM candidates.
In the presence of the interaction term, the Maxwell equations are modified to Wilczek (1987)
where we abbreviate as , here and following. In particular, a uniform oscillating field has the same effects as an oscillating current density in the direction of , while other contributions are suppressed by DM velocity. Accordingly, the ideal ‘target’ for axion to photon conversion is a strong magnetic field; in common with other axion detection experiments, we will use an approximately uniform field from a large magnet.
The remaining difficulty in accomplishing conversion is achieving a setup in which the interaction does not cancel out when integrated over the target. For resonant cavity experiments such as ADMX, modes above the few lowest-lying ones will have small overlap with the effective current . In target materials with small-scale periodicity, another formulation is that DM-photon conversion process must conserve (pseudo)momentum, up to the material’s reciprocal lattice vectors, analogously to Bragg scattering. As Figure 2 schematically illustrates, this means that the target must have structure on scales set by the Compton wavelength of the DM.
A simple way to realize the appropriate structure is with photonic
materials, which have spatially non-uniform optical properties.
The simplest situations to analyze correspond to periodic layered structures. Photon modes in an infinitely extended periodic medium are Bloch modes; if the layers are uniform in the plane, then
with , where is the periodicity in of the material. Taking to lie within the first Brillouin zone, , we can label different modes at the same by band number . Figure 3 illustrates part of this band structure for some simple periodic materials. While we are dealing with finite numbers of periods, rather than an infinite material, the modes in a large but finite stack will be similar to those in the infinite case. As noted above, the DM momentum is small compared to its mass, so we are interested the modes for which is small compared to . In particular, a DM mode with frequency with small momentum can convert to a photon of the frequency , if the Bloch mode at that frequency has Bloch momentum close enough to .
The DM absorption rate of any 1D stack configuration can be calculated via transfer matrices, as presented in Millar et al. (2017). However, in terms of developing a physical picture, it can be useful to present the calculation, at least around the points, in terms of the unforced photon modes in the layers. The simplest to analyse periodic configurations have layers of alternating refractive indices , with thicknesses such that their phase depths are equal, .
We start by ignoring the DM velocity, and treating the DM field as a classical background oscillating at frequency , with . This induces an electric field in the material, resulting in an instantaneous absorbed power of
where is the background magnetic field (this is the instantaneous energy flow from the DM field to the SM target).
At points in the cycle when , the electric field is continuous at the layer interfaces. The EM fields, at these moments, must correspond to those of a free photon mode. The out-of-phase component of the electric field does not contribute to the absorbed power, so we can calculate this taking only the free mode into account. For a half-wave stack with refractive indices , and layer thicknesses , there is a mode with electric field (parallel to the layers)
as illustrated in Figure 4 (there is also the out-of-phase mode, for which averages to zero across a period). If is the amplitude in this mode excited by the DM oscillation, then the instantaneous absorbed power is
where is the cross-sectional area of the layers,
assuming a uniform background magnetic field parallel to
is the energy stored in this mode Joannopoulos et al. (2008), we obtain a cycle-averaged power loss of
where is the local dark matter density and is the axion mass. This corresponds to solving for such that . The ‘quality factor’ depends on the stack’s surroundings. For an ‘open cavity’ setup corresponding to a stack of layers in air (i.e. none of the photons emitted by the layers are reflected back), the power loss is proportional to the area of the end-caps, so . However, the precise value depends on the form of the stack-air interfaces. For example, if the left-hand end of the stack were an layer, and the right-hand end an layer, as in Figure 4, then .
The ‘half-wave stack’ configurations considered above
have many useful special properties.
For generic layer profiles, there are ‘bandgaps’ around
each point, as illustrated in Figure 3.
At frequencies within the bandgap,
there are no periodic mode solutions — for a finite set of layers,
incident modes at these frequencies are exponentially attenuated
inside the material. Since the converted photons
produced by the DM must be close to these bandgap edges, this can lead
to complicated behavior (in particular, to very narrow
frequency range coverage).
However, the bandgaps around the points for half-wave stacks
vanish (right-hand panel of Figure 3).
A related advantage of this configuration is that such a stack has
high transparency at frequencies close to zero Bloch momentum.
Consequently, it is easy for photons produced by one stack
to pass through another at a nearby frequency (Section II.5).
Another point is that, for a stack terminated on one end by a mirror (as per Figure 1), the mirror’s boundary condition means that there is only one free mode at a given frequency, which is a standing wave. The absorbed power will then be controlled by the integral of this mode. However, while this can be very small at the central frequency, if the mirror selects the ‘wrong’ Bloch mode, we can go to a close-by frequency, , and recover an order- overlap. If the mirror selects out the correct Bloch mode, e.g. if the left-hand side of Figure 4 were replaced by a mirror, then equation 8 applies. For a stack-air interface at an layer, .
For a half-wave stack, it is easy to modify further by placing reflecting surfaces (e.g. other dielectric layers) at the ends of the stack. However, as Section II.3 shows, increased is always compensated for by decreased frequency coverage.
The DM in our galaxy is expected to have a virialized velocity distribution, with typical velocity . For (see below), the effects of the finite number of layers will generally be more important than the small spread in DM momenta, with regard to the conversion rate. One important point is that the transverse momentum of a converted photon is the same as that of the DM quantum, so the emitted photons will be distributed within a cone of opening angle around the axis. This is important in terms of focussing the emitted photons (Section III.2), and could potentially be used post-discovery to determine the velocity distribution of the DM Arvanitaki et al. (2017).
Another effect of the DM velocity distribution is that, as mentioned in the Introduction, the DM field has a coherence length . This means that photon emission from points further apart than adds incoherently, when averaged over long times. Thus, stacks of periods will no longer have peak conversion power .
ii.2 Dark photon conversion
In addition to axion-like particles, light vector DM is another natural candidate for dielectric haloscope searches. A dark photon coupled to the SM through kinetic mixing with the photon has an unusually large window of open parameter space, since plasma effects suppress its emission from stars An et al. (2013a); Hardy and Lasenby (2017); Chang et al. (2017), and it does not mediate long-range forces between neutral matter. Also, unlike spin-0 DM candidates, vector DM has a polarization direction, and can convert to transverse photons even in a homogeneous target without velocity suppression. As discussed in Section V, a dark matter abundance of such vectors can naturally be produced in the early universe via inflation.
Suppose that dark matter consists of a dark photon , with
this is equivalent, after field redefinition, to the usual ‘kinetic mixing’ interaction . Solving for the fields in a uniform dielectric, an oscillatory background field induces a corresponding Standard Model (visible) electric field , where is the dark electric field, and is the polarizability. This is in comparison to the axion-induced electric field Millar et al. (2017). Since the boundary conditions at a dielectric interface are the same in both cases, the dark-photon induced power (per unit area, from a single interface) is given by
where denotes the part parallel to the interface. Thus, the conversion rate for a dark photon amplitude is equivalent to that for an axion amplitude with . For a half-wave stack,
where is the angle of from the layer normals. This angle will be approximately constant over the coherence timescale of the DM field, , and will vary over longer timescales; for an isotropic DM velocity distribution, the long-time average of is . One difference between the dark photon and axion cases is that, for the latter, the polarization of the emitted photons is set by the field, while for a dark photon, it is set by the DM itself.
From equation 11, the dark photon to photon conversion rate is independent of , for a given target volume (since the length of an -period stack is ). This may naively be worrying, since the dark photon has to decouple from the SM in the limit. For cavity experiments such as those proposed in Chaudhuri et al. (2015), this manifests itself as a suppression of the conversion rate, where is the scale of the experiment, for . However, in our case, the width of the layers is set by , so when our expressions are valid, the experiment is automatically much larger than the DM Compton wavelength.
ii.3 Frequency-averaged power absorption
A natural expectation is that the fractional range in frequencies over which we get converted power (eqns (8),(11)) is . For example, a change leads to a change in phase across layers, destroying the coherent addition. Figure 5 shows this scaling for example values of . Thus, while the peak conversion power is , the averaged power over an range in frequencies is , and is not coherently enhanced. If we do not know the DM mass, which could a priori be anywhere in a wide interval, then the frequency-averaged power controls how fast we can scan over a range of masses (for a low-background experiment).
We can see the scaling in more generality from an ‘impulse response’ argument (for clarity, we take the example of axion DM here; the same arguments apply to a dark photon). For a target consisting of a set of thin interfaces between uniform layers, a sufficiently short ‘pulse’ of the DM field must, immediately after its arrival, result in only local field disturbances around the interfaces: influences have not had time to propagate further, and a uniform lossless dielectric does not absorb any energy. In the linear regime, the response of the target to a superposition of DM signals is the superposition of the responses to each signal; accordingly, we can decompose the pulse into sinusoidal components. Analyzing the response of a single interface to a sinusoidal DM oscillation, we find that the cycle-averaged power absorbed, per unit area, is
where and are the refractive indices on either side of the interface Millar et al. (2017). So, for a pulse shorter than , for which the power absorbed should sum incoherently across different interfaces, the total energy absorbed (per unit area) is
where is the Fourier transform of the pulse , and the sum runs over interfaces . Since, in the linear regime, the power absorbed depends only on the power spectrum of the DM signal, and not on the relative phases of different frequency components, we can infer that for a general DM signal, with power spectral density , the long-time average power absorbed, per unit area, is
if is a broad distribution over the appropriate frequency
range, and the refractive indices do not appreciably change
over this frequency range.
where we take the average to be over . For a half-wave stack, most of this power is in the peak, at frequencies within of , as illustrated in Figure 5.
If we do not know the DM mass, then since the DM is non-relativistic, this corresponds to not knowing the oscillation frequency of . Suppose that we have some number of different experimental configurations which, in combination, provide sensitivity over a range of . Then, the total energy converted by all of the experiments, at the for which this is lowest, is at most the average energy converted over the whole range of . This corresponds to taking approximately constant over the given frequency range, and summing across all of the experimental configurations. Thus, the minimum time-averaged powered converted by the set of configurations is at most
where is the upper end of the mass range covered, and runs over the different stacks. For a dark photon, the corresponding expression with replaced by applies. If the power converted as a function of frequency is too ‘spiky’, this bound maybe be difficult to attain; for example, Figure 6 shows how random layer spacings result in sharp peaks at a fairly random set of frequencies, despite having similar frequency-averaged power (see Section II.4). However, simple frequency profiles, such as that obtained from a half-wave stack, can easily be added together to obtain smooth coverage over an order-1 frequency range, as illustrated in Figure 7.
If an experiment is not background free,
the frequency-averaged conversion power will not be the only relevant
quantity. Spending shorter times searching more, but narrower,
frequency ranges will result in the same total number
of signal photons produced, over the whole experiment, but
these all come within a shorter time,
improving the signal to noise.
However, as we will review in Section III,
close to background-free photon detectors are possible over
most of the frequency range we are considering.
Consequently, in searching for a DM signal of unknown frequency over a broad
range, frequency-averaged power is a useful figure of merit.
The simplest way to cover a large frequency range is either to construct a set of stacks,
each covering a small part of the range, or to construct
a single stack with segments of different periodicities.
In the latter case, the transparency properties of the half-wave
stack configuration discussed above are useful, as discussed
in Section II.5.
As discussed above, periodic layers have the advantage, compared to more random configurations, that their converted power is a smoother function of frequency (i.e. of DM mass), especially in the case of a half-wave stack. However, imperfections in the manufacturing process will result in some unintended variation in layer properties.
Considering flat, parallel interfaces, if the total accumulated phase error across all of the layers in a half-wave stack is , then we expect the frequency profile to change significantly (though, as per above, the frequency-integrated power will stay approximately constant). Such deviations could arise from a combination of modified layer thicknesses and refractive indices. If deviations in different layers are uncorrelated, per-layer fractional deviations of up to will not significantly affect the profile. This is illustrated in Figure 6, which also shows an example of the highly modified ‘spiky’ profile resulting from larger random deviations.
If we allow position-dependent thickness and/or index variations, resulting in non-planar layers, then the same condition on uncorrelated fractional deviations is required at each position. This also ensures that the emitted photons are kept within the cone set by the DM velocity, with opening angle , as required for optimum focussing; while the layers may be ‘bumpy’, the bumps have sub-wavelength height, and diffraction ensures that the overall emission is still collimated. As an additional point, we do not necessarily require that the fractional deviation condition applies strictly over the whole area of the stack. Emission from areas of the stack separated by more than a DM coherence length adds incoherently — therefore, to avoid a spiky frequency profile or uncollimated emission, we only need the fractional deviation condition to hold within these small areas. If layer thicknesses and/or indices change smoothly over larger distance scale, then different cross-sectional pieces of the stack effectively have different central frequencies, and add incoherently (as illustrated in the blue curve of Figure 7).
The complicated frequency profiles of randomly-spaced layers occur because of their effect on photon propagation. In a random medium, instead of the definite bandgaps of a periodic medium, the frequency range corresponding to the inverse scale of variation becomes a ‘pseudogap’, in which photons propagate diffusively Anderson (1985); John (1987). The very long ‘effective path length’ for a photon to escape the layers means that a very small change in frequency can significantly change the photon mode, leading to a very quickly-varying absorption rate with frequency.
So far, we have used the approximation of a lossless dielectric. However, a real material will absorb some of the light passing through it. Considering a photon mode in the layers, absorption will become important when the damping rate is , where is the mode’s quality factor. The half-wave stacks considered above have , so if the imaginary part of the refractive index is , then absorption will not be important. Configurations with narrower frequency peaks will be correspondingly more affected.
As we increase the number of layers in a half-wave stack, the frequency range covered decreases as . It is possible to employ stacks of different periodicities stacked on top of each other in order to build fewer separate configurations but still take advantage of a larger number of layers. Since a half-wave stack is not completely transparent to photons at different frequencies, there will be some interference between the emission from different stacks. As illustrated in Figure 7, which shows the results of stacking 11 30-period half-wave stacks at closely-spaced frequencies, this will generally result in a spikier frequency profile than adding the emission incoherently. However, even at the troughs in this distribution, the power is only a factor smaller than the incoherent combination, so if more volume can be filled in this way, it is still beneficial.
Iii Experimental Setup
|Pathfinder||Phase I||Phase II|
|Signal||Dark Photon||Dark Photon & Axion||Dark Photon & Axion|
|Number of periods ()|
|Stacks per -fold|
|Detector Dark Count ()||(e.g. CCD)||(e.g. TES)||(e.g. TES)|
|Detector Efficiency ()|
|Magnetic Field (Axion)||N/A|
Figure 1 shows a sketch of the experimental setup outlined above; a stack of dielectric layers is placed in a shielded volume, and then photons emitted normal to the layers are focused onto a small detector. A dielectric stack in free space will emit equally in both directions; to optimize detection, one end is terminated with a mirror. For dark photon DM, this setup is sufficient, while for an axion-photon coupling, a large background field parallel to the layers would be introduced. In this Section, we develop this outline into a more detailed illustration of how such an experiment might be implemented.
To recap the most important properties of such a setup, the emitted photon power for periods of alternating dielectrics, with refractive indices , at the frequency for which they form a half-wave stack (see Figure 4),
where is the quality factor of the mode in the layers, and is the volume of the stack. This expression is valid as long as the depth of the stack is smaller than a DM coherence length, corresponding to (See Section II.1.1). The quality factor depends on how the layers are terminated, but is . For example, the setup drawn in Figure 1, with a mirror on one side and air on the other, has . For a dark photon, the expression is
after averaging over many DM coherence times. These expressions apply over a fractional frequency range ; the converted power as a function of frequency is non-Lorentzian (see Figure 5), and has peak power at frequencies away from the center frequency .
The frequency-averaged conversion power from a half-wave stack, over the DM mass range , is
where DP stands for dark photon. If we construct different half-wave stacks, spaced to cover a frequency range with central frequency , then the converted power averaged over the frequency range is (if all of the stacks have the same number of layers , and ). Since stacks are required to obtain a smooth frequency profile, . For mirror-backed stacks, with , this gives the signal power in our experiment,
In the following, we consider three illustrative stages for the experiment (see Table 1). The first, a ‘pathfinder’, would be the simplest that could still probe new parameter space; it would aim to detect dark photon DM, could be run at room temperature, and would use readily-available detectors and layer fabrication methods. Phase I would aim to explore significant new dark photon parameter space, and start gaining sensitivity to new axion parameter space. This would require cooling the target to cryogenic temperatures, using cryogenic detectors, and using high-contrast layers, as well as operating with a large background field for the axion search. Phase II would aim to cover significant new axion and further dark photon parameter space, using a larger volume of layered material.
iii.1 Dielectric materials
|Range ()||Range (eV)||()|
|1.46||(0.13, 3.5)||(0.35, 9.5)||0.55|
|GaAs||3.8||(1, 15)||(0.08, 1.2)||5.7|
|NaCl||1.49||(0.2, 20)||(0.06 , 6.2)||44||Strauch (2014); Patrignani et al. (2016)|
|2.00||(0.25, 8)||(0.15, 5)||3.3|
|Si||3.42||(1.1, 9)||(0.12, 1.0)||2.55|
|1.41||(0.12, 9.5)||(0.13 ,10)||13.7|
|1.43||(0.15, 9)||(0.14, 8.3)||18.85|
|2.4||(0.55, 20)||(0.06, 2.3)||7.1||Patrignani et al. (2016)|
|2.3||(0.37, 13.6)||(0.09, 3.4)||3.17|
As discussed above, the proposed experiment requires periodic structures with a significant refractive index contrast that can be produced at room temperature and subsequently cooled to operate at liquid nitrogen and helium temperatures. In this section, we summarize the requirements on material properties necessary for our setup.
The layers should be transparent at the relevant energy to avoid losses; transparency windows, typically extending up to bandgap energy, are listed in table 2. For example, transmittance in excess of at wavelengths above 250 nm is demonstrated for silica with thickness Gao et al. (2013). Transparency is particularly important in the case where we consider running several stacks in parallel so the total thickness can be larger than a cm. An interesting phenomenological property of crystal dielectrics is that the refractive index decreases with increasing bandgap energy : Moss (1950, 1985). Thus with common dielectrics it is easier to achieve high sensitivity at lower frequencies.
Another requirement comes from the fact that, in order to suppress thermal backgrounds at frequencies , it will be necessary to cool the dielectric stack to cryogenic temperatures (see Section III.4.1). For the layers to be stable under this temperature change, the materials should ideally have similar thermal expansion properties. One possibility would be to use the same host material — for example, the widely used silica () — with index-raising (e.g. Germania ()) and index-lowering (e.g. Boron trioxide () and Fluorine ()) dopants for the alternating layers Dudley et al. (2006); Butov et al. (2002); Hermann and Wiechert (1989). A refractive index contrast of up to can be achieved without significantly altering the mechanical and thermal properties of the material Bachmann et al. (1988); Sinha et al. (1978); this procedure has been studied extensively e.g. to improve the performance of optical fibers at low temperatures Hashimoto and Shimizu (2008), which rely on a significant core-cladding refractive index contrast created by adding dopants to the host material in the fiber core Ziemann et al. (2008); M (1992); Dudley et al. (2006).
For an improved version of the experiment, materials with different refractive indices should be employed: maximum dark matter to photon conversion is achieved when the dielectric materials have indices of refraction (eq. (20)). The highest power results when alternating air or vacuum with materials with high refractive index such as silicon (Si, ); layering alternating dielectrics may be more mechanically robust, in which case pairs such as silica () and gallium arsenide (GaAs, ), commonly used in the semiconductor industry Sinha et al. (1978) can achieve of maximum power. It will be necessary to demonstrate that alternating layers can be constructed and withstand the thermal stresses of cooling to cryogenic temperatures Soma et al. (1982); Wang and Pirouz (2007).
The relevant technologies to create layered structures vary depending on the scale of the desired spatial periodicity, set by the dark matter mass. For DM in the mass range, corresponding to layer thicknesses , possible production processes include chemical vapor deposition (CVD), physical vapor deposition (PVD), spin coating, epitaxy, and sputtering (for a review, see Seshan (2012)). These methods are well-established for such length scales and are employed in producing optics such as mirror and lens coatings, and semiconductor diode lasers (surface-emitting laser and the vertical-cavity surface-emitting laser (VCSEL)) Iga (2000); Princeton Optronics (); Zhang et al. (2014), with tens of layers in the optical wavelength range commercially available. Acid-etching can potentially be used to achieve alternating dielectric/air structures Russell (2003); Tonucci et al. (1992).
As discussed in Section II.4, the structure should be close to flat and periodic. Each stack, once built, can be tested with an broadband laser beam, for example a Ti:Sapphire laser Moulton (1986). By measuring the transmitted and reflected wave from the stack, the resonant frequency as well as the normal direction of the layers can be measured to very high precision Joannopoulos et al. (2008); Noda (2016); Johnson and Joannopoulos (2001).
iii.2 Photon detection
As discussed in Section III.5, the existing constraints on axion or dark photon couplings mean that, even if they make up all of the dark matter, the photon conversion rate from a reasonably-sized target will be small. Accordingly, we will require a sensitive photon detector with a sufficiently low energy threshold, high photon efficiency, and low noise.
From Section II.1, the DM momentum spread means that converted photons are emitted in a narrow cone (opening angle ) around the normal to the layers. Consequently, the photons from a stack of cross-sectional area can be optically focused down to an area (for stack radii , this is larger than a wavelength, for the DM mass range we are considering). For a stack of area , the detector must have area to intercept of the signal photons. The required area could be decreased by various techniques, such as using a high-refractive-index concentrator on top of the detector, or having the signal photons bounce multiple times within a cavity.
The most commonly used photon detectors in our frequency range are charge-coupled devices (CCDs), which are found in a wide range of astronomical and laboratory applications. A close analogue to our low-signal-flux, long-integration-time setting is the ALPS ‘light shining through wall’ experiment Bähre et al. (2013), which looks for very rare photon-axion-photon conversion events at optical frequencies. The PIXES CCD camera Princeton Instruments (), planned for the ALPS-II upgrade, operates at and has pixels with per-pixel area of , detection efficiency of , and dark count rate for wavelengths shorter than . Accordingly, we adopt similar parameters for the pathfinder stage.
Detectors with similar frequency coverage (), better efficiency () and lower dark count rate in cryogenic environments ( per pixel) pri (), are being developed for dark matter direct detection experiments based on liquid xenon. The per-pixel area of this detector is , and arrays of pixels have been demonstrated Arneodo et al. (2018). With optimization, these detectors can be ideal for transitioning between pathfinder and phase I of our experiment.
To reach our Phase I and Phase II sensitivities, dark count rates of are required. These rates have been demonstrated for multiple detector technologies, including Transition Edge Sensors (TES) Dreyling-Eschweiler (2014); Cabrera et al. (1998); Karasik et al. (2012); Lita et al. (2008); Bastidon et al. (2015) (for a review, see Irwin and Hilton (2005)), Microwave Kinetic Inductance Detectors (MKIDs)Mazin (2009); Day et al. (2003); Gao et al. (2012), and nanowires Rosfjord et al. (2006). The efficiency of these detectors can be quite high: a TES can be coupled to photon modes of the system with specially designed coatings in a narrow range of frequencies, reaching efficiencies of , with demonstrated detection efficiency for wavelengths between and Lita et al. (2008); Karasik et al. (2012).
Achieving such low dark count rates generally requires small detectors. For example, a TES has an exponentially suppressed dark count rate above its energy resolution, but this energy resolution increases with the size of the TES; thus, to keep the dark count rate low, it is crucial that we are able to focus the signal to a small area, of order tens of microns on a side Cabrera et al. (1998); Irwin and Hilton (2005); Lita et al. (2008). One possibility to achieve a larger total detector area is to multiplex multiple TES pixels; arrays of more than 200 pixels have been demonstrated Irwin and Hilton (2005).
For most of the energy range we cover, we assume that the DCR and other backgrounds (see Section III.4) can be controlled to below for the pathfinder phase, and below for phases I and II. At DM masses below , the energy of a signal photon is close to the currently achievable detector energy resolution. For the low-frequency regions of Figures 8 and 9 (shown in dotted lines), we assume a TES-type noise curve with , with . This normalization gives a reach that approximately matches to the reach of a bolometer with noise equivalence power of NEP demonstrated at lower energies Farrah et al. (2017); Bradford et al. (2015); Irwin and Hilton (2005) at around . We are not aware of whether such sensitivities have been demonstrated in the near- to mid-IR regime we are considering.
The simplest scanning mechanism would be if it were possible to change the refractive indices of the layers by an external perturbation. If the refractive indices could be changed by , this would enable a single stack to cover an entire decade of frequency range. However, the optical materials of the types we have been considering typically have very small refractive index changes in response to reasonable external perturbations. For example, the rate of change of refractive index with temperature for silica, at wavelengths, is Toyoda and Yabe (1983). Especially at the lower end of our frequency range, where the layers’ temperature needs to be low to suppress blackbody radiation, this does not allow significant scanning. Similarly, the refractive index changes due to applied electric fields (Kerr/Pockels effect), magnetic fields (Verdet effect) or strain (photoelastic/piezooptic effect) are generally too small for our purposes.
While birefringent materials can have different refractive indices for different propagation directions, there are only two distinct polarizations corresponding to a given propagation direction. Since the direction of photon emission is set (to within ) by the layered structure of the material, changes that do not affect the layered structure itself (e.g. rotating the material relative to the field) will not result in significant scanning.
The opposite question, of whether we can tunably narrow the bandwidth of an existing stack, is also of interest. When backgrounds are important, narrowing the bandwidth improves the signal to noise, and it would also allow us to home in on a tentative signal. The simplest way to reduce the bandwidth is to increase the quality factor by placing e.g. a half-wave stack inside a ‘cavity’. For example, if we place a quarter-wave stack of periods above a half-wave stack, these increase the factor by , up to manufacture and alignment accuracy. By changing the separation between the quarter-wave and half-wave stacks over a distance wavelengths, we can scan this narrowed bandwidth across the entire original bandwidth of the half-wave stack. This procedure would demand improved tolerances so as not to smear out the narrowed peaks – in particular, the separation between the plates should be the same across the whole area, to within of a wavelength – and accurate positioning of the quarter-wave stack, again, to within wavelengths.
iii.4 Environmental backgrounds
There is a range of sources of noise from the environment; as we are interested in very weak signals, it is important to be able to discriminate signal and background. In this section, we discuss the background from radioactive decays in the detector and its vicinity and thermal blackbody radiation at finite temperature, as well as the requirements on experimental design to reduce these backgrounds.
If the detector’s field of view is at temperature , the rate at which thermal photons within a small energy range of hit the detector is
for , where is the area of the detector. For the pathfinder, a room-temperature field of view () gives small ( dark-count rate for . Since , we are well into the blackbody tail, and the minimum frequency giving the desired dark count rate depends only logarithmically on detector size. For phases I and II, we are interested in photon energies down to , and similarly require . Since we are focusing the signal photons from the layers onto the detector, we require that at least the dielectric layers, and the surrounding shielding, are cooled to these temperatures. It would likely be practical to cool the target volume to using liquid helium.
The photodetectors we require for phases I and II are generally cryogenic, and need to be operated at temperatures ; for examples, low-noise TESs are operated at . To help achieve this, a cold filter can be placed between the layers and the detector, which is transparent to photons at the signal frequency, but blocks the lower-frequency thermal radiation.
At sea level cosmic rays are dominated by muons, which have a flux of , and deposit an average of of energy in a stack through ionization or electron excitation, depending on the thickness of the layers Grieder (2001). The rate of background photons from this process is largely reduced by the small solid angle acceptance of the detector, and by using neighboring detector pixels to veto shower products that hit multiple pixels in a TES or CCD array, as in Bähre et al. (2013). These measures should be sufficient to reduce backgrounds below a mHz, the requirement for the prototype. In the event that the background photon rates are above , the experiment can be run deep underground or scintillation detectors can be used for an active muon veto in phases I and II.
Solar neutrinos scatter with rate in materials used to construct the layers. The scintillation photons from these scatterings will not be normal to the surface further reducing this negligible background.
Radioactive decays from natural or cosmogenic abundances of unstable elements in the materials used to manufacture the layers, the detector, and the experimental setup produce showers of hard photons, while the signal consists of single eV-energy photons that emerge perpendicular to the layer surface. This suggests two background reduction mechanisms; first, focusing the collimated signal onto a small detector area will reduce the isotropic background rate by . In the rare case where the decay happens close to the layer surface and the shower does not fully develop as a result, the few hard photons will most likely be not in the direction towards the much smaller photon detector. Second, an active veto system of detectors (with higher DCR and worse energy resolution than the signal detector) can be used, as background showers will be simultaneously detected in multiple detectors and can be vetoed. A detector simulation of the experimental setup will be needed to optimize the spatial coverage, temporal resolution, and DCR of the veto system.
In addition, pure materials can be selected for the experiment to reduce the total background rate. For example, a potential element used in the layers is chlorine, with radioactive isotope , natural fractional abundance and half life of Bhat (1992). This gives rise to a background rate of
A much lower ratio of can be found in old ground water Strauch (2014), which will reduce the background event rate below . On the other hand, Potassium () has a half life of and natural fractional abundance of Bhat (1992), producing background events at the rate of , so dielectrics such as KCl should be avoided in layer construction; see Table 2 for details of other relevant materials. For a discussion of other elements relevant to the rest of the experimental setup, see Arvanitaki et al. (2017); Arpesella et al. (2002); Iga (2000).
Another type of radioactivity is cosmogenic. For example, high energy electrons in cosmic rays can produce radioactive from stable . This may require the experiment to take place underground to reduce cosmic ray flux. Similarly, layer synthesis processes which employ high energy electrons or ions should be avoided in order to avoid accidental production of radioactive elements (e.g., from stray electrons in e.g. sub-methods of pulsed laser deposition).
Typical decays will have energies of MeV, far above the signal region, so a combination of pure materials, shielding, and active vetoes can be used Phases I and II. Avoiding materials with high radioactive isotope abundances and the small angular acceptance of the detector can be sufficient for the Pathfinder phase.
Figures 8 and 9 show the projected reach of our different experimental phases (Table 1) for dark photon and axion DM. The mass range we consider is a well-motivated target for bosonic dark matter searches: dark photon dark matter is naturally produced by inflationary perturbations in the early universe, while for axions, a combination of inflationary perturbations and decays of non-perturbative defects can produce dark matter densities (Section V).
For the sensitivity plots, we assume that the dark photon or axion make up all of the local DM density, . To obtain smooth coverage over an -fold range in DM mass, we require half-wave stacks with quality factor , at fractional frequency spacings . Equation (21) gives the converted power from the incoherent sum of these stacks. The sensitivity curves take an exposure time of for each stack. For the 30-layer pathfinder configurations shown in Figure 8, covering an -fold mass range requires stacks; a 1 year total integration time would require running five stacks simultaneously. For the 100-layer (1000-layer) configurations in phases I and II, () stacks can cover an -fold reasonably smoothly. As discussed in Section II.5, stacks at different frequencies can be run on top of each other, at the expense of a slightly spikier frequency profile.
The different curves within each phase of Figures 8 and 9 represent different material pairs for the dielectric stacks. Section III.1 describes how a range of dielectric materials could be used to achieve close-to-optimal conversion rates over our entire range of frequencies, to (see Table 2 for examples). The lower end of this frequency range is limited by the threshold energy of low dark count single-photon detectors (Section III.2; see Section VI for a brief discussion of bolometric detectors). The upper end is limited by the availability of simple dielectrics with low losses and high refractive indices at high frequencies, and is already well-constrained by other experiments and astrophysical observations.
Given the stringent constraints on axion-like particles, Phase I parameters, with the addition of a Tesla magnetic field applied parallel to the layer surface, are necessary to improve on current bounds, Fig. 9. Phase II can significantly improve on axion-like particle dark matter which couples to photons in the mass range of . The reach in coupling scales directly with applied B field, so large fields are required; however, we do not require a large volume: one stack can occupy as little as . Large magnetic fields with the above-mentioned volumes have been demonstrated Apollinari et al. (2015) for both a resistive DC magnet (, bore diameter ) Laboratoire National des Champs Magnetiques Intenses - Grenoble () as well as a superconducting magnet (, bore diameter ) National high magnetic field laboratory (). The volume versus B field magnitude should be optimized to achieve the largest reach; in particular, if larger fields are available with smaller bore diameters, the area of an individual stack can be reduced and multiple stacks of smaller area can be run at the same time. To maximize the B field volume, mirror optics can be used to guide the light out of the magnet prior to focusing.
The axion couplings that we could probe in Phases I and II are well below the KSVZ and DFSZ axion-photon couplings for this mass range, respectively; however, the QCD axion-nucleon couplings in this mass range are already constrained by the lack of SN1987A energy loss to axions Chang et al. (2018). We briefly discuss the possible extension of our experimental setups to frequencies in Section VI.
In the event of a signal, the relevant mass range can be studied extensively by creating layers with higher factors, either by adding a quarter waveplate to the end of an existing stack or by creating stacks at the signal frequency with a larger number of layers (Section III.3). Doing so increases the signal strength as well as pins down the value of the signal frequency, allowing for excellent discrimination from background, and, eventually, characterization of the signal properties.
Iv Other Dark Matter Candidates
Spin-1 DM candidates, such as the dark photon, have an intrinsic
polarization direction, so can convert to photons even through a
locally isotropic target material. For spin-0 DM candidates,
there must be some intrinsic direction to the target to
determine the polarization of the converted photon, or else
the conversion rate is suppressed by at least , where
is the DM velocity.
In the case
of an coupling, a uniform background field provides
this direction — however, for couplings to SM fermions,
the target material itself must have some directionality to avoid
Here, we give a brief overview of how dielectric haloscopes, using different target materials, can absorb some other DM candidates.
Apart from the electromagnetic current, the other conserved current in the SM is (if neutrinos are Dirac). This means that it could be consistently gauged, and that a new spin-1 particle coupled to could naturally be light without running into strong constraints from non-renormalizable couplings. Phenomenologically, in situations where interactions with nuclei are subdominant (for example, refraction in a dielectric), a vector with coupling will behave like a dark photon with . Consequently, our experiments will absorb DM in the same way as they would dark photon DM.
A difference from the dark photon case is that the couplings of a vector to neutrons and neutrinos result in stronger constraints. The coupling to neutrons results in a fifth force between neutral matter Adelberger et al. (2003), while a DM abundance with can decay to neutrinos, resulting in cosmological bounds Gong and Chen (2008); Wang and Zentner (2010). These constraints mean that even our Phase II experiment could only just reach new parameter space.
iv.2 Scalar couplings
New light scalar particles (i.e. with couplings to even-parity SM operators) can arise from UV physics in various ways, e.g. Higgs portal models Piazza and Pospelov (2010), dilatons / radions Damour and Polyakov (1994); Taylor and Veneziano (1988); Arkani-Hamed et al. (2000), or moduli fields Dimopoulos and Giudice (1996). For low-energy interactions, the important couplings are those to EM via , and fermion mass couplings to the electron and nucleons.
The coupling modifies the Maxwell equations to
Compared to an axion-like coupling, the non-velocity-suppressed term now couples to the background electric field, rather than the magnetic field. The electric fields that can be applied to standard materials are , and even in polar materials, the volume-averaged electric fields are . As a consequence, our experiments would give sensitivities weaker than current constraints from fifth force tests and stellar cooling Adelberger et al. (2003); Raffelt (1996).
The scalar couplings to fermion masses, , give a non-relativistic interaction Hamiltonian (to first order in )
where is the charge of . The first term gives rise to a force in the presence of a gradient, giving velocity-suppressed absorption rates. The second gives rise to an oscillating magnetic dipole moment, and to velocity- and acceleration-dependent forces on the fermion. The coupling to electron mass, , and the couplings to nucleon masses, , are subject to strong constraints from fifth force tests and stellar cooling Hardy and Lasenby (2017). These mean that the velocity-suppressed absorption rate will not probe new parameter space. The other terms could lead to coherent absorption in directional target materials, but we leave such calculations to future work.
V Dark Matter Production Mechanisms
Light bosonic DM can be produced through a range of early-universe mechanisms, the simplest of which is purely gravitational production during inflation. For a spin-0 field, inflation ‘stretches’ quantum fluctuations to super-horizon scales; with many -folds of inflation, this results in the observable universe having, at the end, approximately the same background value of the field everywhere. This background value persists until the universe cools down enough that the Hubble rate is less than the mass of the particle, after which the scalar field starts oscillating, and behaves like matter.
In many models, axion-like particles have a periodic potential. For a field with a cosine potential, , and a mass that does not change with temperature, its DM abundance today is Irastorza and Redondo (2018)
with , where is the post-inflationary value of the field over the observable universe. The scale is generally associated with the same physics that result in axion-SM couplings, which are then suppressed by . Hence, the mass and couplings ranges that our experiments will probe can naturally result in the correct DM abundance, through this ‘misalignment mechanism’.
The QCD axion is slightly different from this case, both because its mass and symmetry-breaking scale are related, and also because its potential is temperature-dependent Grilli di Cortona et al. (2016). Consequently, misalignment production can only account for all of the DM if , well below the mass range we have been considering. However, there are post-inflationary production mechanisms that can increase this abundance. In particular, if the axion only becomes an effective degree of freedom post-inflation, then the phase transition in which this occurs will generically create a network of strings, whose cores contain the unbroken phase. This string network will evolve until around the QCD phase transition, when the temperature-dependent axion mass becomes comparable to the Hubble rate. At this point, it is expected to decay through the formation of domain walls.
Since the string network is expected to evolve towards an attractor solution,
with eventual statistical properties almost independent of the
the axion DM abundance should theoretically
depend only on the QCD axion mass.
Spin-1 DM can also be produced during inflation, through a process similar to the spin-0 misalignment mechanism. Once again, quantum fluctuations are blown up by inflation to super-horizon scales. These do not start oscillating until the Hubble rate becomes smaller than the DM mass. The difference from the scalar case is that, during the time when the field is not oscillating, the magnitude of the vector field still decreases as , where is the FRW scale factor, and so the energy density in the field decreases as . In contrast, the potential energy density in the scalar field does not change until it starts oscillating. Consequently, the vector modes which are redshifted least are those which enter the horizon just as they are becoming non-relativistic. Larger-scale modes spent more time being redshifted before starting to oscillate, while smaller-scale modes spend time red-shifting rapidly as radiation. At late times, this results in a DM population that is dominated by modes at that special comoving scale, giving an overall density of Graham et al. (2016)
where is the Hubble scale during inflation. Bounds on the CMB tensor to scalar ratio constrain Ade et al. (2016), so this production mechanism can produce all of the DM at masses .
In this work we have detailed the theoretical description and experimental proposal for a light bosonic dark matter search around the eV mass range. Periodic, layered dielectrics paired with modern low dark-count-rate detectors enable significant reach into new parameter space for dark photon and axion dark matter. Advantages of our proposal include simple target materials, small volumes, and signal power emitted into collimated IR-UV photons, which can be efficiently detected.
Such dielectric haloscopes can achieve near-optimal dark matter absorption rates per unit volume (and, for axions, at a given background field), averaged over a range of possible dark matter frequencies. Achieving these high rates relies on constructing robust dielectric layers with high index of refraction contrasts and large numbers of layers. At optical frequencies, it is relatively simple to operate in the regime where (thermal) backgrounds are small, so our fairly low factors allow for broad frequency coverage without sacrificing too much signal-to-noise.
The search for new light particles is ongoing on many fronts, including non-dark matter probes of new particles around the scale. Improved solar observations Raffelt (1996) and dark photon helioscopes could probe some of the same parameter space. The proposed IAXO axion helioscope experiment Armengaud et al. (2014) and the ALPS II light shining through walls experiment Bähre et al. (2013) would improve on current axion-photon coupling bounds at masses and , respectively. For axion couplings to fermions, the ARIADNE Arvanitaki and Geraci (2014) experiment aims to search for new spin-dependent forces, and could probe axion-nucleon couplings for masses . A signal found in a DM experiment would strongly motivate searches of these kinds, while conversely, signals in non-DM experiments can inform the dark matter search program.
The frequency coverage of our proposal is complementary to existing dark matter direct detection searches: at DM masses above , absorptions have enough energy to ionize atoms in convenient target materials such as liquid xenon. Such detectors have the advantage of large target volumes, and easy detection of ionizations, enabling them to place strong constraints on dark photon DM An et al. (2015); Bloch et al. (2017); Agnese et al. (2016); Aprile et al. (2016); Hochberg et al. (2017). Direct detection experiments with lower thresholds are becoming possible, extending future reach toward the scale de Mello Neto et al. (2016); Tiffenberg et al. (2017). At DM masses , a number of other types of collective low-lying excitations have been proposed for DM absorption Hochberg et al. (2017, 2016, 2018); Arvanitaki et al. (2017); Knapen et al. (2017); Bunting et al. (2017); Horns et al. (2013).
As reviewed in the previous section, the isotropic target materials we have considered are generally not optimal for detecting DM particles with different couplings. In recent work, Arvanitaki et al. (2017) put forward a DM-to-photon conversion scheme based on a gas-phase target, which can probe a range of DM candidates and interactions through molecular excitations in the range.
A clear extension of our proposal is toward lower frequencies. The ultimate aim would be to close the sensitivity gap in axion-photon couplings between microwave setups such as MADMAX, and the optical-frequency experiments we have described. In particular, the best-motivated candidate for light bosonic dark matter is the QCD axion; the mass range which can give the full dark matter abundance is very broad given large theoretical uncertainties and different cosmological histories. Astrophysical constraints imply that the PQ scale for a generic QCD axion is Chang et al. (2018), corresponding to masses . Similarly, dark photon DM at lower frequencies is currently poorly constrained, and as discussed in Section V, could be produced during high-scale inflation.
Photon detection at these mid- to far-IR frequencies is more difficult than in the optical, but even with bolometric detectors, photonic haloscopes could reach interesting new parts of parameter space. For example, axion dark matter with the KSVZ photon coupling would produce a signal power of (independent of ) given a half-wave stack of 1000 high-contrast layers with area in a 10 Tesla field. The same layers, in the absence of a field, would produce a signal power of at an meV from dark photon DM with , which is the approximate astrophysical bound in the range. For comparison, bolometers with noise equivalent powers of are achievable with current technology Farrah et al. (2017); Bradford et al. (2015). Blackbody radiation backgrounds become more important at these frequencies; if the layers are cooled to , then most of the blackbody power is below the signal frequency and could be filtered out before reaching the detector. For example, only of the emitted blackbody power from a stack at is at frequencies ; for the parameters from above, this corresponds to . At smaller , better than fractional frequency selectivity would be necessary. Other experimental challenges include materials, optics, layer construction, and background rejection. We leave detailed consideration of experiments at these frequencies to future work.
Acknowledgements.We thank Francesco Arneodo, Aaron Chou, Adriano Di Giovanni, Savas Dimopoulos, Shanhui Fan, Andrew Geraci, Giorgio Gratta, Edward Hardy, Kent Irwin, Sae Woo Nam, Maxim Pospelov, Leslie Rosenberg, and Gray Rybka for helpful discussions. We are grateful to Asimina Arvanitaki, Saptarshi Chaudhuri, Ken Van Tilburg and Xu Yi for helpful discussions and comments on the manuscript, and to Asimina Arvanitaki, Matthew Kleban, and Ken Van Tilburg for introductions to experimental colleagues and collaborators. MB is supported in part by the Banting Postdoctoral Fellowship. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation.
- Another possibility would be to use a spatially non-uniform magnetic field, as per the ORPHEUS experiment Rybka et al. (2015), but this is harder to implement with a multi-tesla field.
- There is an ambiguity coming from the integral outside the stack, which cancels over a period, but this is only a fractional contribution Millar et al. (2017).
- These configurations are referred to by the MADMAX collaboration as ‘transparent mode’ Millar et al. (2017).
- This corresponds to the ‘area law’ derived in Millar et al. (2017), but extends it by giving an explicit expression for the frequency-averaged power in terms of the refractive indices of the layers.
- MADMAX Caldwell et al. (2017); Millar et al. (2017) addresses the problem of using a small number of slabs to scan and to achieve narrow frequency coverage by physically repositioning the slabs. They are limited by thermal backgrounds, so narrow frequency coverage improves the signal to noise. In our case, where such tuning may be difficult, but many more layers fit into a reasonable volume, the problem instead becomes covering a broad frequency range without having to construct an enormous number of stacks; hence our emphasis on the half-wave stack.
- For axion-like couplings to fermions, there is the additional feature that the target spins must be polarized to get appreciable coherent conversion.
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