Sensitivity improvement in hidden photon detection using resonant cavities

In the 1980s, a new species of photon was proposed to explain the spectral shape of the cosmic background radiation by extending the Standard Model (SM) with an additional U(1) gauge symmetry [1]. The corresponding spin-1 boson, called the hidden photon (HP), is assumed to act as a force mediator among particles in the hidden sector (HS), hence the name. Even if the strength is feeble, the coupling of this gauge boson to the electric charge of the SM particles via kinetic mixing with the SM counterpart, ordinary photons, could provide a portal to the invisible hidden sector [2, 3]. For instance, if it interacts with particles in the dark sector, it would shed light on the nature of dark matter, or, if lightly massive, the dark photon itself could account for a significant fraction of cold dark matter. In addition, the tantalizing tension between theory and experiment regarding the anomalous magnetic moment of the muon has drawn attentions to the possible existence of this gauge particle [4]. A variety of search activities have been conducted considering mass and experimental signatures. Searches for heavy mass HPs in the GeV scale have utilized particle colliders [5, 6], beam dumps [7], and fixed targets [8, 9]. On the other hand, light HPs typically in the sub-eV range are sought at smaller-scale labs employing the photon regeneration scheme [10, 11, 12].

In general, the Lagrangian describing the unified framework of the visible and hidden sectors at low energies is given by

where $F_{\mu\nu}$ ($F^{\prime}_{\mu\nu}$) is the field strength tensor of the SM (HS) gauge field $A_{\mu}(A^{\prime}_{\mu})$, $m_{A^{\prime}}$ is the the mass of the HS gauge boson, and $J_{\mu}^{em}$ is the electromagnetic (EM) charged current that $A_{\mu}$ couples to with coupling strength $e$. The kinetic mixing parameter $\chi$ in the third term is responsible for effective mixing between the ordinary and hidden gauge fields, leading to the interactions between the visible and invisible sectors. To the first-order approximation of $\chi$, Eq. 1 can be reformulated based on the mass eigenstate basis by linear transformations, $A_{\mu}\rightarrow A_{\mu}-\chi A^{\prime}_{\mu}$ and $A^{\prime}_{\mu}\rightarrow{A^{\prime}}_{\mu}$. On this basis, the kinetic term and mass term of these fields are diagonalized and the Lagrangian becomes

The last term implies that the SM photon ($\gamma$) can be converted to the HS photon ($\gamma^{\prime}$) or vice versa via the current of charged fermions.

The HP as dark matter in our Galactic halo can be detected using the conventional axion haloscope technique at no cost to a strong magnetic field. However, unlike the axion, which is a pseudo-scalar, the vector property of the gauge particle makes a dependence of detection sensitivity on orientation of the cavity resonant mode with respect to the unknown spin polarization. On the other hand, as a mediator of interactions between HS particles, HPs can be generated in the laboratory and thus the spin polarization can be under control (at least known) depending on how they are generated. Inspired by the concept of photon regeneration for axion-like particles (ALPs), e.g., “light shining through wall” (LSW) experiments, a detection scheme to search for HPs was proposed that employed two electromagnetically isolated conducting cavities tuned to the same frequency [13, 14]. Hidden photons converted from high-density EM photons in one cavity resonator by aid of the surface current propagate in space until reaching the other resonator where they convert back into EM photons. With $\gamma^{\prime}$ serving as a mediator, this schematic configuration forms a $\gamma$-$\gamma^{\prime}$-$\gamma$ oscillation induced by charged currents, analogous to the $\gamma$-$a$-$\gamma$ oscillation induced by strong magnetic fields. We propose a new cavity configuration based on the fact that the generated HPs propagate isotropically in space.

LSW-type HP search experiments typically import a high-power RF signal generator to supply EM photons to one cavity (emission cavity) and an RF antenna to receive the reconverted EM photon signal in the other (detection cavity) [11]. The expected oscillation power can be generally expressed as

where $Q_{d,\ e}$ is the cavity quality factor with subscripts $d$ and $e$ denoting the detector and emitter, and $P_{\rm in}$ is the RF power injected into the emission cavity. The dimensionless parameter $\rho\equiv k_{\gamma^{\prime}}/k_{\gamma}$, with $k=2\pi/\lambda$ denoting the wave number of a photon of wavelength $\lambda$, is bounded between 0 and 1 to satisfy the condition $k_{\gamma}^{2}=k_{\gamma^{\prime}}^{2}+m_{\gamma^{\prime}}^{2}$ on the mass shell. The factor $\mathcal{G}$ encodes the geometric configuration of the pair of cavities:

where $\vec{A}$ is the normalized eigen-vector of the vector potential corresponding to the cavity resonant mode.

Similar to cavity-based axion experiments, the signal-to-noise ratio (SNR) provides a measure of experimental sensitivity:

where $P_{\rm sig}$ $(P_{\rm sys})$ is the detected signal (system noise) power. The system noise, $\delta P_{\rm sys}$, is defined as fluctuations in system noise power obtained within integration time $\tau$ over frequency bandwidth $b$ and dictated by the Dicke radiometer equation [15]:

using the Boltzmann constant $k_{B}$ and the system noise temperature $T_{\rm sys}$. The major sources of system noise are the thermal noise of the cavity and the shot noise of the receiver chain. The former is attributed to black-body radiation due to the cavity’s physical temperature and represented by the effective noise temperature $T_{\rm eff}$, which includes the zero-point quantum fluctuations, as

where $h$ is the Plank constant and $\nu$ is the photon frequency. The latter accounts for the noise added by the RF readout chain, particularly the first stage amplifier, and is represented by the equivalent noise temperature $T_{\rm add}$. These two noise components linearly contribute to the total system noise, i.e., $T_{\rm sys}=T_{\rm eff}+T_{\rm add}$ [16].

An in-depth study of the geometric configuration and electromagnetic properties of cavity modes was performed to determine the optimal setup [12]. They demonstrated the viability of the LSW-type experiment at room temperature using two cylindrical copper cavities stacked axially with a choice of the TE₀₁₁ resonant mode to excite the cavities. ADMX reported a HP search result from their axion dark matter search detectors relying on the TM₀₁₀ resonant mode [11]. By bringing the detection cavity to a cryogenic temperature, they improved the experimental sensitivity and placed more stringent limits on kinetic couplings for masses $<3\,\mu$eV. In this article, we perform extensive studies to find the optimal cavity configurations and then tune the experimental parameters in Eq. 2 to enhance the sensitivity even further.

The geometric factor has a strong dependence on cavity configuration and the resonant mode of interest. The Green’s function in Eq. 3 implies that the generated HP field will propagate isotropically, and thus it is inferred that a geometric configuration consisting of the detection cavity surrounding the emission cavity would be beneficial. In addition, the choice of cavity mode could be important, because relative field alignment between the two cavities influences the geometry factor, as manifested by the inner product of the vector fields in Eq. 3.

As indicated by Eq. 2, the signal power also depends on the quality factor of the individual cavities and the input power of the RF photons input into the emitter. The dependency can be addressed in a more straightforward manner than the complicated geometric effect discussed in Sec. 2. For normal conductors, the quality factor is obtained using a relation with the cavity geometry factor $G$ and the surface resistance $R_{s}$ as

where $\mu$ is the permeability inside the cavity, $H$ is the magnetic field strength of the resonant mode. The surface resistance is determined by the conductivity $\sigma$ and the skin depth $\delta=\sqrt{2/\mu\sigma\omega}$ in the radio frequency regime. In Eq. 6, $Q$ was numerically calculated by taking both the cavity geometry and surface resistance into account.

A natural approach to improving the cavity quality factor would be to employ superconducting radio-frequency (SRF) technology. Various characteristics of SRF cavities, particularly those made of niobium branch superconductors, have been extensively studied [20] and successfully described by the BCS theory of superconductivity [21]. For these superconductors, the surface resistance can be decomposed into two components – the BCS resistance $R_{s}^{BCS}$ and the residual resistance $R_{s}^{res}$. The former contribution has a strong dependence on temperature and frequency, which is given by

where $\lambda$ is the London penetration depth, $\sigma_{n}$ is the electrical conductivity in normal state. The energy gap $\Delta/k_{B}T$ is a temperature-dependent parameter with respect to the critical temperature $T_{c}$ and it is known that $\Delta(0)/k_{B}T_{c}=1.76$ for pure Nb [22]. The latter contribution to surface resistance, on the other hand, is a residual component attributed to extraneous effects such as material properties and fabrication methods, which can be cured to some degree by heat treatments [20, 23]. The residual resistance accounts for losses, which are unrelated to the superconducting state, and is approximately independent of temperature and frequency in the absence of a magnetic field [20]. As a result, the total surface resistance is expressed as $R_{s}(\nu,T)=R_{s}^{BCS}(\nu,T)+R_{s}^{res}$.

A study of RF properties in niobium cavities in the GHz range has reported that, particularly at temperatures $<2$ K, there is a very slow increment in $R_{s}$ with surface field up to roughly 100 mT, beyond which $R_{s}$ suddenly rises, indicating entrance to the normal state [24]. This implies that the quality factor of an niobium emission cavity will endure the input power of EM waves up to a certain level. Here we show that the surface field can be analytically determined for a given cavity geometry and injected RF power. For instance, for the TM₀₁₀ mode of a cylindrical cavity, the quality factor (Eq. 7) and resonant frequency are determined respectively by

If we assume that all the input power is dissipated by loss on the surface, the $Q$-factor is given by definition as

where $U=\mathcal{C}B_{s}^{2}$ is the stored energy expressed by the surface magnetic field $B_{s}$ and the coefficient $\mathcal{C}$, which can be determined by simulation. Combining the relations above, we derive a formula for $B_{s}$ as a function of $P_{\rm in}$ as

For the ADMX cavity geometry, we obtain $\mathcal{C}=3.7\times 10^{4}$ J/T² from the simulation, which gives $B_{s}=0.42\,\mu$T for an input power of 150 mW assuming $R_{s}=6.14\times 10^{-3}\,\Omega$ for copper.

However, the dissipated power on the surface will cause the cavity temperature to rise. For superconducting cavities, in particular, this may substantially increase the surface resistance and thus the maximum input power will be limited by the cooling capacity of the cryogenic system. Now, if we define a function of frequency and temperature, including the noise temperature $T_{\rm sys}$, as

Eq. 4 can be reformulated as

with the coefficient $\mathcal{D}$ encompassing all other parameters which are assumed to remain unchanged for a given experimental setup. For the moment, the surface resistance is also assumed to be independent of the input power. Since $\mathcal{H}(\nu,T)$ is a numerically calculable function, the optimal $P_{\rm in}(T)$ can be estimated for a given cooling power $P_{c}$. For example, commercially available cryogen-free coolers have a typical cooling power of $\sim 500\,\mu{\rm W}$ at 100 mK with a quadratic temperature dependence, i.e.,

Assuming minimal added noise, represented by a half of the standard quantum limit (SQL), $T_{\rm SQL}=h\nu/k_{B}T=48\,{\rm mK}\times\nu/$GHz, and $R_{s}^{res}=3$ n$\Omega$ for Nb [20], the function $\mathcal{H}(\nu,T)$ is numerically obtained, as represented by the two-dimensional distribution in Fig. 9. We notice that the optimal system temperature asymptotically approaches a particular value, $\sim 1.1$ K, at high frequencies, at which Eq. 8 returns the maximum input power to the emitter of $\sim 60$ mW. This corresponds to a surface field of $2.9\,\rm{m}$T, which falls well within the range where the quality factor is not noticeably degraded. Finally, a combination of these parameter values will yield the maximum SNR for the given cryogenic system in the GHz range.

Unlike axion or ALP searches where the particle mass is determined by the searching frequency of the photon in the non-relativistic limit, for HP searches, the particle mass is uncertain within the range $0<m_{\gamma^{\prime}}<k_{\gamma}$ due to its unknown momentum. Therefore the $\gamma$-$\gamma^{\prime}$ mixing depends on the HP mass for a given photon frequency. By plugging Eqs. 2 and 5 into Eq. 4, we express the kinetic mixing parameter $\chi$ as

from which we estimate the projected sensitivity as a function of HP mass $m_{\gamma^{\prime}}=k_{\gamma}\sqrt{1-\rho^{2}}$. It is found that heavy HPs have higher couplings than light ones. It is also noted that the mass must be massive, i.e., $\rho<1$, such that the HP becomes distinguishable from the massless SM photon and the coupling constant in Eq. 9 does not diverge.

Adopting our cylindrical geometric configuration, consisting of a cylinder surrounded by a hollow cylinder, the mixing parameter $\chi$ is computed as a function of HP mass, using the same cavity dimensions and measurement parameters as for ADMX, i.e., copper cavities with $P_{\rm{in}}=150$ mW, $T_{\rm sys}=6$ K, $b=125$ Hz, and $\tau=2$ h. After choosing the TM₀₁₀ mode, the expected sensitivities for several scenarios are projected in Fig. 10, where the ADMX result is also compared. It is remarkable that our geometric configuration alone improves the sensitivity by about two orders of magnitude. An additional improvement is apparent with the application of quantum science and SRF technology for low noise and high $Q$-factor, respectively. Upgrading the cryogenic refrigeration systems with higher cooling capacity will also allow for high power injection into the emitter. It was also observed that the scale of the cavity, which alters the resonant frequency, does not affect the geometry factor as long as the aspect ratio remains unchanged, implying that the experimental sensitivity is weakly dependent on the searching frequency in contrast to axion cavity haloscope experiments.

We propose a geometric configuration for LSW-type experiments which employs a pair of cavities to search for a new gauge boson in the hidden sector via kinetic mixing with the SM gauge boson. The cylindrical configuration, which consists of a circular cylinder as an emitter and a hollow cylinder as a detector, remarkably enhances the geometric factor for an isotropically propagating HP field. The performance dependency on the aspect ratio of the cavity was examined for various resonant modes. Exploiting higher-order modes was found to be beneficial when searching for high-mass HPs using a detector designed for the EM solution with the lowest-order field variation. In addition, the experimental setup does not requires an external magnetic field, which allows for a direct application of the SRF technology to improve the cavity quality factor. The optimal operating temperature is determined by the given cooling capacity, which limits the RF power injected into the emitter. By putting all these features together, the experimental sensitivity to the mixing parameter can be improved by multiple orders of magnitude in the $\mu$eV mass range.

This work was supported by the Institute for Basic Science (IBS-R017-D1-2020-a00/IBS-R017-Y1-2020-a00).