1. Introduction
Dark matter is one of the most fascinating puzzles in modern physics. Cosmological and astrophysical observations indicate that it constitutes around 27% of the total energy density of the Universe. In addition, dark matter is expected to be cold, with a non-relativistic speed on the order v ~ 10-3c [1]. Axions and dark photon dark matter are among the most promising candidates. Both are thought to be light, with masses m $\lesssim$ 10-4 eV [2, 3]. Furthermore, their coupling to Standard Model (SM) particles is extremely weak. One of the effective methods to detect them involves the use of a resonant cavity with a high quality factor Q [4, 5].
Axion and dark photon waves have a large de Broglie wavelength λ ≈ 10 m for m ≈ 10-5 eV. The spatial correlation length could degrade to some extent due to uncertainties in the wave vector directions, but it can generally be treated as spatially constant on the laboratory scale. Temporal decoherence, however, poses a greater challenge. Bosonic dark matter can be created from non-thermal mechanisms such as [6-9]. However, the cosmic structure formation may result in coherence or non-coherent dark matter halos, and there are still many uncertainties. Assuming that each particle has a random phase Φi, the wave function could oscillate completely randomly: $D={D}_{0}{{\rm{e}}}^{{\rm{i}}[{\omega }_{0}t+\phi (t)]}$, where Φ(t) is a random function. The estimation of coherent time Γ ≈ 1/δν is valid when the waves have fixed relative phases. However, even if the waves are monofrequency δν = 0, but composed of random phases, their coherent time is finite.
In this paper, we find that the classical picture allows the signal from noncoherent waves to be boosted by the quality factor Q. This is consistent with the quantum perspective [10], where the cavity boost is similar to the Purcell effect, resulting in a higher density of states near resonance compared to free space [11]. Certainly, the resulting signal is not coherent in nature if dark matter wave is noncoherent and may therefore require a single-photon detector for readout.
In the following discussion, we use the standard classical derivation [12], with necessary modifications to accommodate the noncoherent wave case.
2. Axion to photon transition in a resonant cavity
The single axion particle wave function can be written as:
$\begin{eqnarray}{a}_{i}(\overrightarrow{{\boldsymbol{x}}},t)={A}_{0}\cos \left[m\left(1+\frac{{v}_{i}^{2}}{2}\right)t+m{\overrightarrow{{\boldsymbol{v}}}}_{i}\cdot \overrightarrow{{\boldsymbol{x}}}+{\phi }_{i}\right],\end{eqnarray}$
where vi is small (e.g. 10-3c). The multiparticle wave function can become decoherent. Spatial decoherence arises primarily from misalignment of ${\overrightarrow{{\boldsymbol{v}}}}_{i}$. Due to the large de Broglie wavelength λ compared to laboratory scales, the spatial component of the wave function is generally negligible if the dark matter particles have a small mass. Temporal decoherence is driven mainly by Φi. If Φi is completely random, the dark matter field can be written as [13]: $\begin{eqnarray}\begin{array}{rcl}a(t) & = & \displaystyle \frac{\sqrt{{\rho }_{\mathrm{DM}}}}{m}\displaystyle \sum _{j}^{}{\alpha }_{j}\sqrt{f({v}_{j}){\rm{\Delta }}v}\\ & & \times \cos \left[m\left(1+\displaystyle \frac{{v}_{j}^{2}}{2}\right)t+{\phi }_{j}\right],\end{array}\end{eqnarray}$
where f(v) is the dark matter particle velocity distribution, ρDM is the local dark matter density, and αj is a random variable following a Rayleigh distribution $P({\alpha }_{j})={\alpha }_{j}{{\rm{e}}}^{-{\alpha }_{j}^{2}}$.The Lagrangian for the coupled axion and SM photon field is: 4 ) can be written as
$\begin{eqnarray}{{ \mathcal L }}_{{\rm{a}}{\rm{p}}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{2}{\partial }_{\mu }a{\partial }^{\mu }a-{g}_{a\gamma \gamma }a{\boldsymbol{E}}\cdot {\boldsymbol{B}}-\frac{1}{2}{m}^{2}{a}^{2},\end{eqnarray}$
where Fμν = ∂μAν - ∂νAμ is the field strength, Aμ is the photon field, and ga$\gamma$$\gamma$ is the axion-photon coupling constant. In addition to the axion interaction, the electromagnetic field in the cavity also experiences a damping effect due to energy dissipation. Near a particular resonance frequency ωn, the damping coefficient can be written as ωn/Qn, where Qn is the cavity quality factor at frequency ωn. The quality factor measures the energy loss of the cavity. There are theoretically infinite resonant modes, but in practice, only a few low-frequency modes may be useful. For example, most axion search experiments use the TM010 mode, which has a higher Qn and larger electric field overlap with the magnetic field. Assuming the applied magnetic field is ${B}_{0}\overrightarrow{{\boldsymbol{z}}}$, the equation of motion for the electromagnetic wave inside the cavity is $\begin{eqnarray}{{\rm{\nabla }}}^{2}{\boldsymbol{E}}-{\partial }_{t}^{2}{\boldsymbol{E}}-\frac{{\omega }_{n}}{{Q}_{n}}{\partial }_{t}{\boldsymbol{E}}=-{g}_{a\gamma \gamma }{B}_{0}{\partial }_{t}^{2}a\overrightarrow{{\boldsymbol{z}}}.\end{eqnarray}$
The electric field can be expanded by the cavity modes en(x) as $\begin{eqnarray}{\boldsymbol{E}}({\boldsymbol{x}},t)=\displaystyle \sum _{n}^{}{E}_{n}(t){{\boldsymbol{e}}}_{n}({\boldsymbol{x}}),\end{eqnarray}$
where en satisfy the boundary condition of the cavity and the equation of motion: $\begin{eqnarray}({\omega }_{n}^{2}+{{\rm{\nabla }}}^{2}){{\boldsymbol{e}}}_{n}({\boldsymbol{x}})=0.\end{eqnarray}$
By performing a Fourier transform, equation ( $\begin{eqnarray}\begin{array}{l}\left({\omega }_{n}^{2}-{\omega }^{2}-{\rm{i}}\frac{\omega {\omega }_{n}}{{Q}_{n}}\right){E}_{n}(\omega )\displaystyle \int {{\rm{d}}}^{3}x{{\boldsymbol{e}}}_{n}({\boldsymbol{x}})\cdot {{\boldsymbol{e}}}_{n}^{* }({\boldsymbol{x}})\\ \quad =\frac{{g}_{a\gamma \gamma }{B}_{0}{\omega }^{2}}{\displaystyle \int {{\rm{d}}}^{3}x{{\boldsymbol{e}}}_{n}({\boldsymbol{x}})\cdot {{\boldsymbol{e}}}_{n}^{* }}a(\omega )\displaystyle \int {{\rm{d}}}^{3}x\overrightarrow{{\boldsymbol{z}}}\cdot {{\boldsymbol{e}}}_{n}^{* }({\boldsymbol{x}}),\end{array}\end{eqnarray}$
which leads to $\begin{eqnarray}{E}_{n}(\omega )=\frac{{g}_{a\gamma \gamma }{B}_{0}}{\int {{\rm{d}}}^{3}x{{\boldsymbol{e}}}_{n}({\boldsymbol{x}})\cdot {{\boldsymbol{e}}}_{n}^{* }({\boldsymbol{x}})}\frac{{\omega }^{2}a(\omega )\int {{\rm{d}}}^{3}x\overrightarrow{{\boldsymbol{z}}}\cdot {{\boldsymbol{e}}}_{n}^{* }({\boldsymbol{x}})}{\left({\omega }_{n}^{2}-{\omega }^{2}-{\rm{i}}\omega {\omega }_{n}/{Q}_{n}\right)}.\end{eqnarray}$
Even if the axion field oscillates with a random phase, we may still assume that the long-time average of energy density repeats: 2 ) to $\left\langle {a}^{2}\right\rangle $, because $j\ne {j}^{{\prime} }$ term cancel each other, one gets: 13 ) arises from the use of a special model, while the equality should be exact in physical context, m2‹a2› = ρDM if dark matter is non-relativistic.
$\begin{eqnarray}\langle {a}^{2}\rangle =\frac{1}{2T}{\int }_{-T}^{T}{a}^{2}(t){\rm{d}}t={\int }_{-\infty }^{\infty }| a(\omega ){| }^{2}\frac{{\rm{d}}\omega }{2\pi }.\end{eqnarray}$
Because the time average of the electric field and magnetic field energy is the same in a cavity, the total energy U in a cavity is $\begin{eqnarray}\begin{array}{rcl}U & = & \frac{{g}_{a\gamma \gamma }^{2}{B}_{0}^{2}| \displaystyle \int {{\rm{d}}}^{3}x\overrightarrow{{\boldsymbol{z}}}\cdot {{\boldsymbol{e}}}_{n}^{* }({\boldsymbol{x}}){| }^{2}}{V\displaystyle \int {{\rm{d}}}^{3}x{\left|{{\boldsymbol{e}}}_{n}({\boldsymbol{x}})\right|}^{2}}\\ & & \times V{\displaystyle \int }_{-\infty }^{\infty }\frac{{\rm{d}}\omega }{2\pi }\frac{{\omega }^{4}| a{(\omega )}^{2}| }{{\left|{\omega }_{n}^{2}-{\omega }^{2}-{\rm{i}}\omega {\omega }_{n}/{Q}_{n}\right|}^{2}}.\end{array}\end{eqnarray}$
For a high-quality factor Q, the integral is predominantly concentrated around the resonant frequency ωn, which yields: $\begin{eqnarray}{\int }_{-\infty }^{\infty }\frac{{\rm{d}}\omega }{2\pi }\frac{{\omega }^{4}| a{(\omega )}^{2}| }{{({\omega }_{n}^{2}-{\omega }^{2})}^{2}+{\omega }^{2}{\omega }_{n}^{2}/{Q}_{n}^{2}}\approx {Q}_{{\rm{c}}}^{2}\left\langle {a}^{2}\right\rangle .\end{eqnarray}$
where Qc is the quality factor of the resonant cavity with detector loads. Inserting equation ( $\begin{eqnarray}\left\langle {a}^{2}\right\rangle =\displaystyle \frac{{\rho }_{\mathrm{DM}}}{2T\cdot {m}^{2}}\cdot \displaystyle \sum _{j}{\alpha }_{j}^{2}f({v}_{j}){\rm{\Delta }}v\cdot {\int }_{-T}^{T}{\cos }^{2}(\omega t+{\phi }_{j}){\rm{d}}t.\end{eqnarray}$
For a long time period, $\int {\cos }^{2}(\omega t+{\phi }_{j})\,{\rm{d}}t\approx T$, consequently: $\begin{eqnarray}\left\langle {a}^{2}\right\rangle \approx \displaystyle \frac{{\rho }_{\mathrm{DM}}}{2{m}^{2}}\left\langle {\alpha }_{j}^{2}\right\rangle \int f(v){\rm{d}}v\approx \displaystyle \frac{{\rho }_{\mathrm{DM}}}{{m}^{2}},\end{eqnarray}$
where $\left\langle {\alpha }^{2}\right\rangle =\int {\alpha }^{2}P(\alpha )\,{\rm{d}}\alpha =2$. αj is a random variable (Rayleigh-distributed) sampled for each particle j and is not time-dependent. The ensemble average is taken over αj and vj. The sum ${\sum }_{j}{\alpha }_{j}^{2}f({v}_{j}){\rm{\Delta }}v$ represents the ensemble average over the random phases and velocities of the dark matter particles. For large particle counts, ${\sum }_{j}{\alpha }_{j}^{2}f({v}_{j}){\rm{\Delta }}v\to \int {\alpha }_{v}^{2}f(v){\rm{d}}v\,={\bar{\alpha }}^{2}\int f(v){\rm{d}}v$. There are differences between ${\bar{\alpha }}^{2}$ and ‹α2› because the dark matter velocity distribution (e.g. truncated Maxwellian) is not exactly the same as the Rayleigh distribution. However, the difference is typically small. Note that the approximation in equation (Define a form factor:
$\begin{eqnarray}C=\frac{{\left|\int {{\rm{d}}}^{3}x\overrightarrow{{\boldsymbol{z}}}\cdot {{\boldsymbol{e}}}_{n}({\boldsymbol{x}})\right|}^{2}}{V\int {{\rm{d}}}^{3}x{\left|{{\boldsymbol{e}}}_{n}({\boldsymbol{x}})\right|}^{2}},\end{eqnarray}$
and assuming the velocity distribution function is sharply peaked near the resonant point $\int$ f(v) dv = 1, then the total energy stored in the cavity is $\begin{eqnarray}U=\displaystyle \frac{{g}_{{\rm{a}}\gamma \gamma }^{2}{B}_{0}^{2}{\rho }_{\mathrm{DM}}{Q}_{{\rm{c}}}^{2}VC}{{m}^{2}},\end{eqnarray}$
which leads to the power of axion-photon conversion: $\begin{eqnarray}{P}_{\mathrm{axion}}=\displaystyle \frac{U}{{Q}_{{\rm{c}}}}m=\displaystyle \frac{{g}_{{\rm{a}}\gamma \gamma }^{2}{B}_{0}^{2}{\rho }_{\mathrm{DM}}VC}{m}{Q}_{{\rm{c}}}.\end{eqnarray}$
The signal power is identical to what would be obtained for a coherent field. This arises because the random phases Φj in the non-coherent field cause cross-terms ($j\ne {j}^{{\prime} }$) to cancel, leaving only the sum of squared amplitudes (diagonal terms). In addition, the ensemble average over αj and velocity distribution f(v) recovers the same mean energy density as a coherent field. Thus axion signal power hold for both coherent and non-coherent cases.If the cavity narrower than dark matter linewidth, i.e. Qc > QDM, the cavity samples only a fraction ∆ωc/∆ωDM = QDM/Qc of dark matter power. In equation (11 ) a(ω)2 is replaced by a(ω)2QDM/Qc, therefore, the conversion power is [14]
$\begin{eqnarray}{P}_{\mathrm{axion}}=\displaystyle \frac{{g}_{{\rm{a}}\gamma \gamma }^{2}{B}_{0}^{2}{\rho }_{\mathrm{DM}}VC}{m}\min ({Q}_{{\rm{c}}},{Q}_{\mathrm{DM}}).\end{eqnarray}$
3. Dark photon to photon transition in a resonant cavity
Dark photons and SM photons can be coupled through ultraviolet quantum processes, leading to a small kinetic mixing between them [15]. In addition, dark photons can be produced in the early universe, for example, through the misalignment mechanism, and thus constituting a substantial portion of dark matter today. The dark matter dark photons are primarily dominated by their electric field component. Therefore, when a resonant cavity is present, the dark electric field can induce a weak electric current that generates photons within the cavity. The effective Lagrangian can be written as: 2 ) $\overrightarrow{{{\boldsymbol{E}}}^{{\prime} }}$ can be written as:
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & = & -\frac{1}{4}\left({F}_{\mu \nu }{F}^{\mu \nu }+{X}_{\mu \nu }{X}^{\mu \nu }\right)+\frac{{m}^{2}}{2}{X}^{\mu }{X}_{\mu }\\ & & +{m}^{2}\chi {X}_{\mu }{A}^{\mu }+{J}^{\mu }{A}_{\mu },\end{array}\end{eqnarray}$
where Aμ and Fμν are the ordinary photon field and its field strength tensor, respectively; Jμ is the ordinary current; Xμν and Xμ are the dark photon tensor and dark photon field, respectively. The time-like polarization of dark photon X0, is generally suppressed, and the dark matter dark photons are dominated by their electric field component. Thus, the equation of motion can be expressed as: $\begin{eqnarray}{\partial }_{\mu }{\partial }^{\mu }\overrightarrow{{\boldsymbol{E}}}=\chi {m}^{2}\overrightarrow{{{\boldsymbol{E}}}^{{\prime} }},\end{eqnarray}$
where χ is the kinetic mixing parameter between dark photon and photon [16, 17], and $\overrightarrow{{{\boldsymbol{E}}}^{{\prime} }}$ denotes the dark electric field. Using equation ( $\begin{eqnarray}\begin{array}{rcl}\overrightarrow{{{\boldsymbol{E}}}^{{\prime} }} & = & -{\rm{i}}\displaystyle \frac{\partial \overrightarrow{{\boldsymbol{X}}}}{\partial t}\\ & \approx & -{\rm{i}}\sqrt{{\rho }_{\mathrm{DM}}}\displaystyle \sum _{j}^{}{\alpha }_{j}\sqrt{f({v}_{j}){\rm{\Delta }}v}\cdot \cos \left(\omega t+{\phi }_{j}\right)\cdot \hat{{\boldsymbol{n}}}.\end{array}\end{eqnarray}$
An additional order-one amplitude factor could appear, depending on whether the polarization of the dark electric field is aligned or points in random directions. Denote $\omega =m(1+{v}_{j}^{2}/2)$. Let us expand the electric field E into the cavity modes: $\begin{eqnarray}\overrightarrow{{\boldsymbol{E}}}({\boldsymbol{x}},t)=\displaystyle \sum _{k}{E}_{k}(t){{\boldsymbol{e}}}_{k}({\boldsymbol{x}}).\end{eqnarray}$
When close to the cavity resonant frequency, the damping coefficient can be written as ωk/Qc, so we have: $\begin{eqnarray}\frac{{{\rm{d}}}^{2}{E}_{k}(t)}{{\rm{d}}{t}^{2}}+\frac{{\omega }_{k}}{{Q}_{{\rm{c}}}}\frac{{\rm{d}}{E}_{k}(t)}{{\rm{d}}t}+{\omega }_{k}^{2}{E}_{k}(t)=\chi {m}^{2}\frac{\int {{\rm{d}}}^{3}x{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\cdot \overrightarrow{{{\boldsymbol{E}}}^{{\prime} }}}{\int {{\rm{d}}}^{3}x{\left|{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\right|}^{2}}.\end{eqnarray}$
The cavity electric field spectrum is then: $\begin{eqnarray}{E}_{k}(\omega )=\frac{\chi {m}^{2}{E}^{{\prime} }(\omega )}{{\omega }_{k}^{2}-{\omega }^{2}-{\rm{i}}{\omega }_{k}\omega /{Q}_{{\rm{c}}}}\frac{\int {{\rm{d}}}^{3}x{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\cdot \hat{{\boldsymbol{n}}}}{\int {{\rm{d}}}^{3}x{\left|{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\right|}^{2}}.\end{eqnarray}$
Assuming the average dark electric field energy density remains periodical, we then have: $\begin{eqnarray}\langle {E}^{{\prime} }{(t)}^{2}\rangle =\frac{1}{2T}{\int }_{-T}^{T}{E{}^{{\prime} }}^{2}(t){\rm{d}}t={\int }_{-\infty }^{\infty }| E(\omega ){| }^{2}\frac{{\rm{d}}\omega }{2\pi }.\end{eqnarray}$
The energy stored in the cavity is $\begin{eqnarray}\begin{array}{rcl}U & = & {\displaystyle \int }_{-\infty }^{\infty }\frac{{\rm{d}}\omega }{2\pi }| {E}_{k}(\omega ){| }^{2}\displaystyle \int {{\rm{d}}}^{3}x| {{\boldsymbol{e}}}_{k}({\boldsymbol{x}}){| }^{2}\\ & = & {\chi }^{2}{m}^{4}\cdot \frac{{\left|\displaystyle \int {{\rm{d}}}^{3}x{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\cdot \hat{{\boldsymbol{n}}}\right|}^{2}}{\displaystyle \int {{\rm{d}}}^{3}x{\left|{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\right|}^{2}}\\ & & \times {\displaystyle \int }_{-\infty }^{\infty }\frac{{\rm{d}}\omega }{2\pi }\frac{| {E}^{{\prime} }(\omega ){| }^{2}}{{({\omega }_{k}^{2}-{\omega }^{2})}^{2}+{\omega }_{k}^{2}{\omega }^{2}/{Q}_{{\rm{c}}}^{2}}.\end{array}\end{eqnarray}$
For a high quality factor, the integral is predominantly concentrated around the resonant frequency ωk, so we have: $\begin{eqnarray}U\approx {\chi }^{2}{m}^{4}\frac{{Q}_{{\rm{c}}}^{2}}{{\omega }_{k}^{4}}\frac{{\left|\int {{\rm{d}}}^{3}x{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\cdot \hat{{\boldsymbol{n}}}\right|}^{2}}{\int {{\rm{d}}}^{3}x{\left|{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\right|}^{2}}\cdot \left\langle {E}^{{\prime} }{(t)}^{2}\right\rangle .\end{eqnarray}$
Taking a similar calculation in the previous section, we obtain: $\begin{eqnarray}\begin{array}{rcl}\left\langle {E}^{^{\prime} }{(t)}^{2}\right\rangle & = & {\rho }_{\mathrm{DM}}\cdot \displaystyle \sum _{j}{\alpha }_{j}^{2}\,f({v}_{j}){\rm{\Delta }}v\cdot \displaystyle \frac{1}{2T}{\int }_{-T}^{T}{\cos }^{2}(\omega t+{\phi }_{j}){\rm{d}}t\\ & \approx & {\rho }_{\mathrm{DM}}.\end{array}\end{eqnarray}$
Let us define the form factor as: $\begin{eqnarray}{ \mathcal G }=\frac{{\left|\int {{\rm{d}}}^{3}x{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\cdot \hat{{\boldsymbol{n}}}\right|}^{2}}{V\int {{\rm{d}}}^{3}x{\left|{{\boldsymbol{e}}}_{k}({\boldsymbol{x}})\right|}^{2}}.\end{eqnarray}$
Thus, the total energy stored in the cavity is: $\begin{eqnarray}\begin{array}{rcl}U & = & {\chi }^{2}{m}^{4}{\rho }_{\mathrm{DM}}\displaystyle \frac{{Q}_{{\rm{c}}}^{2}}{{\omega }_{k}^{4}}V{ \mathcal G }\\ & \approx & {\chi }^{2}{\rho }_{\mathrm{DM}}{Q}_{{\rm{c}}}^{2}V{ \mathcal G }.\end{array}\end{eqnarray}$
Finally, the signal power from dark photon-to-photon conversion is $\begin{eqnarray}{P}_{{\rm{d}}{\rm{p}}}=\displaystyle \frac{U}{{Q}_{{\rm{c}}}}{\omega }_{k}={\chi }^{2}m{\rho }_{\mathrm{DM}}V{ \mathcal G }\min ({Q}_{{\rm{c}}},{Q}_{\mathrm{DM}}),\end{eqnarray}$
where the Qc > QDM case has been incorporated, as discussed in the previous section.4. Scanning rate
The signal-to-noise ratio (SNR) is given by [14]:
$\begin{eqnarray}\mathrm{SNR}=\displaystyle \frac{{P}_{{\rm{r}}}}{{k}_{{\rm{B}}}{T}_{\mathrm{sys}}\sqrt{{\rm{\Delta }}{\nu }_{\mathrm{sign}}/{\rm{\Delta }}t}},\end{eqnarray}$
where Pr is the signal power coupled to the receiver, kB is the Boltzmann constant and Tsys is the equivalent system noise temperature. The denominator represents the noise of the system, which is described by the Dicke radiometer equation as fluctuations in the noise power over an integration time ∆t within a frequency bandwidth ∆νsign. In current axion or dark photon experiments, the frequency bandwidth is typically set as ∆νsign ≈ ν0/106 [18], where ν0 is the central frequency of the dark matter field.The total power from the cavity can be divided into two components: (1) dissipation and (2) useful power received by the detector. The quality factors satisfy: 31 ) and (33 ), we obtain:
$\begin{eqnarray}\frac{1}{{Q}_{{\rm{c}}}}=\frac{1}{{Q}_{0}}+\frac{1}{{Q}_{{\rm{r}}}},\end{eqnarray}$
where Q0 represents the power loss due to dissipation, and Qr represents the power loss due to the receiver. A dimensionless cavity coupling parameter can be defined as β = Q0/Qr, and the signal power coupled to the receiver is given by: $\begin{eqnarray}{P}_{{\rm{r}}}=\displaystyle \frac{\beta }{1+\beta }{P}_{\mathrm{DM}}.\end{eqnarray}$
In the case that Qc < QDM, combining equations ( $\begin{eqnarray}\displaystyle \frac{{\rm{\Delta }}{\nu }_{\mathrm{sign}}}{{\rm{\Delta }}t}=\displaystyle \frac{{\nu }_{0}/{Q}_{\mathrm{DM}}}{{\rm{\Delta }}t}=\displaystyle \frac{{P}_{\mathrm{axion}/{\rm{d}}{\rm{p}}}^{2}}{{k}_{{\rm{B}}}^{2}{T}_{\mathrm{sys}}^{2}{\mathrm{SNR}}^{2}}\displaystyle \frac{{\beta }^{2}}{{(1+\beta )}^{2}}.\end{eqnarray}$
Typical haloscope experiments use a copper cavity, where Qc ~ 105 « QDM ~ 1/δv2 ~ 106. Each frequency pin covers a frequency range ∆νcavi = ν0/Qc. Depending on the particular scanning setups, the limit of scanning rate is then:
$\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{axion}} & = & \displaystyle \frac{{\rm{\Delta }}{\nu }_{\mathrm{cavi}}}{{\rm{\Delta }}t}=\displaystyle \frac{{\nu }_{0}/{Q}_{\mathrm{DM}}}{{\rm{\Delta }}t}\displaystyle \frac{{Q}_{\mathrm{DM}}}{{Q}_{{\rm{c}}}}\\ & = & {g}_{{\rm{a}}\gamma \gamma }^{4}\displaystyle \frac{{\rho }_{{\rm{a}}}^{2}}{{m}_{{\rm{a}}}^{2}}\displaystyle \frac{1}{{\mathrm{SNR}}^{2}}\displaystyle \frac{{B}_{0}^{4}{V}^{2}{C}^{2}}{{k}_{{\rm{B}}}^{2}{T}_{\mathrm{sys}}^{2}}\displaystyle \frac{{\beta }^{2}}{{(1+\beta )}^{2}}{Q}_{\mathrm{DM}}{Q}_{{\rm{c}}},\end{array}\end{eqnarray}$
for axions, and $\begin{eqnarray}{R}_{{\rm{d}}{\rm{p}}}={\chi }^{4}{\rho }_{{\rm{D}}}^{2}{m}_{{\rm{D}}}^{2}\displaystyle \frac{1}{{\mathrm{SNR}}^{2}}\displaystyle \frac{{V}^{2}{{ \mathcal G }}^{2}}{{k}_{{\rm{B}}}^{2}{T}_{\mathrm{sys}}^{2}}\displaystyle \frac{{\beta }^{2}}{{(1+\beta )}^{2}}{Q}_{\mathrm{DM}}{Q}_{{\rm{c}}},\end{eqnarray}$
for dark photons. In case Qc > QDM, the signal bandwidth becomes the same as the cavity bandwidth: $\begin{eqnarray}\displaystyle \frac{{\rm{\Delta }}{\nu }_{\mathrm{sign}}}{{\rm{\Delta }}t}=\displaystyle \frac{{\nu }_{0}/{Q}_{{\rm{c}}}}{{\rm{\Delta }}t}=\displaystyle \frac{{P}_{\mathrm{DM}}^{2}}{{k}_{{\rm{B}}}^{2}{T}_{\mathrm{sys}}^{2}{\mathrm{SNR}}^{2}}\displaystyle \frac{{\beta }^{2}}{{(1+\beta )}^{2}}.\end{eqnarray}$
The scanning rate is then [14]: $\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{axion}} & = & \displaystyle \frac{{\nu }_{0}/{Q}_{{\rm{c}}}}{{\rm{\Delta }}t}\\ & = & {g}_{{\rm{a}}\gamma \gamma }^{4}\displaystyle \frac{{\rho }_{{\rm{a}}}^{2}}{{m}_{{\rm{a}}}^{2}}\displaystyle \frac{1}{{\mathrm{SNR}}^{2}}\displaystyle \frac{{B}_{0}^{4}{V}^{2}{C}^{2}}{{k}_{{\rm{B}}}^{2}{T}_{\mathrm{sys}}^{2}}\displaystyle \frac{{\beta }^{2}}{{(1+\beta )}^{2}}{Q}_{\mathrm{DM}}^{2},\end{array}\end{eqnarray}$
for axions, and $\begin{eqnarray}{R}_{{\rm{d}}{\rm{p}}}={\chi }^{4}{\rho }_{{\rm{D}}}^{2}{m}_{{\rm{D}}}^{2}\displaystyle \frac{1}{{\mathrm{SNR}}^{2}}\displaystyle \frac{{V}^{2}{{ \mathcal G }}^{2}}{{k}_{{\rm{B}}}^{2}{T}_{\mathrm{sys}}^{2}}\displaystyle \frac{{\beta }^{2}}{{(1+\beta )}^{2}}{Q}_{\mathrm{DM}}^{2},\end{eqnarray}$
for dark photons. Thus, a higher cavity quality factor improves the scanning rate but saturates when Qc becomes larger then QDM. Therefore, a better understanding of the dark matter wave properties, such as its effective bandwidth, can optimize the experimental setups. It is important to emphasize that a non-coherent wave does not necessarily result in a larger bandwidth. 'Non-coherent' here means the relative phases of the particles are not fixed, but their velocity distributions can still be small.5. Conclusions
Bosonic dark matter with sub-eV mass, such as axions, axion-like particles, and dark photons, is of great interest from both theoretical and experimental perspectives. These particles could constitute a substantial portion of dark matter today, with non-thermal production mechanisms that lead to a broad range of possible parameter spaces. Their extremely weak interaction with SM particles makes cavity resonance enhancement essential to achieve the required sensitivity.
This paper demonstrates that the assumption of a coherent wave is not necessary for cavity resonance experiments, as long as an appropriate readout mechanism is employed. The resulting power in the cavity is the same for both coherent and non-coherent waves. From a classical perspective, this occurs because the random phases in the non-coherent field cause cross-terms to cancel, leaving only the sum of squared amplitudes. The ensemble average of the particles recovers the same mean energy density as in the coherent case. Therefore, the signal power holds for both coherent and non-coherent waves.
This concept becomes clearer from a quantum perspective [10, 11], where the cavity's boost effects is similar to the Purcell effect, enhancing the density of states near the resonance frequency. Thus, transitions are boosted regardless of the coherence of the dark matter wave. Experimentally, this allows for longer integrate times without concern for the decoherence of dark matter waves.
It is important to note that bandwidth and coherence should be considered as independent features in cavity experiments. The bandwidth of dark matter waves is crucial to these experiments, while coherence is not. Naturally, a non-coherent dark matter wave will produce a non-coherent microwave signal in the cavity, meaning the readout mechanism may need to be adapted (e.g. using a single-photon detector). This makes cavity experiments applicable to a broader range of scenarios, and the integration time could be extended. Combining this understanding with quantum theory and quantum measurement techniques presents significant potential for future development in cavity experiments.