#### The Model

The axion Compton frequency, can be expressed as a function of its mass, as:

A model is presented here for some fixed mass values of axions. The parameters of this model (following both the DFSZ and KSVZ specifications) and detection regimen are enumerated in

Table 1. Unlike a variable mass scan range technique (the broadband approach), a fixed axion mass search is devised here, searching only for some chosen values of axion mass which may have high likelihood (in view of excluded regions and a few theoretical suggestions, as shall be discussed later in the discussions section) by repeating the experiment with cavities with dimensions corresponding to the mass of interest. The masses are chosen in a symmetric manner, corresponding to harmonics of the base frequency of an initial mass value.

An arbitrary axion mass value of 11.25 μeV has been chosen to start with, selected from the window of some likely axion mass values, followed by its multiples as 22.5, 33.75, 45, 56.25, 67.5, 78.75, 90, and 112.5 μeV. These mass values correspond to the associated axion Compton frequencies of 2.7, 5.4, 8.1, 10.8, 13.5, 16.2, 18.9, 21.6, 24.3, and 27 GHz, respectively (the initial value f_{0} and its nine higher harmonics or multiples). However, in view of the exclusion of some of these values by existing experimental searches or cosmological constraints, the initial values to be probed in the proposed experiments are the eight values from 22.5 to 112.5 μeV with an error of ±0.5 μeV (corresponding to frequencies from 5.4 to 27 GHz, respectively, with an error of ±0.12 GHz). Out of these, four values, viz. 22.5, 56.25, 90 and 112.5 μeV, are chosen for a pilot experiment to probe, as our chosen values for an axion or ALP particle, if at all it exists. However, these values are just some chosen ad hoc values (however, in a symmetric manner), which we believe may have strong likelihood for coupling to an axionic/ALP field, but any other values may also have equal likelihood within the mass windows suggested by various theoretical estimates.

As suggested by contemporary cosmological models, the overall galactic dark matter density around earth (

${\mathsf{\Omega}}_{DM}{h}^{2}$) is estimated at values such as 0.1143 to 0.12 [

22,

23,

24] (where

h is the Hubble parameter), although it is possible that it may have a higher value if dark matter has more profound abundance. Based on those models, we choose a unanimously accepted and recent value of

${\mathsf{\Omega}}_{DM}$~0.3 ± 0.1 GeV·cm

^{−3} [

25] as the total DM density for our model.

The average contribution of axions to the overall DM density is estimated at [

15]:

Using the inverse relation between

${m}_{a}$ and

${f}_{a}$, this can be alternatively expressed as:

Thus, the corresponding axion density around the Earth for our suggested axion mass window is approximately on the order of:

An axion with mass

${m}_{a}$ and momentum

$\overrightarrow{p}$ incident on an earth-bound terrestrial detector, whether in the form of a purported “axion wind” or as part of a surrounding galactic halo, would have a Lorentz-boosted total energy (with

${\beta}_{a}={v}_{a}/c$):

We assume the velocity and direction of axions coming from the galactic halo to Earth as ${v}_{a}~2.3\times {10}^{5}$ ms^{−1} in orthogonal direction.

Most of the energy content of the axion is concentrated in its mass and a small value is carried by its kinetic energy.

An important factor to consider here is a dispersion in the axion kinetic energy owing to the time-dependent velocity dispersion ($\Delta v\left(t\right)$) of axions coming from the galactic halo and reaching a terrestrial detector. The axion velocities are thus expected to follow an appropriate velocity distribution profile, an important variable to be taken care of in realistic calculations.

The probability of axion photonic decay in a magnetic field (

$\overrightarrow{B}$) under an inverse coherent Primakoff effect, can be written down in a generic form, as per Sikivie’s model, as:

Here, ${\beta}_{a}$ is the Lorentz-boosted velocity for the axions, l is the length of the detector, and $\overrightarrow{q}$ is the axion-photon momentum transfer (which is $({m}_{\gamma}^{2}-{m}_{a}^{2})/2\omega $ in vacuo). The last term expresses the degree of coherence between the two particle wavefunctions involved with the axion-photon conversion and is constant for a fully-coherent conversion. Assuming a coherent conversion and weak coupling of axions, one can appreciate from Equation (10) that the probability for the axion-two-photon conversion is quite small and the most significant factors contributing to it are the axion coupling, detector length and the ambient magnetic field.

The power registered within the cavity for a particular mode (

$\phi \rho z$) may be expressed as:

This is the most important expression in any cavity-based axion search, as it gives an estimate of the powers involved with an axion signal. Here V and Q_{L} are the cavity volume and loaded Quality factor, respectively, and ${C}_{\phi \rho z}$ is the cavity axion-photon coupling form factor (normalized to 1) which determines the coupling of an axion to a particular mode of the electromagnetic field within the cavity.

The second most important factor other than power is the Signal to Noise Ratio (SNR) of any measurement scheme.

The SNR of a radiometer, or a microwave signal measurement scheme, can be estimated by an adapted form of the Dicke Radiometer Equation [

26]:

Here, P refers to the signal power (as expressed in the Equation (12)), T the collective physical noise temperature of the system (which is a sum of the ambient, amplifier and post-amplification signal processing stages temperatures), t the integration time (i.e., the time over which a measured sample is averaged) and Δf the integrated bandwidth (i.e., the measurement bandwidth, over which a single measurement is made). This equation is simply a consequence of the Central Limit theorem which specifies how the noise temperature measurement uncertainty scales with the square root of the number of samples. Thus, SNRs can significantly be improved by integration over long periods of time, often recovering an otherwise impossible signal which lies beneath the noise floor baseline of a measurement scheme. Typically, the integration time is on the order of minutes and it is set between 10^{3} to 10^{4} for our experiments. The integrated bandwidth is on the order of few KHz (which is 20 KHz in our data acquisition system, as set by the system parameters and our DAQ). The expected SNR for axion searches is normally taken as equal to greater than 5, which is a challenge, especially for the DFSZ model axion searches. However, after long integration times, it is possible to achieve SNRs of up to 10 in realistic conditions. This is a great advantageous factor to overcome hardware limitations in axion, or any weak quantum measurement, schemes.

Figure 2a illustrates our theoretical estimation of power from both KSVZ and DFSZ axion events, plotted above the noise floor, arising from the cavity detector, for a range of center frequencies of 2.7, 5.4, and 8.1 GHz in our model, whereas

Figure 2b provides the corresponding raw signal to noise ratios (SNRs) based upon the radiometer formula (without any signal processing or integration). This gives a rough measure of the difficulties faced with detecting true axionic events and heralds the need for unconventional detection techniques.

Based on the Equations (10)–(12), the tenet of resonant cavity detection schemes, in general, is to couple an incoming axion to a resonant electromagnetic mode (

ϕρz) in a carefully chosen volume of a high Q-factor cavity, permeated by a strong and highly uniform magnetic field, while working at low physical temperatures and taking a large number of samples, over a sufficient (but not too large) a bandwidth. Such a strategy may, in principle, facilitate the resonant detection of an axion-photon conversion event beyond the thermal noise. Although the axion signal power depends on a multitude of factors, it has substantial dependence on the strength and uniformity of the ambient magnetic field, the resonant frequency mode and cavity properties (such as volume, finesse and form factor). It is important to note, however, that the loaded Q factor of cavity (Q

_{L}) cannot be much larger than the axion signal quality factor (Q

_{a}) within the cavity [

27] (the two are typically expected to be on the order of approximately 10

^{5} to 10

^{6}). The secondary factors which aid in the detection strategy are signal processing and analysis techniques, which lie in the domain of software.

In a similar fashion, we attempt to measure a resonant frequency weak signal corresponding to a probable axion-two-photon conversion. An overview of our proposed experiment, which is a based upon the conventional Primakoff effect-based resonant cavity axion detection experiments, is illustrated in

Figure 3a. A resonant cavity made with copper or niobium, with a center frequency corresponding to the axion mass of interest, is enclosed in a cryostat under the influence of a strong solenoidal magnetic field to facilitate the conversion of an incident axion into a photon. Since we probe pre-defined frequencies corresponding to specific axion masses, there is no tuning mechanism involved, thus substantial noise and interference are reduced (in addition to a lot of hardware burden). Any possible photon arising from an axionic conversion within the cavity is detected by an antenna (here we have devised a special tuning fork antenna geometry). The ultra-weak signal is amplified by a three-stage cryogenic amplification cascade which increases the signal power to be read out by a conventional data acquisition and read-out scheme.

Figure 3b provides a coarse, false-color, visualization of the z-axis electric field vector component

$\left({\overrightarrow{\mathit{E}}}_{\mathit{z}}\right)$ distribution in the cavity, based on a usual Finite-Element simulation, over which a cartoon of a magnetic field-mediated axion field’s conversion into a photon (γ) going outward is depicted.

In order to assess the form of axionic signals in cavity and intrinsic noise, including its spectral distribution in the frequency range of our interest, some calculations were performed and simulations were carried out. A 15,000-point simulation was written in a computer program and results were accumulated in a data file.

Figure 4 and

Figure 5 illustrate the time-domain and frequency-domain results of these simulations, respectively, whereas

Figure 6 depicts a histogram of the simulated power’s spectral distribution, constructed from the simulation of power as a function of frequency in the 1–12 GHz spectral region.