Signal photon flux and background noise in a coupling electromagnetic detecting system for high-frequency gravitational waves

A coupling system among Gaussian-type microwave photon flux, a static magnetic field, and fractal membranes (or other equivalent microwave lenses) can be used to detect high-frequency gravitational waves (HFGWs) in the microwave band. We study the signal photon flux, background photon flux, and the requisite minimal accumulation time of the signal in the coupling system. Unlike the pure inverse Gertsenshtein effect (G effect) caused by the HFGWs in the gigahertz band, the electromagnetic (EM) detecting scheme proposed by China and the U.S. HFGW groups is based on the composite effect of the synchroresonance effect and the inverse G effect. The key parameter in the scheme is the first-order perturbative photon flux (PPF) and not the second-order PPF; the distinguishable signal is the transverse first-order PPF and not the longitudinal PPF; the photon flux focused by the fractal membranes or other equivalent microwave lenses is not only the transverse first-order PPF but the total transverse photon flux, and these photon fluxes have different signal-to-noise ratios at the different receiving surfaces. Theoretical analysis and numerical estimation show that the requisite minimal accumulation time of the signal at the special receiving surfaces and in the background noise fluctuation would be $\sim {10}^3–{10}^5$ seconds for the typical laboratory condition and parameters of $h_{\mathrm{rms}}\sim {10}^{-26}–{10}^{-30}/\sqrt{\mathrm{Hz}}$ at 5 GHz with bandwidth $\sim 1\mathrm{Hz}$. In addition, we review the inverse G effect in the EM detection of the HFGWs, and it is shown that the EM detecting scheme based only on the pure inverse G effect in the laboratory condition would not be useful to detect HFGWs in the microwave band.

Figure 1

If a HFGW passes through a static magnetic field ${\widehat{\overrightarrow{B}}}_y^{(0)}$, the interaction of the HFGW with the static magnetic field will produce an EMW, where $L$ is the interacting dimension between the HFGW and the static magnetic field. The ${\mathrm{EMW}}_2$ has a maximum in the terminal position ($Z=L$) of the interacting volume due to the space accumulation effect in the propagating direction (the $z$ direction).

Figure 2

If the HFGW and the ${\mathrm{EMW}}_0$ pass simultaneously through the transverse static magnetic field, under the resonant state (${\omega }_e={\omega }_g$), the first-order perturbative EMW (${\mathrm{EMW}}_1$, i.e., “the interference term”) and the second-order perturbative EMW (${\mathrm{EMW}}_2$) can be generated. However, because the ${\mathrm{EMW}}_1$ and the ${\mathrm{EMW}}_0$ have the same propagating direction and distribution, and ${\mathrm{EMW}}_1$ is often much less than the ${\mathrm{EMW}}_0$, the ${\mathrm{EMW}}_1$ will be swamped by the ${\mathrm{EMW}}_0$.

Figure 3

In the coupling system of the static magnetic field and the plane EMW, $|{\overrightarrow{E}}_x^{(0)}|$ and $|{\overrightarrow{B}}_y^{(0)}|$ denote the background EM fields, $|{\overrightarrow{E}}^{(1)}|$ and $|{\overrightarrow{B}}^{(1)}|$ express the perturbative EM fields generated by the direct interaction of the HFGW with the static magnetic field, and ${\overrightarrow{n}}_{\gamma }$ is the total photon flux density.

Figure 4

When the HFGW propagates along the $z$ direction in the coupling system of the GB and the transverse static magnetic field ${\widehat{\overrightarrow{B}}}_y^{(0)}$, the resonant interaction (${\omega }_e={\omega }_g$) of the HFGW with the EM fields will generate not only the longitudinal perturbative photon flux $n_z^{(1)}$ but also the transverse perturbative photon fluxes ($n_x^{(1)}$ and $n_y^{(1)}$) in the $x$ and $y$ directions due to the spread property of the GB itself. This is an important difference between Figs. 2, 4. Moreover, unlike $n_z^{(1)}$ and $n_z^{(0)}$, $n_x^{(1)}$ and $n_x^{(0)}$ have very different distribution and decay rates.

Figure 5

The first-order PPF density $n_z^{(1)}$ and the BPF density $n_z^{(0)}$ have the same propagating direction and similar distribution. Thus $n_z^{(0)}$ is much larger than $n_z^{(1)}$ in most of the regions.

Figure 6

Schematic diagram of strength distribution of $n_x^{(0)}$ and $n_x^{(1)}$ in the “outgoing wave” region of the GB (another one is the “imploding wave” region. For an optimum GB, such properties of the transverse BPFs in two such regions would be “antisymmetric”). Unlike Fig. 5, here $n_x^{(0)}{|}_{x=0}=0$ while $n_x^{(1)}{|}_{x=0}=n_x^{(1)}{|}_{\max }$. Therefore, $n_x^{(1)}\Delta t$ can be effectively larger than the background noise photon flux fluctuation $(n_x^{(0)}\Delta t{)}^{1/2}$; i.e., $n_x^{(1)}\Delta t>(n_x^{(0)}\Delta t{)}^{1/2}$ at the $yz$ plane and at the parallel surfaces near the $yz$ plane, and $n_x^{(1)}$ will be a major fraction of the total transverse photon flux passing through the $yz$ plane, provided thermal photon flux and other noise photon fluxes passing through the surface can be effectively suppressed. Clearly, the EM response of the coupling system between the plane EMW and the static magnetic field has no such characteristic. Moreover, the propagating directions of $n_x^{(1)}$ are opposite in the regions of $y>0$ and $y<0$ for our scheme. Thus, the total momentum of the PPF in the $x$ direction vanishes. In other words, such a property ensured conservation of the total momentum in the coherent resonance interaction (see Ref. 32).

Figure 7

Unlike the photon fluxes $N_z^{(0)}$ and $N_z^{(1)}$, $N_x^{(1)}{|}_{x=0}=N_x^{(1)}{|}_{\max }$, where $N_x^{(0)}{|}_{x=0}=0$. This means that $N_x^{(0)}$ and $N_x^{(1)}$ focused by the FM at the $yz$ plane or at the parallel planes very near the $yz$ plane would have a good focusing effect and SNR.

Figure 8

If the FM is just laid at the $yz$ plane or at the parallel planes very near the $yz$ plane, then the wave fronts of the photon fluxes passing through the planes would be the plane or the pseudoplane, and it is possible to obtain an effective focusing effect.

Figure 9

If the FM is laid at an obvious nonsymmetrical plane, then it is difficult to focus the photon fluxes due to the spread property of the GB.