Electromagnetic response of gravitational waves passing through an alternating magnetic field: A scheme to probe high-frequency gravitational waves

Searching for high-frequency gravitational waves by ground high field magnetic resonant sweepings

H. Zheng and L. F. Wei, School of Physics, State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou; L. F. Wei, Information Quantum Technology Laboratory, School of Information Science and Technology, Southwest Jiaotong University, Chengdu; H. Wen and F. Y. Li, Institute of Gravitational Physics and Department of Physics, Chongqing University, Chongqing.

Abstract

With laser interferometers, LIGO-Virgo collaboration has recently realized the direct detections of the intermediate-frequency (i.e., from dozens to hundreds of Hertz) gravitational waves (GWs) by probing their mechanically-tidal responses. Alternatively, in this letter we propose a feasible approach to actively search for the high-frequency GWs by probing their electromagnetic responses (EMRs) in a high alternating magnetic field. Differing from the original Gertsenshtein-Zeldovich configuration (in which the EMRs are proportional to the square of the amplitudes of the GWs, and consequently are too weak to be detected experimentally), the EMRs of the GWs passing through the present configuration are linearly related to the amplitudes of the GWs and thus the relevant signals are detectable with the current weak-light detection technique. As the wave impedances of the GWs-induced electromagnetic signals (EMSs) are very different from those of the EM radiations in flat space-time, i.e., 377 ohms, the stronger background noises (without any GWs information) could be effectively filtered out by using wave-matching technique. Given the frequency of the applied alternating magnetic field is conveniently adjustable, the configuration proposed here could be utilized to actively search for the GWs (if they really exist) in a sufficiently-wide frequency band (e.g., could be ten to the seventh to ten to the twelfth hertz), once the scale of the cavity and the sweeping frequency of the applied alternating magnetic field are experimentally achievable.

Introduction

It is well-known that the existence of gravitational waves (GWs), i.e., the ripples of the curved space-time, is one of the most important predictions in general relativity. Therefore, the direct detections of the GWs not only provided new evidences to verify Einstein’s gravitational theory, but also begun the era of gravitational wave astronomy. Yet, it is very difficult to verify such a prediction, due to the significantly-minute amplitudes of the GWs. Interestingly, with the precise laser interferometer the LIGO-Virgo collaboration has realized recently the direct detections of the GWs by probing their minute mechanical displacements with the peak strains being at the order of ten to the minus twenty-one and the frequencies being in the low- and intermediate frequency band (i.e., dozens and hundreds of Hertz). Still, probing the GWs in wider bands, such as the high frequency ones expected by a series of models for cosmology and high-energy astrophysical processes, is a challenge.

Historically, many approaches have been tried to directly probe the GWs or observe their certain indirect effects. In fact, before the recent LIGO-Virgo’s direct detections, an indirect experimental evidence of the existence of the GWs was obtained by observing the change in the orbital period of the PSR B1913+16 binary, i.e., a pair of stars with one of them being a pulsar. Also, observing the B-mode polarizations in the cosmic microwave background is regarded as another approach to indirectly verify the existence of the GWs in very low frequency band. Since the 1980s the laser interferometers, including the typical LIGO-Virgo setup and the proposed Laser Interferometer Space Antenna (LISA), and Einstein telescope, etc., have been demonstrated to implement the detections of the GWs-induced mechanical displacements. Similarly, the pulsar timings and pulsar timing arrays are believed as another astronomic candidate to probe the GWs in low-frequency band.

Besides the mechanically-tidal effects used in the above GW-detectors, it is believed that the electromagnetic responses (EMRs) of the GWs could be also observed. This is based on the so-called Gertsenshtein-Zeldovich (GZ) effects, i.e., the mutual conversions between EMWs and GWs in a high stationary magnetic field. Unfortunately, the amplitude of such a GW-induced EMW is too minute to be experimentally detected, as the relevant detectable quantities are proportional to the square of the perturbative metric of the passing GWs. To overcome such a difficulty, a series of modified schemes have been proposed by introducing certain ground auxiliary EMWs — the plane EMWs and the Gaussian beams — to enhance the observable effects of the perturbative EMW signals.

Continuously, in this letter we propose a feasible approach to actively search for the desired first-order EM perturbative signals generated by the high-frequency GWs passing through a high magnetic field. Besides a high stationary magnetic field utilized in the original GZ configuration, a relatively-weak alternating magnetic field is additionally applied. By exactly solving the relevant Einstein-Maxwell equation in a curved space-time, we show that the GWs-induced EMRs in the present setup are linearly related to the amplitudes of the passing GWs. These signals could be detected with the current weak-signal probing technique. As the amplitude and the frequency of the applied weak alternating magnetic field are locally controllable, the scheme proposed here could be applied to actively search for the probably-existing high-frequency GWs (predicted by a series of theoretical models) in a sufficiently wide frequency band. Also, due to the wave impedances of the GWs-induced EMW signals, satisfying the Einstein-Maxwell equation in the curved space-time, being very different from those of the background EM noises (which obey the flat space-time Maxwell equation and thus always take the value of 377 ohms in vacuum), the well-developed wave impedance matching technique could be utilized to safely filter out the background EM noises. As a consequence, only the GWs-induced EM signals, i.e., the signal photon fluxes, are left to be conducted into the detectors for detections.

Model and solutions

A simplified configuration of our setup to actively search for the GWs is shown in Figure 1, wherein a background magnetic field is confined in region II: a high static component along the y axis, plus an alternating component of amplitude B-prime-y and frequency omega-B, the alternating amplitude being smaller than the static one. For simplicity, the transmission of the cavity photons along the minus-z direction is neglected.

If a monochromatic circular polarized plane GW passes through the cavity — a perturbed ripple whose plus-polarization and cross-polarization components are the amplitudes A-plus and A-cross multiplying the propagation factor exp of i omega-g times z over c minus t (Equation 1) — then the background flat Minkowski space-time will be perturbed into the metric of Equation 2, the Minkowski metric plus the perturbation h, whose entries carry the two polarization amplitudes. The electromagnetic field described by that tensor obeys the relevant Einstein-Maxwell equation (Equation 3): the divergence of the metric-weighted field tensor vanishes, together with the cyclic identity on the field tensor. As the perturbations of the plus- and cross-polarizations of the GW are individual, we just discuss the plus-polarization modulations specifically; the cross-polarization modulations can also be treated similarly.

It is well-known that in a curved space-time only local measurements made by the observer traveling in the world-line are physical, and all the observable quantities are just the projections of the relevant tensors on the tetrads of the observer’s world-line. Obviously, in the present system the observer should be at rest in the static magnetic field, i.e., only the zeroth component of the four-velocity is non-vanishing. Furthermore, with the simplified boundary conditions for the proposed configuration shown in Figure 1, the solution of Equation 3 can be conveniently divided into three parts (Equation 4): for each of the electric and magnetic components, a zero-order term plus two first-order perturbative terms.

The zero-order solution (Equation 5) describes the usual electromagnetic induction without the GW perturbation: an electric field proportional to the alternating amplitude times c times the sine of omega-B times z over c minus t, with the magnetic partner equal to that electric field divided by c. The perturbative electromagnetic field, up to the first order of the amplitude A-plus, is then given region by region.

In region I, before the cavity, all the first-order fields vanish (Equation 6).

In region II, inside the cavity, the first branch of the first-order field grows with the distance travelled into the cavity, being proportional to the product of the GW amplitude, the static magnetic field, the GW frequency and that distance, with the magnetic partner again equal to the electric one divided by c (Equation 7). The second branch (Equation 8) carries the combination frequencies: it is a sum over the two sidebands, with coefficients set by the GW amplitude, the alternating field amplitude, the GW frequency and the detunings — the detuning being the GW frequency plus or minus the sweep frequency — each term modulated by the factor exp of minus i times the sweep frequency times z, minus one.

In region III, beyond the cavity at distances greater than the cavity length, the transmitted first-order fields (Equations 9 and 10) are fixed by that length, which is chosen as an odd multiple of pi times c divided by the sweep frequency.

It is seen that, besides the usual electro-magnetic inductions in the local flat space-time, the perturbed electromagnetic field induced by the GWs passing through the alternating magnetic field includes three frequencies: the GW frequency itself, and the GW frequency plus or minus the sweep frequency. Distinguishing these signals from ones without any GW information, rather than the mechanical one utilized usually in most GW detecting setups (such as the LIGO-Virgo installations), the GWs could be detected electromagnetically.

Observable effects of GWs-induced perturbed electromagnetic signals

Physically, the averaged power densities — the time average over the period of the EMWs of the Poynting vector, the cross product of the electric and magnetic fields divided by the permeability of free space — could be used to describe the strengths of the electromagnetic signals. For a detector at the plane z equal to z-nought with the receiving surface spanning x from x1 to x2 and y from y1 to y2, the averaged number of the photons with the frequency omega being detected per second is that averaged flux integrated over the receiving surface and divided by the reduced Planck constant times the frequency (Equation 11).

The zero-order perturbative energy flow density, generated by the usual electro-magnetic induction of the alternating magnetic field, is the product of the zero-order electric and magnetic fields divided by the permeability. The corresponding averaged photon number (Equation 12) is proportional to the receiving area and to the square of the alternating field amplitude, divided by the permeability, the reduced Planck constant and the sweep frequency; it is certainly very large.

The energy flow densities of the first-order perturbative field, which is linearly related to the amplitude of the GWs, are the cross terms between the zero-order and first-order fields. They correspond to two kinds of perturbed EM signals with the photon fluxes: the first flux (Equation 13), proportional to the receiving area, the static field, the alternating field, the cavity length and the GW amplitude, generated by the perturbations of the GWs whose frequency equals the sweep frequency; and the second flux (Equation 14), proportional to the receiving area and the square of the alternating field and the GW amplitude, divided by the GW frequency, produced by the perturbations of the GWs whose frequency is twice the sweep frequency.

Specifically, Figure 2 shows how the detectable numbers of the photons of the two channels vary with the amplitude of the local alternating magnetic field for the selected GWs signals. It is seen clearly that, without the alternating magnetic field, the first-order perturbative photons vanish. The lower frequency of the GWs, which corresponds to the higher amplitude of the GWs, induces the stronger signal photon fluxes. Additionally, the higher amplitude of the alternating magnetic field, applied locally to search for the GWs, yields the stronger electromagnetic responses of the GWs and consequently larger signal photon fluxes. Anyway, with the single-photon detectors developed well in recent years, the induced photon fluxes could be detected, at least theoretically.

Next, with the usual frequency matching filtering technique, all the electromagnetic noises except the ones with the sweep frequency could be filtered out. However, the left signals with that same frequency still comprise three components: the photon flux without any GWs information, and the desired first-order perturbative ones carrying the information of the GWs. Fortunately, the wave impedance of the zero-order perturbative signal, generated by the local electromagnetic induction and satisfying the Maxwell equation in the flat space-time, always reads the square root of the permeability divided by the permittivity of free space, that is 377 ohms (Equation 15). The wave impedances of the first-order perturbed signals (Equations 16 and 17) are instead set by the ratios of the perturbed fields, involving the static field, the sweep frequency, the cavity length and the GW amplitude. Figure 3 shows how the wave impedances of the first-order perturbative electromagnetic signals vary with the amplitude of the local alternating magnetic field. It is seen that two of them are far greater than 377 ohms and the other two far smaller. Therefore, with the wave impedance matching filtering technique, the electromagnetic signal without any GWs information could be robustly filtered out, and consequently only the signals carrying the information of the GWs could be effectively conducted into the weak light detectors to implement the desired detections.

Conclusions and discussions

In summary, a theoretical proposal to search for the high-frequency GWs with semi-open side cavity biased locally by an alternating high magnetic field is proposed. By analytically solving the Einstein-Maxwell equation in the curved space-time, the electromagnetic responses of the GWs passing through the cavity are obtained. With the present setup the significantly stronger first-order perturbations of the GWs are generated and could be detected experimentally with the current developed-well weak-light detectors. Technically, the GWs-induced first-order perturbative EM signals could be selected out for detections, as the noises with the different frequencies and wave impedances could be filtered out by using the usual frequency matching and wave impedance matching filtering wave techniques.

It is emphasized that the proposed ground setup to search for the GWs works in a sufficiently wide band, as the frequency of the applied alternating magnetic field is adjustable within a very large regime, once the size of the cavity is experimentally achievable. Table I lists the typical frequency band of the detectable GWs with the proposed configuration. Here, a generic argument on the relation between the amplitude and frequency of the GWs is used: the amplitude scales as the inverse of the frequency. One can see that the proposal should work for the GWs with the frequency from ten to the seventh hertz to ten to the twelfth hertz, within the size of the cavity being experimentally reachable.

Certainly, the setup proposed here is still very imperfect for the realistic experiments. Its sensitivity should be further analyzed, in detail, in the presence of various noises and imperfect factors. Anyway, the present proposal provides a potential approach to locally deliver the observable first-order perturbative effects of the GWs passing through a high magnetic field. Probably, it can serve as an effective complementary of the very successful detections of the GWs based on their mechanically-tidal effects with the usual laser interferometers.

Figures

Figure 1. A simplified configuration of a ground setup to search for the high frequency GWs along the z axis: a single-side transmission cavity, i.e., region II with z running from 0 to the cavity length l, is biased by an alternating magnetic field, and only the desired GWs-induced EM signals — those whose GW frequency equals the sweep frequency or twice the sweep frequency — can pass through the wave-impedance matchers for being detected.

Figure 2. The detectable photon number of the one-order perturbative signals of both channels versus the amplitude of the locally applied alternating magnetic field. The relevant parameter is chosen as a static field of 10 tesla. Panel (a) is drawn for a GW amplitude of ten to the minus twenty-seven at a frequency of 2 pi times ten to the eighth hertz in a cavity 3 metres long; panel (b) for an amplitude of ten to the minus twenty-nine at 2 pi times ten to the tenth hertz in a cavity 0.03 metres long.

Figure 3. The wave impedances of the first-order perturbative signals of the first channel versus the amplitude of the applied alternating magnetic field, in a high static magnetic field of 10 tesla, for typical frequencies and amplitudes of the GWs.

Table I

Various parameters for probing the electromagnetic responses of GWs: amplitude and frequency of the GWs, the size of the high magnetic field regime, and the number of signal photons at unit time. Here the alternating field amplitude is 0.005 tesla.

| GW amplitude | Sweep frequency (Hz) | Cavity length l (m) | First-channel photon flux | Second-channel photon flux | |---|---|---|---|---| | ten to the minus 26 | 2 pi times ten to the 7 | 30 | 2.83 times ten to the 11 | 6.76 times ten to the 7 | | ten to the minus 27 | 2 pi times ten to the 8 | 3 | 2.83 times ten to the 9 | 6.76 times ten to the 5 | | ten to the minus 28 | 2 pi times ten to the 9 | 0.3 | 2.83 times ten to the 7 | 6.76 times ten to the 3 | | ten to the minus 29 | 2 pi times ten to the 10 | 0.03 | 2.83 times ten to the 5 | 67.6 | | ten to the minus 30 | 2 pi times ten to the 11 | 0.003 | 2.83 times ten to the 3 | 0.68 | | ten to the minus 31 | 2 pi times ten to the 12 | 0.0003 | 28.3 | 0.07 |

Acknowledgements

We thank Profs. Z. B. Li, X. G. Wu and Z. Y. Fang for useful discussions. This work is partly supported by the National Natural Science Foundation of China (NSFC) under Grant No. U1330201.

(The 35-item reference list is omitted; the complete text, with the displayed equations and the figures, is at the source.)