We investigate the coherent control of the transmission spectrum in a cavity magnetomechanical system consisting of microwave photon, magnon, and phonon modes, where the microwave cavity is driven by a strong pump field and a weak probe field, and the magnon is driven by a weak microwave source. Different from a single transparency window in the absence of the phonon–magnon interaction, two transparency windows and three absorption dips can be observed in the presence of the phonon–magnon interaction, which originates from the joint interaction of phonon–magnon and photon–magnon. In addition, two absorption dips located at both sides of the central absorption dip can be modulated asymmetrically into amplification and absorption by varying the magnetic field amplitude of the magnon driving field. Interestingly enough, the relative phase of applied fields could have profound effects on both the transmission spectrum and the group delay of the output field by choosing the appropriate magnetic field amplitude of the magnon driving field. The transmission group delay can be switched between positive to negative and vice versa by adjusting the relative phase between the applied fields. The present results illustrate the potential to utilize the relative phase for controlling the microwave signal in the cavity magnomechanical system, as well as guidance in the design of information transduction and quantum sensing.

## I. INTRODUCTION

Cavity magnomechanics,^{1} similar to the cavity optomechanics,^{2} has emerged as an important new frontier in quantum optics and attracted extensive attention recently^{3–14} due to the great potential for building a highly adjustable hybrid coherent information processing system.^{15–21} Different from the cavity optomechanics, the magnetostrictive forces instead of the radiation pressure forces play an important role in the cavity magnomechanics, and the magnon oscillator can be flexibly adjusted through the magnetic field.^{1,4,6,7}

In particular, the cavity magnomechanical system consisting of magnons in a single-crystal yttrium iron garnet (YIG) sphere strongly coupled to the cavity mode has been theoretically proposed and experimentally demonstrated in the past few years, in which the phonon–magnon and photon–magnon interactions are introduced and observed.^{1,17,18} The magnon-induced absorption, as well as the magnetomechanically induced transparency, has been investigated,^{1,22,23} which originates from the internal constructive (destructive) interference that can be interpreted by analogy with the optomechanical induced absorption (transparency)^{24–26} in the cavity optomechanics. Meanwhile, the modification in the light response of the system resulting from the coupling of the magnon and the cavity microwave photon brings a significant transformation in the phase dispersion of the transmitted field.^{22,27,28} Tunable slow and fast light has also been demonstrated in the cavity magnomechanics,^{22,23,29,30} which provide a potential application value for quantum memories,^{31} quantum information processing,^{32} tunable microwave filters and amplifiers,^{33} signal sensing, and microwave-to-optical conversion.^{34}

In this paper, we investigate the coherent control of the transmission spectrum in a cavity magnetomechanical system consisting of microwave photon, magnon, and phonon modes, where the microwave cavity is driven by a strong pump field and a weak probe field and the magnon is driven by a weak microwave source. In order to clearly show the dependence of the transmission spectrum on the driving parameters of the magnon, we provide a comparison of the transmission spectra of the cavity magnetomechanical system with and without including the phonon–magnon interaction. Different from a single transparency window in the absence of the phonon–magnon interaction, it finds that two transparency windows and three absorption dips can be observed in the presence of the phonon–magnon interaction. Furthermore, the effect of the relative phase of applied fields and the magnetic field amplitude of the magnon driving field on the transmission characteristics is explored. It demonstrates that two absorption dips located at both sides of the central absorption dip can be modulated asymmetrically into amplification and absorption by adjusting the magnetic field amplitude of the magnon driving field. More interestingly, it is found that the relative phase of applied fields could have profound effects on both the transmission spectrum and the group delay of the output field by choosing the appropriate magnetic field amplitude of the magnon driving field. As a result, the transmission of the output signal can be switched between slow to fast light and vice versa by modulating the relative phase between the applied fields.

## II. MODEL

As depicted in Fig. 1, we consider a cavity magnomechanical system consisting of a 3D microwave cavity with the frequency $\omega a$ and a YIG sphere with the diameter 250 $\mu $m. The cavity is driven by a strong pump field and a weak probe field with the driving frequencies $\omega l$, $\omega p$, and the phases $\varphi l$, $\varphi p$, respectively. The YIG sphere is directly driven by a weak microwave source with the driving frequency $\Omega m$ and the phase $\varphi m$. When the fields are applied to the system, due to the excellent magnetic properties of the ferrimagnet, the magnon with the frequency $\omega m$ is embodied by a large number of collective spin motions in the YIG and can be tuned by adjusting the external bias magnetic field $H$.^{17,38} Meanwhile, the variation of magnetization induced by the magnon excitation leads to the geometric deformation of the YIG sphere so that the mechanical mode with the frequency $\omega b$ is produced in the system. In this system, the magnon mode can couple the photon mode through the magnetic dipole interaction with the magnon–photon coupling strength $gma$. The mechanical mode couples the magnon mode through magnetostrictive interaction, and the magnomechanical coupling strength is $gmb$. We consider the dimensions of the YIG sphere, in this work, is much smaller than the wavelength of the microwave so that the influence of light radiation pressure in the system is negligible.^{17,18} The total Hamiltonian of the system reads^{1,17,22}

where

Here, $a(a\u2020)$, $m(m\u2020)$, and $b(b\u2020)$ are the annihilation(creation) operators of the cavity mode, the magnon mode, and the mechanical mode, respectively. $H0$ represents the free energy of the cavity, the magnon, and the mechanical oscillator. $H1$ represents the interaction Hamiltonian where the first term is the coupling term of the magnon and the cavity field, and the second term is the coupling term of the magnon and the mechanical oscillator. $Hd$ represents the coupling Hamiltonian of the cavity field with the pump field where the amplitude $\epsilon in=P1\u210f\omega l$. $\eta $ is the coupling efficiency of the cavity and the fields that can be continuously adjusted in the experiment,^{39,40} and $\kappa a$ is the cavity decay rate. $Hp$ represents the coupling Hamiltonian of the cavity field with the weak probe field where the amplitude $\epsilon p=Pp\u210f\omega p$. $P1$ and $Pp$ are the pump and probe field powers, respectively. $Hpm$ describes the interaction of the magnon mode with the weak microwave source where $\epsilon m=54\gamma NB0$ is the coupling strength of the magnon and the microwave field; $\gamma $ denotes the gyromagnetic ratio; $N$ is the total number of spins inside YIG sphere; and $B0$ is the drive magnetic field amplitude.

Furthermore, we consider the rotating-wave approximation of the system, i.e., $gma(a+a\u2020)(m+m\u2020)\u2192gma(am\u2020+a\u2020m)$ (which is valid when $\omega a$, $\omega m\u226bgma$, $\kappa a$, $\kappa m$^{1,17}), and combine the transformation $H\u2192U\u2020HtU\u2212iU\u2020\u2202tU$ written into the interaction picture with respect to $U=exp[\u2212i\omega l(a\u2020a+m\u2020m)t]$, the Hamiltonian can be obtained afresh,

where $\Delta a=\omega a\u2212\omega l$ and $\Delta m=\omega m\u2212\omega l$ represent the detunings of the cavity-pump and magnon-pump fields, respectively. $\delta pl=\omega p\u2212\omega l$ represents the detuning of the probe and pump fields, and $\delta ml=\Omega m\u2212\omega l$ represents the detuning of magnon driving field and pump field. $\varphi pl=\varphi p\u2212\varphi l$ represents the relative phase of the probe and pump fields.

## III. QUANTUM DYNAMICS AND FLUCTUATIONS

According to the Hamiltonian of the system and taking into account the decay of the cavity mode and the dissipation of magnon mode and the mechanical mode existing in the system, we can derive the Heisenberg–Langevin equations of the motion as follows:

where $\kappa a$, $\kappa b$, $\kappa m$ are the total loss rates of the cavity field, the mechanical oscillator and the magnon, respectively. The quantum input noise and thermal noise terms of the system have been eliminated due to we are interested in the mean response of this system to the probe field.^{41,42}

Assuming that the amplitudes of the applied fields satisfy ${\epsilon p,\epsilon m}\u226a\epsilon in$ and setting the value of the difference equation in the right half of Eq. (4) equal to 0, the steady state equations of the system can be obtained

where $\Delta ~m=\Delta m+gmb(bs+bs\u2217)$ is the effective detuning between the magnon and the pump field due to the magnomechanical interaction.

Then, we divide each operator of Eq. (4) into a small fluctuation and its steady state expectation value, i.e., $O=Os+\delta O(O=a,b,m)$. Through a series of mathematical calculations and ignoring all the higher order terms($\delta O\delta O$), the fluctuation equations of the system can be written as

with $G=gmbms$ is the effective magnomechanical coupling strength of the system.

Considering the perturbation produced by the probe field, the fluctuation terms of the system can be expanded as^{22,43}

with $\delta =\delta pl=\delta ml$. By substituting Eq. (7) into Eq. (6), we obtain the amplitude of the first-order sideband of the cavity magnomechanical system,

with $h1\xb1=\xb1i\Delta a+\kappa a\u2212i\delta $, $h2\xb1=\xb1i\omega b+\kappa b\u2212i\delta $, $h3\xb1=\xb1i\Delta ~m+\kappa m\u2212i\delta $, $\beta 1=|G|2(h2+\u2212h2\u2212)$, $\beta 2=h3+(gma2+h1\u2212h3\u2212)$, $\beta 3=h1\u2212\beta 1(h3+\u2212h3\u2212)$, $\beta 4=gma2\u2212h3+h1\u2212+h1\u2212h3\u2212$.

By using input–output relation $Sout=Sin\u2212\eta \kappa aa$,^{44} in the rotating frame with respect to the pump frequency $\omega l$, the output of the probe field can be given by

where $c0=\epsilon in\u2212as\eta \kappa a$, $c1\u2212=\epsilon pe\u2212i\varphi pl\u2212\eta \kappa aA1\u2212$, and $c1+=\u2212\eta \kappa aA1+$, respectively. We define the transmission of the probe field as $tp=c1\u2212/(\epsilon pe\u2212i\varphi pl)$.^{45} The transmission rate of the cavity magnomechanical system can be derived as

It can be seen that the transmission rate $|tp|2$ depends on the amplitude of magnon driving field $\epsilon m$ and the phases of fields $\varphi pl$, $\varphi m$, where $\epsilon m$ is closely related to the drive magnetic field amplitude $B0$ and the relative phase of applied fields $\Phi =\varphi m\u2212\varphi pl$ can be controlled in the experimental platform.^{21,46,47} In the following, we focus on analyzing the influence of these parameter values on the characteristics of transmission of the system.

## IV. RESULTS AND DISCUSSION

In this section, we numerically evaluate the transmission rate of the output field based on Eqs. (8) and (10) and, in detail, analyze the effect of the magnon driving parameters, such as the magnetic field amplitude $B0$ of magnon driving field and the relative phase $\Phi =\varphi m\u2212\varphi pl$, on the transmission spectrum. Here, we choose the parameters of the cavity field,^{1,19} i.e., the frequency of the microwave cavity $\omega a/2\pi =7.86$ GHz, the cavity decay rate $\kappa a/\pi =3.35$ MHz, and the coupling efficiency of the cavity and the fields $\eta =1/2$. We consider the parameters of the YIG sphere^{1} with the diameter $D=250$ $\mu m$, the spin density $\rho =4.22\xd71027$ $m\u22123$, the magnon linewidth $\kappa m/\pi =1.12$ MHz, the gyromagnetic ratio $\gamma /2\pi =28$ GHz/T, and the magnon–photon coupling strength $gma/2\pi =3.2$ MHz.^{17} In addition, we set the phonon frequency $\omega b/2\pi =11.42$ MHz,^{1} the mechanical dissipation rate $\kappa b/\pi =300$ Hz,^{1} and the magnon–phonon coupling strength $gmb/2\pi =1$ Hz.^{15} In terms of the parameters of the fields applied in the system, we choose the pump power of cavity $P1=10$ mW, the amplitude of the probe field $\epsilon p=0.0001\epsilon in$. The detuning between the cavity field and pump field $\Delta a=\omega b$, the detuning between the magnon driving field and pump field $\Delta m=\omega b$.

In order to clearly display the influence of magnon driving parameters on the characteristics of the output field of the system, in the absence of the additional magnon driving field of the system, we first plot in Fig. 2 the transmission rate $|tp|2$ as a function of the probe detuning $\delta $ with several different values of the magnon–phonon coupling strength $gmb$. From Fig. 2, one can find that in the absence of the magnon–phonon coupling $gmb=0$, there is a transparency window in the position of $\delta =\omega b$ and the bilateral symmetric absorption dips in the positions of $\delta =0.71\omega b$(left) and $\delta =1.29\omega b$(right) correspond to the magnon-induced transparency and the magnon-induced absorptions,^{22,23} respectively, which results from the formation of magnon–polaritons between magnon and photon coupling in the system.^{48} In contrast, considering the magnon–phonon coupling $gmb\u22600$, an absorption dip appears at the point $\delta =\omega b$ and becomes deep as the increasing of $gmb$. Thus, two transparency windows and three absorption dips can be observed in the presence of the magnon–phonon coupling, which is different from the conventional magnetiomechanically induced transparency (MMIT). The physical interpretation of this phenomenon shown in Fig. 2 is rather clear. The ultimate physical mechanism of this phenomenon can be explained by the energy level diagram in Fig. 3. In the absence of the magnon–phonon coupling, i.e., $gmb=0$, under the influence of the pump field and the cavity decay, the photon emission transition can be achieved via exciting the magnon from the state $|Na,Nm+1,Nb\u27e9$ $\u2192$ $|Na+1,Nm,Nb\u27e9$ $\u2192$ $|Na,Nm,Nb\u27e9$, which degenerates with the probe photon generated from the state $|Na,Nm,Nb\u27e9$ $\u2192$ $|Na+1,Nm,Nb\u27e9$ $\u2192$ $|Na,Nm,Nb\u27e9$ excited by the probe field. In this case, the destructive interference effect emerges in the system resulting in a magnon-induced transparency window and the bilateral symmetric absorption dips. When the magnon–phonon coupling is included ($gmb\u22600$), there is an additional transition path $|Na,Nm+1,Nb\u27e9$ $\u2192$ $|Na,Nm,Nb+1\u27e9$ $\u2192$ $|Na,Nm+1,Nb\u27e9$ $\u2192$ $|Na+1,Nm,Nb\u27e9$ $\u2192$ $|Na,Nm,Nb\u27e9$. This transition induced by magnon–phonon interaction, along with the transition caused by photon–magnon coupling, forms a constructive interference and leads to a sharp dip of the magnetomechanically induced absorption at the point $\delta =\omega b$.

In the view of the discussion of Fig. 2, the positions of the magnon-induced absorption and the magnetomechanically induced absorption of the system are determined, respectively. We impose an additional magnon driving field to investigate, in detail, the effect of the magnon driving parameters on the characteristics of output filed of the cavity magnomechanical system. In the absence of the phase modulation, i.e., $\Phi =0$, the transmission rate $|tp|2$ as a function of the probe detuning with the different drive magnetic field amplitude is plotted in Fig. 4. It is shown that the central absorption at the point $\delta \u2243\omega b$ becomes deeper slightly with the increasing magnetic field amplitude $B0$ of magnon driving field, while the bilateral symmetric absorption dips located at both sides of the central absorption dip coupling become asymmetry. In the red-sideband regime ($\delta \u22430.71\omega b$), it is found that with the increase of magnetic field amplitude, the dip induced by the magnon–photon interaction evolves into transparency. Furthermore, when the magnetic field amplitude becomes a little larger, the transmission rate of the output field is greater than 1, and thus, the amplification of the weak probe field can be obtained. On the contrary, it can find that in the blue-sideband regime($\delta \u22431.29\omega b$), with the increasing magnetic field amplitude, there is a lower absorption dip, which decreases gradually and reaches saturation. The result we have shown is analogous to the Autler–Townes effect where the coupled transition in a three-level $\u2227$-type system is perturbed by a microwave field.^{49–51} In the system, the modulation of the magnon driving field amplitude can control the magnon-induced absorption into amplification, which has potential applications in quantum computation and quantum sensing.

It is worth noting that the relative phase of applied fields has been used for the coherent control of OMIT in the cavity optomechanics systems,^{43} coined as the phase control technology. Furthermore, we analyze the influence of the relative phase between the cavity and the magnon driving fields on the transmission rate of the system. We choose the magnetic field amplitude $B0=10.3\xd710\u221210$ T and plot the transmission rate of the output field as a function of the probe detuning with the different relative phase $\Phi $ in Fig. 5. The curves of Fig. 5 demonstrate that the bilateral asymmetric absorption induced by magnon–polaritons interaction has a sensitive dependence on the relative phase. On the contrary, the central absorption dip corresponding to the magnetomechanically induced absorption at the point $\delta =\omega b$ remains almost unchanged when the relative phase varies from $0$ to $\pi $. The relative phase dependence of the transmission spectrum in Fig. 5 can readily be explained physically in the following. According to the energy level diagram depicted in Fig. 3, when the magnon driving field is applied in the system, there are three transition paths can form a closed-loop $\Delta $-type transition structure in the system. Specifically, the transition path caused by the magnon driving field $|Na,Nm,Nb\u27e9$ $\u2194$ $|Na,Nm+1,Nb\u27e9$ joins the transition paths caused by the cavity probe and the pump fields, i.e., $|Na,Nm,Nb\u27e9$ $\u2194$ $|Na+1,Nm,Nb\u27e9$ and $|Na,Nm+1,Nb\u27e9$ $\u2194$ $|Na+1,Nm,Nb\u27e9$, lead to the phase-sensitive effect of the cavity magnomechanical system. It is similar to the phase-dependent gain absorption spectra when three coherent driving fields simultaneously drive a three-level atomic system.^{52,53} The present results suggest that when an additional magnon driving field is applied in the cavity magnomechanical system, the magnon-induced absorption and the magnon-induced transparency of the cavity magnomechanical system are switchable by adjusting the value of the magnetic field amplitude of magnon driving field and the relative phase of the applied fields.

From Figs. 4 and 5, it is clear that the transmission rate of the system is significantly dependent on the amplitude and phase of the magnon driving field. To demonstrate the feasibility of our scheme, we present a sensitive tool for direct measurement of the perturbing magnetic field amplitude $B0$. By setting $d(|tp|2)/d\delta =0$ in the probe detuning $\delta \u22430.71\omega b$, the maximum values of absorption $tmaxAB$ and amplification $tmaxAM$ of transmission rate corresponding to the unit magnetic field amplitude $B0$ are plotted in Fig. 6. It is well known that the sensitivity of the measurement can be evaluated by the slope of maximum values of the transmission spectrum with respect to the perturbing magnetic field, which shows the identification ability for the change of magnetic field.^{41,54} In the absorption and amplification regime, we define sensitivity $SAB=|d(tmaxAB)/dB0|$ and $SAM=|d(tmaxAM)/dB0|$, respectively. From Fig. 6, it is evident that the variation of maximum values of the transmission rate is distinct with different relative phases. In the case of $\Phi =0$, the maximum values of the transmission rate increases linearly with increasing magnetic field amplitude $B0$, which experiences absorption and amplification regions. Correspondingly, $S1AB<S1AM$ indicates the sensitivity is higher in weak perturbing magnetic field. In contrast to $\Phi =0$, when the relative phase $\Phi =\pi $, the maximum values of transmission rate decreases linearly with increasing perturbing magnetic field amplitude and only appear in the absorption region. With $S3AB<S2AB$, it demonstrates that sensitivity is higher in strong perturbing magnetic field. Compared with traditional sensors in cavity optomechanics, we calculate sensitivity coefficient $\eta =tmax\u2212tmax\u2032tmax+tmax\u2032\xd7100%$, where $tmax$ and $tmax\u2032$ represent two adjacent maximum values of transmission rate when perturbing magnetic field amplitude changes in units of $10\u221210$ T. It verifies in present system $\eta S1AB=38.46%$, $\eta S1AM=21.38%$, $\eta S2AB=63.02%$, and $\eta S3AB=97.85%$, which is a significant improvement over the Refs. 55 and 56. These results provide a new possibility to improve the detection sensitivity of the magnetic field strength by adjusting the value of the relative phase of the applied fields.

As discussed above, we have investigated how to manipulate the transmission spectrum of the present system by adjusting appropriately parameters of magnon and driving fields and have found that the transmission has sensitive dependence on the relative phase of the applied fields. As is known that the transmission in OMIT and MMIT system experiences a considerable group delay. Thus, it reminds us of another question: whether the group delay can be observed in the present cavity magnomechanical system with magnon driving. In the following, we study the effect of the group delay of the cavity magnomechanical system. The group delay of the output signal field $\tau g$^{22,57,58} can be defined as

where $\varphi t(\omega p)=arg\u2061[tp(\omega p)]$ is the phase dispersion of the transmitted light field. When the group delay is a positive value, i.e., $\tau g>0$, there is a slow light in the system. When the group delay is a negative value, i.e., $\tau g<0$, a fast light can be observed in the system.

In the case of $\Phi =0$, we show in Fig. 7, the group delay $\tau g$ as a function of probe detuning $\delta $ with the different drive magnetic field amplitude $B0$. Clearly, it can be found that the group delay can be tuned by changing the magnetic field amplitude of magnon driving field. In the probe detuning $\delta \u22430.71\omega b$ and $\delta \u22431.29\omega b$ corresponding to the magnon-induced absorption dips of Fig. 4, either slow light or fast light can be achieved with the appropriate value of the magnetic field amplitude $B0$ in the present system. Moreover, Fig. 7 also shows that the group delay can be switched from a negative value to a positive value so that the fast light in the probe detuning $\delta \u22430.71\omega b$ and $\delta \u22431.29\omega b$ can be converted to the slow light. The maximal value of the group delay induced by the magnon–polaritons interaction can reach $3.7\mu s$ in the probe detuning $\delta \u22431.29\omega b$. In the probe detuning $\delta \u22431.00\omega b$, a negative group delay ($\tau g<0$) is achievable, which implies that the magnon–phonon interaction leads to fast light. Furthermore, the effect of negative group delay becomes pronounced with increasing the value of magnetic field amplitude $B0$.

Figure 8(a) shows that the group delay can be switched by modulating the relative phase of the applied fields. Compared with the curve of group delay in the case of $\Phi =0$, Fig. 8(a) shows that the positive group delay in the probe detuning $\delta \u22430.71\omega b$ corresponding to the mganon-induced absorption has a giant increase in the case of $\Phi =\pi $. However, in the probe detuning $\delta \u22431.29\omega b$, the positive group delay decreases dramatically when the relative phase $\Phi $ is tuned from $0$ to $\pi $. Interestingly enough, it can be seen that the curve of the group delay in the case of $\Phi =\pi $ is the mirror image symmetric with regard to the case of $\Phi =0$. In the probe detuning $\delta \u22431.00\omega b$, Fig. 8(b) displays that the group delay vs the magnetic field amplitude of the magnon driving field with the different relative phase of the applied fields. It demonstrates that for a fixed value of the magnetic field amplitude $B0$, the dispersion property of the system can be modulated and the slow and the fast light can be achievable and switchable in the cavity magnomechanical system by using the phase modulation of the applied fields. Besides, the slow and the fast light can be significant boosted with the increasing magnetic field amplitude. Thus, we note that the present work provides an efficient scheme for controlling dispersion as well as slow light in the specific frequency. The results we investigated in the cavity magnomechanical system may provide a potential application for the fields of telecommunications and microwave data processing.

## V. CONCLUSIONS

In conclusion, we investigate the coherent control of the transmission spectrum in a cavity magnetomechanical system consisting of microwave photon, magnon, and phonon modes, where the microwave cavity is driven by a strong pump field and a weak probe field and the magnon is driven by a weak microwave source. We mainly focus on the relative phase of the applied fields and the magnetic field amplitude of magnon driving field on the transmission spectrum and the dispersion property of the system. The present results clearly show the dependence of the transmission spectrum on the driving parameters of the magnon by comparing the transmission spectra of the cavity magnetomechanical system with and without including the magnon–phonon coupling. It is worth noting that, different from a single transparency window in the absence of the magnon–phonon interaction, two transparency windows and three absorption dips can be observed in the presence of the magnon–phonon interaction, which originates from the joint interaction of magnon–phonon and magnon–photon. The present work also illustrates that two absorption dips located at both sides of the central absorption dip can be modulated asymmetrically into amplification and absorption by varying the magnetic field amplitude of magnon driving field. More interestingly, the relative phase of applied fields could have profound effects on both the transmission spectrum and the group delay of output field by choosing the appropriate magnetic field amplitude of magnon driving field so that the transmission spectra and the dispersion property could be controllable. To demonstrate the feasibility of our scheme, we present a sensitive tool for direct measurement of the perturbing magnetic field strength by adjusting the value of the relative phase of the applied fields, which demonstrate a significant improvement over previous research in cavity optomechanics. These present results illustrate that manipulating the relative phase of the applied fields in the cavity magnetomechanical system has a potential application in the field of quantum sensing and microwave signal processing.

## ACKNOWLEDGMENTS

The research is supported in part by National Natural Science Foundation of China under Grant Nos. 11774054 and 12075036.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.