Quantum illumination based on cavity-optomagnonics system with Kerr nonlinearity

Cheng-Zhang Du; Da-Wei Wang; Cheng-Song Zhao; Junya Yang; Ling Zhou

doi:10.1364/OE.496693

1. Introduction

Quantum illumination is a quantum optical sensing technique to detect low reflectivity targets [1–7] , where one of a pair of entangled photons is sent into a target region, and the other photons are retained at the receiver; a joint measurement between the reflected back from target region and the retained photon can discriminate whether the object is present or not. Microwave quantum illumination utilizes entanglement between microwave and optical mode, and the microwave photons are sent to target region and reflected back to joint measurement [8–10]. Compared with the traditional radar, quantum illumination can use microwave signals to achieve target detection with low detection error rate and high signal-to-noise ratio, but at the same time it is difficult to generate and detect in microwave wavelengths [11].

Recently, magnons (i.e., spin wave quanta) with flexible controllability and high spin density have received much theoretical and experimental attention [12–16] . Many aspects of the cavity magnonics such as magneton dark modes [17], spin currents [18], entanglement [19–26] , magnon blockade [27–30], non-reciprocity [31], magnon-induced absorption [32], microwave-to-optical conduction [33,34], and removing asymmetric (Einstein-Podolsky-Rosen) EPR steering [35] have been investigated. Moreover, the Kerr nonlinearity of magnons arising from the magnetocrystalline anisotropy of yttrium iron garnet (YIG) spheres in cavity magnonics system [36,37] is of tunability through controlling the external magnetic field. This nonlinearity has been demonstrated experimentally [38] and has been used to study bistability and multistability [39–41] , quantum entanglement [42], quantum phase transitions [43], and magneton-mediated long-range spin-spin coupling [44]. Although the microwave quantum illumination scheme utilizing cavity optomagnonics system (COMs) [45], the Kerr nonlinearity of magnons is not considered. Since the YIG sphere is of intrinsic Kerr nonlinearity, the improvable nonlinearity should have non-negligible effect on microwave quantum illumination.

In this paper, we investigate the microwave quantum illumination based on cavity optomagnonics system by considering the Kerr nonlinearity of the magnon. Considering the magnons driven by a microwave field [39,44], the Kerr nonlinearity results in a classical multi-stable and offers us a degenerate-parametric term of magnon mode. Consequently, the entanglement between microwave field and optical field can be improved. We carefully compare the different effect between improving the optomagnonical coupling and adjusting degenerate-parametric term of magnon mode on the enhancement of the entanglement and find that the larger the Kerr coefficient the better entanglement, while the optomagnonical coupling on affecting entanglement is not. Our results show that increasing the Kerr coefficient K leads to enhancing entanglement between the optical and microwave fields, improving signal-to-noise ratio, and reducing error probability of detection. Compared with the case [45] without considering the Kerr nonlinearity, our scheme has advantages in improving the entanglement and reducing the error probability.

2. Hamiltonian of the system and its dynamic equation

As shown in Fig. 1 (a), the COMs works as microwave-optical converter, which is consisted of microwave cavity and the YIG sphere coupling with an optical nanofiber. The magnon modes of the YIG excited by an external bias magnetic field $B_{0}$ couples to microwave cavity mode through magnetic dipole interaction. Meanwhile, the magnon modes interact with the optical TE and TM whispering gallery modes (WGMs) in the YIG sphere through a three-wave magnon-induced Brillouin scattering process. Assuming $B_{0}$ along z-direction and the TE WGMs defined as optical mode whose polarization direction is parallel to the x-y plane, TM WGMs polarization is along the z-direction [45]. Considering the magnons driven by a microwave field with frequency $\omega _{d}$ and amplitude $\Omega _{d}$ [39,44], we can write the Hamiltonian of the system $H=H_{0}+H_{1}+H_{d}$[20,39,45,46] with

(1)$$\begin{aligned}{H}_{0} &=\omega _{TE}a_{TE}^{{\dagger} }a_{TE}+\omega _{TM}a_{TM}^{{\dagger}}a_{TM}+\omega _{b}b^{{\dagger} }b+\omega _{m}m^{{\dagger} }m,\\ {H}_{1} &=g_{ma}(a_{TE}^{{\dagger} }a_{TM}m+a_{TE}a_{TM}^{{\dagger} }m^{{\dagger}})+g_{mb}(m^{{\dagger} }+m)(b^{{\dagger} }+b)+K(m^{{\dagger} }m)^{2},\\ H_{d} &=\Omega _{d}(m^{{\dagger} }e^{{-}i\omega _{d}t}+me^{i\omega _{d}t}), \end{aligned}$$

where $a_{j}$ ($a_{j}^{\dagger }$ with $j=TE,TM$) , $b$ ($b^{\dagger }$) and $m$ ($m^{\dagger }$) are the annihilation (creation) operators of the $\pi$-polarized TE ($\sigma ^{-}$-polarized TM) WGM, microwave mode and magnon mode with the frequency $\omega _{TE}$ ($\omega _{TM}$), $\omega _{b}$ and $\omega _{m}$, respectively. $H_{0}$ is the free energy of the cavity optomagnonics system. The first term in $H_{1}$ describes the three-wave-mixed process with strength $g_{ma}$, the second term represents the coupling between microwave mode and magnon mode with rate $g_{mb}$, and the third term is the Kerr nonlinear induced by the magneto-crystal anisotropy where the nonlinear coefficient $K$ can be of the positive $(K>0)$ or negative $(K<0)$ value by adjusting the orientation between the YIG sphere crystal axis and the bias magnetic field $B_{0}$ [39]. $H_{d}$ describes the magnon mode driven by microwave field.

Fig. 1. (a) The sketch of the magnonical reversible COMs.A YIG sphere is placed in a microwave cavity and holds the magnon mode excited by an external bias magnetic field, which establishes the electromagnonical coupling via magnetic dipole interaction. The optical mode in nanofiber can evanescently couple to TE WGMs in the YIG sphere, which produces and establishes the optomagnonical coupling between TM WGM and magnon mode. (b) The sketch of cavity-optomagnonics quantum illumination. Microwave photon of a pair of entangled photon is sent into a target region, and the optical photon is retained at the receiver; a joint measurement between the reflected back from target region and the retained photon can discriminate whether the object is present or not.

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In the frame rotating with $H_{0}^{\prime }=\omega _{TE}a_{TE}^{\dagger }a_{TE}+\omega _{TM}a_{TM}^{\dagger }a_{TM}+\omega _{b}b^{\dagger }b+\omega _{d}m^{\dagger }m$, the Hamiltonian can be changed into time-independent form. Under rotational wave approximation, when the frequency of pumping field magnon mode and cavity mode is near resonant, i.e., $\omega _{d}=\omega _{b}$, the interaction $\hbar g_{mb}(bm^{\dagger }+b^{\dagger }m)$ is remained, and the term $\hbar g_{mb}(b^{\dagger }m^{\dagger }e^{2i\omega _{d}t}+h.c.)$ can be ignored if $2\omega _{d}$ $\gg g_{mb}$. Then, under the condition $\omega _{TE}=\omega _{TM}+\omega _{d}$, we rewrite the Hamiltonian as

(2)$$\begin{aligned} H &=\Delta m^{{\dagger} }m+K(m^{{\dagger} }m)^{2}+g_{ma}(a_{TE}^{{\dagger} }a_{TM}m+a_{TE}a_{TM}^{{\dagger} }m^{{\dagger} })+g_{mb}(m^{{\dagger} }b+mb^{{\dagger} })\\ &\quad +\Omega _{d}(m^{{\dagger} }+m), \end{aligned}$$

where the frequency detuning $\Delta =\omega _{m}-\omega _{d}$. Considering the TE mode driven by a coherent tone leading to $a_{TE}\rightarrow \alpha$, we can rewrite $a_{TM}$ as $a$, then the Hamiltonian (2) can be linearized as

(3)$$H =\Delta m^{{\dagger} }m+K(m^{{\dagger} }m)^{2}+G_{ma}(am+a^{{\dagger}}m^{{\dagger} })+g_{mb}(m^{{\dagger} }b+mb^{{\dagger} })+\Omega _{d}(m^{{\dagger} }+m),$$

where $G_{ma}=\alpha g_{ma}$. One can see clearly that the effective coupling between optical mode $a$ and magnon mode $m$ is of nondegenerated-parametric type interaction, which will result in entanglement between them. We will show that it is the entanglement that makes the quantum illumination smoothly.

The quantum Langevin equations are given by

(4)$$\begin{aligned} \dot{a} &={-}iG_{ma}m^{{\dagger} }-\kappa _{a}a+\sqrt{2\kappa _{a}}a_{in},\\ \dot{b} &={-}ig_{mb}m-\kappa _{b}b+\sqrt{2\kappa _{b}}b_{in},\\ \dot{m} &=({-}i\Delta -iK-\kappa _{m})m-2iKm^{{\dagger} }mm-ig_{mb}b-iG_{ma}a^{{\dagger} }-i\Omega _{d}+\sqrt{2\kappa _{m}}m_{in}, \end{aligned}$$

where $\kappa _{a}$, $\kappa _{b}$, and $\kappa _{m}$ represent the dissipation rates of the optical, microwave, and magnon modes, respectively. $a_{in}$, $b_{in}$, and $m_{in}$ are the input noise operators for the optical, microwave, and magnon mode. The noise operators satisfy $\langle \mu _{in}(t)\mu _{in}^{\dagger }(t^{\prime })\rangle =(n_{\mu }^{T}+1)\delta (t-t^{\prime })$, where $n_{\mu }^{T}=(e^{\hbar \omega _{\mu }/k_{B}T}-1)^{-1}$ $(\mu =a,b,m)$ is the thermal equilibrium mean numbers. One can safely assume $n_{a}^{T}\approx 0$ since $\hbar \omega _{a}/k_{B}T\gg 1$ for the high optical frequencies, while thermal microwave photons and magnons cannot be neglected in general, even at very low temperatures because of its low frequency.

Considering the magnon mode driven with strong field, then each operator of the cavity optomagnonics system can be written as its expectation value plus the corresponding fluctuation, i.e., $O=O_{s}+\delta O$ ($O=a,b,m$). For simplicity, the fluctuation operator $\delta O$ ($O=a,b,m$) is written as $O$, and symbols $\delta$ is ignored. The semi-classical steady-state value of magnon mode can be obtained as

(5)$$m_{s}=\frac{i\Omega _{d}}{-i(\Delta +K+2K|m_{s}|^{2})+B},$$

with $B=\frac {G_{ma}^{2}}{\kappa _{a}}-\frac {g_{mb}^{2}}{\kappa _{b}}-\kappa _{m}$. Due to Kerr nonlinearity , $m_{s}$ in (5) exhibits multiple values. In Fig. 2 (a), the multistability $|m_{s}|^{2}$ is plotted, which clearly show that the multistability is sourced from the Kerr nonlinearity. Only when $K$ is larger than $6.5\times 10^{-12}\kappa _{m}$ (see black dotted line), multistability $|m_{s}|^{2}$ can be observed for the group of the parameters. If we increase $G_{ma}$ to region $G_{ma}\approx 6.5\kappa _{m}\sim 15\kappa _{m}$, the multistability values of $|m_{s}|^{2}$ disappear, and value of $|m_{s}|^{2}$ is in the monostable region. In Fig. 2 (b), we plot $K|m_{s}|^{2}$ varying with $G_{ma}$ in the range of $6.5\kappa _{m}\sim 15\kappa _{m}$ for three different Kerr coefficients. The red, blue and black lines are all shown us monostable values. Hereafter, we will choose the parameters for $m_{s}$ in single value region. The YIG spheres chosen for the experiments are of millimeter or even micron magnitude. This will limit the strength of the Kerr coefficients. The Kerr nonlinear coefficient $K/2\pi \approx 10^{-10}$Hz can be gained in a 1-mm-diameter YIG sphere [39]. The linearized Langevin equations are

(6)$$\begin{aligned} \dot{b} &={-}ig_{mb}{m}-\kappa _{b}b+\sqrt{2\kappa _{b}}b_{in},\\ \dot{a} &={-}iG_{ma}{m^{{\dagger} }}-\kappa _{a}a+\sqrt{2\kappa _{a}}a_{in},\\ \dot{m} &=[{-}i\Delta ^{\prime }-\kappa _{m}]m-2i\Lambda m^{{\dagger} }-ig_{mb}{b}-iG_{ma}{a^{{\dagger} }}+\sqrt{2\kappa _{m}}m_{in}, \end{aligned}$$

where $\Delta ^{\prime }=\Delta +K+4K|m_{s}|^{2}$, $\Lambda =Km_{s}^{2}$. The Hamiltonian corresponding to (6) is of the form

(7)$$H=\Delta ^{\prime }m^{{\dagger} }m+\Lambda m^{{\dagger} 2}+\Lambda ^{{\ast} }m^{2}+(g_{mb}mb^{{\dagger} }+G_{ma}am+h.c.),$$

which means that the Kerr nonlinearity induces a degenate-parametric term for the magnon mode with a enhanced strength $\Lambda$ through improving the classical pumping. For negative value of $\Delta$, the effective detuning $\Delta ^{\prime }$ is also reduced due to the Kerr nonlinearity. We can foresee that the term $\Lambda m^{\dagger 2}+\Lambda ^{\ast }m^{2}$ results in entanglement improving, which will be discussed in next section. To facilitate the calculation, we rewrite the Eq. (6) in compact form

(8)$$\dot{u}(t)=Au(t)+\xi (t),$$

where $u(t)=(b,b^{\dagger },a,a^{\dagger },m,m^{\dagger })^{T}$, $\xi (t)=( \sqrt {2\kappa _{b}}b_{in},\sqrt {2\kappa _{b}}b_{in}^{\dagger },\sqrt {2\kappa _{a}}a_{in},\sqrt {2\kappa _{a}}a_{in}^{\dagger },\sqrt {2\kappa _{m}}m_{in},$

Fig. 2. (a) The steady-state magnon number $|m_{s}|^{2}$ as a function of the Kerr coefficient $K/ \kappa _{m}$. (b) The steady value $K|m_{s}|^{2}$ as a function of the optical-magnon coupling constant $G_{ma}/ \kappa _{m}$. The parameters are $ \kappa _{m}/2\pi =1\rm {MHz}$, $ \kappa _{b}=2.4 \kappa _{m}$, $ \kappa _{a}=0.24 \kappa _{m}$, $g_{mb}=10 \kappa _{m}$, $\Omega _{d}=6\times 10^{8} \kappa _{m}$, $ \omega _{m}=9\times 10^{3} \kappa _{m}$, $ \omega _{d}=9.3\times 10^{3} \kappa _{m}$, $T_{env}=30$mK, $\Delta =-300 \kappa _{m}$.

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$\sqrt {2\kappa _{m}}m_{in}^{\dagger })^{T}$, the superscript T means transposition. The coefficient matrix is given by

(9)$$A=\left( \begin{array}{cccccc} -\kappa _{b} & 0 & 0 & 0 & -ig_{mb} & 0 \\ 0 & -\kappa _{b} & 0 & 0 & 0 & ig_{mb} \\ 0 & 0 & -\kappa _{a} & 0 & 0 & -iG_{ma} \\ 0 & 0 & 0 & -\kappa _{a} & iG_{ma} & 0 \\ -ig_{mb} & 0 & 0 & -iG_{ma} & -i\Delta ^{\prime}-\kappa _{m} & -2i\Lambda \\ 0 & ig_{mb} & iG_{ma} & 0 & 2i\Lambda^{{\ast}} & i\Delta ^{\prime}-\kappa _{m} \end{array} \right) .$$

Due to the existence of the terms $\Lambda m^{\dagger 2}+\Lambda ^{\ast }m^{2}$ and $G_{ma}am+h.c.$, the system might be unstable. Only when the real part of all eigenvalues of the matrix $A$ is negative (satisfy the Routh-Hurwitz criterion), the system attains a steady state. Beside guaranteeing $\Lambda$ in monostable region, we should also keep the system in the stable region. In Fig. 3(a) and (b), we show the maximum value of the real part of the eigenvalues of $A$ as a function of $K/\kappa _{m}$ and $G/\kappa _{m}$, respectively. $G_{ma}=6.5\kappa _{m}$ and $G_{ma}=15\kappa _{m}$ are the boundary of monostable region in Fig. 2(a). It can be seen that the stable region in Fig. 3(a) is also monostable region. Similar result is also shown in Fig. 3(b). Because the maximum value of the real part of the eigenvalues of matrix $A$ is less than zero, all the real part of the eigenvalues of matrix $A$ is negative, which means that the system is stable. In next section, we will choose the parameters in the stable region.

Fig. 3. (a) Re$(A)_{max}$ (maximum value of all the real parts of the eigenvalues of the matrix $A)$ as function of the Kerr coefficient $K/ \kappa _{m}$ (a) and the optical-magnon coupling constant $G_{ma}/ \kappa _{m}$ (b), respectively. The other parameters are the same as in Fig. 2.

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3. Microwave-optical output entanglement

Since it has been shown that the entanglement benefits the quantum illumination [38,45,47], we now investigate the property of the output entanglement between microwave and optical modes in current scheme. By performing the Fourier transform $O(\omega )=\frac {1}{ \sqrt {2\pi }}\int dtO(t)e^{i\omega t}$, we can solve Eq. (8) in frequency domain. Considering the input-output relation $d_{j}=j_{out}=\sqrt { 2\kappa _{j}}j-j_{in}$ ($j=a,b$), we can obtain the output microwave and optical field as

(10)$$\begin{aligned} & d_{b}=A_{1}b_{in}+A_{2}b_{in}^{{\dagger} }+A_{3}a_{in}^{{\dagger} }+A_{4}a_{in}^{{\dagger} }+A_{5}m_{in}+A_{6}m_{in}^{{\dagger} },\\ & d_{a}=B_{1}b_{in}+B_{2}b_{in}^{{\dagger} }+B_{3}a_{in}^{{\dagger} }+B_{4}a_{in}^{{\dagger} }+B_{5}m_{in}+B_{6}m_{in}^{{\dagger} }. \end{aligned}$$

The coefficients $A_{j}$ and $B_{j}$ ($j=1,2\cdots 6$) are given in Appendix A, where for simplicity we consider $\omega =0$ as [45]. We see clearly that $d_{b}$ is contributed partly by optical field ($d_{b}$ contains $a_{in}$ and $a_{in}^{\dagger }$). Similarly, $d_{a}$ is contributed partly by microwave field ($d_{a}$ contains $b_{in}$ and $b_{in}^{\dagger }$), which exhibits that there is quantum correlation between the optical field and microwave field. We define $n(i|j)$ ($i,j=a,b$) to describe $i$ contributed by $j$, which is also stand for the correlation between $i$ and $j$. Then $n(a|b)$ express the mean number of microwave field contributed by the output optical field, and $n(b|a)$ describes the mean number of optical field contributed by the output microwave field. From Eqs. (20) and (10), we have

(11)$$\begin{aligned} n(b|a) &=(|A_{3}|^{2}+|A_{4}|^{2})\bar{n}_{a}^{T}+|A_{4}|^{2},\\ n(a|b) &=(|B_{1}|^{2}+|B_{2}|^{2})\bar{n}_{b}^{T}+|B_{2}|^{2}. \end{aligned}$$

Now we introduce the covariance matrix (CM) $V$ in the frequency domain. The matrix element of the $4\times 4$ CM can be expressed as

(12)$$V_{ij}=\frac{1}{2}\langle f_{i}f_{j}+f_{j}f_{i}\rangle ,$$

where

(13)$$\mathbf{f}=[X_{b},Y_{b},X_{a},Y_{a}]^{T} .$$

with $X_{j}=(j_{out}+j_{out}^{\dagger })/\sqrt {2}$ and $Y_{j}=(j_{out}-j_{out}^{\dagger })/(i\sqrt {2})$ ($j=a,b$) denoting the quadratures fluctuation of coordinate and momentum.

With the help of CM, we can easily quantify the output entanglement between the microwave field and optical field using logarithmic negativity [48]

(14)$$E_{N}=\mathrm{max}\{0,-\mathrm{ln}2\xi ^{-}\},$$

where $\xi ^{-}=2^{-1/2}\left [ \Sigma (V)-\sqrt {\Sigma (V)^{2}-4\mathrm {det}V }\right ] ^{1/2}$ is the lowest symplectic eigenvalue of the partial transpose of the CM, and $\Sigma (V)=$det$A+$det$B-2$det$C$ with $A$, $B$ and $C$ being the $2\times 2$ matrix, taking from CM

(15)$$V=\left( \begin{array}{cc} A & C \\ C^{T} & B \end{array} \right) .$$

The quantum correlation $n(a|b)$ ($n(b|a$) in subgraph) as functions of Kerr nonlinearity $K$ and $G_{ma}$ is plotted in Figs. 4(a) and (b) respectively, meanwhile the entanglement between microwave and optical field as functions of Kerr nonlinearity $K$ and $G_{ma}$ is shown in Figs. 4(c) and (d). We can observe that the correlation $n(a|b)$ ($n(b|a)$) and $E_{N}$ are of same varying trend with $K$ and $G_{ma}$, which had been shown in [47,49]. The key point is that the $E_{N}$ or correlation is not zero when $K=0$. It is exactly the case which have been investigated in [45]. However, the entanglement increases with the increasing of $K$ for the groups of parameters in Figs. 4(c) and (d). Although $G_{ma}$ indicates the coupling strength between optical field and magnon mode, entanglement $E_{N}$ between the microwave field and optical field does not monotonously increases with the increasing $G_{ma}$ in the range of $G_{ma}\approx 6.5\sim 15\kappa _{m}$ (see Figs. 4(b) and (d)). The phenomenon also can be seen in [45]. To understand the mechanism of the Kerr nonlinearity on improving the entanglement, we now introduce Bogoliubov transformation $m_{s}=m\rm {cosh}r+m^{\dagger }\rm {sinh}r$ for the Hamiltonian (7) where $\tanh 2r=\frac {2\rm {Re}(\Lambda )}{\Delta ^{\prime }}$. Then Hamiltonian (7) is transformed as

(16)$$H_{eff}=\widetilde{\Delta }_{m}m_{s}^{{\dagger} }m_{s}+g_{+}(m_{s}^{{\dagger}}b+m_{s}b^{{\dagger} })+g_{-}(m_{s}b+m_{s}^{{\dagger} }b^{{\dagger} })+G_{+}(m_{s}a+m_{s}^{{\dagger} }a^{{\dagger} })+G_{-}(m_{s}^{{\dagger}}a+m_{s}a^{{\dagger} }),$$

where $\widetilde {\Delta }_{m}=\sqrt {\Delta ^{\prime 2}-4\rm {Re}(\Lambda )^2}$, $g_{+}=g_{mb}\rm {cosh}r$, $g_{-}=-g_{mb}\rm {sinh}r$, $G_{+}=G_{ma}\rm {cosh}r$, and $G_{-}=-G_{ma}\rm {sinh}r$. Thus, the participation of the Kerr nonlinearity can simultaneously increase $g_{+}$ and $G_{+}$ and also induce additional coupling terms denoted by $g_{-}$ and $G_{-}$. The addition coupling terms provide us another channel to generate entanglement. In addition, for negative value $\Delta$, the reduced detuning $\Delta ^{\prime }$ offers us relative large number of squeezing coefficient $r$ and decrease the detuning $\widetilde {\Delta }_{m}$. Therefore, increasing the coefficients $K$ on improving entanglement is different from enhancement $G_{ma}$. The large value $G_{ma}$ does not mean the large entanglement, nevertheless, the large value of $K$ does improve the entanglement for our group of parameters.

Fig. 4. The quantum correlation of microwave-optical fields $n(a|b)$ ($n(b|a)$ in subgraph) versus $K$ (a) and $G_{ma}$ (b), respectively. (c) Entanglement $E_{N}$ as function of $K$ for several values of $G_{ma}$. (d) Entanglement $E_{N}$ as function of $G_{ma}$ for several values of $K$. The other parameters are the same as in Fig. 2.

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4. Error probability of the quantum illumination

As shown in Fig. 1(b), the microwave emitted by the COMs shines on the surface of the target object, and the reflected back signal by the object is the input signal to the COMs receiver. The optical output of the receiver, similar with Eq. (10), can be written as

(17)$$d_{\eta ,a} =B_{1}\,a_{R}+B_{2}\,a_{R}^{{\dagger} }+B_{3}\,a_{in}+B_{4}\,a_{in}^{{\dagger} }+B_{5}\,m_{in}+B_{6}\,m_{in}^{{\dagger} },$$

where we have defined a new input operator of microwave cavity $a_{R}$. Under the hypothesis that the target region contain a low-reflectivity object (hypothesis $H_{1}$), $a_{R}=\sqrt {\eta }d_{b}+\sqrt {1-\eta }a_{B}$, under the hypothesis that the target region do not contain a low-reflectivity object (hypothesis $H_{0}$), $a_{R}=a_{B}$, $a_{B}$ is the annihilation operator of the noise which is in a thermal state with the photon number $\bar {n}_{B}$, and $\eta$ is the reflectivity of the target object. The microwave quantum illumination is a joint detection between the remain optical field of the transmitter and the output field of the receiver. As shown in Fig. 1(b), the converted optical signal $(d_{\eta,\mathrm {a}}^{(k)})$ and the remained optical idler $(d_{\mathrm {a}}^{(k)})$ are mixed by a $50:50$ beam splitter whose outputs is $a_{\eta,\pm }=\frac {d_{\eta,a}\pm d_{a}}{\sqrt {2}}$.

For quantum illumination target detection, the analysis of signal patterns must be extended to a continuous wave system, where the output field with a bandwidth of $B$ $\rm {Hz}$ contains a number of patterns in time period $t_{m}$ of $M=t_{m}B\gg 1$. Photo detection of $M$ fully independent patterns, comparing the difference in the total number of photons on the two detectors, can be used to determine the presence or absence of objects in the target area. $N_{\eta }=\sum _{k=1}^{M}\left ( N_{\eta,+}^{(k)}-N_{\eta,-}^{(k)}\right ) ,$ where $N_{\eta,\pm }^{(k)}=a_{\eta,\pm }^{(k)\dagger }\,a_{\eta,\pm }^{(k)}$ is corresponding to the photon-counts. The error probability of quantum illumination can be expressed as

(18)$$\mathrm{P}=\frac{{\mathrm{erfc}}[\sqrt{\mathrm{SNR/8}}]}{2}, $$

where the $M$-Mode signal-to-noise ratio (SNR) is

(19)$$\mathrm{SNR}=\frac{4M[(\langle N_{\eta ,+}\rangle _{H_{1}}-\langle N_{\eta ,-}\rangle _{H_{1}})-(\langle N_{\eta ,+}\rangle _{H_{0}}-\langle N_{\eta ,-}\rangle _{H_{0}})]^{2}}{\left( \sqrt{\langle (\Delta N_{\eta ,+}-\Delta N_{\eta ,-})^{2}\rangle _{H_{0}}}+\sqrt{\langle (\Delta N_{\eta ,+}-\Delta N_{\eta ,-})^{2}\rangle _{H_{1}}}\right) ^{2}}.$$

Equations (18) (19) can be calculated easily, since we have calculated $\langle N_{\eta }\rangle _{H_{j}}$ and $\langle (\Delta N_{\eta,+}-\Delta N_{\eta,-})^{2}\rangle _{H_{j}}$ in Appendix B (20–23).

Now we discuss the properties of quantum illumination. We numerically simulate the variation of the signal-to-noise ratio SNR and error probability $P$ with the parameter $G_{ma}$, which is shown in Fig. 5(a) and Fig. 5(b). Fig. 5(a) shows that SNR changing with $G_{ma}$ is in accord with the entanglement varying with $G_{ma}$ for three value of $K$ shown in Fig. 4(d), which is shown again that SNR is really relative to entanglement [38,45,47], and corresponding error probability $P$ consents with SNR. We notice that when the Kerr coefficient is non-negligible or relative large,in order to improve SNR, the coupling strength $G_{ma}$ should be chosen a reasonable value, the large value of $G_{ma}$ reduces SNR and depresses the effect of the Kerr nonlinearity. Therefore, with assistance of Kerr nonlinearity, we can improve signal-to-noise ratio and decrease error rate.

Fig. 5. Signal-to-noise ratio $\mathrm {SNR}$ (a) and error probability $\mathrm {P}$ (b) as a function of $G_{ma}/ \kappa _{m}$, where $ \eta =0.07$, $M=10^{6.4}$, $n_{B}=610$, and the other parameters are the same as that in Fig. 2.

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For the conventional quantum illumination, signal-to-noise ratio $\mathrm {SNR_{coh}}=\frac {4\eta Mn_{b}}{2n_{B}+1}$ where $n_{b}=\langle d_{b}^{\dagger }d_{b}\rangle$. To further investigate the advantages of quantum illumination based on cavity-optomagnonics system with Kerr nonlinearity, we simulate the error probability $P$ varying with the $\rm {Log_{10}}$M of the entanglement mode and the effective reflectivity $\eta$ of the object under different Kerr coefficient $K$ in Fig. 6. As shown in Fig. 6(a), comparing the conventional radar with the quantum illumination based on the cavity optomagnonics system, it can be clearly seen that the error probability of the quantum illumination based on the cavity optomagnonics system is lower than that of the conventional radar. This means that the Kerr nonlinearity participation in quantum illumination based on cavity-optomagnonics system can reduce the requirement for entanglement resource. As shown in Fig. 6(b), for a fixed reflectivity $\eta$, the higher the value of Kerr coefficient $K$, the lower the error rate in detecting objects even if their reflectivity is very low. This means that the Kerr nonlinearity of the current system can be used to detect low reflectivity target objects. Considering the above, current quantum illumination scheme is distinct superior to traditional radar.

Fig. 6. Error probability P versus mode pairs M (a) and efficient reflectivity $ \eta$ (b). For (a), $ \eta =0.07$. For (b), $M=10^{6.4}$. For both (a) and (b) $n_{B}=610$, $T_{env}=30$mK. The other parameters are the same as Fig. 2.

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5. Discussion and conclusion

We briefly discuss the feasibility of the experimental setup that may be used in the experiment. The microwave-magnon coupling can reach strong coupling regime $g_{mb}/2\pi =10.8$MHz, with $\kappa _{m}=1.06$MHz even ultrastrong coupling strength $g_{mb}/2\pi =2.5$GHz [50], which obviously satisfies the requirement of the scheme. The coupling between magnon mode and whispering gallery mode relies on the Faraday effect, such that the coupling strength is still weak, which fortunately can be solved by coupling higher-order magneton modes or developing new micro-nano structures [51]. With these improvements, it is possible to enhance the coupling strength to the $G_{ma}/2\pi =10$MHz level [52]. The Kerr nonlinear coefficient resulting from by the magneto-crystal anisotropy $K/2\pi =-6.5\times 10^{-9}$nHZ and the decay rate $\kappa _{m}=2.2$MHz [53], which is still smaller than what we have used above, while when the diameter of the YIG sphere decreases [54], the Kerr nonlinearity can be further enhanced. As a whole system, the present scheme may be experimental feasible in near future.

In this paper, we consider the Kerr nonlinearity of magnon and investigate the effects of nonlinearity on quantum illumination based on the cavity optomagnonics system. We demonstrate that the entanglement between the optical and microwave fields can be improved by reasonable adjustment of the Kerr coefficients, and the mechanism of the effect of Kerr nonlinearity on quantum entanglement is analyzed. Consequently, the Kerr nonlinearity of the magnon can improve the signal-to-noise ratio of detected low-reflectivity objects and reduce the probability of detection errors. This provides us with an effective method to improve quantum illumination.

Appendix A: the expression of output fields

The coefficients $A_{j}$ and $B_{j}$ of Eq. (10) can be expressed as

\begin{aligned} A_{1} &={-}\frac{2\Delta ^{-}\kappa _{m}\Lambda _{b}}{D}+1,A_{2}={-}\frac{ 4i\Lambda \kappa _{m}\Lambda _{b}}{D}, A_{3}={-}\frac{4i\Lambda \kappa _{m}\sqrt{\Lambda _{b}\Lambda _{a}}}{D} ,A_{4}={-}\frac{2\Delta ^{-}\kappa _{m}\sqrt{\Lambda _{b}\Lambda _{a}}}{D},\\ A_{5} &={-}\frac{2i\Delta ^{-}\kappa _{m}\sqrt{\Lambda _{b}}}{D},A_{6}={-}\frac{ 4\Lambda \kappa _{m}\sqrt{\Lambda _{b}}}{D}, B_{1}={-}\frac{4i\Lambda ^{{\ast} }\kappa _{m}\sqrt{\Lambda _{b}\Lambda _{a}} }{D},B_{2}=\frac{2\Delta ^{+}\kappa _{m}\sqrt{\Lambda _{b}\Lambda _{a}}}{D},\\ B_{3} &=\frac{2\Delta ^{+}\kappa _{m}\Lambda _{a}}{D}+1,B_{4}={-}\frac{ 4i\Lambda ^{{\ast} }\kappa _{m}\Lambda _{a}}{D}, B_{5}=\frac{4\Lambda ^{{\ast} }\kappa _{m}\sqrt{\Lambda _{a}}}{D},B_{6}={-} \frac{2i\Delta ^{+}\kappa _{m}\sqrt{\Lambda _{a}}}{D}, \end{aligned}

where $D=\Delta ^{+}\Delta ^{-}-4|\Lambda |^{2}$, $\Delta ^{\pm }=(\pm i\Delta ^{\prime }+\kappa _{m}-\kappa _{m}\Lambda _{a}+\kappa _{m}\Lambda _{b}),$ $\Lambda _{a}=G_{ma}^2/\kappa _{a}\kappa _{m}$, $\Lambda _{b}=g_{mb}^2/ \kappa _{b}\kappa _{m}$.

Appendix B: the calculation of Signal-to-noise ratio

In order to calculate Eq. (18) and (19), firstly we calculate $\langle N_{\eta,\pm }\rangle _{H_{j}}$ as follows

(20)$$\begin{aligned} \langle N_{\eta ,\pm }\rangle _{H_{1}} &=\frac{1}{2}(D_{1}\bar{n}_{b}^{T}+D_{2}\bar{n}_{a}^{T}+D_{3}\bar{n}_{m}^{T}+D_{4}\bar{n} _{B}+F_{1}+F_{2}+F_{3}+F_{4}), \\ \langle N_{\eta ,\pm }\rangle _{H_{0}} &=\langle N_{\eta ,\pm }\rangle_{H_{1}}(\eta \rightarrow 0), \end{aligned}$$

where the coefficients of Eq. (20) is

\begin{aligned} D_{1} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{1}+\sqrt{\eta }B_{1}^{{\ast} }A_{2}^{{\ast} }\pm B_{2}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{2}+\sqrt{\eta }B_{2}A_{1}^{{\ast} }\pm B_{2}]\\ &\quad +[\sqrt{\eta }B_{2}^{{\ast} }A_{2}+\sqrt{\eta }B_{1}^{{\ast} }A_{1}^{{\ast} }\pm B_{1}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{1}+\sqrt{\eta }B_{2}A_{2}^{{\ast} }\pm B_{1}],\\ &\\ D_{2} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{3}+\sqrt{\eta }B_{1}^{{\ast} }A_{4}^{{\ast} }+B_{4}^{{\ast} }\pm B_{4}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{4}+\sqrt{\eta }B_{2}A_{3}^{{\ast} }+B_{4}\pm B_{4}]\\ &\quad +[\sqrt{\eta }B_{2}^{{\ast} }A_{4}+\sqrt{\eta }B_{1}^{{\ast} }A_{3}^{{\ast} }+B_{3}^{{\ast} }\pm B_{3}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{3}+\sqrt{\eta }B_{2}A_{4}^{{\ast} }+B_{3}\pm B_{3}],\\ &\\ D_{3} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{5}+\sqrt{\eta }B_{1}^{{\ast} }A_{6}^{{\ast} }+B_{6}^{{\ast} }\pm B_{6}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{6}+\sqrt{\eta }B_{2}A_{5}^{{\ast} }+B_{6}\pm B_{6}]\\ &\quad +[\sqrt{\eta }B_{2}^{{\ast} }A_{6}+\sqrt{\eta }B_{1}^{{\ast} }A_{5}^{{\ast} }+B_{5}^{{\ast} }\pm B_{5}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{5}+\sqrt{\eta }B_{2}A_{6}^{{\ast} }+B_{5}\pm B_{5}],\\ &\\ D_{4} &=(1-\eta )(B_{2}^{{\ast} }B_{2}+B_{1}^{{\ast} }B_{1}), \end{aligned}

\begin{aligned} F_{1} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{1}+\sqrt{\eta }B_{1}^{{\ast} }A_{2}^{{\ast} }\pm B_{2}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{2}+\sqrt{\eta }B_{2}A_{1}^{{\ast} }\pm B_{2}],\\ &\\ F_{2} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{3}+\sqrt{\eta }B_{1}^{{\ast} }A_{4}^{{\ast} }+B_{4}^{{\ast} }\pm B_{4}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{4}+\sqrt{\eta }B_{2}A_{3}^{{\ast} }+B_{4}\pm B_{4}],\\ &\\ F_{3} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{5}+\sqrt{\eta }B_{1}^{{\ast} }A_{6}^{{\ast} }+B_{6}^{{\ast} }\pm B_{6}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{6}+\sqrt{\eta }B_{2}A_{5}^{{\ast} }+B_{6}\pm B_{6}],\\ &\\ F_{4} &=(1-\eta )B_{2}^{{\ast} }B_{2}. \end{aligned}

Then $\langle (\Delta N_{\eta,+}-\Delta N_{\eta,-})^{2}\rangle _{H_{j}}$ can be calculated as

(21)$$\begin{aligned} \langle (\Delta N_{\eta ,+}-\Delta N_{\eta ,-})^{2}\rangle _{H_{j}} &=\langle N_{\eta ,+}\rangle _{H_{j}}(\langle N_{\eta ,+}\rangle _{H_{j}}+1)+\langle N_{\eta ,-}\rangle _{H_{j}}(\langle N_{\eta ,-}\rangle _{H_{j}}\\ &\quad +1)-\frac{(\langle d_{\eta ,\mathrm{a}}^{{\dagger} }d_{\eta ,\mathrm{a} }\rangle _{H_{j}}-\langle d_{\mathrm{a}}^{{\dagger} }{d_{\mathrm{a}}\rangle } )^{2}}{2}, \end{aligned}$$

with $\langle d_{\mathrm {a}}^{\dagger }d_{\mathrm {a}}\rangle$ and $\langle d_{\eta,\mathrm {a}}^{\dagger }d_{\eta,\mathrm {a}}\rangle _{H_{j}}$ in Eq. (21) being

(22)$$\begin{aligned} \langle d_{\mathrm{a}}^{{\dagger} }d_{\mathrm{a}}\rangle &=(B_{2}^{{\ast} }B_{2}+B_{1}^{{\ast} }B_{1})\bar{n}_{\mathrm{b}}^{T}+(B_{4}^{{\ast} }B_{4}+ B_{3}^{{\ast} }B_{3})\bar{n}_{\mathrm{a}}^{T}+(B_{6}^{{\ast} }B_{6}+B_{5}^{{\ast} }B_{5})\bar{n}_{\mathrm{m}}^{T}+\\ &B_{2}^{{\ast} }B_{2}+B_{4}^{{\ast} }B_{4}+B_{6}^{{\ast} }B_{6},\\ \langle d_{\eta ,\mathrm{a}}^{{\dagger} }d_{\eta ,\mathrm{a}}\rangle _{H_{1}} &=K_{1}\bar{n}_{\mathrm{b}}^{T}+K_{2}\bar{n}_{\mathrm{a}}^{T}+K_{3}\bar{n}_{ \mathrm{m}}^{T}+K_{4}\bar{n}_{B}+\sum_{i=1}^{4}T_{i}, \\ \langle d_{\eta ,\mathrm{a}}^{{\dagger} }d_{\eta ,\mathrm{a}}\rangle _{H_{0}} &=\langle d_{\eta ,\mathrm{a}}^{{\dagger} }d_{\eta ,\mathrm{a}}\rangle _{H_{1}}(\eta \rightarrow 0), \end{aligned}$$

where the coefficient of Eq. (22) is

\begin{aligned} K_{1} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{1}+\sqrt{\eta }B_{1}^{{\ast} }A_{2}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{2}+\sqrt{\eta }B_{2}A_{1}^{{\ast} }]\\ &\quad +[\sqrt{\eta }B_{2}^{{\ast} }A_{2}+\sqrt{\eta }B_{1}^{{\ast} }A_{1}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{1}+\sqrt{\eta }B_{2}A_{2}^{{\ast} }],\\ &\\ K_{2} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{3}+\sqrt{\eta }B_{1}^{{\ast} }A_{4}^{{\ast} }+B_{4}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{4}+\sqrt{\eta }B_{2}A_{3}^{{\ast} }+B_{4}]\\ &\quad +[\sqrt{\eta }B_{2}^{{\ast} }A_{4}+\sqrt{\eta }B_{1}^{{\ast} }A_{3}^{{\ast} }+B_{3}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{3}+\sqrt{\eta }B_{2}A_{4}^{{\ast} }+B_{3}], \end{aligned}

\begin{aligned} K_{3} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{5}+\sqrt{\eta }B_{1}^{{\ast} }A_{6}^{{\ast} }+B_{6}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{6}+\sqrt{\eta }B_{2}A_{5}^{{\ast} }+B_{6}]\\ &\quad +[\sqrt{\eta }B_{2}^{{\ast} }A_{6}+\sqrt{\eta }B_{1}^{{\ast} }A_{5}^{{\ast} }+B_{5}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{5}+\sqrt{\eta }B_{2}A_{6}^{{\ast} }+B_{5}],\\ &\\ K_{4}&=(1-\eta )[B_{2}^{{\ast} }B_{2}+B_{1}^{{\ast} }B_{1}], \end{aligned}

\begin{aligned} T_{1} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{1}+\sqrt{\eta }B_{1}^{{\ast} }A_{2}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{2}+\sqrt{\eta }B_{2}A_{1}^{{\ast} }],\\ &\\ T_{2} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{3}+\sqrt{\eta }B_{1}^{{\ast} }A_{4}^{{\ast} }+B_{4}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{4}+\sqrt{\eta }B_{2}A_{3}^{{\ast} }+B_{4}],\\ &\\ T_{3} &=[\sqrt{\eta }B_{2}^{{\ast} }A_{5}+\sqrt{\eta }B_{1}^{{\ast} }A_{6}^{{\ast} }+B_{6}^{{\ast} }] \times \lbrack \sqrt{\eta }B_{1}A_{6}+\sqrt{\eta }B_{2}A_{5}^{{\ast} }+B_{6}],\\ &\\ T_{4} &=(1-\eta )B_{2}^{{\ast} }B_{2}. \end{aligned}

Funding

National Key Research and Development Program of China (No. 2021YFE0193500.); National Natural Science Foundation of China (No.12274053).

Disclosures

The authors declare no conflict of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Quantum illumination based on cavity-optomagnonics system with Kerr nonlinearity

Abstract

1. Introduction

2. Hamiltonian of the system and its dynamic equation

3. Microwave-optical output entanglement

4. Error probability of the quantum illumination

5. Discussion and conclusion

Appendix A: the expression of output fields

Appendix B: the calculation of Signal-to-noise ratio

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (28)

Optics Express