Generation of robust optical entanglement in cavity optomagnonics

Hong Xie; Le-Wei He; Chang-Geng Liao; Chang-Geng Liao; Chang-Geng Liao; Zhi-Hua Chen; Zhi-Hua Chen; Zhi-Hua Chen; Xiu-Min Lin; Xiu-Min Lin; Xiu-Min Lin; Xiu-Min Lin

doi:10.1364/OE.478963

1. Introduction

In recent years, hybrid systems based on magnons have attracted great attentions for the perspective of novel quantum technologies [1–3]. Magnons, the quanta of the collective spin excitations, can coherently interact with microwave photons in the strong-coupling [4–10] and even the ultrastrong-coupling regime [11,12]. This can in turn engineer a microwave-mediated effective coupling between magnons and superconducting qubits [13,14]. Magnons can also couple with optical photons through magneto-optical effect [15], as well as with phonons via magnetostrictive interaction [16]. The diverse set of interactions makes magnon-based hybrid systems very attractive for applications such as microwave-to-optical transducers [17,18], magnon spintronics [19], and high-precision measurements [20].

The generation of highly entangled states is an important task for implementing quantum information processing. In the magnon-based hybrid systems, great progresses have been reported on the study of continuous variable entanglement, e.g., tripartite magnon-photon-phonon entanglement [21,22], magnon-magnon entanglement [23–28], and entanglement between two microwave fields mediated by magnons [29]. However, the entanglement generations are usually hampered by the thermal noise since magnons are inherently coupled to its environment. A common method to address the thermal noise of magnons is to cool the magnon mode to its ground state [30], but the ground-state cooling of magnon mode is still a challenge in current experiments.

In this paper, we propose a scheme based on cavity optomagnonics [31–50] for realizing optical entanglement. The entanglement generation is robust against the thermal noise of the magnon mode. Our model involves two optical whispering gallery modes (WGMs) and one magnon mode in a magnetic insulator yttrium iron garnet (YIG) sphere. The interaction among the three modes is intrinsically nonlinear, where an incoming WGMs photon will be scattered by a magnon into a photon in another optical mode. We reveal that the beam-splitter-like and two-mode squeezing magnon-photon interactions can be realized simultaneously by driving two optical WGMs. Entanglement between two optical modes is then generated via their coupling with magnons. Inspired by the Bogoliubov-mode-based method developed in cavity optomechanical system [51], we show that the effects of initial magnon occupations can be eliminated at selected time by employing the destructive quantum interference between the two bright modes. The effects of magnonical thermal noise can also be highly suppressed by exploring the excitation of the Bogoliubov dark mode, which is always decoupled with magnon modes. Therefore, the optical entanglement can be generated in high-temperature thermal bath and without the requirement of initially cooling magnon modes to the ground state. This greatly reduces the difficulty of experimental implementation. Moreover, entanglement generated in our scheme is that between two orthogonally polarized optical modes, i.e., transverse-electric (TE) mode and the transverse-magnetic (TM) mode. Such entanglement is different from that usually studied in optomechanical system [52], where the optical entanglement does not involve the degrees of freedom of polarization. The entanglement generated in our scheme have potential applications in polarization-dependent quantum information tasks.

2. Model

Our model is composed of two optical WGMs and one magnetostatic mode, which are supported by a YIG sphere, as shown in Fig. 1(a). The Hamiltonian of the system reads ($\hbar =1$)

(1)$${H}_s= \omega_1a_1^{\dagger}a_1+\omega_2a_2^{\dagger}a_2+\omega_mm^{\dagger}m + g(a_1^{\dagger}a_2m+a_1a_2^{\dagger}m^{\dagger}),$$

where $\omega _i$ and $a_i$ ($i=1,2$) are the frequencies and the annihilation operators for the $i$th optical WGMs, $m$ is the annihilation operator for the magnon mode with frequency $\omega _m$. The two optical WGMs are confined close to the equator of the YIG sphere and discriminated into TE and TM modes, corresponding to a polarization out-of-plane and in-plane with respect to the equator. The frequency of magnon can be fine-tuned by an external magnetic field to meet the TE-TM photon modes splitting, i.e., $\omega _m=\omega _1-\omega _2$. The last term in the right side of Eq. (1) describes the optomagnonic interaction (with coupling strength $g$), which has been demonstrated in various experiments in terms of Brillouin light scattering [31–34,43]. The optomagnonic interaction denotes a three-wave mixing process, where the creation (annihilation) of a photon in cavity mode 1 is accompanied by the annihilation (creation) of a magnon and a photon in cavity mode 2.

Fig. 1. (a) Schematic of cavity optomagnonical system for optical entanglement generation. The YIG sphere supports two optical WGMs (TE and TM modes, which are respectively denoted as $a_1$ and $a_2$ modes) and one magnon mode. The frequency of the magnon mode can be tuned by an external magnetic field $\textbf {B}$. The two optical modes are driven by two external lasers simultaneously. (b) The two nondegenerate cavity modes $a_1$ and $a_2$ are detuned by a magnon resonance frequency $\omega _m$. Two lasers drive respectively the corresponding cavity modes at resonance.

Download Full Size | PDF

Consider that the two cavity modes are driven by two external fields with frequency $\omega _{L1}$ and $\omega _{L2}$, respectively, the total driving Hamiltonian is

(2)$${H}_d= \Omega_1(a_1e^{i\omega_{L1}t}+a_1^{\dagger}e^{{-}i\omega_{L1}t})+\Omega_2(a_2e^{i\omega_{L2}t}+a_2^{\dagger}e^{{-}i\omega_{L2}t}),$$

where $|\Omega _i| = \sqrt {\kappa _iP_\text {in,i}/\hbar \omega _i}$ are the driving amplitudes with cavity dissipation rates $\kappa _i$ and pump power $P_\text {in,i}$ for the $i$th cavity mode. For simplicity, we consider the total cavity dissipation rate is dominated by the external in/out-coupling rate, i.e., the cavity to be overcoupled. In our scheme, both cavity modes are driven at resonance, i.e., $\omega _{L1}=\omega _1$ and $\omega _{L2}=\omega _2$, as shown in Fig. 1(b). In the rotating frame defined by the unitary operator $U=e^{i(\omega _{L1}a_1^{\dagger}a_1+\omega _{L2}a_2^{\dagger}a_2+\omega _mm^{\dagger}m)t}$, the full Hamiltonian $H_s+H_d$ can be transformed to

(3)$${H}=g(a_1^{\dagger}a_2m+a_1a_2^{\dagger}m^{\dagger})+\Omega_1(a_1+a_1^{\dagger})+\Omega_2(a_2+a_2^{\dagger}).$$

Following the standard linearization process, the cavity modes can be split into an average amplitude and a fluctuation term $a_i\rightarrow \alpha _i+a_i$, and the linearized Hamiltonian is given by

(4)$$H_{\text{lin}}=G_1(a_1^{\dagger}m+a_1m^{\dagger})+G_2(a_2m+a_2^{\dagger}m^{\dagger}),$$

where $G_1=g\alpha _2$ and $G_2=g\alpha _1$ are the enhanced effective coupling strengths, the average amplitude $\alpha _i=-2i\Omega _i/\kappa _i$ can be chosen as real-valued without loss of generality, the small term $g(a_1^{\dagger}a_2m+a_1a_2^{\dagger}m^{\dagger})$ has been omitted, and the term $g\alpha _1\alpha _2(m+m^{\dagger})$ has also been omitted by applying an appropriate shift of the magnon mode. Such linearization process has been widely adopted in cavity optomechanics [52]. The second term in the right side of Eq. (4) describes a two-mode squeezing interaction which is responsible for entangling cavity mode 2 and the magnon mode, and the first term denotes a beam-splitter interaction that exchanges the states of magnon mode and cavity mode 1. When the two terms are acting together, we anticipate that cavity mode 1 and mode 2 are entangled.

3. Evolutions of the Bogoliubov modes

To discuss the two-mode squeezed entanglement between the two cavity modes, it is convenient to introduce the Bogoliubov mode operators

(5a)$$\beta_A=\cosh{(r)}a_1+\sinh{(r)}a_2^{\dagger},$$

(5b)$$\beta_B=\cosh{(r)}a_2+\sinh{(r)}a_1^{\dagger},$$

with the squeezing parameter $r=\frac {1}{2}\ln {\frac {G_1+G_2}{G_1-G_2}}$. In terms of the Bogoliubov mode operators, the linearized Hamiltonian Eq. (4) can be expressed as

(6)$$H_{\text{lin}}=G_0(\beta_A^{\dagger}m+\beta_Am^{\dagger})$$

with $G_0=\sqrt {G_1^2-G_2^2}$. Note that the mode $\beta _B$ which is often called Bogoliubov dark mode, is always decoupled from the magnon mode. This means that the dynamical evolution of the mode $\beta _B$ is intrinsically exempted from the magnon noise. By introducing the hybridized modes $\beta _\pm =(\beta _A\pm m)/\sqrt {2}$, which are subject to the magnon noise and often called bright modes, the Hamiltonian can be further diagonalized as

(7)$$H_{\text{lin}}=G_0(\beta_+^{\dagger}\beta_+{-} \beta_-^{\dagger}\beta_-),$$

where the eigenvalues of the bright modes $\beta _\pm$ are obtained as $\pm G_0$.

The evolutions of the dark mode $\beta _B$ and the bright modes $\beta _\pm$ are straightly

(8a)$$\beta_B(t)=\beta_B(0), $$

(8b)$$\beta_\pm(t)=e^{{\mp} iG_0t}\beta_\pm(0). $$

An important feature of the bright modes $\beta _\pm$ is that their superposition gives rise to the Bogoliubov mode $\beta _A=(\beta _++\beta _-)/\sqrt {2}$, where the magnon mode $m$ is eliminated. The evolution of the Bogoliubov mode $\beta _A$ is then obtained as

(9)$$\beta_A(t)=\cos{(G_0t)}\beta_A(0)-i\sin(G_0t)m(0),$$

which is related to the initial states of the Bogoliubov mode $\beta _A$ and the magnon mode $m$. However, at time $t_n=n\pi /G_0$ for integer $n$, one has $\sin (G_0t_n)=0$ and the effect of initial thermal occupation of magnon can be eliminated, hence $\beta _A(t)=(-1)^n\beta _A(0)$. At this time, the time dependences of both Bogoliubov modes $\beta _A$ and $\beta _B$ only contain the cavity mode components. Based on Eqs. (5a) and (5b), the cavity mode operators at time $t_n$ are

(10a)$$a_1(t_n)=({-}1)^n\cosh{(r)}\beta_A(0)-\sinh(r)\beta_B^{\dagger}(0), $$

(10b)$$a_2(t_n)=\cosh{(r)}\beta_B(0)-({-}1)^n\sinh(r)\beta_A^{\dagger}(0). $$

For odd number $n$, we have

(11a)$$a_1(t_n)={-}\cosh{(2r)}a_1(0)-\sinh(2r)a_2^{\dagger}(0), $$

(11b)$$a_2(t_n)=\cosh{(2r)}a_2(0)+\sinh(2r)a_1^{\dagger}(0), $$

which clearly denote an optical two-mode squeezed state with the squeezing parameter $2r$. Note that photon entanglement is generated by exploring the excitation of the Bogoliubov dark mode $\beta _B$ decoupled from magnons, and the destructive interference between the two bright modes to eliminate the thermal fluctuation of initial magnonical states. Hence, the entanglement is robust against the thermal noise of the magnon mode.

4. Covariance matrix and quantification of entanglement

When the dissipations of the system are included, the quantum Langevin equations of the system corresponding to Eq. (4) are as follows

(12a)$$\frac{da_1}{dt} ={-}iG_1m-\frac{\kappa_1}{2}a_1+\sqrt{\kappa_1}a_{1,\text{in}}, $$

(12b)$$\frac{dm}{dt} ={-}iG_1a_1-iG_2a_2^{\dagger}-\frac{\gamma}{2}m+\sqrt{\gamma}m_{\text{in}}, $$

(12c)$$\frac{da_2}{dt} ={-}iG_2m^{\dagger}-\frac{\kappa_2}{2}a_2+\sqrt{\kappa_2}a_{2,\text{in}}, $$

where $\kappa _i$ and $\gamma$ are the decay rates of cavity and the magnon mode, $a_{i,\text {in}}$ and $m_{\text {in}}$ represent input noise operators for cavity and the magnon mode, respectively. The noise correlators associated with the input fluctuations are $\langle a_{i,\text {in}}(t)a^{\dagger}_{i,\text {in}}(t')\rangle =\delta (t-t')$, $\langle a^{\dagger}_{i,\text {in}}(t)a_{i,\text {in}}(t')\rangle =0$, $\langle m_{\text {in}}(t)m^{\dagger}_{\text {in}}(t')\rangle =(\bar {n}_\text {th}+1)\delta (t-t')$, and $\langle m^{\dagger}_{\text {in}}(t)m_{\text {in}}(t')\rangle =\bar {n}_\text {th}\delta (t-t')$. Here the thermal occupation of the cavity field has been chosen zero, which is valid for the optical field with high frequency, and ${\bar n}_{\text {th}}=1/[\exp {(\hbar {\omega _m}/{k_B}T)}-1]$ denotes the thermal magnon number at the environmental temperature $T$.

We introduce the dimensionless quadrature operators of the cavity modes $X_i=(a_i+a_i^{\dagger})/\sqrt {2}$ and $Y_i=(a_i-a_i^{\dagger})/i\sqrt {2}$, and of the magnon mode $X_m=(m+m^{\dagger})/\sqrt {2}$ and $Y_m=(m-m^{\dagger})/i\sqrt {2}$. Then the Langevin equation Eq. (12) can be rewritten in a compact matrix form

(13)$$\frac{du}{dt}=Mu+N,$$

where $u=(X_1,Y_1,X_m,Y_m,X_2,Y_2)^T$, and

(14)$$N=(\sqrt{\kappa_1}X_{1,\text{in}},\sqrt{\kappa_1}Y_{1,\text{in}},\sqrt{\gamma}X_{m,\text{in}},\sqrt{\gamma}Y_{m,\text{in}},\sqrt{\kappa_2}X_{2,\text{in}},\sqrt{\kappa_2}Y_{2,\text{in}})^T.$$

Here, $X_{i,\text {in}}=(a_{i,\text {in}}+a^{\dagger}_{i,\text {in}})/\sqrt {2}$, $Y_{i,\text {in}}=(a_{i,\text {in}}-a^{\dagger}_{i,\text {in}})/i\sqrt {2}$, $X_{m,\text {in}}=(m_{\text {in}}+m^{\dagger}_{\text {in}})/\sqrt {2}$, and $Y_{m,\text {in}}=(m_{\text {in}}-m^{\dagger}_{\text {in}})/i\sqrt {2}$ are corresponding input noise quadratures. The drift matrix $M$ is given by

(15)$$M = \begin{pmatrix} -\frac{\kappa_1}{2} & 0 & 0 & G_1 & 0 & 0 \\ 0 & -\frac{\kappa_1}{2} & -G_1 & 0 & 0 & 0 \\ 0 & G_1 & -\frac{\gamma}{2} & 0 & 0 & -G_2 \\ -G_1 & 0 & 0 & -\frac{\gamma}{2} & -G_2 & 0 \\ 0 & 0 & 0 & -G_2 & -\frac{\kappa_2}{2} & 0 \\ 0 & 0 & -G_2 & 0 & 0 & -\frac{\kappa_2}{2} \\ \end{pmatrix}.$$

The linearized Hamiltonian Eq. (4) and Gaussian nature of the quantum noise ensure that the state of the system is Gaussian, and its information-related properties can be fully characterized by covariance matrix $V$ with the components defined as $V_{ij}=\langle u_iu_j+u_ju_i \rangle /2$. The time evolution of the covariance matrix can be derived as

(16)$$\frac{d{V}}{dt}={MV}+VM^T+D,$$

where the expression of the diffusion matrix $D$ is

(17)$$D=\text{diag}[\frac{\kappa_1}{2}, \frac{\kappa_1}{2}, \frac{\gamma}{2}(2\bar{n}_{\text{th}}+1), \frac{\gamma}{2}(2\bar{n}_{\text{th}}+1), \frac{\kappa_2}{2}, \frac{\kappa_2}{2}].$$

It is convenient to use the logarithmic negativity $E_N$ to quantify the continuous-variable entanglement. For a two-mode Gaussian state, the logarithmic negativity $E_N$ can be directly computed from the reduced $4\times 4$ covariance matrix

(18)$$V_{a_1a_2}=\begin{pmatrix} A & C \\ C^T & B \end{pmatrix},$$

which is obtained by tracing out rows and columns correlated with magnon mode. Here, $A$, $B$, and $C$ are $2\times 2$ matrices. Then the logarithmic negativity $E_N$ can be calculated as

(19)$$E_N=\text{max} \{0,-\ln2\eta^-\},$$

with $\eta ^-=[(\Sigma -\sqrt {\Sigma ^2-4\det V})/2 ]^{1/2}$ and $\Sigma =\det A+ \det B - 2\det C$.

In the optical systems, two-mode Gaussian states are entangled if and only if $E_N>0$. Since entanglement between two cavity modes is generated via their coupling with magnon mode, generally speaking, both effects of initial magnonical state and thermal noise of the bath may strongly impair entanglement. Assuming that the magnon mode is initially in the thermal state with average magnon number $\bar {n}_0$ and the magnon occupation of the bath is $\bar {n}_\text {th}$, the dynamics of the entanglement is plotted in Fig. 2. The negativity $E_N$ oscillates as a function of time, where the maxima of $E_N$ locates at $G_0t_n=n\pi$ for odd number $n$. This result agrees well with the theoretical anticipation in the Sec. 3, where the damping processes are ignored. The evolution of negativity $E_N$ for different values of $\bar {n}_0$ is shown in Fig. 2(a). It is clear that the maxima of $E_N$ is almost unchanged when initial thermal magnon number $\bar {n}_0$ increases from $0$ to $10^3$. This is because the thermal fluctuation of initial magnon states is eliminated by the destructive interference between the two bright modes at $t_n=n\pi /G_0$, as discussed below Eq. (9). Figure 2(b) plots the dynamics of $E_N$ for different values of $\bar {n}_\text {th}$, the maxima of $E_N$ decrease gradually with the increasing of thermal magnon occupation $\bar {n}_\text {th}$. Note the Bogoliubov mode $\beta _A$ is coupled to the magnon mode via a beam-splitter interaction (see Eq. (6)). This induces a mixing between $\beta _A$ and the magnon mode and the coherent evolution of $\beta _A$ is affected by the bath noise of the magnon mode, which leads to the decreasing of $E_N$ with increasing of $\bar {n}_\text {th}$. However, the Bogoliubov mode $\beta _B$ is always decoupled from the magnon mode, making that the two-mode squeezing entanglement is still robust against the thermal magnon noise. Even for $\bar {n}_\text {th}=10^3$, the entanglement $E_N\approx 1.43$ at $t_1$ remains larger than the entanglement limit (i.e., $\ln 2$) based on the coherent parametric coupling [21,53].

Fig. 2. Entanglement $E_N$ as a function of $G_0t/\pi$. Parameters in units of $\kappa _1$ are chosen as: $\kappa _2/\kappa _1=1$, $G_1/\kappa _1=20$, $G_2//\kappa _1=15$, $\gamma /\kappa _1=0.01$. (a) the thermal magnon occupation of the bath $\bar {n}_\text {th}=0$, (b) the thermal number of initial magnon state $\bar {n}_0=0$.

Download Full Size | PDF

For comparison, the maxima of $E_N$ and the steady-state entanglement as a function of $\bar {n}_\text {th}$ are plotted in Fig. 3. The steady-state entanglement is obtained under the same driving conditions as our Bogoliubov-based method. It is found that entanglement based on steady-state scheme quickly approaches to zero with increasing of $\bar {n}_\text {th}$. Our scheme based on the excitation of Bogoliubov dark mode [51,54] displays much higher robustness against bath noise of the magnon mode. In experiments, the maximum entanglement can be obtained by switching off the two driving fields at time $t_1$. When the optomagnonic interactions are turned off at this specific time, the optical entanglement in the cavity is larger than steady-state entanglement. The manipulation of entanglement at specific times is a well stablished methods, which has been adopted in cavity QED for studying quantum gate [55] and optomechanical system for entangling mechanical motion with microwave fields [56].

Fig. 3. Maxima entanglement of our scheme (solid line) and steady-state entanglement (dash line) as a function of the thermal magnon occupation $\bar {n}_\text {th}$. The thermal magnon number of initial state is chosen as $\bar {n}_0=10^3$, other parameters are the same as in Fig. 2.

Download Full Size | PDF

The maximum entanglement as a function of $G_2/G_1$ is displayed in Fig. 4(a). Apparently, the amount of entanglement is a nonmonotonic function of $G_2/G_1$ (with fixed $G_1$). The phenomenon is similar to that discussed in the dissipative entanglement generation in optomechanical system [54] , and it can be explained as follows. The increasing of the ratio $G_2/G_1$ will increase the squeezing parameter $r=\frac {1}{2}\ln {\frac {G_1+G_2}{G_1-G_2}}$, which is beneficial for the enhancement of entanglement. However, the increasing of $G_2/G_1$ is accompanied by the decreasing of $G_0=\sqrt {G_1^2-G_2^2}$, which is the coupling strength between mode $\beta _A$ and magnon mode $m$. The decreasing of $G_0$ reduces the amount of entanglement, since the optical entanglement is generated via the coupling between optical modes and magnon mode. The competition of the two opposing effects denotes there exist a balance, where the maximum entanglement is achieved for the optimal $G_2/G_1$. For the parameters chosen in the numerical calculations, the optimal value of $G_2/G_1$ is about 0.75. In addition, the system becomes unstable when $G_2/G_1>1$.

Fig. 4. (a) Maxima entanglement as a function of $G_2/G_1$. (b) Maxima entanglement as a function of $G_2/\kappa _2$, the coupling strength are chosen as $G_2/G_1=0.75$. The parameters are $\bar {n}_0=0$ and $\bar {n}_\text {th}=10^2$, other parameters are the same as in Fig. 2.

Download Full Size | PDF

Figure 4(b) shows the maximum entanglement as a function of $G_2/\kappa _2$ for different values of $\bar {n}_\text {th}$, the ratio $G_2/G_1$ is set to 0.75 to optimize the entanglement. The maximum of $E_N$ increase with the increasing of coupling $G_2/\kappa _2$, as expected. In the strong coupling regime, the entanglement still survives even for large magnon thermal occupations. In current YIG-based experiments with single photon coupling strength $g/2\pi =10$ Hz [31], the strong coupling conditions are not yet available because of the cavity can not support too strong pump light. For a YIG sphere with diameter of the order the optical wavelength $1$ $\mu$m, it was estimated the single photon coupling rate would be $g/2\pi =0.1$ MHz [36]. With the driving field, the coupling strength can be enhanced to $G_i=g\sqrt {\bar {n}_i}$, where the the intracavity photon number $\bar {n}_i=|\alpha _i|^2=4P_\text {in,i}/\hbar \omega _L \kappa _i$ when the cavity is resonantly driven. Here $P_\text {in,i}$ is the pump power for the $i$th cavity mode. For the parameters $\omega _{Li}/2\pi =300$ THz, $\kappa _i/2\pi =1$ GHz [31], and $g/2\pi =0.1$ MHz, the pump power $P_\text {in,i}=\{10, 50, 100\}$ mW yielding the photon-number-enhanced coupling strength $G_i=\{3.6, 7.9, 11.2\}$ GHz respectively. For a smaller optical dissipation rate $\kappa _i/2\pi =100$ MHz, the pump power $P_\text {in,i}=\{0.1, 1, 10\}$ mW leading to the enhanced coupling strength $G_i=\{1.1, 3.6, 11.2\}$ GHz respectively. Hence, the cavity with high Q factor and strong single photon coupling strength is beneficial for achieving the strong coupling regime.

5. Conclusions

In summary, we have proposed a scheme to generate entanglement between two optical modes in cavity optomagnonics. The entanglement is generated through the coupling of two optical modes with a magnon mode. Based on the use of Bogoliubov dark mode and the destructive quantum interference between the optical bright modes, both effects of initial thermal occupation of magnon and bath noise can be highly suppressed. The scheme requires the strong optomagnonic coupling $G\,>\,\kappa$. Though the strong coupling condition is still an experimental challenge, the coupling strength can be enhanced by maximizing the overlap of magnon and photon modes [40], reducing the optical mode volume [18,38,47], or designing an optomagnonic crystal [57]. Noticeably, very recent studies predict that in the optically dispersive magnetic media even the single photon-magnon coupling strength can be enhanced close to the magnon frequency in the gigahertz range [58,59]. It is to be expected that strong optomagnonic coupling can be realized in the forthcoming future. Our results can be extended to realize entanglement between light and microwave in a hybrid opto-electro-magnonical system [17,18], where microwave photons couple with magnons via beam-splitter-like interaction and optical photons couple with magnons through two-mode squeezing interaction. Compared with the steady-state scheme [60], a more strongly and robustly entanglement between light and microwave can be obtained. Our scheme suggest that cavity optomagnonics could be a promising platform for the applications of magnon-based quantum information processing.

The entanglement generated in our scheme is that between two orthogonally polarized optical modes. This is different from that usually studied in optomechanical system, where the optical entanglement does not involve the degrees of freedom of polarization. It seems that a frequency-bin entangled photon pairs [61–64] can be generated in cavity optomechanics and converted to polarization entanglement, however, the frequency-bin photon qubit has not yet been experimental implemented in cavity optomechanics, as far as we know. The transformation of frequency-bin qubit to polarization qubit will also increase the complexity of experimental set-up, hence reduce the efficiency of entanglement generation. Although the polarization entanglement can be generated in some other systems such as quantum dots [65,66], nonlinear crystals [67], and atomic ensemble [68]. These studies (include the frequency-bin entangled photons in optomechanical system) are mainly focus on the generation of polarization entanglement photon pairs, i.e., a discrete-state entanglement. In contrast, our scheme proposes the generation of continuous-variable polarization entanglement [69], which has some valuable features in quantum communications compare to the discrete-state case.

Funding

National Natural Science Foundation of China (12174054, 12074067, 12004336); Natural Science Foundation of Fujian Province (2021J011228, 2020J01191).

Acknowledgments

We acknowledge support from the National Natural Science Foundation of China and the Natural Science Foundation of Fujian Province of China.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, “Hybrid quantum systems based on magnonics,” Appl. Phys. Express 12(7), 070101 (2019). [CrossRef]

2. B. Zare Rameshti, S. Viola Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M. Hu, H. X. Tang, G. E. Bauer, and Y. M. Blanter, “Cavity magnonics,” Phys. Rep. 979, 1–61 (2022). [CrossRef]

3. H. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, “Quantum magnonics: when magnon spintronics meets quantum information science,” Phys. Rep. 965, 1–74 (2022). [CrossRef]

4. O. O. Soykal and M. E. Flatté, “Strong field interactions between a nanomagnet and a photonic cavity,” Phys. Rev. Lett. 104(7), 077202 (2010). [CrossRef]

5. H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, and S. T. B. Goennenwein, “High cooperativity in coupled microwave resonator ferrimagnetic insulator hybrids,” Phys. Rev. Lett. 111(12), 127003 (2013). [CrossRef]

6. Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113(8), 083603 (2014). [CrossRef]

7. M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity qed with magnons at microwave frequencies,” Phys. Rev. Appl. 2(5), 054002 (2014). [CrossRef]

8. Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120(5), 057202 (2018). [CrossRef]

9. J. T. Hou and L. Liu, “Strong coupling between microwave photons and nanomagnet magnons,” Phys. Rev. Lett. 123(10), 107702 (2019). [CrossRef]

10. Y. Li, T. Polakovic, Y.-L. Wang, J. Xu, S. Lendinez, Z. Zhang, J. Ding, T. Khaire, H. Saglam, R. Divan, J. Pearson, W.-K. Kwok, Z. Xiao, V. Novosad, A. Hoffmann, and W. Zhang, “Strong coupling between magnons and microwave photons in on-chip ferromagnet-superconductor thin-film devices,” Phys. Rev. Lett. 123(10), 107701 (2019). [CrossRef]

11. X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Strongly coupled magnons and cavity microwave photons,” Phys. Rev. Lett. 113(15), 156401 (2014). [CrossRef]

12. N. Kostylev, M. Goryachev, and M. E. Tobar, “Superstrong coupling of a microwave cavity to yttrium iron garnet magnons,” Appl. Phys. Lett. 108(6), 062402 (2016). [CrossRef]

13. Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Coherent coupling between a ferromagnetic magnon and a superconducting qubit,” Science 349(6246), 405–408 (2015). [CrossRef]

14. D. Lachance-Quirion, Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, and Y. Nakamura, “Resolving quanta of collective spin excitations in a millimeter-sized ferromagnet,” Sci. Adv. 3(7), e1603150 (2017). [CrossRef]

15. S. Viola Kusminskiy, “Cavity optomagnonics,” in Optomagnonic Structures: Novel Architectures for Simultaneous Control of Light and Spin Waves, (World Scientific, 2021), pp. 299–353.

16. X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,” Sci. Adv. 2(3), e1501286 (2016). [CrossRef]

17. R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, “Bidirectional conversion between microwave and light via ferromagnetic magnons,” Phys. Rev. B 93(17), 174427 (2016). [CrossRef]

18. N. Zhu, X. Zhang, X. Han, C.-L. Zou, C. Zhong, C.-H. Wang, L. Jiang, and H. X. Tang, “Waveguide cavity optomagnonics for microwave-to-optics conversion,” Optica 7(10), 1291–1297 (2020). [CrossRef]

19. A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, “Magnon spintronics,” Nat. Phys. 11(6), 453–461 (2015). [CrossRef]

20. C. Potts, V. Bittencourt, S. V. Kusminskiy, and J. Davis, “Magnon-phonon quantum correlation thermometry,” Phys. Rev. Appl. 13(6), 064001 (2020). [CrossRef]

21. J. Li, S.-Y. Zhu, and G. S. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121(20), 203601 (2018). [CrossRef]

22. H. Tan, “Genuine photon-magnon-phonon einstein-podolsky-rosen steerable nonlocality in a continuously-monitored cavity magnomechanical system,” Phys. Rev. Res. 1(3), 033161 (2019). [CrossRef]

23. J. Li and S.-Y. Zhu, “Entangling two magnon modes via magnetostrictive interaction,” New J. Phys. 21(8), 085001 (2019). [CrossRef]

24. Z. Zhang, M. O. Scully, and G. S. Agarwal, “Quantum entanglement between two magnon modes via kerr nonlinearity driven far from equilibrium,” Phys. Rev. Res. 1(2), 023021 (2019). [CrossRef]

25. S.-S. Zheng, F.-X. Sun, H.-Y. Yuan, Z. Ficek, Q.-H. Gong, and Q.-Y. He, “Enhanced entanglement and asymmetric epr steering between magnons,” Sci. China Phys. Mech. Astron 64(1), 210311 (2021). [CrossRef]

26. H. Yuan, S. Zheng, Z. Ficek, Q. He, and M.-H. Yung, “Enhancement of magnon-magnon entanglement inside a cavity,” Phys. Rev. B 101(1), 014419 (2020). [CrossRef]

27. V. Azimi Mousolou, A. Bagrov, A. Bergman, A. Delin, O. Eriksson, Y. Liu, M. Pereiro, D. Thonig, and E. Sjöqvist, “Hierarchy of magnon mode entanglement in antiferromagnets,” Phys. Rev. B 102(22), 224418 (2020). [CrossRef]

28. J. M. Nair and G. Agarwal, “Deterministic quantum entanglement between macroscopic ferrite samples,” Appl. Phys. Lett. 117(8), 084001 (2020). [CrossRef]

29. M. Yu, H. Shen, and J. Li, “Magnetostrictively induced stationary entanglement between two microwave fields,” Phys. Rev. Lett. 124(21), 213604 (2020). [CrossRef]

30. S. Sharma, Y. M. Blanter, and G. E. W. Bauer, “Optical cooling of magnons,” Phys. Rev. Lett. 121(8), 087205 (2018). [CrossRef]

31. X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, “Optomagnonic whispering gallery microresonators,” Phys. Rev. Lett. 117(12), 123605 (2016). [CrossRef]

32. J. A. Haigh, A. Nunnenkamp, A. J. Ramsay, and A. J. Ferguson, “Triple-resonant brillouin light scattering in magneto-optical cavities,” Phys. Rev. Lett. 117(13), 133602 (2016). [CrossRef]

33. A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, “Cavity optomagnonics with spin-orbit coupled photons,” Phys. Rev. Lett. 116(22), 223601 (2016). [CrossRef]

34. A. Osada, A. Gloppe, R. Hisatomi, A. Noguchi, R. Yamazaki, M. Nomura, Y. Nakamura, and K. Usami, “Brillouin light scattering by magnetic quasivortices in cavity optomagnonics,” Phys. Rev. Lett. 120(13), 133602 (2018). [CrossRef]

35. T. Liu, X. Zhang, H. X. Tang, and M. E. Flatté, “Optomagnonics in magnetic solids,” Phys. Rev. B 94(6), 060405 (2016). [CrossRef]

36. S. Viola Kusminskiy, H. X. Tang, and F. Marquardt, “Coupled spin-light dynamics in cavity optomagnonics,” Phys. Rev. A 94(3), 033821 (2016). [CrossRef]

37. S. Sharma, Y. M. Blanter, and G. E. W. Bauer, “Light scattering by magnons in whispering gallery mode cavities,” Phys. Rev. B 96(9), 094412 (2017). [CrossRef]

38. J. Graf, H. Pfeifer, F. Marquardt, and S. Viola Kusminskiy, “Cavity optomagnonics with magnetic textures: Coupling a magnetic vortex to light,” Phys. Rev. B 98(24), 241406 (2018). [CrossRef]

39. J. A. Haigh, N. J. Lambert, S. Sharma, Y. M. Blanter, G. E. W. Bauer, and A. J. Ramsay, “Selection rules for cavity-enhanced brillouin light scattering from magnetostatic modes,” Phys. Rev. B 97(21), 214423 (2018). [CrossRef]

40. S. Sharma, B. Z. Rameshti, Y. M. Blanter, and G. E. W. Bauer, “Optimal mode matching in cavity optomagnonics,” Phys. Rev. B 99(21), 214423 (2019). [CrossRef]

41. V. A. S. V. Bittencourt, V. Feulner, and S. Viola Kusminskiy, “Magnon heralding in cavity optomagnonics,” Phys. Rev. A 100(1), 013810 (2019). [CrossRef]

42. W.-J. Wu, Y.-P. Wang, J.-Z. Wu, J. Li, and J. Q. You, “Remote magnon entanglement between two massive ferrimagnetic spheres via cavity optomagnonics,” Phys. Rev. A 104(2), 023711 (2021). [CrossRef]

43. R. Hisatomi, A. Noguchi, R. Yamazaki, Y. Nakata, A. Gloppe, Y. Nakamura, and K. Usami, “Helicity-changing brillouin light scattering by magnons in a ferromagnetic crystal,” Phys. Rev. Lett. 123(20), 207401 (2019). [CrossRef]

44. S. Sharma, V. A. S. V. Bittencourt, A. D. Karenowska, and S. V. Kusminskiy, “Spin cat states in ferromagnetic insulators,” Phys. Rev. B 103(10), L100403 (2021). [CrossRef]

45. F.-X. Sun, S.-S. Zheng, Y. Xiao, Q. Gong, Q. He, and K. Xia, “Remote generation of magnon schrödinger cat state via magnon-photon entanglement,” Phys. Rev. Lett. 127(8), 087203 (2021). [CrossRef]

46. F. Šimić, S. Sharma, Y. M. Blanter, and G. E. W. Bauer, “Coherent pumping of high-momentum magnons by light,” Phys. Rev. B 101(10), 100401 (2020). [CrossRef]

47. J. A. Haigh, R. A. Chakalov, and A. J. Ramsay, “Subpicoliter magnetoptical cavities,” Phys. Rev. Appl. 14(4), 044005 (2020). [CrossRef]

48. J. A. Haigh, A. Nunnenkamp, and A. J. Ramsay, “Polarization dependent scattering in cavity optomagnonics,” Phys. Rev. Lett. 127(14), 143601 (2021). [CrossRef]

49. H. Xie, Z.-G. Shi, L.-W. He, X. Chen, C.-G. Liao, and X.-M. Lin, “Proposal for a bell test in cavity optomagnonics,” Phys. Rev. A 105(2), 023701 (2022). [CrossRef]

50. H. Xie, L.-W. He, X. Shang, G.-W. Lin, and X.-M. Lin, “Nonreciprocal photon blockade in cavity optomagnonics,” Phys. Rev. A 106(5), 053707 (2022). [CrossRef]

51. L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110(23), 233602 (2013). [CrossRef]

52. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]

53. D. Vitali, S. Gigan, A. Ferreira, H. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007). [CrossRef]

54. Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013). [CrossRef]

55. J.-M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001). [CrossRef]

56. T. Palomaki, J. Teufel, R. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013). [CrossRef]

57. J. Graf, S. Sharma, H. Huebl, and S. V. Kusminskiy, “Design of an optomagnonic crystal: Towards optimal magnon-photon mode matching at the microscale,” Phys. Rev. Res. 3(1), 013277 (2021). [CrossRef]

58. V. A. S. V. Bittencourt, I. Liberal, and S. Viola Kusminskiy, “Optomagnonics in dispersive media: magnon-photon coupling enhancement at the epsilon-near-zero frequency,” Phys. Rev. Lett. 128(18), 183603 (2022). [CrossRef]

59. V. A. S. V. Bittencourt, I. Liberal, and S. Viola Kusminskiy, “Light propagation and magnon-photon coupling in optically dispersive magnetic media,” Phys. Rev. B 105(1), 014409 (2022). [CrossRef]

60. Q. Cai, J. Liao, and Q. Zhou, “Stationary entanglement between light and microwave via ferromagnetic magnons,” Ann. Phys. 532(12), 2000250 (2020). [CrossRef]

61. P. Imany, J. A. Jaramillo-Villegas, O. D. Odele, K. Han, D. E. Leaird, J. M. Lukens, P. Lougovski, M. Qi, and A. M. Weiner, “50-ghz-spaced comb of high-dimensional frequency-bin entangled photons from an on-chip silicon nitride microresonator,” Opt. Express 26(2), 1825–1840 (2018). [CrossRef]

62. M. Kues, C. Reimer, J. M. Lukens, W. J. Munro, A. M. Weiner, D. J. Moss, and R. Morandotti, “Quantum optical microcombs,” Nat. Photonics 13(3), 170–179 (2019). [CrossRef]

63. H.-H. Lu, E. M. Simmerman, P. Lougovski, A. M. Weiner, and J. M. Lukens, “Fully arbitrary control of frequency-bin qubits,” Phys. Rev. Lett. 125(12), 120503 (2020). [CrossRef]

64. M. Clementi, F. A. Sabattoli, M. Borghi, L. Gianini, N. Tagliavacche, H. El Dirani, L. Youssef, N. Bergamasco, C. Petit-Etienne, E. Pargon, J. E. Sipe, M. Liscidini, C. Sciancalepore, M. Galli, and D. Bajoni, “Programmable frequency-bin quantum states in a nano-engineered silicon device,” Nat. Commun. 14(1), 176 (2023). [CrossRef]

65. T. Seidelmann, F. Ungar, A. M. Barth, A. Vagov, V. M. Axt, M. Cygorek, and T. Kuhn, “Phonon-induced enhancement of photon entanglement in quantum dot-cavity systems,” Phys. Rev. Lett. 123(13), 137401 (2019). [CrossRef]

66. M. Müller, S. Bounouar, K. D. Jöns, M. Glässl, and P. Michler, “On-demand generation of indistinguishable polarization-entangled photon pairs,” Nat. Photonics 8(3), 224–228 (2014). [CrossRef]

67. D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8(10), 801–807 (2014). [CrossRef]

68. J. Park, H. Kim, and H. S. Moon, “Polarization-entangled photons from a warm atomic ensemble using a sagnac interferometer,” Phys. Rev. Lett. 122(14), 143601 (2019). [CrossRef]

69. S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005). [CrossRef]

Generation of robust optical entanglement in cavity optomagnonics

Abstract

1. Introduction

2. Model

3. Evolutions of the Bogoliubov modes

4. Covariance matrix and quantification of entanglement

5. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (25)

Optics Express