Nonreciprocal Microwave-Optical Entanglement in Kerr-Modified Cavity Optomagnomechanics

Magnons Rameshti et al. (2022); Yuan et al. (2022); Prabhakar and Stancil (2009); Van Kranendonk and Van Vleck (1958), the collective excitations of spins in magnetically ordered materials, are attracting increasing interest in quantum information science. Among these materials, yttrium-iron-garnet (YIG, $\mathrm{Y}_{3}\mathrm{Fe}5\mathrm{O}{12}$)Schmidt et al. (2020); Mallmann et al. (2013); Geller and Gilleo (1957) is particularly notable for its high spin density and low lossLi et al. (2020). In YIG, magnons can strongly couple with microwave photons in cavities Huebl et al. (2013); Tabuchi et al. (2014); Li et al. (2020); Zhang et al. (2014), enabling the exploration of phenomena such as dark modes Zhang et al. (2015), non-Hermitian exceptional points Zhang et al. (2017, 2019a); Zhao et al. (2020); Sadovnikov et al. (2022); Liu et al. (2019); Cao and Yan (2019), one-way quantum steering Yang et al. (2021), dissipative coupling Wang et al. (2019); Harder et al. (2021), spin-photon interaction Hei et al. (2021), and near-perfect absorption Rao et al. (2021). Recently, cavity magnonics has entered the true quantum regime with the realization of single-magnon manipulation through coupling YIG and a superconducting qubit to a three-dimensional microwave cavity Xu et al. (2023, 2024).

Magnons can also couple with phonons via the magnetostrictive effect, forming the basis of cavity magnomechanics Zhang et al. (2016). This coupling mirrors the radiation pressure mechanism in optomechanics Aspelmeyer et al. (2014), allowing the direct transfer of optomechanical phenomena Aspelmeyer et al. (2014); Vitali et al. (2007); Xiong et al. (2022a, 2021, 2016); Kani et al. (2022) into the magnomechanical domain. Magnomechanics further enables the study of tripartite entanglement Li et al. (2018), squeezing Li et al. (2019), and non-Hermitian physics Lu et al. (2021); He et al. (2024). Extending this framework, the mechanical motion in magnomechanics can couple to an optical cavity via radiation pressure, giving rise to cavity optomagnomechanics (COMM) Fan et al. (2023, 2022). This hybrid platform provides a versatile foundation for exploring diverse quantum phenomena Fan et al. (2023, 2022); Zuo et al. (2024).

In addition, the Kerr effect of magnons (i.e., Kerr magnons), stemming from the magnetocrystalline anisotropy in YIG Zhang et al. (2019b), was experimentally observed Wang et al. (2016, 2018), paving the way for investigating Kerr-modified quantum phenomena Zheng et al. (2023); Shen et al. (2022). This nonlinearity facilitates the study of multistability Shen et al. (2022); Bi et al. (2021), spin-spin interactions Xiong et al. (2022b, 2023); Peng et al. (2023), and quantum phase transitions Zhang et al. (2021); Liu et al. (2023). These findings emphasize the potential of magnons in hybrid quantum systems, where multiple quantum subsystems can be coupled for advanced quantum control and information processing. Additionally, Kerr magnons may be used to explore continuous variable entanglement Chen et al. (2024, 2023), although such entanglement has been largely confined to the microwave frequency domain. For broader quantum information applications, optical photons (or flying qubits) are preferred. Thus, achieving microwave-optical entanglement becomes critical. The directionality dependence of the magnon Kerr effect in experiments Wang et al. (2016, 2018) suggests that different magnetic field orientations can lead to opposite Kerr effects, which may introduce nonreciprocal behaviors in magnon-related hybrid systems Chen et al. (2024, 2023); Fan et al. (2024). Previous studies have demonstrated that nonreciprocity can enhance and protect continuous variable entanglement Jiao et al. (2020). This raises the possibility that nonreciprocal magnon Kerr effects could enhance microwave-optical entanglement, although this has not yet been explored.

In this work, we propose a scheme for utilizing the magnon Kerr effect to generate and control nonreciprocal microwave-optical entanglement in COMM. The proposed system consists of a magnonic nanoresonator that is simultaneously coupled to both an optical and a microwave cavity via the magnetostrictive effect and radiation pressure, respectively. The magnonic resonator is embedded within the microwave cavity, facilitating photon-magnon interactions. The optical cavity is constructed using a fixed mirror and a highly reflective, small, and lightweight mirror attached to the magnonic nanoresonator. Under strong driving fields, the magnon Kerr effect induces a frequency shift in the magnons, which can be either positive or negative, similar to the Sagnac effect Maayani et al. (2018); Malykin (2000). This effect also gives rise to a pair-magnon interaction that regulates the values of all entanglement. Both the frequency shift and the coefficients of the pair-magnon interaction are sensitive to the direction of the external magnetic field, leading to nonreciprocal behavior in the entanglement. By tuning system parameters, such as magnon frequency detuning, we find that nonreciprocal enhancement of magnon-phonon, microwave-optical, and optical photon-magnon entanglements is possible when the magnetic field is changed. Furthermore, we demonstrate that ideal nonreciprocity of microwave-optical and optical photon-magnon entanglements is achievable through the defined bidirectional contrast ratio. This type of entanglement, transitioning from the microwave to the optical band, has broad applications in quantum information processing, particularly in quantum networks Schoelkopf and Girvin (2008); Pirandola and Braunstein (2016); Krastanov et al. (2021); Agustí et al. (2022) and hybrid quantum systems Xiang et al. (2013); Clerk et al. (2020). Our work offers a promising avenue to employ the magnon Kerr effect for designing nonreciprocal quantum devices that operate across both microwave and optical bands in COMM.

This paper is organized as follows. In Sec. II, the model and the system Hamiltonian are described. Then the dynamics of the system and the quantity of the bipartite entanglement are given in Sec. III. In Sec. IV, the generation of nonreciprocal microwave-optical entanglement is studied. Finally, a conclusion is given in Sec. V.

We consider a Kerr-modified COMM system comprising a microwave LC resonator (resonance frequency $\omega_{a}$), a magnonic nanoresonator, and an optical cavity (resonance frequency $\omega_{c}$), as illustrated in Fig. 1(a). The magnonic nanoresonator is modeled as a micro-scale bridge supporting both a magnon mode $m$ (resonance frequency $\omega_{m}$) with Kerr nonlinearity $K_{0}$ and a mechanical mode (resonance frequency $\omega_{b}$). Through the magnetostrictive effect Zhang et al. (2016), the magnon mode dispersively couples with the mechanical mode, forming a magnomechanical interaction. Additionally, the magnon mode is coupled to the microwave LC resonator, while the mechanical mode interacts dispersively with the optical cavity through radiation pressure Aspelmeyer et al. (2014). The coupling mechanisms are schematically represented in Fig. 1(b). Under the application of two driving fields, one to the optical cavity and the other to the magnon mode, the total Hamiltonian of the hybrid system can be expressed as (assuming $\hbar=1$):

$\displaystyle H=H_{0}+H_{\rm om}+H_{\rm mm}+H_{\rm km}+H_{\rm drive},$

(1)

where

	$\displaystyle H_{0}=$	$\displaystyle\omega_{a}a^{\dagger}a+\omega_{c}c^{\dagger}c+\omega_{m}m^{\dagger}m+\frac{\omega_{b}}{2}\left(q^{2}+p^{2}\right),$
	$\displaystyle H_{\rm om}=$	$\displaystyle-g_{c}c^{\dagger}cq,~{}~{}H_{\rm mm}=g_{m}m^{\dagger}mq,$
	$\displaystyle H_{\rm km}=$	$\displaystyle g_{\rm am}(a^{\dagger}m+m^{\dagger}a)+K_{0}(m^{\dagger}m)^{2},$
	$\displaystyle H_{\rm drive}=$	$\displaystyle i\mathcal{E}_{m}m^{\dagger}e^{-iw_{m}t}+i\mathcal{E}_{c}c^{\dagger}e^{-iw_{c}t}+{\rm H.c.}.$

The first term, $H_{0}$, represents the free energy of the system. The second term, $H_{\rm om}$, denotes the optomechanical interaction between the mechanical mode and the optical cavity, characterized by the single-photon coupling strength $g_{c}$. The third term, $H_{\rm mm}$, describes the magnomechanical interaction between the mechanical mode and the magnon mode, with the single-magnon coupling strength $g_{m}$. The fourth term, $H_{\rm km}$, accounts for the coupling between the microwave LC resonator and the Kerr magnon mode, with a coupling strength $g_{\rm am}$. Experimentally, $g_{\rm am}$ can reach the strong coupling regime, while the Kerr nonlinearity $K_{0}$ is tunable from negative to positive by varying the direction of the magnetic field Wang et al. (2018). The final term, $H_{\rm drive}$, represents the Hamiltonian of the two driving fields, where $\mathcal{E}_{c}$ and $\mathcal{E}_{m}$ are the amplitudes and $w_{c}$ and $w_{m}$ are the frequencies. In Eq. (1), $q$ and $p$ denote the two quadratures of the mechanical resonator.

When dissipation is included, the dynamics of the system can be governed by the quantum Langevin equations. In the rotating frame with respect to two driving fields, we have

	$\displaystyle\dot{q}=$	$\displaystyle\omega_{b}p,$
	$\displaystyle\dot{p}=$	$\displaystyle-\omega_{b}q-\gamma_{b}p+g_{c}c^{\dagger}c-g_{m}m^{\dagger}m+\xi,$
	$\displaystyle\dot{a}=$	$\displaystyle-\left(\kappa_{a}+i\Delta_{a}\right)a-ig_{\rm am}m+\sqrt{2\kappa_{a}}a_{\rm in},$
	$\displaystyle\dot{c}=$	$\displaystyle-(\kappa_{c}+i\Delta_{c})c+ig_{c}cq+\mathcal{E}_{c}+\sqrt{2\kappa_{c}}c_{\rm in},$		(2)
	$\displaystyle\dot{m}=$	$\displaystyle-(\kappa_{m}+i\Delta_{m})m-ig_{m}mq+\mathcal{E}_{m}+\sqrt{2\kappa_{m}}m_{\rm in}$
		$\displaystyle-ig_{\rm am}a-2iK_{0}m^{\dagger}mm,$

Here, $\Delta_{a(m)}=\omega_{a(m)}-w_{m}$ represents the detuning of the LC resonator (magnon mode) from the driving field with frequency $w_{m}$, while $\Delta_{c}=\omega_{c}-w_{c}$ is the detuning of the optical cavity from the driving field with frequency $w_{c}$. The decay rates of the LC resonator, optical cavity, magnon mode, and mechanical resonator are denoted by $\kappa_{a}$, $\kappa_{c}$, $\kappa_{m}$, and $\gamma_{b}$, respectively. $\sigma_{\text{in}}$ ($\sigma=a,c,m$) are the input noise operators satisfying

	$\displaystyle\langle\sigma_{\text{in}}(t)\rangle=$	$\displaystyle 0,$
	$\displaystyle\langle\sigma_{\text{in}}^{\dagger}(t)\sigma_{\text{in}}(t^{\prime})\rangle=$	$\displaystyle n_{\sigma}(\omega_{\sigma})\delta(t-t^{\prime}),$		(3)
	$\displaystyle\langle\sigma_{\text{in}}(t)\sigma_{\text{in}}^{\dagger}(t^{\prime})\rangle=$	$\displaystyle\left[n_{\sigma}(\omega_{\sigma})+1\right]\delta(t-t^{\prime}),$

where $n_{\sigma}(\omega_{\sigma})=\left[\exp\left(\hbar\omega_{\sigma}/k_{B}T\right)-1\right]^{-1}$ is the mean thermal excitation number at frequency $\omega_{\sigma}$, $k_{B}$ is the Boltzmann constant, and $T$ is the bath temperature.

The mechanical resonator is also subject to a Brownian stochastic force $\xi(t)$, which is inherently non-Markovian. However, for a high mechanical quality factor $Q_{b}=\omega_{b}/\gamma_{b}\gg 1$, the force can be approximated as Markovian Benguria and Kac (1981); Giovannetti and Vitali (2001). In this regime, $\xi(t)$ obeys

$\displaystyle{\langle\xi(t)\xi(t^{\prime})+\xi(t^{\prime})\xi(t)\rangle}\simeq 2\gamma_{b}\left[2n_{b}(\omega_{b})+1\right]\delta(t-t^{\prime}),$

(4)

where $n_{b}(\omega_{b})$ is the mean thermal excitation number of the mechanical resonator, given by $n_{b}(\omega_{b})=\left[\exp\left(\hbar\omega_{b}/k_{B}T\right)-1\right]^{-1}$.

For long-time evolution, the system reaches its steady state, which implies that all time derivatives in Eq. (III) vanish. By assuming that $\langle\cdot\rangle$ denotes the steady-state value of the operator, we then have $\langle a\rangle=-{g_{\rm am}\langle m\rangle}/{\Delta_{a}}$, $\langle c\rangle=-{i\mathcal{E}_{c}}/{\tilde{\Delta}_{c}}$, $\langle p\rangle=0$, $\langle q\rangle={(g_{c}|\langle c\rangle|^{2}-g_{m}|\langle m\rangle|^{2})}/{\omega_{b}}$, and

$\displaystyle\langle m\rangle=$

$\displaystyle\frac{i\mathcal{E}_{m}\Delta_{a}}{g_{\rm am}^{2}-\left(\tilde{\Delta}_{m}+\Delta_{k}\right)\Delta_{a}},$

(5)

where $\tilde{\Delta}_{m}=\Delta_{m}+g_{m}\langle q\rangle$ and $\tilde{\Delta}_{c}=\Delta_{c}-g_{c}\langle q\rangle$ are effective detunings induced by the displacement of the mechanical resonator, $\Delta_{k}=2K_{0}|\langle m\rangle|^{2}$ represents the frequency shift induced by the magnon Kerr effect. In particular, $\Delta_{k}$ can be positive or negative, depending on the sign of $K_{0}$, which can be easily tuned by changing the magnetic field. To obtain above results, we have assumed that the decay rate is much smaller than its corresponding effective frequency detuning.

With steady-state values in hand, we expand each operator around its steady-state value, i.e.,

$\displaystyle\sigma=\langle\sigma\rangle+\delta\sigma,~{}~{}q=\langle q\rangle+\delta q,~{}~{}p=\langle p\rangle+\delta p,$

(6)

where the symbol $\delta$ denotes the meaning of fluctuation. Substutiting these expressions back into Eq. (III) and neglecting higher-order fluctuation terms ensured by the strong driving fields (e.g., $\left|\langle m\rangle\right|,\left|\langle c\rangle\right|\gg 1$) Vitali et al. (2007), the linearized dynamics for the fluctuations can be reduced to

	$\displaystyle\delta\dot{q}=$	$\displaystyle\omega_{b}\delta p,$
	$\displaystyle\delta\dot{p}=$	$\displaystyle-\omega_{b}\delta q-\gamma_{b}\delta p-iG_{c}\left(\delta c^{\dagger}-\delta c\right)/\sqrt{2}$
		$\displaystyle-iG_{m}\left(\delta m^{\dagger}-\delta m\right)/\sqrt{2}+\xi,$
	$\displaystyle\delta\dot{a}=$	$\displaystyle-\left(\kappa_{a}+i\Delta_{a}\right)\delta a-ig_{\rm am}\delta m+\sqrt{2\kappa_{a}}a_{\rm in},$		(7)
	$\displaystyle\delta\dot{m}=$	$\displaystyle-\left[\kappa_{m}+i\left(\tilde{\Delta}m+2\Delta_{k}\right)\right]\delta m-ig_{\rm am}\delta a$
		$\displaystyle+G_{m}\delta q/\sqrt{2}+i\Delta_{k}\delta m^{\dagger}+\sqrt{2\kappa_{m}}m_{\rm in},$
	$\displaystyle\delta\dot{c}=$	$\displaystyle-\left(\kappa_{c}+i\tilde{\Delta}c\right)\delta c+G_{c}\delta q/\sqrt{2}+\sqrt{2\kappa_{c}}c_{\rm in},$

where $G_{c}=i\sqrt{2}g_{c}\langle c\rangle$ is the enhanced optomechanical coupling by multi-photon effect, and $G_{m}=-i\sqrt{2}g_{m}\langle m\rangle$ is the enhanced magnomechanical coupling by multi-magnon effect.

By further introducing quadratures for the LC resonator, optical cavity and the magnon mode, i.e., $\delta X_{\sigma}=\left(\sigma+\sigma^{\dagger}\right)/\sqrt{2}$ and $\delta Y_{\sigma}=\left(\sigma-\sigma^{\dagger}\right)/i\sqrt{2}$, the dynamics in Eq. (III) can be equivalently written as the matrix form,

$\displaystyle\dot{u}(t)=Au(t)+n(t),$

(8)

where $u(t)=[\delta X_{a}(t),\delta Y_{a}(t),\delta X_{m}(t),\delta Y_{m}(t),\delta q(t),\delta p(t)$, $\delta X_{c}(t),\delta Y_{c}(t)]^{T}$ is the vector operator of the system, $n(t)=[\sqrt{2\kappa_{a}}X_{a}^{\rm in}(t),\sqrt{2\kappa_{a}}Y_{a}^{\rm in}(t),\sqrt{2\kappa_{m}}X_{m}^{\rm in}(t)$, $\sqrt{2\kappa_{m}}Y_{m}^{\rm in}(t),0,\xi(t),\sqrt{2\kappa_{c}}X_{c}^{\rm in}(t),\sqrt{2\kappa_{c}}Y_{c}^{\rm in}(t)]^{T}$ is the vector operator of the input noise, and

$\displaystyle A=\left(\begin{array}[]{cccccccc}-\kappa_{a}&\Delta_{a}&0&g_{\rm am}&0&0&0&0\\ -\Delta_{a}&-\kappa_{a}&-g_{\rm am}&0&0&0&0&0\\ 0&g_{\rm am}&-\kappa_{m}&U&G_{m}&0&0&0\\ -g_{\rm am}&0&-V&-\kappa_{m}&0&0&0&0\\ 0&0&0&0&0&\omega_{b}&0&0\\ 0&0&0&-G_{m}&-\omega_{b}&-\gamma_{b}&0&-G_{c}\\ 0&0&0&0&G_{c}&0&-\kappa_{c}&\tilde{\Delta}_{c}\\ 0&0&0&0&0&0&-\tilde{\Delta}_{c}&-\kappa_{c}\end{array}\right)$

(17)

is the drift matrix, where $U=\tilde{\Delta}_{m}+3\Delta_{k}$ and $V=(\tilde{\Delta}_{m}+\Delta_{k})$. According to the Routh-Hurwitz criterion DeJesus and Kaufman (1987), when all eigenvalues of the matrix $A$ have negative real parts, the system is stable. In what follows, the stable condition is ensured numerically.

When the system is stable, it reaches a unique steady state, independently of the initial condition. Since quantum noises $\xi(t)$ and $\sigma_{\rm in}(t)$ are zero-mean quantum Gaussian noises and the dynamics is linearized [see Eq. (III)], the quantum steady state for fluctuations is a zero-mean continuous variable Gaussian state, fully characterized by an $8\times 8$ covariance matrix $\mathcal{V}_{lk}(\infty)=\frac{1}{2}\langle u_{l}(\infty)u_{k}\left(\infty\right)+u_{k}\left(\infty\right)u_{l}(\infty)\rangle(l,k=1,2,\ldots,8)$. The matrix $\mathcal{V}$ can be obtained by directly solving the Lyapunov equation Vitali et al. (2007)

$\displaystyle A\mathcal{V}+\mathcal{V}A^{T}+\mathcal{D}=0,$

(18)

where $D=\operatorname{diag}\left[\kappa_{a}\left(2n_{a}+1\right),\kappa_{a}\left(2n_{a}+1\right),\kappa_{m}\left(2n_{m}+1\right)\right.\\ \left.,\kappa_{m}\left(2n_{m}+1\right),0,\gamma_{b}\left(2n_{b}+1\right),\kappa_{a}\left(2n_{c}+1\right),\kappa_{a}\left(2n_{c}+1\right)\right]$ is defined by $\langle{n}_{l}(t){n}_{k}(t^{\prime})$ $+{n}_{k}(t^{\prime}){n}_{l}(t)\rangle=2\mathcal{D}_{lk}\delta(t-t^{\prime})$. Once the matrix $\mathcal{V}$ is obtained by solving Eq. (18), arbitrary bipartite entanglement can be quantified by the logarithmic negativity Vidal and Werner (2002); Adesso et al. (2004); Plenio (2005),

$\displaystyle E_{N}\equiv\max[0,-\ln 2{\eta^{-}}]$

(19)

with

$\displaystyle\eta^{-}=2^{-1/2}[\Sigma-(\Sigma^{2}-4\rm det\mathcal{V}_{4})^{1/2}]^{1/2},$

(20)

where $\Sigma=\rm{det}\mathcal{A}+\rm{det}\mathcal{B}-2\rm{det}\mathcal{C}$ and $\mathcal{V}_{4}=\begin{pmatrix}\mathcal{A}&\mathcal{C}\\ \mathcal{C^{T}}&\mathcal{B}\\ \end{pmatrix}$ is the $4\times 4$ block form of the correlation matrix, associated with two modes of interest Simon (2000). $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are the $2\times 2$ blocks of $\mathcal{V}_{4}$. A positive logarithmic negativity ($E_{N}>0$) denotes the presence of bipartite entanglement of the two modes of interest in the considered system.

In our setup, there are two kinds of microwave-optical entanglement, i.e., entanglement between photons in the optical cavity and photons in the microwave LC resonator ($E_{N}^{\rm ac}$) or magnons in the magnon mode ($E_{N}^{\rm mc}$).

In summary, we propose a novel scheme that leverages the magnon Kerr effect to generate and control nonreciprocal microwave-optical entanglement in cavity optomagnomechanical (COMM) systems. By utilizing the strong coupling between a magnonic resonator, microwave cavity, and optical cavity, we demonstrate how the direction of the external magnetic field can be exploited to induce nonreciprocal behavior in entanglement dynamics. We show that tuning parameters, such as magnon frequency detuning, allows for enhanced nonreciprocal entanglement across multiple domains, including magnon-phonon, microwave-optical, and photon-magnon entanglements. Moreover, our findings highlight the potential for achieving ideal nonreciprocity in microwave-optical and photon-magnon entanglements, which is crucial for quantum information processing and quantum networks. This work provides a promising path for designing nonreciprocal quantum devices capable of operating across both microwave and optical bands, contributing to the development of hybrid quantum systems and advanced quantum technologies.

This work is supported by Zhejiang Provincial Natural Science Foundation of China under Grants No. LY24A040004, No. LY23A40002, and Wenzhou Science and Technology Plan Project (No. L20240004).