General Theory of Stable Microwave-Optical Quantum Resources in Hybrid-System Dynamics

TL;DR

This work develops a general theory for stable microwave-optical quantum resources in multipartite hybrid systems by deriving an effective two-mode squeezing Hamiltonian $H_{\rm eff}=g_{\rm eff}(a^\dagger c^\dagger+ac)$ created via chain-type intermediate modes and solving the Gaussian open-system dynamics with a covariance-matrix formalism. It provides closed-form expressions for MO entanglement $E_{ac}$ and Gaussian steering measures, revealing steady-state and unsteady-state regimes determined by the relation between $g_{\rm eff}^2$ and $\kappa_a\kappa_c$, and shows that unsteady dynamics can yield enhanced resources. The theory is validated in electro-optomechanical (EOM) and cavity optomagnomechanical (COMM) platforms, with excellent agreement between the analytical expressions and full-system simulations, and it highlights monogamy constraints shaping resource distribution. By enabling precise, parameter-tuned control over MO resources, the work offers analytic tools for robust MO interfaces in converters and distributed quantum networks, and provides a versatile framework for future MO-based quantum technologies.

Abstract

We develop a general theoretical framework for characterizing stable quantum resources between microwave and optical modes in the dynamics of multipartite hybrid quantum systems with intermediary modes. The effective Hamiltonian for microwave-optical (MO) squeezing is formulated via strong interactions in the microwave-intermediary-optical hybrid system, and based on which rigorous solutions for the dynamics of MO entanglement and quantum steering are derived analytically. Remarkably, it is found that stable MO quantum resources can survive in the unsteady evolution beyond the steady one, and the unsteady evolution can exhibit the enhanced quality over the limit of quantum resources in the steady-state case. Furthermore, the stable MO entanglement as well as one-way and two-way quantum steerings are efficiently controllable by modulating the effective coupling strength. The validity of our theory is demonstrated by applying it to the typical models of electro-optomechanical and cavity optomagnomechanical hybrid systems.

Figures (18)

• Figure 1: Diagram of a multipartite hybrid system consisting of two target modes (microwave mode $a$ and optical mode $c$) and chain-type $N$-intermediate modes ($b_1, b_2,\cdots,b_N$). The adjacent couplings are represented by $g_a$, $g_c$, and $g_s$ with $s=1, 2,\dots, N-1$.
• Figure 2: The control of MO quantum resources via effective parameters. (a) The MO entanglement $E_{ac}$ (the blue line), asymmetric quantum steerings $S_{a\rightarrow c}$ (the red line) and $S_{c\rightarrow a}$ (the purple line) along with the relative coupling $g_{\rm eff}/\kappa_a$ for the decay rates $\kappa_c=2\kappa_a=1$. (b) The regional diagram of MO entanglement $E_{ac}$ for the steady-state (the teal area) and the unsteady-state (the golden-yellow area) cases. (c) The regional diagram of asymmetric quantum steering for the steady-state case (the two teal areas for $S_{a\to c}$ and $S_{c\to a}$ with the white boundary being zero value) and the unsteady-state case (the two beige areas for the asymmetric one-way steerings and the golden-yellow area for the two-way steering $S_{a\leftrightarrow c}$). The black dot-dashed line indicates the boundary between the steady-state and unsteady-state dynamics in the three panels, and the effective coupling strength is $g_{\rm{eff}}=1$ in (b) and (c).
• Figure 3: The stable MO entanglement and quantum steerings in the EOM system. (a) The steady-state (the teal region) and unsteady-state (the beige and golden-yellow regions) dynamics of MO quantum resources for three typical values of coupling strength $g_a/\omega_b$ with the star symbols indicating the characteristic time $\tau$. (b) The stationary values of MO quantum resources along with the coupling $g_a$, coinciding with the results of the full system dynamics at $t=\tau$ and $t=2\tau$. The MO entanglement is represented by $E$, the steering from mode $a$ to $c$ is denoted $S$, and the one from $c$ to $a$ is represented by $S'$. The quantities with a tilde are the results of full system dynamics. The system parameters are $g_c=0.12\omega_b$, $\Delta_a=5\omega_b$, $\kappa_c=0.5\kappa_a=10^{-3}\omega_b$, $\kappa_b=10^{-6}\omega_b$, and the thermal occupation numbers are $N_a=N_c=0$, $N_b=10$.
• Figure 4: The regional diagram of stable MO quantum resources in the COMM system. (a) The stable MO entanglement $\tilde{E}_{ac}(\tau)$ in the steady-state and unsteady-state evolutions with the white dot-dashed line being the boundary. (b) The stable MO quantum steering $\tilde{S}_{a\to c}(\tau)$ in two kinds of dynamical processes, where the regional map derived by the effective Hamiltonian method is well reproduced by the numerical results of full COMM-system dynamics. The parameters are set as $g_a=g_c=0.12\omega_b$, $g_m=0.1\omega_b$, $\Delta_a=3\omega_b$, $\kappa_m= 10^{-3}\omega_b$, $\kappa_b=10^{-6}\omega_b$, $N_a=N_c=N_m=0$, and $N_b=10$.
• Figure 5: The efficient quantum control over the MO entanglement (a) and quantum steering (b) in the COMM system, where the decay rates are $\kappa_a=0.5\kappa_c=10^{-4}\omega_b$ and other parameters have the same values as those in Fig. \ref{['comm']}.