Dynamically stable two-mode squeezing in cavity optomechanics

TL;DR

This work addresses generating two-mode squeezed states (TMSS) between two photon modes via a mechanically mediated three-mode cavity optomechanical system. It derives an effective two-mode squeezing Hamiltonian $H_{ m eff}$ with coupling $g_{\rm eff}$ under large-detuning and near-resonant conditions, and validates it by diagonalizing the full transition matrix in the Heisenberg picture. Using a quantum Langevin/open quantum systems framework, it shows that stable TMSS can be achieved even when $g_{\rm eff}^2>\kappa_a\kappa_b$ by optimizing the squeezing quadrature, with explicit expressions for the asymptotic variance $\Delta X(\infty)$ and squeezing level $S$. It further demonstrates robustness to systematic errors in driving strengths and detunings and resilience to thermal noise, and discusses anti-two-mode-squeezing effects and an experimentally feasible parameter regime. The work offers a scalable route to generate high-quality TMSS between Gaussian modes without reservoir engineering $($e.g.$, via phonon mediation$)$, with potential impact on quantum information processing and metrology.

Abstract

Bosonic two-mode squeezed states are paradigmatic entangled states with broad applications in quantum information processing and quantum metrology. In this work, we propose a two-mode squeezing scheme in a hybrid three-mode cavity optomechanical system, where a mechanical resonator couples to two microwave (or optical) photon modes. By applying and modulating strong driving pulses to the photon modes, we construct an effective Hamiltonian that describes two-photon squeezing mediated by the mechanical mode. This effective Hamiltonian is validated through diagonalization of the full system's transition matrix in the Heisenberg picture. With the effective Hamiltonian, we provide a rigorous theoretical solution for the dynamical process of squeezing generation within the framework of open quantum system. Our analysis reveals that stable two-mode squeezing can be obtained by optimizing the squeezing quadrature operator, even in unsteady system states. Remarkably, the squeezing level can exceed the maximum achievable under system stability conditions. Furthermore, we show that our protocol is robust against systematic errors in both driving intensity and frequency, as well as against thermal Markovian noises. Our work provides an extendable approach for generating two-mode squeezed states between indirectly coupled Gaussian modes.

Figures (8)

• Figure 1: Schematic diagram of the hybrid three-mode cavity optomechanical system. (a) A mechanical interface acts as an intermediate mode $m$, coupling with the optical cavity $a$ and $b$. (b) A mechanical resonator $m$ is capacitively coupled to two superconducting microwave resonators, $a$ and $b$. The photon modes $a$ and $b$ are driven by strong fields $\Omega_a$ and $\Omega_b$, respectively. (c) The frequencies and linewidths of the system are adopted to generate TMSS.
• Figure 2: (a) Dynamics of the CM elements using the effective Hamiltonian \ref{['Heff']} or the full system Hamiltonian \ref{['Hamline']}. (b) Dynamics of the $\Delta X(t)$ and $\Delta\tilde{X}(t)$ with the effective Hamiltonian \ref{['Heff']} or the full system Hamiltonian \ref{['Hamline']}. Here, the parameters are set as $g=G=0.1\omega_m,\Delta_b=\omega_m+10g,\kappa_a=\kappa_b=10^{-3}\omega_m,\kappa_m=10^{-6}\omega_m$, and the thermal numbers $N_a=N_b=0,N_m=10$.
• Figure 3: (a) Time evolution of the SL $S$ for operator $X$\ref{['quadratureX']} under the full system Hamiltonian \ref{['Hamline']} at varying coupling strengths. (b) Time evolution of the SL $\tilde{S}$ for numerically optimized quadrature operator $\tilde{X}$ with the full system Hamiltonian \ref{['Hamline']} at different coupling strengths. [(c), (d), and (e)] Comparison of the analytical SL $S_{\rm eff}$ with the numerically calculated results $S_{\rm lin}$ and $\tilde{S}_{\rm lin}$ under different coupling strengths $g/\omega_m$. (f) The relative derivations of SL, $\epsilon$ and $\tilde{\epsilon}$ as functions of the coupling strength $g/\omega_m$. All other parameters are the same as those in Fig. \ref{['vtsqueeze']}.
• Figure 4: [(a),(b)] The SL $S_{\rm lin}$ and $\tilde{S}_{\rm lin}$ at time $\tau$ evaluated under the system Hamiltonian \ref{['Hamexpsys']} as a function of the systematic error associated with the optomechanical coupling strengths, respectively. (c) The numerical optical SL $\tilde{S}_{\rm lin}(\tau)$ under the space spanned by the detuning $\Delta_b$ and coupling error magnitude $\gamma$. [(d), (e)] The SL $S_{\rm lin}(\tau)$ and $\tilde{S}_{\rm lin}(\tau)$ under the system Hamiltonian \ref{['Hamexpsys']} as a function of the systematic error associated with the detuning frequencies, respectively. (f) The SL $\tilde{S}_{\rm lin}(\tau)$ under the space spanned by the detuning $\Delta_b$ and detuning error magnitude $\eta$. For panels (a),(b),(d), and (e), the detuning is fixed at $\Delta_b=\omega_m+10g$. For panels (c) and (f), the coupling strengths are set as $g\equiv G=0.1\omega_m$. All other parameters are identical to those in Fig. \ref{['vtsqueeze']}.
• Figure 5: [(a), (b)] Time evolution of the variances $\Delta X(t)$ and $\Delta Y(t)$ using the system Hamiltonian \ref{['Hamline']} under various coupling strengths. [(c), (d)] Squeezing level $S_{\rm lin}$ and anti-squeezing level $S'_{\rm lin}$ computed from the full system Hamiltonian \ref{['Hamline']} at selected moments under various coupling strengths. All other parameters are identical to those in Fig. \ref{['vtsqueeze']}.