Enhancing Optomechanical Entanglement and Mechanical Squeezing by the Synergistic Effect of Quadratic Optomechanical Coupling and Coherent Feedback

TL;DR

This work addresses generating macroscopic quantum states in a membrane-in-the-middle cavity by marrying linear and quadratic optomechanical couplings with a coherent feedback loop. The authors develop a theoretical framework, deriving linearized quantum Langevin equations and a Lyapunov equation for the covariance matrix, to quantify entanglement via $E_{ math{N}}$ and squeezing via $S_j$, while showing that feedback can dramatically reduce the effective cavity decay rate. The main findings are that positive quadratic coupling with feedback substantially enhances optomechanical entanglement (up to $E_{ math{N}} \sim 0.2$), and negative quadratic coupling with feedback enables strong mechanical squeezing beyond $10$ dB, with maxima controlled by the feedback phase and loop reflectivity. These results demonstrate an all-optical, tunable route to robust nonclassical states in massive mechanical systems, with potential extensions to diverse hybrid optomechanical platforms.

Abstract

Quantum entanglement and squeezing associated with the motions of massive mechanical oscillators play an essential role in both fundamental science and emerging quantum technologies, yet realizing such macroscopic nonclassical states remains a formidable challenge. In this paper, we investigate how to achieve strong optomechanical entanglement and mechanical squeezing in a membrane-embedded cavity optomechanical system incorporating a coherent feedback loop, where the membrane interacts with the cavity mode through both linear and quadratic optomechanical couplings. This hybrid optomechanical architecture offers a flexible tunability of intrinsic system parameters, thus allowing the membrane to be stiffened or softened through tuning the sign of quadratic optomechanical coupling and the cavity decay rate to be reduced via feedback control. Exploiting these unique features, we demonstrate that optomechanical entanglement can be substantially enhanced with positive coupling sign and suitable feedback parameters, while strong mechanical squeezing beyond the 3dB limit is simultaneously achieved over a broad parameter range with negative coupling sign, reaching squeezing degree above 10dB under optimized conditions. Our proposal, establishing an all-optical method for generating highly entangled or squeezed states in cavity optomechanical systems, opens up a new route to explore macroscopic quantum effects and to advance quantum information processing.

Figures (4)

• Figure 1: Schematic diagram of a membrane-embedded COM system coupled with a coherent feedback loop. (a) The COM system comprises a FP cavity formed by two fixed mirrors with optical decay rates $\kappa_{1}$ and $\kappa_{2}$, inside which a partially reflecting membrane with reflectivity $R_{m}$ is located near the cavity center. The FP cavity with resonance frequency $\omega_{c}$ is driven by a coherent laser field of frequency $\omega_{d}$ and amplitude $\varepsilon_{d}$. The membrane, characterized by resonance frequency $\omega_{m}$ and damping rate $\gamma_{m}$, couples to the cavity mode through both linear ($g_{1}$) and quadratic ($g_{2}$) optomechanical couplings. (b) The coherent feedback loop consists of three highly reflected mirrors (HRMs) and a controllable beam splitter (CBS) with tunable reflection coefficient $r_B$. The blue arrows indicate that the optical output field $\hat{a}^{\rm out}$ transmitted from the right-hand mirror are fed back to the cavity through the left-hand mirror. $\hat{a}_{1}^{\mathrm{in}}$ and $\hat{a}_{2}^{\mathrm{in}}$ describe the optical input noise arising from the zero-point fluctuations in the vacuum entering the cavity from the CBS and the right-hand mirror.
• Figure 2: The influence of QOC and coherent feedback on system parameters. (a) The effective membrane frequency $\Omega_{m}$ versus the dimensionless QOC strength $g_{2}/g_{1}$. When changing the sign of QOC, $\Omega_{m}$ increases for $g_{2}<0$ and decreases for $g_{2}>0$, which thus provides an efficient method for manipulating quantum effect. (b) The decay ratio $\eta$, characterizing the reduction efficiency of the cavity decay rate, is plotted as a function of the feedback parameters $r_{B}$ and $\theta$. By properly adjusting $r_{B}$ and $\theta$, $\eta$ is effectively reduced, leading to a significant suppression of cavity decay rate. The parameters used here are provided in the main text.
• Figure 3: Enhancement of optomechanical entanglement via the synergistic effect of QOC and coherent feedback. (a,b) The logarithmic negativity $E_{\mathcal{N}}$ versus the scaled optical detuning $\Delta/\omega_{m}$ for different values of $g_{2}/g_{1}$: (a) without coherent feedback ($r_{B}=0$) and (b) with coherent feedback ($r_{B}=0.2$, $\theta=3\pi/2$). (c-f) The logarithmic negativity $E_{\mathcal{N}}$ under different parameter settings: (c) as a function of the scaled optical detuning $\Delta/\omega_{m}$ and optical phase shift $\theta$ with $r_{B}=0.5$ and $g_{2}/g_{1}=3\times10^{-5}$; (d) as a function of the dimensionless QOC strength $g_{2}/g_{1}$ and reflection coefficient $r_{B}$ with $\Delta/\omega_{m}=0.25$ and $\theta=1.5\,\pi$; (e) as a function of the reflection coefficient $r_{B}$ and optical phase shift $\theta$ with $\Delta/\omega_{m}=0.25$ and $g_{2}/g_{1}=1.5\times10^{-5}$; (f) as a function of the dimensionless QOC strength $g_{2}/g_{1}$ and optical phase shift $\theta$ with $\Delta/\omega_{m}=0.25$ and $r_{B}=0.7$. The other parameters are as in the main text.
• Figure 4: Enhancement of mechanical squeezing via the synergistic effect of QOC and coherent feedback. The mechanical quadrature squeezing degree $S_{j}$ versus the scaled optical detuning $\Delta/\omega_{m}$: (a) in the absence of coherent feedback ($r_{B}=0$) with $g_{2}/g_{1}=0$ (top panel) and $g_{2}/g_{1}=-10^{-3}$ (bottom panel); (b) in the presence of coherent feedback ($r_{B}=0.8$, $\theta=0$) with $g_{2}/g_{1}=0$ (top panel) and $g_{2}/g_{1}=-10^{-3}$ (bottom panel). (c) The mechanical quadrature squeezing degree $S_{j}$ as a function of dimensionless QOC strength $g_{2}/g_{1}$ for $\Delta/\omega_{m}=0.1$ with $r_{B}=0$ (top panel) and $r_{B}=0.8$, $\theta=0$ (bottom panel). The blue solid and red dashed lines correspond to $S_{q}$ and $S_{p}$, respectively. (d-f) The Q-quadrature squeezing degree $S_{q}$ under different parameter settings: (d) as a function of the reflection coefficient $r_{B}$ and optical phase shift $\theta$ with $\Delta/\omega_{m}=0.1$ and $g_{2}/g_{1}=-10^{-3}$; (e) as a function of the scaled optical detuning $\Delta/\omega_{m}$ and optical phase shift $\theta$ with $r_{B}=0.8$ and $g_{2}/g_{1}=-10^{-3}$; (f) as a function of the scaled optical detuning $\Delta/\omega_{m}$ and dimensionless QOC strength $g_{2}/g_{1}$ with $r_{B}=0.8$ and $\theta=0$. The other parameters are as in the main text except for $\kappa_1/2\pi=2.25\,\mathrm{MHz}$, $\kappa_2/2\pi=0.75\,\mathrm{MHz}$, and $T=1\,\mathrm{mK}$.