Full dynamics of two-membrane cavity optomechanics

TL;DR

This work addresses the full nonlinear dispersive interaction in a two-membrane cavity optomechanical system (the sandwich-in-the-middle setup) and shows that higher-order couplings can be crucial in parameter regimes where the standard first-order, linearized model fails. The authors derive exact quantum Langevin equations from the full dispersive Hamiltonian, compare them to the conventional linear model, and demonstrate significant differences in both classical (self-sustained oscillations and synchronization) and quantum (Gaussian entanglement under bichromatic driving) phenomena. Using stochastic Langevin simulations and covariance-matrix analysis, they identify regimes where high-order terms enhance or suppress entanglement and alter synchronization, highlighting the importance of including all orders in $\delta\omega$ for accurate predictions. The results offer a more accurate framework for predicting OMS dynamics and have implications for quantum information processing and precision measurement in structured optomechanical environments.

Abstract

In a two-membrane cavity optomechanical setup, two semi-transparent membranes placed within an optical Fabry-Pérot cavity yield a nontrivial dependence of the frequency of a mode of the optical cavity on the membranes' positions, which is due to interference. However, the system dynamics is typically described by a radiation pressure force treatment in which the frequency shift is expanded stopping at first order in the membrane displacements. In this paper, we study the full dynamics of the system obtained by considering the exact nonlinear dependence of the optomechanical interaction between two membranes' vibrational modes and the driven cavity mode. We then compare this dynamics with the standard treatment based on the Hamiltonian linear interaction, and we find the conditions under which the two dynamics may significantly depart from each other. In particular, we see that a parameter regime exists in which the customary first-order treatment provides distinct and incorrect predictions for the synchronization of two self-sustained mechanical limit-cycles, and for Gaussian entanglement of the two membranes in the case of two-tone driving.

Figures (7)

• Figure 1: Schematic diagram of the system. Two movable dielectric membranes are placed inside a Fabry-Pérot cavity of length $L$, and the driving laser field is applied from one side of the cavity. $Q_1$ and $Q_2$ are the locations where the two membranes are placed. The origin of the coordinates is set at the center of the cavity.
• Figure 2: Coupling coefficients map in the $Q_1$-$Q_2$ plane. (a) Renormalized frequency $\delta\omega$ of the cavity field induced by the membranes. (b), (c) First-order coupling coefficients, corresponding to the single-photon coupling strength $g_j$ of the resonator $j$. (d), (e) Quadratic coupling coefficients $g_{j,2}$ of resonator $j$. (f) Cross coupling coefficient $g_t$. Here we set $\delta=1$ kHz and the other parameters are the same as those in Tab. \ref{['tab']}.
• Figure 3: (a) First-order coupling strength $\vert g_1\vert$ (dashed line), second-order coupling strength $\vert g_{1,2}\vert$ (dot line), and their ratio $\vert g_{1,2}/g_1\vert$ (solid line) corresponding to oscillator $1$ as a function of $Q_1/\lambda$, with fixed $Q_2/\lambda=0.5$. (b) Contour lines of the coupling ratio on the $Q_1$-$Q_2$ plane. The blue solid lines and dashed lines represent $g_{1,2}/g_1=10^{-7.5}$ and $g_{2,2}/g_2=10^{-7.5}$, respectively. The red lines represent both the cases for $g_{t}/g_1=10^{-7.5}$ and $g_{t}/g_2=10^{-7.5}$. Parameters are the same as those of Fig. \ref{['fig:2']}.
• Figure 4: Synchronization phase diagrams in terms of the measure $\mathcal{P}$[(a) and (d)] and the criterion $\delta \mathcal{P}$[(b) and (e)] in the driving-detuning plane. (c), (f) The mode competition order parameter $R_c$ is represented in the driving-detuning plane. Here the phase diagrams (a), (b) and (c) are obtained from the full dynamics equations \ref{['eq:quantum langevin WA']}, while the phase diagrams (d), (e) and (f) are calculated from the first-order equations \ref{['eq:quantum langevin tradition WA']}. Here we set $\{Q_1,Q_2\}/\lambda=\{0.562,0.440\}$, $\bar{\omega}t_s= 2\times 10^6$, $\bar{\omega}\Delta t= 5\times 10^5$, the other parameters are the same as those in Fig. \ref{['fig:2']}.
• Figure 5: The synchronization measure $\bar{\mathcal{P}}$ as a function of the dimensionless parameter $Eg/\bar{\omega}^2$. Here five locations for the membranes are chosen as follows (1): $\{Q_1,Q_2\}/\lambda=\{0.3040,0.3330\}$, (2): $\{Q_1,Q_2\}/\lambda=\{0.3020,0.3310\}$, (3): $\{Q_1,Q_2\}/\lambda=\{0.2980,0.3270\}$, (4): $\{Q_1,Q_2\}/\lambda=\{0.2840,0.3131\}$ and (5): $\{Q_1,Q_2\}/\lambda=\{0.3800,0.4090\}$. In each case, the condition $g_1=g_2=g$ is satisfied. The positions in the $Q_1$-$Q_2$ plane are marked in the inset, which is a zoom of Fig. \ref{['fig:3']}(b). The blue identical lines are obtained from the first-order equations \ref{['eq:quantum langevin tradition WA']}, the red lines are obtained from the full dynamics equations \ref{['eq:quantum langevin WA']}. The other parameters are the same as those in Fig. \ref{['fig:2']}.