Cauchy-Schwarz and Bell Inequality Violations in Coupled Optomechanical Systems

TL;DR

The paper investigates how two weakly driven, coupled optomechanical cavities can exhibit nonclassical correlations strong enough to violate the classical Cauchy–Schwarz inequality and Bell–CHSH nonlocality. Using a Kerr-like nonlinear effective model with inter-cavity hopping, it combines numerical master-equation simulations and analytical truncated-Fock-space methods to demonstrate robust photon antibunching and photon blockade arising from multi-path interference. The study defines a CSI witness and computes a Bell-CHSH parameter, finding regions where nonclassical correlations and nonlocality coexist with sub-Poissonian statistics, closely linked to interference between direct and tunnel pathways. While non-deterministic, the system supports heralded entanglement prospects and on-demand photon–phonon pair generation, offering a feasible route toward integrated quantum information processing in hybrid optomechanical platforms.

Abstract

Destructive interference-based photon-phonon antibunching can lead to violations of classical inequalities in optomechanical cavity systems. In this paper, we explore the violation of the classical Cauchy-Schwarz inequality by examining second-order auto-correlation and cross-correlation functions, as well as Bell's nonlocality, to analyze the quantum correlations of single photon-phonon excitations when the system is driven by two weak probe fields. We propose that the violation of the Cauchy-Schwarz inequality can serve as an indicator for the stronger nonclassical tests associated with Bell's theorem. Our system reveals strong quantum correlations of photon-phonon pairs with distinctive antidiagonal patterns of photon filtering. For numerical analysis, we consider a weak effective optomechanical coupling strength and various optical-to-mechanical field amplitude ratios that enable unconventional photon (phonon) blockades at resonance. The findings are significant for producing sub-Poissonian signals under optimal conditions and have potential applications in hybrid systems for generating on-demand single photon-phonon pairs.

Figures (6)

• Figure 1: (a) Schematic of two coupled optomechanical systems consisting of a thin membrane in the middle. The cavity is excited by a control pump with amplitude $E_{j}$ with phases $\theta_{j}$. The corresponding damping rates of the cavity and membrane are $\kappa_j$ and $\gamma_j$. $J$ denotes the optical coupling between the two systems. (b) Schematic representation of transition pathways of multiple Fock states yielding single photon blockade due to destructive quantum interference.
• Figure 2: (a) The equal-time second-order correlation function $g^{(2)}_{a_1}(0)$ (Log scale) as a function of normalized detuning $\Delta/\kappa$ for three different parameter regimes: $J = 1.5\kappa, U = 0.09\kappa$ (blue), $J = \kappa, U = 0.5\kappa$ (green), and $J = 0.75\kappa, U = \kappa$ (red). The black dashed line marks the classical threshold $g_{a_1}^{(2)} = 1$. Panels (b)--(d) display the corresponding photon blockade with the phase difference of the coherent fields $\theta$ in polar coordinates for each case.
• Figure 3: The steady-state photon numbers as functions of normalized detuning $\Delta/\kappa$, for parameters (a) $J = 1.5\kappa, U = 0.09\kappa$ and (b) $J = 0.75\kappa, U = \kappa$, respectively. Both cases exhibit two distinct resonance peaks due to hybridization via the inter-cavity coupling $J$. (c) Time evolution of the second-order correlation functions, revealing the violation of the second classical inequality.
• Figure 4: Analytical results of the second-order correlation functions (logarithmic scale) $g^{(2)}_{a_1}(0)$ (red), $g^{(2)}_{a_2}(0)$ (blue), and the non-classicality witness CS violation $\mathcal{C}$ (yellow) as a function of normalized detuning $\Delta/\kappa$, for three different regimes of Kerr nonlinearity $U$ and inter-cavity coupling $J$: (a) $U = 0.09\kappa$, $J = 1.5\kappa$; (c) $U = 0.5\kappa$, $J = \kappa$; and (e) $U = \kappa$, $J = 0.75\kappa$. Corresponding contour plots in (b), (d), and (f) show the dependence of $\log_{10}g^{(2)}_{a_1}(0)$ on the detuning $\Delta/\kappa$ and Kerr nonlinearity $U/\kappa$, illustrating regions of strong photon antibunching while keeping other parameters the same.
• Figure 5: Contour plots of the non-classicality witness $\mathcal{C}$ (left of each block) and Bell-CHSH parameter $\mathcal{B}$ (right of each block) as functions of normalized detuning $\Delta/\kappa$ and Kerr nonlinearity $U/\kappa$, for different regimes of inter-cavity coupling (a) $J = 0.5\kappa$, (b) $J = \kappa$, (c) $J = 1.5\kappa$ and (d) $J = 2\kappa$. The cyan contour lines in the plots represent the classicality threshold $\mathcal{C} = 1$ and $\mathcal{B} = 2$.