Cauchy-Schwarz and Bell Inequality Violations in Coupled Optomechanical Systems

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.