Quantum Optomechanic Systems: Geometric analysis of stability and Schrödinger cat states transfer

Abstract

This thesis consists of two main topics on optomechanical systems: Suggesting the geometric methodology how to analyze stability and proposing the quantumstate transfer particularly to transfer macroscopic quantum superposed states such as Schrödinger cat states (SCSs). In the first part, we present a novel geometric approach for determining the unique structure of a Hamiltonian and establishing an instability criterion for quantum quadratic systems. Our geometric criterion provides insights into the underlying geometric perspective of instability: A quantum quadratic system is dynamically unstable if and only if its Hamiltonian is hyperbolic. By applying our geometric method, we analyze the stability of two-mode and three-mode optomechanical systems. Remarkably, our approach demonstrates that these systems can be stabilized over a wider range of system parameters compared to the conventional approach with rotating wave approximation (RWA) assumption. Furthermore, we reveal that the systems transit their phases from stable to unstable, when the system parameters cross specific critical boundaries. The results imply the presence of multistability in the optomechanical systems.

In the second part, we introduce a novel model to transfer macroscopic quantum superposed states such as SCSs between two optical cavities and a mechanical oscillator, which works in a non-linear optomechanical interaction regime. Our approach relies on phonon-induced dynamic resonance, which is at the heart of successful state transfer. Our method works under conditions, including the high amplitude limit of the mechanical oscillator mode, the weak coupling strength between the two cavity modes, and the adiabatic regime. We show that, under these conditions, SCSs can be transferred with nearly perfect fidelity in a deterministic process.