Coupled harmonic oscillators and their quantum entanglement

Abstract

A system of two coupled quantum harmonic oscillators with the Hamiltonian $\hat{H}=\frac{1}{2}\left(\frac{1}{m_1}{\hat{p}}_1^2+\frac{1}{m_2}{\hat{p}}_2^2+A{x}_1^2+B{x}_2^2+C{x}_1{x}_2\right)$ can be found in many applications of quantum and nonlinear physics, molecular chemistry, and biophysics. The stationary wave function of such a system is known, but its use for the analysis of quantum entanglement is complicated because of the complexity of computing the Schmidt modes. Moreover, there is no exact analytical solution to the nonstationary Schrodinger equation $\hat{H}\Psi =i\hslash \frac{\partial \Psi }{\partial t}$ and Schmidt modes for such a dynamic system. In this paper we find a solution to the nonstationary Schrodinger equation; we also find in an analytical form a solution to the Schmidt mode for both stationary and dynamic problems. On the basis of the Schmidt modes, the quantum entanglement of the system under consideration is analyzed. It is shown that for certain parameters of the system, quantum entanglement can be very large.

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• Entanglement in continuous variable systems: Recent advances and current perspectives
• Dynamics of the entanglement between two oscillators in the same environment
• Bringing entanglement to the high temperature limit
• Quantum Limit for Driven Linear Non-Markovian Open-Quantum-Systems

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