Coupled harmonic oscillators and their quantum entanglement

TL;DR

This paper derives analytical solutions for the nonstationary Schrödinger equation and Schmidt modes of two coupled quantum harmonic oscillators, enabling detailed analysis of their quantum entanglement, which can be significantly large under certain conditions.

Contribution

It provides the first analytical solutions for both stationary and dynamic Schmidt modes of coupled oscillators, facilitating entanglement analysis in complex quantum systems.

Findings

Quantum entanglement can be very large for certain system parameters.

Analytical solutions for nonstationary Schrödinger equation and Schmidt modes are obtained.

The approach simplifies entanglement analysis in coupled quantum harmonic oscillators.

Abstract

A system of two coupled quantum harmonic oscillators with the Hamiltonian ${\hat H}=\frac{1}{2}\left(\frac{1}{m_1}{\hat p}^{2}_1 + \frac{1}{m_2}{\hat p}^{2}_2+A x^2_1+B x^2_2+ C x_1 x_2\right)$ 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 ${\hat H}\Psi=i\hbar\frac{\partial \Psi}{\partial t}$ 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.