Coupled harmonic oscillators and their quantum entanglement

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

Figure 1

Results of calculations of quantum entanglement for the Von Neumann entropy $S$ and the Schmidt parameter $K$. The parameter $\mu \in (0,1)$ varies from the minimum to the maximum possible value, and the pairs of quantum numbers are denoted as ($m$, $n$) [for example, for $m=5$ and $n=10$, denoted as (5, 10)].

Figure 2

The results of calculating the Von Neumann entropy $S=S\left(\delta t\right)$ for $\mu =(1,3/4,1/2,1/10,1/100)$ and a) $s_1=0,s_2=10$; b) $s_1=5,s_2=10$; c) $s_1=10,s_2=10$; d) $s_1=20,s_2=10$.