Linearly Coupled Quantum Harmonic Oscillators and Their Quantum Entanglement

In many applications of quantum optics, nonlinear physics, molecular chemistry and biophysics, one can encounter models in which the coupled quantum harmonic oscillator provides an explanation for many physical phenomena and effects. In general, these are harmonic oscillators coupled via coordinates and momenta, which can be represented as $\widehat{H}={\sum }_{i=1}^2\left(\frac{{\widehat{p}}_i^2}{2m_i}+\frac{m_i{\omega }_i^2}{2}{x}_i^2\right)+{\widehat{H}}_{int}$, where the interaction of two oscillators ${\widehat{H}}_{int}=i{k}_1{x}_1{\widehat{p}}_2+i{k}_2{x}_2{\widehat{p}}_1+k_3{x}_1{x}_2-k_4{\widehat{p}}_1{\widehat{p}}_2$. Despite the importance of this system, there is currently no general solution to the Schrödinger equation that takes into account arbitrary initial states of the oscillators. Here, this problem is solved in analytical form, and it is shown that the probability of finding the system in any states and quantum entanglement depends only on one coefficient $R\in (0,1)$ for the initial factorizable Fock states of the oscillator and depends on two parameters $R\in (0,1)$ and $\varphi$ for arbitrary initial states. These two parameters $R\in (0,1)$ and $\varphi$ include the entire set of variables of the system under consideration.