Linearly coupled quantum harmonic oscillators and their quantum entanglement

Quantum harmonic oscillators linearly coupled through coordinates and momenta, represented by the Hamiltonian ${\hat H}=\sum^2_{i=1}\left( \frac{ {\hat p}^{2}_i}{2 m_i } + \frac{m_i \omega^2_i}{2} x^2_i\right) +{\hat H}_{int}$, where the interaction of two oscillators ${\hat H}_{int} = i k_1 x_1 { \hat p }_2+ i k_2 x_2 {\hat p}_1 + k_3 x_1 x_2-k_4 {\hat p}_1 {\hat p}_2$, found in many applications of quantum optics, nonlinear physics, molecular chemistry and biophysics. Despite this, there is currently no general solution to the Schr\"{o}dinger equation for such a system. This is especially relevant for quantum entanglement of such a system in quantum optics applications. Here this problem is solved and it is shown that quantum entanglement depends on only one coefficient $R \in (0,1)$, which includes all the parameters of the system under consideration. It has been shown that quantum entanglement can be very large at certain values of this coefficient. The results obtained have a fairly simple analytical form, which facilitates analysis.