A system of two coupled quantum harmonic oscillators with the Hamiltonian H[over ̂]=1/2(1/m_{1}p[over ̂]_{1}^{2}+1/m_{2}p[over ̂]_{2}^{2}+Ax_{1}^{2}+Bx_{2}^{2}+Cx_{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 H[over ̂]Ψ=iℏ∂Ψ/∂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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It is well known that a beam splitter (BS) can be used as a source of photon quantum entanglement. This is due to the fact that the statistics of photons changes at the output ports of the BS. Usually, quantum entanglement and photon statistics take into account the constancy of the reflection coefficient R or the transmission coefficient T of the BS, where $$R + T = 1$$ R + T = 1 . It has recently been shown that if BS is used in the form of coupled waveguides, the coefficients R and T will depend on the photon frequencies. In this paper, it is shown that the quantum entanglement and statistics of photons at the output ports of a BS can change significantly if a BS is used in the form of coupled waveguides, where the coefficients R and T are frequency-dependent.