The generation of squeezed states of light in a two-mode Kerr nonlinear directional coupler (NLDC) was investigated using two different methods in quantum mechanics. First, the analytical method, a Heisenberg-picture-based method where the operators are evolving in time but the state vectors are time-independent. In this method, an analytical solution to the coupled Heisenberg equations of motion for the propagating modes was proposed based on Baker–Hausdorff (BH) formula. Second, the phase space method, a Schrödinger-picture based method in which the operators are constant and the density matrix evolves in time. In this method, the quantum mechanical master equation of the density matrix was converted to a corresponding classical Fokker-Planck (FP) equation in positive-P representation. Then, the FP equation was converted to a set of stochastic differential equations using Ito rules. The strength and weaknesses of each method are discussed. A good agreement between both methods was achieved, especially at early evolution stages and lower values of linear coupling coefficient. On one side, the analytical method seems insensitive to higher values of nonlinear coupling coefficients. Nevertheless, it demonstrated better numerical stability. On the other side, the solution of the stochastic equations resulting from the phase space method is numerically expensive as it requires averaging over thousands of trajectories. Besides, numerically unstable trajectories appear with positive-P representation at higher values of nonlinearity.
Exact density matrix elements for a driven dissipative system described by a quadratic Hamiltonian
