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Acta Optica Sinica

The effect of random mode coupling on intentional inverse mode coupling

S Zhang

Published Jan 1, 1986• 0 citations
OpenAlex

Abstract

In this paper the method of density matrix is used to describe two coupled modes. In the case of inverse coupling the density matrix equation of motion is modified by power coupling coefficients caused by random imperfections. The exact analytical solution is obtained.

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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 the 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 strengths and weaknesses of each method are discussed. Good agreement between both methods was achieved, especially at early evolution stages and lower values of linear coupling coefficient. On one hand, the analytical method seems insensitive to higher values of nonlinear coupling coefficients. Nevertheless, it demonstrated better numerical stability. On the other hand, 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.
An analytic expression for the inversion and intensity of the Jaynes-Cummings model with cavity damping is derived in the rotating-wave approximation for vanishing thermal photon numbers. Using the s-parametrized quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)], the equation of motion for the density operator is transformed into c-number equations for the quasiprobability functions. By a suitable expansion into a Fourier series and into Laguerre functions, we obtain ordinary tridiagonal coupled differential equations for the expansion coefficients. By an appropriate choice of a scaling parameter and by a proper elimination procedure, it is shown that the coefficients that determine the inversion and the mean intensity are only coupled to coefficients with the same index and to coefficients with the next upper index. Because of this coupling the Laplace transform can be given analytically. Furthermore, it is shown that the eigenvalues and eigenvectors can also be calculated, thus leading to an analytic solution for the inversion of the Jaynes-Cummings model.