EIGENVALUES FOR THE QUANTUM PARAMETRIC OSCILLATOR IN THE OVERDAMPED REGIME

By an adiabatic elimination of the fundamental mode, the equation of motion for the density operator describing the model of Drummond, McNeil, and Walls [Opt. Acta 28, 211 (1981)] of the quantum parametric oscillator can be reduced to an equation of motion for the subharmonic field for large damping constants of the second-harmonic field. Introducing the quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)], it is shown next that the equation of motion for the density operator is reduced to an equation of motion of the quasiprobability distributions. Because of symmetry relations the eigenfunctions can be divided into four classes according to whether the eigenfunctions are symmetric or antisymmetric in the real or imaginary part of the complex field variable. By using intensity and phase variables, the quasiprobability distributions are then expanded into Laguerre functions with respect to the intensity and into a Fourier series with respect to the phase. With a proper notation the equation of motion for the expansion coefficients can be cast into a tridiagonal recurrence relation, which is then solved by the matrix continued-fraction method. The results are compared with one-dimensional approximations.