Abstract

Reynaud and Heidmann have shown¹ that the squeezing spectra of linearized parametric processes can be computed by the use of classical Langevin equations and classical input-output relations if the obtained averages are interpreted as expectation values of symmetrically ordered operators. This is possible because the master equation for dissipative linear or linearized quantum systems is a Fokker-Planck equation in the Wigner representation. The stochastic process determined by this Fokker-Planck equation can be expressed by the equivalent linear classical Langevin equations, the equations used by Reynaud and Heidmann. For nonlinear quantum optical systems the Wigner representation of the master equation involves derivatives higher than second order; therefore, no classical stochastic process can be constructed from this equation. However, in the limit of weak nonlinearities,^2,3 realized in quantum optics, these higher-order derivatives may be negligible. This is not a systematic approximation; however, it works well even in the neighborhood of bifurcation points. It is shown that the nonlinear Langevin equations equivalent to this Fokker–Planck equation and the classical input output relation can be used again to compute the quantum mechanically correct squeezing spectrum within this Fokker-Planck approximation. This can be seen by a general computation of the autocorrelation function of the quadrature components of the outgoing field in terms of averages over quasiprobabilities.

© 1991 Optical Society of America