We consider a vector c-number Langevin equation describing a single mode of the radiation field under the influence of a quadratic nonlinear interaction. The nonlinearity of the interaction is found to give rise to a nonzero diffusion matrix, which may be non-positive-definite. Using this diffusion matrix, we derive and solve an equation for the variance matrix of the system. An analysis of the solution then yields information about the production and properties of squeezing. In particular, we find that pulsed pumping of the medium, as compared with continuous pumping, can significantly enhance the maximum level of squeezing. Finally, we delineate the role of the non-positive-definite diffusion matrix through a discussion of the physical source of squeezing.
© 1989 Optical Society of America
N. R. Davis and C. D. Cantrell, "Squeezed states in the context of Langevin equations and non-positive-definite diffusion coefficients," J. Opt. Soc. Am. B 6, 74-81 (1989)