Cumulant expansion for studying damped quantum solitons

The quantum statistics of damped optical solitons is studied using cumulant-expansion techniques. The effect of absorption is described in terms of ordinary Markovian relaxation theory, by coupling the optical field to a continuum of reservoir modes. After introduction of local bosonic field operators and spatial discretization pseudo-Fokker-Planck equations for multidimensional s-parameterized phase-space functions are derived. These partial differential equations are equivalent to an infinite set of ordinary differential equations for the cumulants of the phase-space functions. Introducing an appropriate truncation condition, the resulting finite set of cumulant evolution equations can be solved numerically. Solutions are presented in Gaussian approximation and the quantum noise is calculated, with special emphasis on squeezing and the recently measured spectral photon-number correlations [Spaelter et al., Phys. Rev. Lett. 81, 786 (1998)].