Cumulant expansion for studying damped quantum solitons

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Abstract
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-parametrized 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 a Gaussian approximation and the quantum noise is calculated, with special emphasis on squeezing and the recently measured spectral photon-number correlations [Spälter et al., Phys. Rev. Lett. 81, 786 (1998)].