  The paper is concerned with open quantum systems whose Heisenberg dynamics
are described by quantum stochastic differential equations driven by external
boson fields. The system-field coupling operators are assumed to be quadratic
polynomials of the system observables, with the latter satisfying canonical
commutation relations. In combination with a cubic system Hamiltonian, this
leads to a class of quasilinear quantum stochastic systems which retain
algebraic closedness in the evolution of mixed moments of the observables.
Although such a system is nonlinear and its quantum state is no longer
Gaussian, the dynamics of the moments of any order are amenable to exact
analysis, including the computation of their steady-state values. In
particular, a generalized criterion is developed for quadratic stability of the
quasilinear systems. The results of the paper are applicable to the generation
of non-Gaussian quantum states with manageable moments and an optimal design of
linear quantum controllers for quasilinear quantum plants.
