A two-fold generalization of results concerning the third-order nonlinear dissipative oscillator is presented. The exact solution to the master equation for an arbitrary initial state is given, and closed formulae for the squeeze variances in the case of initial Gaussian pure and mixed states are derived.
The statistical properties of radiation passing through a nonlinear medium modelled as a third-order nonlinear dissipative oscillator interacting with squeezed light have been investigated. Whereas the photon statistics, being insensitive to the nonlinearity, are determined exactly, an approximative approach has been adopted in the description of squeezing in order to involve fluctuations. As a consequence of the general treatment, new results for a damped linear oscillator with initial squeezed light are provided.
Recently the statistical properties of the third-order nonlinear dissipative oscillator have been determined exactly to an approximation respecting non-zero reservoir temperatures for a most general input state.1 Later the assumption of zero-'temperature heat bath has been removed, but the initial coherent state has been considered. In this coinmunication we present the statistical characteristics of the third-order non linear dissipative oscillator responsive to the requirements of quantum optics and established without any restrictions on the initial state.
We derive a closed-form analytical expression for the expectation values of a damped Kerr nonlinear oscillator initialized in a coherent state. Starting from the exact Liouville-space solution of the Lindblad master equation, we specialize to coherent-state initial conditions, obtaining explicit time-dependent expressions for arbitrary normally ordered expectation values. The resulting expressions depend only on the system parameters and the initial coherent-state amplitude, requiring neither numerical integration nor evolution in a truncated Fock space. We verify the analytical expressions against master-equation simulations over a range of nonlinear interaction strengths. The closed-form solution provides an efficient alternative to numerical simulation while remaining free of Fock-space truncation error. Finally, we note an extension of the underlying Lie-algebraic structure that may provide a useful starting point for obtaining exact solutions to a broader class of open quantum optical systems.
The analytic solution of the time evolution of a dissipative nonlinear quantum oscillator in the generalized s-ordered representation is derived. The analytical solution allows for comparsion of the various representations with the classical and semiclassical dynamics. It is shown by the eigenvalue analysis of the evolution operators that in the case of a small nonlinearity a semiclassical analysis is an excellent approximation to the exact quantum dynamics. This is further demonstrated by using the Green's function of the anharmonic oscillator to solve the initial-value problem for a superposition of coherent states.