Gaussian approximation and its corrections for driven dissipative Kerr model

We develop a systematic projection-operator technique for constructing Gaussian approximations and their perturbative corrections in bosonic nonlinear models. As a case study, we apply it to the driven dissipative Kerr oscillator. In the absence of external driving, the model can be solved exactly within a low-dimensional Fock subspace, leading to strongly non-Gaussian states. Nevertheless, we demonstrate that the evolution of first- and second-order moments is captured by our Gaussian scheme with high accuracy even in this regime, providing a natural benchmark. For the general case with external driving, our approach reduces the equations of motion to a closed system for means and covariances and allows one to compute systematic corrections beyond the Gaussian level in closed form. We also calculate the dynamics of linear and quadratic combinations of creation and annihilation operators in both weak- and strong-drive regimes.