# Gaussian Quantum Trajectories for the Variational Simulation of Open Quantum-Optical Systems
## Quantum Trajectories for Expectation Values and Gaussianity
### Closing at the Gaussian Level  
Equation (5) or (8) will generally introduce an infinite hierarchy of correlation functions. This is a similar situation to classical mechanics, where the Liouville equation introduces the BBGKY hierarchy, which must be truncated at some point. At the level of evolution for the full ensemble (the master equation), a systematic derivation of proper truncation schemes (also known as cumulant expansions) for driven-dissipative quantum systems has been performed in [61]. Generally, one expects the resulting predictions to converge as more correlation functions of higher order are taken into account. Nevertheless, it is only at the mean-field or at the Gaussian level that it is possible to close the hierarchy instead of truncating, ensuring that the state remains physical by construction. When closing Equations (5), (8) at the mean-field level, the dynamics are trivial and coincide with the mean-field solution of the master equation. At the Gaussian level on the other hand, the closing of the equations can be performed by applying [PERSON]'s theorem and reflects the stochasticity of the unravelings.  
As an ansatz, we thus approximate the state to be contained in a Gaussian subspace of the Hilbert space. These Gaussian states are entirely characterized by their first and second moments, keeping the amount of independent variables, for which the evolution needs to be studied, limited. This provides a large reduction of complexity from the whole Hilbert space, which scales exponentially with system size (the number of modes). Because of the presence of the second moments, a Gaussian ansatz is by nature one order higher than the mean-field approach that corresponds to a coherent ansatz. Formally then, the Gaussian methods are of the same order as Bogoliubov theory. Nevertheless, we can expect these Gaussian trajectory methods to provide a more refined description of the whole state because the second-order expansion is done at the level of individual trajectories instead of on the level of the ensemble. In the next section, we will illustrate this with the example of bistability in a Kerr nonlinear cavity. Whereas the [PERSON] theory is only able to describe the fluctuations around one of the two stable branches, the variational Gaussian method is also capable of describing the switching between the branches.  
The fact that our method is variational can be seen from the following argument. By using the Wick theorem, we clearly restrict to the manifold of Gaussian states. The variational principle consists of evolving the state (by which it will leave the manifold) and then projecting is back. For such a state, the time evolution of the first and second order correlators over a small time interval can be computed by applying [PERSON]'s theorem to the right-hand side of the Heisenberg equation. The next step is to project back to the Gaussian manifold. This projected state will have to a good approximation the same first and second order correlation functions as the evolved state.