# Gaussian Quantum Trajectories for the Variational Simulation of Open Quantum-Optical Systems
## 4 Phase Diffusion and the \(N\Theta\)-Gaussian Method
### N\(\Theta\)-Gaussian States
The update rules for the quantum jumps are all exact except the one for \(\left\langle\hat{\delta}_{\theta}\hat{\delta}_{\theta}\right\rangle\) where an expansion in orders of \(\hat{n}^{-1}\) must be performed. However, in case of a pure state (or more generally, constant purity), it can be omitted and, similar to the \(XP\)-Gaussian case, computed from the other expectation values by the relation:  
\[\left\langle\hat{\delta}_{n}\hat{\delta}_{n}\right\rangle\,\left\langle\hat{ \delta}_{\theta}\hat{\delta}_{\theta}\right\rangle-\left\langle\hat{\delta}_{n }\hat{\delta}_{\theta}\right\rangle_{\text{sym}}^{2}=\frac{1}{4} \tag{25}\]  
which is the equivalent of Equation (17). For a state that is initially pure, Relation (25) again remains satisfied up to order \(\left\langle\hat{\mathsf{n}}\right\rangle^{-5}\) after a jump. Equations of motion for heterodyne measurement of density-phase Gaussian states are also given in Appendix B, for example for the density,  
\[d\left\langle\hat{\mathbf{n}}\right\rangle =\left[2\operatorname{Im}\left[F\left\langle\sqrt{\hat{\pi}}e^{-i \hat{\theta}}\right\rangle\right]-\gamma\left\langle\hat{\mathbf{n}}\right\rangle \right]dt\] \[\quad+2\operatorname{Re}\left[(\left\langle\hat{\delta}_{n} \sqrt{\hat{\pi}}e^{-i\hat{\theta}}\right\rangle-\left\langle\sqrt{\hat{\pi}}e^ {-i\hat{\theta}}\right\rangle)\sqrt{\gamma}dZ\right] \tag{26}\]  
is obtained. In Figure 4a, the expected evolution of quadrature variable \(\hat{X}\) is shown. The right panels show the intra-sample variance \(\operatorname{Var}_{1}\), inter-sample variance \(\operatorname{Var}_{2}\) and total variances for the evolution of a state that was originally coherent. We see here that the \(XP\)-Gaussian methods for both the photon-counting and the heterodyne unraveling strongly underestimate the phase diffusion, the description of which is worse for the photon-counting description. Only in regimes where the importance of losses is much higher than the importance of dephasing can \(XP\)-Gaussian states provide a suitable description. On the other hand, we see that the \(N\Theta\)-methods are able to capture the phase diffusion on the level of the ensemble, as well as TWA. This relative success of the \(N\Theta\)-methods with respect to the \(XP\)-methods is somewhat reminiscent of a similar observation for number-phase phase-space methods of monitored quantum systems [68; 69]. What distinguishes the \(N\Theta\)-Gaussian method from TWA, however, is that the \(N\Theta\)-Gaussian method is able to show the composition of the ensemble: it maintains information of individual trajectories, which is lost in TWA (we note that in practice, under appropriate conditions, a single TWA sample may still be representative of experimental realizations [56; 70]). Figure 4: Mean (**left** panel) and inner-sample, between-sample and total variances (**right** panel) of the \(X\)-quadrature obtained by the numerically-exact trajectory methods (in a truncated Fock space) and TWA, as well as \(XP\)-Gaussian and \(N\Theta\)-Gaussian variational methods as a function of time after free evolution from a coherent initial state \(\left|\alpha\right\rangle=\left|10\right\rangle\) (same parameters as Figure 3). Averages are taken over 1e3 samples (1e4 for TWA). It is clear that the \(N\Theta\) variational method provides an accurate description of the true dynamics, both on the level of the whole ensemble as on the level of its constitution in pure states, whereas the latter cannot be obtained from the TWA method. Very similar behavior is present regarding \(P\) and its variances, as well as for the \(X,P\)-covariances.