# Gaussian Quantum Trajectories for the Variational Simulation of Open Quantum-Optical Systems
## 4 Phase Diffusion and the \(N\Theta\)-Gaussian Method  
### Phase Space Evolution  
As another example, we look at the time-evolution of a freely-evolving Kerr cavity: we envision an initial state present in the sytem with Hamiltonian Equation (9), where at \(t=0\), the pump is turned  
Figure 1: Numerically exact trajectory (up to particle number truncation) with its corresponding \(XP\)-Gaussian approximation for a photon-counting process with parameters \(U/\gamma=0.05,\ \Delta/\gamma=1\) and \(F/\gamma=2.235\). Figure 2: Stationary ensemble expectations of photon number (**left**) and second-order correlation (**right**) as a function of \(F\) through the bistability regime for parameters \(U/\gamma=0.05,\Delta/\gamma=1\). Results were obtained by averaging over \(10^{4}\) samples after \(t=100\gamma^{-1}\) of evolution from the vacuum. The exact solution is the analytical result from [59]. off, i.e., \(F=F_{on}\Theta(-t)\). Without the pump, \(\mathcal{U}(1)\) symmetry is restored, allowing the phase to diffuse freely. The challenging nature of this problem for our Gaussian variational ansatz can be appreciated from the Wigner distributions of a single realization after some time, shown in Figure 3. Panel (a) shows that the phase space distribution has almost spread out over a full circle, implying the loss of phase coherence. The difference between the left- and right-hand panels, obtained with photon counting and heterodyne detection, respectively, shows how the phase space distribution is kept more concentrated under heterodyne measurement as compared to the photon counting. This is expected, because only the heterodyne measurement gives phase information. In this example, we clearly see the importance of the unraveling on the applicability of a Gaussian approximation. Unfortunately for the parameters of Figure 3, even for the heterodyne measurement, a Gaussian approximation is very crude. The applicability of the Gaussian method is directly related to the intra-sample variance of the phase. In general, in a quantum trajectory simulation, the variance of an observable can be written as:  
\[\text{Var}\left(\hat{\text{O}}\right)=\text{Var}_{1}\left(\hat{\text{O}}\right) +\text{Var}_{2}\left(\hat{\text{O}}\right), \tag{19}\]  
where the intra- and inter-trajectory variances are respectively defined as: [8]  
\[\text{Var}_{1}(\hat{\text{O}}) =\frac{1}{N_{\text{traj}}}\sum_{\alpha}\left[\left\langle\hat{ \text{O}}^{2}\right\rangle_{\alpha}-\left\langle\hat{\text{O}}\right\rangle_{ \alpha}^{2}\right], \tag{20}\] \[\text{Var}_{2}(\hat{\text{O}}) =\sum_{\alpha}\frac{\left\langle\hat{\text{O}}\right\rangle_{ \alpha}^{2}}{N_{\text{traj}}}-\left[\sum_{\alpha}\frac{\left\langle\hat{\text {O}}\right\rangle_{\alpha}}{N_{\text{traj}}}\right]^{2}, \tag{21}\]  
where index \(\alpha\) labels the trajectories. Only \(\text{Var}(\hat{\text{O}})\) can be measured without unraveling the dynamics, and it is the only one that is accessible within the master equation description (and hence, the TWA). The parameter regime where the width of the phase space distribution is expected to become small can be found by requiring that the phase diffusion rate be much smaller than the rate at which phase information is obtained: \(U\sqrt{\left\langle\hat{\text{a}}^{\text{T}}\hat{\text{a}}\right\rangle}\ll \gamma\left\langle\hat{\text{a}}^{\text{T}}\hat{\text{a}}\right\rangle\). For the parameters of Figure 3, we have \(U/(\sqrt{\left\langle\hat{\text{a}}^{\text{T}}\hat{\text{a}}\right\rangle}) =0.1\ll 1\), but still, the phase space distribution cannot be accurately approximated by a Gaussian. Figure 3: Wigner function snapshots of a typical single exact trajectory (obtained by numerical integration in truncated Fock space) after \(t=0.1\gamma^{-1}\) phase diffusion out of an initial coherent \(\left|\alpha\right\rangle=\left|10\right\rangle\) state (\(F/\gamma=0\), \(U/\gamma=1\), \(\Delta/\gamma=100\)), for photon-counting (**left**) and heterodyne detection (**right**).