# Gaussian Quantum Trajectories for the Variational Simulation of Open Quantum-Optical Systems
## 3 Kerr Bistability and the _Xp_-Gaussian Methods
From Equation (5) and after applying [PERSON]'s theorem to close the set of equations, we obtain for photon-counting the deterministic evolution:  
\[\partial_{t}\alpha =\left(\frac{-\gamma}{2}+i\Delta\right)\alpha-Ui\left(\left| \alpha\right|^{2}\alpha+2\alpha\left\langle\hat{\mathtt{a}}^{\dagger}\,\hat{ \delta}\right\rangle+\alpha^{*}\left\langle\hat{\delta}\hat{\delta}\right\rangle \right)-iF-\gamma\left(\alpha\left\langle\hat{\mathtt{a}}^{\dagger}\,\hat{ \delta}\right\rangle+\alpha^{*}\left\langle\hat{\delta}\hat{\delta}\right\rangle\right) \tag{10}\] \[\partial_{t}\left\langle\hat{\delta}\delta\right\rangle =\left(2i\Delta-\gamma\right)\left\langle\hat{\delta}\delta \right\rangle-Ui\left(\alpha^{2}(1+2\left\langle\hat{\mathtt{a}}^{\dagger}\, \hat{\delta}\right\rangle)+\left\langle\hat{\delta}\delta\right\rangle(1+4| \alpha|^{2}+6\left\langle\hat{\mathtt{a}}^{\dagger}\,\hat{\delta}\right\rangle )\right)-2\gamma\left\langle\hat{\mathtt{a}}^{\dagger}\,\hat{\delta}\right\rangle \left\langle\hat{\delta}\delta\right\rangle\] (11) \[\partial_{t}\left\langle\hat{\delta}^{\dagger}\,\hat{\delta}\right\rangle =2U\,\mathrm{Im}\left[\alpha^{2}\left\langle\hat{\delta}\delta \right\rangle^{*}\right]-\gamma\left(\left\langle\hat{\mathtt{a}}^{\dagger} \,\hat{\delta}\right\rangle+\left\langle\hat{\mathtt{a}}^{\dagger}\,\hat{ \delta}\right\rangle^{2}+\left|\left\langle\hat{\delta}\delta\right\rangle \right|^{2}\right). \tag{12}\]  
This deterministic evolution is propagated until the norm, evolving through (4), becomes \(\widetilde{\left\langle 1\right\rangle}=\mathds{R}\). Then, according to Equation (6), jumps occur as:  
\[\alpha \xrightarrow{J} \tag{13}\] \[\left\langle\hat{\delta}\hat{\delta}\right\rangle \xrightarrow{J}\] (14) \[\left\langle\hat{\delta}^{\dagger}\,\hat{\delta}\right\rangle \xrightarrow{J} \tag{15}\]  
In the heterodyne unraveling on the other hand, the evolution of the mean-field Equation (8) is given by:  
\[d\alpha=\left[\left(\frac{-\gamma}{2}+i\Delta\right)\alpha-Ui\left(\left|\alpha \right|^{2}\alpha+2\alpha\left\langle\hat{\mathtt{a}}^{\dagger}\,\hat{\delta} \right\rangle+\alpha^{*}\left\langle\hat{\delta}\hat{\delta}\right\rangle \right)-iF\right]dt+\sqrt{\gamma}\left(\left\langle\hat{\mathtt{a}}^{\dagger} \,\hat{\delta}\right\rangle dZ+\left\langle\hat{\delta}\hat{\delta}\right\rangle dZ ^{*}\right). \tag{16}\]  
Generally, one would expect similar stochastic equations for \(\left\langle\hat{\delta}\hat{\delta}\right\rangle\) and \(\left\langle\hat{\mathtt{a}}^{\dagger}\,\hat{\delta}\right\rangle\). Remarkably, they are entirely deterministic and, moreover, are identical to the deterministic part of photon-counting Equations (14), (15). It should be noted, however, that the the equations of motion are different for more general homodyne detection schemes. For all unravelings, as a slightly more efficient and stable alternative to evolving \(\left\langle\hat{\delta}^{\dagger}\,\hat{\delta}\right\rangle\) along explicitly through Equation (15), one can obtain it from \(\left\langle\hat{\delta}\hat{\delta}\right\rangle\) by asserting that the state remains pure, corresponding to the condition [60]:  
\[\left\langle\hat{\delta}^{\dagger}\,\hat{\delta}\right\rangle+\left\langle \hat{\delta}^{\dagger}\,\hat{\delta}\right\rangle^{2}=\left|\left\langle\hat{ \delta}\hat{\delta}\right\rangle\right|^{2}. \tag{17}\]  
For an initially pure state, Relation Equation (17) remains exactly satisfied through heterodyne measurement, as well as through the deterministic evolution between photon jumps, although the purity:  
\[\text{tr}\Big{[}\hat{\rho}^{2}\Big{]}=\left(1+4\left(\left\langle\hat{\delta }^{\dagger}\,\hat{\delta}\right\rangle+\left\langle\hat{\delta}^{\dagger}\, \hat{\delta}\right\rangle^{2}-\left|\left\langle\hat{\delta}\hat{\delta} \right\rangle\right|\right)\Big{)}^{-1/2} \tag{18}\]  
briefly decreases at a jump, after which it tends to relax back to one.