# Self-Organized PT-Symmetry of Exciton-Polariton Condensate in a Double-Well Potential
## 4 Phase Fluctuations  
As mentioned above, the coherence of the polariton condensate will decay due to the phase fluctuations that are always present in realistic quantum systems. We, therefore, incorporate the phase noise in our numerical calculations and investigate how it modifies the dynamics of the polaritons and their coherence. We model the phase fluctuations as the standard Wiener process for stochastic differential equations. Thus, the single-particle energies \(\epsilon_{L,R}\) in Equation (1) become Gaussian stochastic variables with the mean \(\langle\epsilon_{L,R}\rangle=0\) and variance \(\sigma^{2}=2\xi/\delta t\), where \(\xi\) is the decoherence rate and \(\delta t\) is the time step used to select a new random energy. In Figure 3 we show the results of our numerical simulations, as obtained upon the ensemble average over \(N=1000\) independent realizations of the system dynamics. Wecompute the first-order correlation functions \(g^{(1)}(t)\), which quantify the coherence for the polaritonic fields, via  
\[g^{(1)}(t)=\frac{\langle\psi(t_{0})\psi(t)\rangle}{\sqrt{\langle|\psi(t_{0})|^{2 }\rangle\langle|\psi(t)|^{2}\rangle}}, \tag{14}\]  
where \(\psi=\psi_{L}\) or \(\psi_{R}\), and \(\langle\ldots\rangle\) denotes the ensemble average. Below the pumping threshold, the polariton fields decay with rate \(\gamma_{L}+\gamma_{R}<0\), but the phase fluctuations with rate \(\xi\) causes an even faster decay of coherence, \(g^{(1)}(t)\propto e^{-(\xi-|\gamma_{L}+\gamma_{R}|/2)t}\), as seen in Figure 3a. For pumping above threshold, the phase noise causes an exponential decay in the correlation function \(g^{(1)}(t)\propto e^{-\xi t}\), while the system approaches the steady state independently of the initial value of coherence \(\mathrm{Re}\Theta(0)\), as seen in Figure 3b. Including the nonlinear interaction \(\eta\
eq 0\) further accelerates the decay of the correlation function and the system approaches the steady state even faster, as seen in Figure 3c. We finally note that, for an ergodic system, the ensemble-averaged and time-averaged correlation functions are equivalent. To verify whether our polariton system is ergodic, we also compute the field correlation function  
\[g^{(1)}(t)=\frac{\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}d\tau\psi(\tau)\psi^{*}( \tau+t)}{\sqrt{\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}d\tau|\psi(\tau)|^{2} \int_{t_{\mathrm{h}}}^{t_{\mathrm{i}}}d\tau|\psi(\tau+t)|^{2}}} \tag{15}\]  
resulting from a single, long-time trajectory with \(t_{\mathrm{f}}-t_{\mathrm{i}}=3000/J\). As seen in Figure 3 (lower panels), the computed ensemble-averaged and time-averaged correlation functions coincide to a very approximation, attesting to the ergodicity of our system. Figure 3: Dynamics of the polariton populations \(|\psi_{L,R}|^{2}\) (upper panels), the coherence \(\Theta\) (middle panels), and the left-well polariton field correlation function \(g^{(1)}(t)\) (lower panels), for the same parameters as in Figure 2 with the addition of phase fluctuations causing decoherence with rate \(\xi=0.05J\), as obtained from the ensemble averaged solution of Equations (1) and (2). (**a**) Linear case (\(\eta=0\)) with the below threshold pumping \(P_{L}=1000J\) and \(P_{R}=990J\) leading to the exponential decay in the correlation function \(g^{(1)}(t)\propto e^{-(\xi-|\gamma_{L}+\gamma_{R}|/2)t}\). (**b**) Same as in (**a**), but for stronger pumping above threshold \(P_{L}=1080\) and \(P_{R}=1020\); independently of the initial value of \(\mathrm{Re}\Theta(0)\), the phase fluctuations cause an exponential decay in \(g^{(1)}(t)\propto e^{-\xi t}\) while the system approaches the steady state. (**c**) Same as in (**b**) but in the presence of nonlinearity \(\eta=0.1J\), causing a slightly accelerated decay of \(g^{(1)}(t)\propto e^{-\xi t}\) with \(\xi^{\prime}>\zeta\), and faster approach of the system to the steady state.