# Self-Organized PT-Symmetry of Exciton-Polariton Condensate in a Double-Well Potential
## 3 Equivalence of the PT-Symmetry and Steady State Conditions  
To understand the dynamics of the system, it is convenient to express Equation (1) in terms of the polariton populations \(|\psi_{L,R}|^{2}\) and coherence \(\Theta\equiv\psi_{L}\psi_{R}^{*}\) as  
\[\partial_{t}|\psi_{L}|^{2} = 2\gamma_{L}|\psi_{L}|^{2}+2J\text{Im}\Theta, \tag{7a}\] \[\partial_{t}|\psi_{R}|^{2} = 2\gamma_{R}|\psi_{R}|^{2}-2J\text{Im}\Theta,\] (7b) \[\partial_{t}\Theta = -i\Big{[}\epsilon_{L}-\epsilon_{R}+\eta\big{(}|\psi_{L}|^{2}-| \psi_{R}|^{2}\big{)}\Big{]}\Theta\] (7c) \[+(\gamma_{L}+\gamma_{R})\Theta-ij\Big{(}|\psi_{L}|^{2}-|\psi_{R}| ^{2}\Big{)}.\]  
Note that, below threshold, \((\gamma_{L}+\gamma_{R})<0\), both the polariton populations and their coherence decay to zero, as mentioned above. Let us set, without loss of generality, \(\epsilon_{L,R}=0\) and consider first the case of vanishing nonlinearity \(\eta=0\). Equation (7c) indicates that the coherence only decays if its real part is nonzero. In turn, the solution for the real part of the coherence is  
\[\text{Re}\Theta(t)=\text{Re}\Theta(0)\ e^{\int_{0}^{t}(\gamma_{L}+\gamma_{R}) dt^{\prime}}. \tag{8}\]  
Hence, if initially \(\text{Re}\Theta(0)=0\), it will remain so at later times, \(\text{Re}\Theta(t)=0\ \forall\ t>0\). Then, the dynamics of the system, if pumped above threshold, will exhibit continuous Rabi-like oscillations with frequency \(J\), while no steady-state will be attained, as in Figure 2b. In practice, however, even if we initially have \(\text{Re}\Theta(0)=0\) (e.g., either \(\psi_{L}(0)=0\) or \(\psi_{R}(0)=0\)), the unavoidable phase fluctuations in the polaritons will eventually lead to the appearance of finite \(\text{Re}\Theta(t)\
eq 0\), which, in turn, will result in the decay of coherence and drive the system to the steady state. Equivalently, if we initially have \(\text{Re}\Theta(0)\
eq 0\), the system can still initially exhibit Rabi-like oscillations, but will eventually attain the  
Figure 2: Dynamics of the polariton populations \(|\psi_{L,R}|^{2}\) (upper panels), the reservoir excitons \(n_{L,R}\) (insets), and the coherence \(\Theta\) (lower panels), as obtained from the numerical solution of Equations (1) and (2) for the parameters \(\epsilon_{L,R}=0\), \(\kappa=10J\), \(\Gamma=2J\), \(R=0.02J\) (time is in units of \(J^{-1}\)), corresponding to the threshold values of pumping \(P_{L}+P_{R}=2\kappa\Gamma/R=2000J\) and the steady-state exciton populations \(n_{L}+n_{R}=2\kappa/R=1000\) (green dashed lines in the insets). The initial exciton populations are always taken as \(n_{L,R}=P_{L,R}/\Gamma\), while the initial polariton amplitudes \(\psi_{L,R}\) are seeded with small (complex) values. (**a**) Linear case (\(\eta=0\)) with the pumping rates \(P_{L}=1000J\) and \(P_{R}=990J\) being slightly below threshold, leading to a decay in the initial polariton populations \(|\psi_{L}(t)|^{2}\) and \(|\psi_{R}(t)|\) and steady-state exciton population \(n_{L}+n_{R}<2\kappa/R\). (**b**) Same as in (**a**), but for stronger pumping rates \(P_{L}=1080\) and \(P_{R}=1020\) above threshold and initial conditions \(\text{Re}\Theta(0)=0\), leading to the initial build-up of the polariton populations and their continuous Rabi-like oscillations, while \(\text{Re}\Theta(t)=0\ \forall\ t\), and the exciton population \(n_{L}+n_{R}\) oscillating slightly above the threshold value \(2\kappa/R=1000\). (**c**) Same as in (**b**), but for the initial conditions \(\text{Re}\Theta(0)\
eq 0\), leading to a steady-state of the system. (**d**) Same as in (**b**) with the initial conditions \(\text{Re}\Theta(0)=0\), but in the presence of nonlinearity \(\eta=0.1\)) that couples \(\text{Re}\Theta\) and \(\text{Im}\Theta\), leading to a steady-state of the system. steady state, as in Figure 2c. Finally, as seen from Equation (7c), the nonlinear interaction couples the real and imaginary parts of the coherence \(\Theta\) with the rate \(\eta(|\psi_{L}|^{2}-|\psi_{R}|^{2})\).