# Self-Organized PT-Symmetry of Exciton-Polariton Condensate in a Double-Well Potential
## 2 The Exciton-Polariton System  
The system under consideration is schematically illustrated in Figure 1. One or more layers of semiconductor quantum wells are placed inside the semiconductor microcavity near the antinode of the resonant cavity field mode. Spatially shaped pumping lasers replenish continuously the exciton reservoirs and simultaneously create confining potentials for the polariton condensate. Assuming a tight-binding double-well potential, the exciton-polariton system can be described by the following coupled non-linear Schrodinger equations for the polariton condensate wavefunctions \(\psi_{L}\) and \(\psi_{R}\) in the left (\(L\)) and right (\(R\)) wells [30]:  
\[i\partial_{t}\psi_{L} = \Big{[}\epsilon_{L}+\eta|\psi_{L}|^{2}\Big{]}\psi_{L}+\frac{i}{2 }[Rn_{L}-\kappa]\psi_{L}-J\;\psi_{R}, \tag{1a}\] \[i\partial_{t}\psi_{R} = \Big{[}\epsilon_{R}+\eta|\psi_{R}|^{2}\Big{]}\psi_{R}+\frac{i}{2 }[Rn_{R}-\kappa]\psi_{R}-J^{*}\psi_{L}, \tag{1b}\]  
where \(\epsilon_{L,R}\) are the single-particle energies, \(\eta\) is the nonlinear interaction strength, \(\kappa\) is the decay rate of the polaritons due to the exciton recombination and cavity photon losses (assumed the same for both wells), and \(J\) is the Josephson (tunnel) coupling between the wells. The polariton equations are supplemented by the equations for the populations \(n_{L,R}\) of the reservoir excitons,  
\[\partial_{t}n_{L} = P_{L}-\Gamma n_{L}-Rn_{L}|\psi_{L}|^{2}, \tag{2a}\] \[\partial_{t}n_{R} = P_{R}-\Gamma n_{R}-Rn_{R}|\psi_{R}|^{2}, \tag{2b}\]  
which are created by laser pumping with rates \(P_{L,R}\), decay with rate \(\Gamma\), and scatter into the polariton condensate with rate \(R\). Figure 1: Schematic top view (**left panel**) and side energy view (**right panel**) of a polariton system in a double quantum well. Spatially shaped pumping lasers populate with rates \(P_{L}\) and \(P_{R}\) the reservoir excitons \(n_{L,R}\), which decay via recombination with rates \(\Gamma\) and energy-relax and scatter into the polariton condensate with rate \(R\). The pumping lasers also create the confining potentials for the polaritons \(\psi_{L}\) and \(\psi_{R}\), which decay with rates \(\kappa\), and are continuously replenished by reservoir excitons with rates \(Rn_{L,R}\), while interacting with each other via the Josephson coupling \(J\). In Appendix A, we briefly outline the PT-symmetry conditions for a condensate in a double well potential. Momentarily neglecting the non-linearity \(\eta\), the PT-symmetry condition is satisfied when \(\epsilon_{L,R}=\epsilon\left(=0\right)\) and the gain in one well exactly compensates the losses in the other,  
\[\gamma_{L}\equiv\frac{1}{2}[Rn_{L}-\kappa]=-\frac{1}{2}[Rn_{R}-\kappa]\equiv- \gamma_{R}, \tag{3}\]  
as per Equations (1a) and (1b), which leads to \(n_{L}+n_{R}=\frac{2\kappa}{R}\). The threshold pumping at which the polariton condensate starts to form can be obtained from the condition that the sum of the gain and loss in both wells is non-negative, \(\gamma_{L}+\gamma_{R}\geq 0\). With Equation (3), this condition is equivalent to  
\[n_{L}+n_{R}\geq\frac{2\kappa}{R}. \tag{4}\]  
Note that, exactly at the threshold, this is the same condition as for the PT-symmetry. In the stationary regime for the reservoir excitons, \(\partial_{t}n_{L,R}=0\), we find from Equation (2) the steady-state values  
\[n_{L,R}=\frac{P_{L,R}}{\Gamma+R|\psi_{L,R}|^{2}}. \tag{5}\]  
Exactly at the threshold for condensate formation, the values of the polariton populations in both wells, \(|\psi_{L,R}|^{2}\), are marginally equal to zero, and we have \(n_{L,R}\simeq P_{L,R}/\Gamma\). Substituting these values into Equation (4), we find the threshold pumping condition  
\[P_{L}+P_{R}\geq\frac{2\kappa\Gamma}{R}, \tag{6}\]  
above which the condensate begins to form, while for \(P_{L}+P_{R}<2\kappa\Gamma/R\) the condensate decays to zero.