# Self-Organized PT-Symmetry of Exciton-Polariton Condensate in a Double-Well Potential
## 1 Introduction  
One of the prominent research directions in semiconductor optics is the study of exciton-polariton condensation in microcavities. Exciton-polaritons are hybrid quasiparticles of strongly coupled quantum well excitons and cavity photons [1, 2], retaining the properties of both matter and light. The excitonic part mediates effective interactions between the polaritons, giving rise to interesting nonlinear properties, whereas the small effective mass of the photonic component enables Bose-Einstein condensation even at ambient temperatures [3, 4, 5], in contrast to their ultracold atomic counterparts [6]. The short lifetime of the polariton condensate renders it an open system that requires continuous replenishing from the excitonic reservoir via external pumping. After their experimental realization [7, 8], the polariton condensates were shown to be an ideal system for studies of many effects at the interface between non-equilibrium physics and nonlinear dynamics.  
The intrinsic nonlinear dynamics of polariton systems lead to a variety of effects, such as the appearance of a Mach-Cherenkov cone in a supersonic flow [9], the formation of quantized vortices [10], and dark solitons [11]. Moreover, the polariton condensates can be engineered with high precision by the external laser fields [8, 12, 13, 14, 15, 16, 17, 18]. Finally, such systems are promising candidates for various applications in photonic devices, such as switches, gates and transistors [3], as well as for simulations of interacting spin models [19, 20].  
The \"open\" nature of a system featuring gain and loss leads to interesting implications when the dissipative dynamics become pseudo-Hermitian. This is the case in parity-time (PT) symmetric setups, where dissipation losses are exactly balanced by the pumping gain. Systems with PT-symmetry have been a flourishing and broad research field, extending from quantum mechanics [21] and field theory [22] to optics [23] and acoustics [24].  
The interplay between the inherent losses and the laser pumping adjusted to preserve the PT symmetry provides an effective framework where a polariton system can exhibit coherent, Hermitian-like dynamics for a relatively long time. Remarkable recent results include permanent Rabi oscillations [25], multistability and condensation below the threshold [26], exceptional points in polaritonic cavities below the lasing threshold [27], dynamic phenomena at a critical exceptional point [28], and coherent oscillations of a two-species polariton mixture in a double well [29]. The latter has been shown to be able to simulatethe dynamics of a pair of spin-1/2 particles (qubits) in the presence of exchange interaction. However, to date, polariton structures have not been extensively studied in the framework of PT symmetry, and more efforts are needed to reveal the rich landscape of phenomena that can emerge from this framework.  
In this work, we study the dynamics of an exciton-polariton condensate in a double well potential, in the presence of time-varying exciton populations and phase fluctuations. We consider the coupled-mode Gross-Pitaevskii equations for the polaritons supplemented by the rate equations for the laser-pumped exciton reservoirs, and analytically derive the steady state solutions for the exciton and polariton populations, as well as their coherence. We find that, when the total pumping rate is above threshold, the system automatically attains the PT symmetric state, independently of the pumping rates of the individual sites. By employing numerical simulations for different pumping rates and initial conditions, we verify our analytical findings. We also study the stability and robustness of our results in the presence of phase noise caused by the unavoidable phase fluctuations in the polaritons.