Abstract：A quantum system in complex potentials obeying parity-time ($\mathcal{P}\mathcal{T}$) symmetry could exhibit all real spectra, starting out in non-Hermitian quantum mechanics. The key physics behind a $\mathcal{P}\mathcal{T}$-symmetric system consists of the balanced gain and loss of the complex potential. We plan to include the nonequilibrium nature (i.e., the intrinsic kinds of gain and loss of a system) to a $\mathcal{P}\mathcal{T}$-symmetric many-body quantum system, with an emphasis on the combined effects of non-Hermitian due to nonequilibrium nature and $\mathcal{P}\mathcal{T}$ symmetry in determining the properties of a system. To this end, we investigate the static and dynamical properties of a dark soliton of a polariton Bose–Einstein condensate under the $\mathcal{P}\mathcal{T}$-symmetric non-resonant pumping by solving the driven-dissipative Gross–Pitaevskii equation both analytically and numerically. We derive the equation of motion for the center of mass of the dark soliton's center analytically with the help of the Hamiltonian approach. The resulting equation captures how the combination of the open-dissipative character and $\mathcal{P}\mathcal{T}$-symmetry affects the properties of the dark soliton; i.e., the soliton relaxes by blending with the background at a finite time. Further numerical solutions are in excellent agreement with the analytical results.
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2020, Vol. 37 
