A quantum system in complex potentials obeying parity-time () symmetry could exhibit all real spectra, starting out in non-Hermitian quantum mechanics. The key physics behind a -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 -symmetric many-body quantum system, with an emphasis on the combined effects of non-Hermitian due to nonequilibrium nature and 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 -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 -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.

中文翻译：

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非共振${ \mathcal P }{ \mathcal T }$-对称泵浦下极化子凝聚的暗孤子

服从奇偶时间 () 对称性的复势量子系统可以展示所有真实的光谱，从非厄米量子力学开始。对称系统背后的关键物理学包括复势的平衡增益和损失。我们计划将非平衡性质（即系统的内在增益和损失类型）包括到对称多体量子系统中，并强调由于非平衡性质和对称性决定的非厄米特的综合效应一个系统的属性。为此，我们通过解析和数值求解驱动耗散 Gross-Pitaevskii 方程，研究了在对称非谐振泵浦下极化子玻色-爱因斯坦凝聚体的暗孤子的静态和动态特性。我们在哈密顿方法的帮助下解析地推导出暗孤子质心的运动方程。由此产生的方程捕捉了开放耗散特性和 - 对称性的组合如何影响暗孤子的特性；即，孤子通过在有限时间内与背景混合而松弛。进一步的数值解与分析结果非常吻合。