Dark Soliton of Polariton Condensates under Nonresonant PT -Symmetric Pumping*

Abstract

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

Interaction between an impurity and nonlinear excitations in a polariton condensate

“…The time evolution of the parameters in Eq. ( 7) can be then obtained via the Euler-Lagrangian equations for the dissipative system [43][44][45][46][47]…”

Interaction between an Impurity and Nonlinear Excitations in a Polariton Condensate

“…The key assumption underlying the ansatz ( 7 ) is that the functional forms of the soliton and the impurity-induced function are preserved in the presence of perturbation, whereas the corresponding parameters become slowly time-dependent. The time evolution of the parameters in Equation ( 7 ) can be obtained via the Euler-Lagrangian equations for the dissipative system [ 47 , 48 , 49 , 50 , 51 ] with and , and labels the real part of the expression. In Equation ( 8 ), the Lagrangian is referred to as the average Lagrangian of Equation ( 6 ) with , where the Lagrangian density is given by …”

$\mathcal{PT}$ symmetry of a square-wave modulated two-level system

“…They found that there existed multiple frequency windows where PT symmetry was broken and restored. The non-Hermitian Rabi model has drawn growing attention due to its especially rich phenomena which are absent in the static counterparts [37][38][39][40][41][42][43][44][45]47]. Lee et al found the PT symmetry breaking at the (2k−1)-photon resonance and derived the boundaries of the PT phase diagram by doing perturbation theory beyond rotating-wave approximation [37].…”