Dynamic Simulation Of Space - Time Inversion Symmetric Quantum System

The parity-time (PT) symmetric quantum systems can have real and positive eigenvalues. The purpose of this thesis is to study and understand the immanent cause and physical mechanism of it. We restrict ourselves in several stability problems of PT-symmetric nonlinear optical systems to simulate PT-symmetric quantum systems. Our results can be concluded as the exchange and balance of energy are the reasons of PT-symmetric systems with gain and loss are more stable than pure dissipative systems.To verify our conjecture, thesis designs and studies the existence and stability of soliton in three kinds of PT-symmetric dissipative gratings, the feasibility demonstration of the realization of these dissipative gratings in laboratory is also included. These dissipative gratings are coupled PT-symmetric nonlinear dissipative grating with constant coupling, coupled PT-symmetric nonlinear dissipative grating with periodically modulated coupling and PT-symmetric nonlinear planar dissipative grating with periodic modulation of gain and loss along transversal direction, respectively. Solitary solutions are obtained analytically in all three kinds of optical systems, variational approximation is adopted to analysis the stabilities of solitons in all systems and results in the details of stable parameter regions, these results are confirmed by split-step method and Runge-Kutta numerical simulation.To get deeper understand of the energy exchange in PT-symmetric systems, thesis discusses the scattering properties of PT-symmetric nonlinear impurities. We focus on the existence and shifting mechanism of local modes in PT-symmetric nonlinear impurities. Stability analysis, collective coordinate approximation and numerical simulation are adopted to study this subject. The results show that the local modes can exist and be stable over a wide parameter region in PT-symmetric nonlinear impurity, and local modes can transfer to transmissive modes or reflection modes with appropriate magnitude of gain and loss. This result is also a representation of the basic role of energy exchange in PT-symmetric systems.