Classical And Quantum Chaotic Dynamics Of The Non-Hermitian Kicked Rotor Model

In recent years,the periodically-driven system has received more and more attention and plays an important role in the branch of physics.Therefore,it is of great significance to extend the periodically-driven system to non-Hermitian quantum mechanics.So far,people have discovered many interesting physical phenomena in non-Hermitian quantum systems with PT symmetry.However,more work is needed to study periodically-driven systems with PT symmetry.We mainly study the Kicked Rotor model with PT symmetry from two aspects of classical dynamics and quantum dynamics.In classical dynamics,we derive the classical mapping equation of non-Hermitian system.And by comparing the Poincaré cross-sections of multiple trajectories under different parameters,it reveals the influence of the system’s regular motion to chaotic motion on the angular coordinates and angular momentum.After that,we select the parameters to make the system in a state of chaotic motion.At this time,the system exhibits a classic diffusion behavior.We quantify classical diffusion by defining the second moment.We find that the real part of the classical complex trajectory will diffuse normally with the increase of time,while the imaginary part of the classical complex trajectory will exponentially diffuse with the increase of time.We analyze and predict the exponential diffusion of the imaginary part of momentum and the time when the diffusion mode transitions to saturation.We proved the validity of our prediction by comparing the numerical results with the theoretical prediction results.In terms of quantum dynamics,we first explained the basic concepts of PT symmetry and PT symmetry breaking and the corresponding properties.After that,we determined the conditions for PT symmetry breaking in the system based on the basic properties of PT symmetry breaking.Based on this research,we modify the imaginary part of the kick potential energy to show the average value of the momentum of the system when the system breaks from PT symmetry to PT symmetry,the average value of the square of momentum,and the change trend of the second-order moment over time.We found that when the selected parameter is near the threshold of PT symmetry breaking,the average momentum of the system and the average of the square of the momentum will increase stepwise with time.When the selected parameter is far greater than the threshold for PT symmetry breaking,the average momentum will increase linearly with time,and the energy of the system will undergo ballistic diffusion.Because the PT symmetry of the system is broken,the average value of momentum increases linearly with time,that is,quantum diffusion occurs in the system.We count the acceleration rate of the real part of the different kick potential energy intensity.We find that the acceleration rate will increase stepwise with the increase of the real part of the kick potential energy intensity.We propose an improved classic acceleration model to explain this phenomenon.And it is basically consistent with the predicted result and the numerical calculation result.Finally,we introduce the OTOC to reflect the signs of the chaotic diffusion of the complex trajectory.In the semi-classical region,the exponential growth rate of quantum OTOC is equal to the growth rate of exponential diffusion of the imaginary part of the second moment of the complex trajectory.