Classical Simulation Of Degenerate Quantum Systems And Its Study On The Adiabatic And Nonadiabatic Dynamics

The birth of quantum mechanics has brought great changes to physics.As the physicists develop and study quantum mechanics,it is closely related to other disciplines such as quantum biology,quantum chemistry and quantum information.There are corresponding relations between the classical Hamilton systems and quantum systems in mathematical structures and physical laws.On one hand,we can make use of the quantization to introduce the laws of classical mechanical systems into the microcosmic field.On the other hand,using the mathematical connections between the classical and quantum systems,we can also map the dynamic or geometric characteristics of quantum systems to classical systems,study and simulate some quantum effects through the laws of classical mechanics.In this paper,we mainly study the classical simulation of quantum degenerate systems and its adiabatic and nonadiabatic cyclic evolution and geometric phases.We also study the nonadiabatic dynamics and energy spectrum characteristics of the non-Hermitian system driven by the time-periodic external field in the means of Floquet theory.This thesis is divided into six chapters,where the third section of the third chapter,of which the third,the forth and the fifth chapters are our main research work.The specific arrangements are as follows:In chapter 1,we introduce the research background and significance of this paper,involving the main developments of classical mechanics and quantum mechanics,the geometric phases,shortcuts to adiabaticity,time-periodic driving quantum systems,and the nonHermitian systems,etc.In chapter 2,the basic concepts and principles involved in our research work are shown,which includes the adiabatic evolution of quantum mechanical systems and the geometric phases of Abelian and non-Abelian systems,and the integrable systems,adiabatic evolution and adiabatic geometric angles of classical Hamiltonian systems.Then the quantum-classical mapping is introduced which can combine quantum systems with classical systems.It is also the main methods of our work.Floquet theory,which can be used to study the properties of nonadiabatic dynamics and energy spectrum of the time-periodic driven quantum systems.In chapter 3,we map the dynamics and geometric phases of the quantum degenerate systems into the classical resonant oscillators by the quantum-classical mapping theory.The energy degeneracy is interpreted as the resonance between the eigenfrequencies of the classical oscillators.Because the averaging principle is not applicable to the classical resonant subspaces,we generalize the averaging principle and define the non-Abelian geometrical angles in the case of classical resonant oscillators.This work shows the process of simulating quantum degenerate systems by the classical resonant oscillators,which provides a good platform to study the non-Abelian dynamics and phases of the quantum degenerate systems.In chapter 4,we study the classical mapping of shortcuts to adiabaticity(STA)in the quantum degenerate systems to accelerate the adiabatic evolution of the classical systems.By using the quantum-classical mapping theory,the transitionless quantum driving methods in quantum degenerate systems is generalized to the classical resonant oscillators.We give the Hamiltonian from transitionless classical driving,which includes the original Hamiltonian from the adiabatic evolution of classical systems and the additional Hamiltonian.The additional Hamiltonian can be understood as the additional coupling terms to offset the nonadiabatic effects(the coupling between oscillators with different eigenfrequencies)and give the classical geometrical angles,so as to achieve the same effects as the adiabatic evolution.The framework of STA can be used to simulate some interesting evolution phenomena of the quantum systems.In chapter 5,the energy spectrum and nonadiabatic dynamics of the non-Hermitian one-dimensional system driven by the time-periodic external field is studied.By the Floquet theory,it shows that there are robust zero-energy states in the real parts of the quasi-energy spectrum,and two additional pairs of conjugated imaginary parts are shown in the imaginary parts compared with the steady case.In terms of dynamics,the evolution of the probability that a quantum state is at the initial lattice is also found to be related to the tunneling amplitude.This work provides a sufficient theoretical basis for simulating the rich effects of the driven quantum systems using classical systems.In chapter 6,the conclusion and outlook are shown.