Lyapunov Control And Non-hermitian Property In One Dimensional Topological Chain

Quantum information is a new inter-discipline subject.Investigation for this subject leads to the development of many related fields.However its development depends on certain physical systems,such as quantum optics,nuclear magnetic resonance system,superconduc-tor circuit system,topological materials and so on.One of the key elements is preserving the quantum characters.However these characters are too fragile against noise.Topological materials are robust against disorder which provide novel platforms for solving this problem.In this thesis,we mainly focus on one dimensional topological models,preparing nontriv-ial topological edge states by Lyapunov control;investigating the influence of localization transition on adiabatic pumping for edge states and preparing the edge state by quantum Lyapunov control and the influence of non-Hermitian dissipation on the topological properties for a dimer chain.The main contents for this thesis are present in detail below.In first chapter,the author briefly introduce the background and development for quan-tum information and the related quantum computation and quantum communication.And then topological 'system is introduced as well as the systems to implement it such as cold atoms trapped in optical lattice and the coupled optical waveguide arrays.Concepts for quantum control,quantum Lyapunov control and non-Hermitian Hamiltonian are also given.In second chapter,we list the knowledge used in this thesis.In third chapter,the author study the one dimensional Harper model of fermions loaded in optical lattice.Starting from a ring structure,by checking the edge state related to the coupling strengths and phases in the system,the author specifies the model to be investigated further and the goal state in Lyapunov control.Then based on electro-optic phase modulator,the author gives the time-varying control Hamiltonian adopted in Lyapunov control.It is shown that by two Lyapunov control schemes,the edge state can be prepared with higher fidelities in a large parameters interval.Then the author verify the validity of the two control schemes by checking a variety of random initial states.It is shown that in a certain interval of amount and intensity for the impurities,the two schemes can finish the goal with high fidelities.For chains with the same periodic character but different lengths,the two control schemes can both finish the control goal.At last,the author introduce the envelope function with the amplitudes decrease with time to optimize the control fields.High fidelities in both control schemes are also obtained.In fourth chapter,the author mainly focuses on the localized transition for an extended one dimensional incommensurate Aubry-Andre-Harper(AAH)model and its influence on two dynamical processes.By exhibiting the average localization transition diagram,the author specifies the model and finds that not only the ground state but also most of the excited ones exhibit localization transition.And the change for the band splitting behavior coincides with the localization transition behavior of the eigenstates.In terms of dynamics,in the nonlocal region,the edge state can be pumped from one end of the chain to the other end by slowly(adiabatically)changing the parameter.However,in the localized region,by similar pumping rate,the pumping process breaks down.This is because in the localized region,the transportation of the excitation would be suppressed but not in the nonlocal region.And then the author prepare the edge sate which performs locally in the nonlocal region with high fidelity by quantum Lyapunov control.But the preparation effects are suppressed obviously in the localized region.This also results from the suppression of the transportation of the excitations in the localized region.Then the author checks the influence of Kerr-type nonlinearity on the localization transition in terms of dynamical process by means of statistical method.The predictions in this chapter can be implemented by coupled optical waveguide arrays or cold atoms trapped in quasiperiodic optical lattice.In fifth chapter,the author investigate the influence of loss on the topological properties and energy band for a dimer chain.This model can be implemented by coupled optical waveguide array.The loss here locates on one component of this chain.It is found that when there is loss,in terms of the phase in hopping potential,the range for nontrivial topological phase does not change when the topological property is represented by product of the two components of an eigenvector.The average displacement for the single excitation initially locates on non-decay sites can indicate the topological property for this chain.The long-lived dark states which locate on the non-loss sites can be observed on the chain with periodic boundary condition.They may result from interference.Considering the phase in the on-site modulation potential,in presence of non-Hermitian loss,the interval for nontrivial topological phase represented by Chern number will shrink.Loss leads the real energy band tend to zero energy in both cases above.In this system,degenerate zero-energy state performs robust against disorder in cell,between cell and of non-Hermitian loss.But it is fragile against disorders on sites.This is because this kind of disorder breaks the particle-hole symmetry.At last,in sixth chapter,we make the summary for this work and the outlook.