Quantum Simulation And Detection Of Topological Invariant In Ultracold Atom System

Simulating the quantum physics is known to be a difficult computational problem,especially when dealing with large system due to the explosive Hilbert space.Due to the same reason,however,Feynman put forward the concept of the quantum comput-er,which simulates the problem with a quantum system.During the past decades,the development of experimental techniques has boosted lots quantum systems for the implementation of the quantum simulation.Among the quantum simulators,neutral atoms in optical lattice has played a central role in mimicking the condensed matter physics.On the one hand,the optical potentials can be adjusted to allow the change of geometry and dimensionality of the lattice.On the other hand,the light-induced artifi-cial gauge field and the tunable atomic interactions through Feshbach resonance have indeed provided versatile tools for the implementation of quantum simulation.Since the first experimental simulation of the quantum phase transition from a superfluid to a Mott insulator with cold atomic gases in the optical lattice,there has been increasing interest on the study of condensed matter physics with cold atoms in optical lattices.The discovery of the quantum Hall effect and topological insulators leads to intense efforts devoted to the exploration of the novel topological quantum matters.The topo-logical band theory is then applied to the optical lattices and the measurements of the topological invariant and edge states hav been developed in cold-atom systems.Lots of methods,such as the semi-classical dynamics,AB effect and the quantum quench,have been implemented experimentally.In this thesis,we put forward an alternative way to characterize the topological phases through the Zitterbewegung and wave pack-et dynamics.The thesis is organized as follows:In the Chapter one,we mainly review the background of cold aotms,topological insulators and the quantum simulation with cold atoms.In the first section,we briefly review the controlling of atoms in cold-atoms systems,which includes the prepara-tion of BEC/degenerate Fermionic gases,artificial gauge field and Feshbach resonance that manipulates the interactions between atoms.In the second section,we briefly introduce the concept of topological band theory with some specific examples—the Su-Schrieffer-Heeger model and Haldane model are reviewed for the sake of latter in-vestigations of the topological phases in the next chapters.Finally we review the Z2 invariant classifying the time-reversal invariant topological insulators.For the two-dimensional Chern insulators,the Dirac point in the Brillouin Zone behaves like a magnetic monopole.Under the application of force,the particle would deflect and acquire a AB phase when passing through the monopole.Based on this,the semi-classical dynamics and AB interferometry have been developed for the exper-imental measurements of the Chern number.However,for a one dimensional system,the above picture does not hold.Inspired by this,in the chapter two we investigate the one dimensional chain Su-Schrieffer-Heeger model using the wave packet dynamics under the application of an external constant force.In the weak force case,the final band population exhibits Stiickelberg oscillation.During the evolution,we implement an quench that turns system into topological distinct phase and the final band popu-lation show a ? phase shift.In the strong force limit,we demonstrate that for a one-dimensional Bravais lattice the topological phases of the system can be characterized by the band population.In the final chapter,we investigate the measurements of the topological invariant in two-dimensional Chern insulators with the Zitterbewegung.For the topological sys-tem,it can be demonstrated that the Chern number can be extracted through the signs of the masses of the Dirac cones in Brillouin Zone.On the other hand,we show that the direction of Zitterbewegung is related to signs of the masses and hence it can be applied to the measurement of the Chern number.With this method,we need not to measure the Berry curvature at lots of quasimomenta while only the directions of Zitterbewe-gung at,for example,two Dirac points is adequate to withdraw the Chern invariant.Following the investigation of the topological quantum matter,we have further stud-ied the Zitterbewegung of the Maxwell quasiparticle,which is emerged in a so-called Maxwell insulator.The Hamiltonian is described by a spin-1 Wely-type Hamiltoni-an and demonstrated that the Zitterbewegung effect shares a similar selection rules as the electric-dipole transition.Along with the study of the Zitterbewegung effect,the characterization of topological phase transition with Zitterbewegung in the Maxwell insulators has also been studied.