Quantum Simulation With Ultracold Atoms

When we use a classical computer to simulate a quantum many-particle system,the exponential growth of the Hilbert-space dimension with the particle number will lead to the exponential slowdown of the simulation speed.In 1982,the famous physicist Richard Feynman proposed to employ quantum simulators to overcome this problem.On the other hand,the development of the techniques to cool and trap atoms with lasers by Steven Chu,Claude Cohen-Tannoudji and William D.Phillips et al.in 1985 led to the realization of Bose-Einstein condensation in dilute atomic gases in 1995,which makes the study of exotic quantum phenomena with ultracold atoms possible.Owing to its high purity and excellent controllability,ultracold-atom system has become the ideal platform for quantum simulations.Currently,the relevant physics of ultracold atoms——ultracold-atom physics has already developed into an independent discipline,of which an important goal is to construct quantum simulators with ultracold atoms.In ultracold-atom physics,the theoretical research is closely connected to experi-ment.In recent years,with the development of experimental techniques,the research of ultracold-atom physics is developing rapidly both in broadness and depth.For example,in atomic systems,Fermi condensate was first created in 2003 and recently Bose-Fermi superfluid mixture has been realized;in interatomic interaction,the Feshbach resonance in a Bose-Einstein condensation was first observed in 1998 and recently orbital Fesh-bach resonance in alkaline-earth(-like)atoms is realized;in external potential,ultracold atoms was first loaded into a optical lattice in 2005 and recently can be loaded into an optical cavity;in synthetic gauge fields,one dimensional spin-orbit coupling is first realized in 2011 and recently two dimensional spin-orbit coupling has been realized.These advances not only inspire a lot of theoretical research,but also enrich our tool-box for constructing quantum simulators.Based on recent experimental progresses,we propose and study several models of special significance in quantum simulation in this thesis.Specifically,this thesis includes the following several works:1.Topological superradiant states of a degenerate Fermi gasBased on recent experimental progress about the ultracold atoms in an optical cav-ity,we propose a cavity-assisted spin-orbit coupling scheme for a degenerate Fermi gas.We predict the existence of a topological superradiant state that is simultaneously char-acterized by nontrivial topological invariant and nonzero local order parameter in this system.We map out the steady-state phase diagram,and reveal the relation between the observable quantities like cavity field and density distribution and the parameters characterizing the topological phase transitions,based on which we propose the mea-surement scheme for the topological superradiant states and relevant phase transitions.At the same time,we also carefully analyze the high-band effect induced by the Raman field in this model.2.Bose-Einstein condensate in an optical lattice with two-dimensional spin-orbit couplingIn this work,we systematically study the basic properties of a Bose-Einstein con-densate in an optical lattice with recently realized two-dimensional spin-orbit coupling.We find that,the ground-state phase diagram and the band-topology phase diagram are strongly modified by the inter-band coupling induced by the Raman field.Typically,the high-band effect gives rise to new phase boundaries in both the ground-state phase diagram and the band-topology phase diagram.Furthermore,we employ the Bogoli-ubov theory to study the excitation spectrum of this system.In the band-topology phase diagram of the excitation spectrum,we find a fine structure near the zero Zeeman field.Finally,by eliminating the high-band degrees of freedom,we derive an effective two-band model including the high-band correction.3.Quantum simulations of symmetry-protected topological state and topological Fulde-Ferrell state with ultracold Alkaline-Earth-like atomic gasesBased on recent experimental progress on the orbital Feshbach resonance,we re-spectively propose two schemes to realize the interacting fermionic symmetry-protected topological state and the topologically nontrivial Fulde-Ferrell state with ultracold Alkaline-Earth-like atomic gases.In the first scheme,we show that the combination of the Raman-assisted spin-orbit coupling on the electronic orbitals and the spin-exchange interactions can give rise to a nontrivial interacting fermionic symmetry-protected topo-logical state.Based on numerical calculations,we identify an interaction-induced topo-logical phase transition,and map out the ground-state phase diagram.Finally,we pro-pose to detect the interaction-induced topological phase transition by measuring local density distribution of the topological edge modes.In the second scheme,in order to create the spin-orbit coupling,we proposed to directly couple the hyperfine states from different orbits with two lasers.By characterizing the Zak phase and the edge states,we show that the interplay between the spin-orbit coupling and pairing interactions can give rise to a topologically nontrivial Fulde-Ferrell state.In experiment,this state can be probed from the momentum-space density distribution obtained from time-of-flight images.4.Vortex-core structure in a mixture of Bose and Fermi superfluidsBased recent experimental progress,we study a single vortex state of a Bose and Fermi superfluid mixture in this work.Due to it has a richer structure,we mainly focus on the case with the vortex in the Fermi component.We find that,when the coupling between the Bose and Fermi components is increased,the vortex core will experience a sharp transition.In the strong-coupling regime,the Bose component will completely localized near the vortex core,and then many properties of the vortex state will be changed dramatically.We systematically characterize the behaviors of this vortex-core transition in the whole the BCS to BEC regimes of the Fermi interaction.In the reso-nance and BEC regimes,as a result of the localization of the Bose component,multiple branches of vortex bound states emerge from the bulk spectrum.