Exotic Superfluids Of Ultracold Fermi Gases In Optical Lattices

The superconductivity was first discovered by Onnes at the beginning of last cen-tury.From then on,the concept of superconductor was widely used in condensed matter physics and the superconductivity have attracted tremendous interest in the past decades.From neutron stars to quark-gluon plasma,researchers have found that the supercon-ductivity appears in various systems.Until 1957,Bardeen,Cooper and Schrieffer pro-posed BCS theory to elucidate the mechanism of the emerging of superconductivity.The BCS theory pointed out that blow the critical temperatures,the electrons with op-posite momentum and different internal spin states close to the Fermi level form into Cooper pairings with zero center-of-mass momentum.A few years later,Fulde,Fer-rell,Larkin and Ovchinnikov predicted that at low temperatures and strong magnetic fields the Cooper pairs have finite center-of-mass momenta and people start to search this unconventional superfluid state in various systems.However,due to the limitation of experiment,so far such states have been observed only in in organic superconductors and heavy-fermion superconductors.At the end of last century,the experimental realization of Bose-Einstein conden-sation in dilute alkali atoms is the beginning of the study of ultracold atom physics,in which people mainly focus on the alkali atoms and alkali earth atoms at very low temperatures.Due to the rapid development in the past decades,the ultracold atomic quantum gases have been considered as one of the most feasible platforms for quan-tum simulation.The realization of Feshbach resonance in ultracold atoms allow us to tune the interaction strength between atoms,and make the simulation of strong corre-lated system in ultracold atomic system become possible.In addition,the realization of spin-orbital coupling and optical lattice make the parameters of ultracold atomic system become more controllable and tunable,which motivate us to explore the unconventional quantum phenomena in low dimensional systems that are hard to reach in condensed matter physics.Due to the high degree of controllability and free of defects,the ultracold atomic gases in optical lattice provide an ideal platform for searching exotic superfluid states.In this thesis,we mainly focus on the properties and realization exotic superfluid phases in optical lattices,and the thesis is organized as follows:1.The Fulde-Ferrell superfluids in two-dimensional optical lattice with spinless ultracold Fermi gases.The early works about preparation of Fulde-Ferrell superfluids are based on Zee-man field.Due to the Zeeman field,the different internal spin states experience dis-tinct energy shifts,which leads to the center-of-mess momentum of Cooper pairs that formed by opposite spin states nonzero.However,because of the strict requirement and instability against thermal fluctuation,the realization of FFLO states is a tough task.In the past few years,an intensive proposals prepare for FFLO phases have been proposed based on ultracold atomic gases by employing the shaking optical lattice and anisotropic optical lattice.In addition,the indirect evidence for FFLO states has been observed in the spin imbalanced ultracold atomic system.On the other hand,the spin-less fermionic systems have attractted tremendous interest,since Kitaev first pointed out that in one-dimensional spinless fermions with p-wave interaction possess a topo-logically nontrivial superfluid.Furthermore,the spinless fermionic optical lattice has been realized in cold-atom experiments.However,to the best of our knowledge,the Fulde-Ferrell superfluid in spinless fermionic system hasn't been noticed yet.For that sake,an interesting question that can be raised:Can Fulde-Ferrell superfluid phases emerge in a spinless system?This work consist four parts:(i)We study the superfluid phases of a spinless Fermi gas in a two-dimensional square optical lattice and explain the mechanism of the emerging of Fulde-Ferrell superfluid phase;(ii)We characterize the topological features of the BCS superfluids and Fulde-Ferrell superfluids by calcu-lating the Chern numbers;(iii)We discuss the stability of BCS superfluid phases and Fulde-Ferrell superfluid phases at finite temperature;(iv)We give a possible experi-mental scheme to realize the spinless lattice model with current cold-atom technology.2.The preparation of pair-density wave states in a multilayer optical latticeMotivated by previous proposals on preparing pair-density wave states,we show that pair-density wave states can emerge in a multilayered lattice system with an out-of-plane Zeeman field accompanied with layer-independent spin-orbital coupling.We present the model Hamiltonian to describe Fermi gases in a multilayer optical lattice with Zeeman field and SO coupling.We shows the numerical results which is obtained by self-consistently solving the BdG equation.We discuss the phase diagram and ex-plain the mechanism of the appearance of PDW phases.By tuning the Zeeman field,we will see the superfluid order parameters acquires a spontaneous layer-modulated phase from inter-band pairing.We show the transition from the BCS to PDW states associates with a change of the Chern number,along with the band gap close and reopen.We extend the study to a trilayer system with Zeeman field and SO coupling and show the phase diagram.