Topological Superfluidity Of Bilayer Fermi Gases With Spin-orbit Coupling

New exotic topological quantum states are among the most fundamental issues in current condensed matter physics.These topics have drawn enormous interest for ultracold atomic gases enabling simulations of many condensed matter phenomena.With a recent experimental progress in synthetic SOC for degenerate atomic gases,diverse new quantum phases due to the SOC have been predicted,such as the stripe phase and vortex structure in the ground states of atomic Bose-Einstein condensates(BECs),as well as the Rashba pairing bound states(Rashbons)and topological superfluidity in degenerate Fermi gases.The synthetic SOC has been successfully implemented and explored by Raman coupling of a pair of atomic hyperfine ground states accompanied by a recoil momentum.This provides the SOC along the recoil direction representing the one-dimension(1D)SOC.The realization of the synthetic SOC for ultracold atoms in two or more dimensions is very desirable.The two dimensional SOC of the Rashba type has a non-trivial dispersion.The lower dispersion branch contains a highly degenerate ground state(the Rashba ring).Additionally,there is a Dirac cone at an intersection point of two dispersion branches,and a band gap can be opened by adding a Zeeman term.This is essential for the topological superfluidity.Majorana fermions(MFs),unlike Dirac fermions,are fermionic particles that are their own antiparticles.These intriguing particles are exotic for their non-Abelian exchange statistics,rather than Fermi or Bose statistics and robust against local perturbations,thus offering an unique platform in fault tolerant topological quantum computation,quantum memory and quantum random-number generation.Therefore in condensed matter physics there have been various theoretical proposals for the realization of Majorana zero modes(MZMs),a special case of Majorana fermions that occur at exact zero energy.Majorana zero energy modes,which have real potentials for quantum computer technology,recently have been hot topic in the condensed matter research.In chapter 1,we first give the fundamental concept of topology and introduce the development of topological matter in condensed matter physics.Recently,new topological state becomes a research hot topic.Based on the ultracold atomic gas platform,we discuss the realization of sin-orbit coupling in ultracold atomic gas.Moreover,we try to find Majorana fermions,which have potential applications in quantum computer.In chapter 2,we investigate the bilayer fermi gas with spin-orbit coupling.Via the laser-induced interlayer tunneling and Raman laser spin-orbit coupling,two-dimensional spin-orbit coupling in ultracold fermi gas has been realized.By increasing the intralayer and interlayer coupling,we observe a non-trivial superfluid phase transition: in the regime of weak-coupling,it will be FFLO superfluid state with finite-momentum pairing.Furthermore,if the system enter the strong coupling regime,we can realize the 2D spin-orbit coupling and the system enter the non-trivial topological phase.In chapter 3,we present the realization of multiple Majorana feremions in double atomic chains.At first,we propose and validate a specific experimental setup for the realization of multiple MZMs-one-dimensional Fermi double wires with Raman laser-induced spin-orbit coupling.By tunneling the inter-wire tunneling,we find that the traditional gap condition induced by the collective effect of spin-orbit coupling and Zeeman field is totally changed.As a consequence,phase transitions between different topological regions characterized by high Winding numbers.We further demonstrate the Majorana zero modes within the bulk energy gap by the self-consistent Bogoliubov-de Gennes(BdG)simulations for a cloud trapped in a harmonic potential.In chapter 4,we give a brief summary and prospect of correlated topological superfluidity.