We study the superfluidity of a spin-orbit coupled Bose-Einstein condensate (BEC) by computing its Bogoliubov excitations, which are found to consist of two branches: one is gapless and phonon-like at long wavelength; the other is typically gapped. These excitations imply a superfluidity that has two new features: (i) due to the absence of the Galilean invariance, one can no longer define the critical velocity of superfluidity independent of the reference frame; (ii) the superfluidity depends not only on whether the speed of the BEC exceeds a critical value, but also on cross helicity that is defined as the direction of the cross product of the spin and the kinetic momentum of the BEC.

The extended Bose-Hubbard model for a double-well potential with pair tunneling is studied through both exact diagonalization and mean field theory (MFT). When pair tunneling is strong enough, the ground state wavefunction predicted by the MFT is complex and doubly degenerate while the quantum ground state wavefunction is always real and unique. The time reversal symmetry is spontaneously broken when the system transfers from the quantum ground state into one of the mean field ground states upon a small perturbation. As the gap between the lowest two levels decreases exponentially with particle number, the required perturbation inducing the spontaneous symmetry breaking (SSB) is infinitesimal for particle number of typical cold atom systems. The quantum ground state is further analyzed with the Penrose-Onsager criterion, and is found to be a fragmented condensate. The state also develops the pair correlation and has non-vanishing pair order parameter instead of the conventional single particle order parameter. When this model is generalized to optical lattice, a pair superfluid can be generated. The mean field ground state can be regarded as effective ground state in this simple model. The detailed computation for this model enables us to offer an in-depth discussion of the relation between SSB and effective ground state, giving a glimpse on how nonlinearity arises in the SSB of a quantum system.

We study the superfluidity of a pure spin current that is a spin current without mass current. We examine two types of pure spin currents, planar and circular, in spin-1 Bose gas. For the planar current, it is usually unstable, but can be stabilized by the quadratic Zeeman effect. The circular current can be generated with spin-orbit coupling. When the spin-orbit coupling strength is weak, we find that the circular pure spin current is the ground state of the system and thus a super-flow. We discuss the experimental schemes to realize and detect a pure spin current.

Liquid helium 4 had been the only bosonic superfluid available in experiments for a long time. This situation was changed in 1995, when a new superfluid was born with the realization of the Bose-Einstein condensation in ultracold atomic gases. The liquid helium 4 is strongly interacting and has no spin; there is almost no way to change its parameters, such as interaction strength and density. The new superfluid, Bose-Einstein condensate (BEC), offers various aspects of advantages over liquid helium. On the one hand, BEC is weakly interacting and has spin degrees of freedom. On the other hand, it is convenient to tune almost all the parameters of a BEC, for example, the kinetic energy by spin-orbit coupling, the density by the external potential, and the interaction by Feshbach resonance. Great efforts have been devoted to studying these new aspects of superfluidity, and the results have greatly enriched our understanding of superfluidity. Here we review these developments by focusing on the stability and critical velocity of various superfluids. The BEC systems considered include a uniform superfluid in free space, a superfluid with its density periodically modulated, a superfluid with artificially engineered spin-orbit coupling, and a superfluid of pure spin current. Due to the weak interaction, these BEC systems can be well described by the mean field Gross-Pitaevskii theory and their superfluidity, in particular critical velocities, can be examined with Landau's theory of superfluid. Experimental proposals to observe these new aspects of superfluidity are discussed.