Exotic superfluidity in spin-orbit coupled Bose-Einstein condensates

We study the superfluidity of a spin-orbit coupled Bose-Einstein condensate (BEC) by computing its Bogoliubov excitations, which are found to have two branches: one is gapless and phonon-like at long wavelength; the other is typically gapped. These excitations imply 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 crosshelicity that is defined as the direction of the cross-product of the spin and the kinetic momentum of the BEC. Copyright c © EPLA, 2012 Superfluidity was first discovered in 1938 and has fascinated physicists ever since. This interesting phenomenon was explained by Landau [1], whose theory has been very successful in explaining many important properties of superfluids. However, Landau’s theory of superfluidity may be facing challenges brought by the recent experimental realization of artificial gauge fields for ultracold bosonic atoms [2–5]. When the artificial gauge field is nonAbelian [6–8], it is effectively spin-orbit coupling (SOC). SOC has played a crucial role in many exotic phenomena such as topological insulators [9]. However, in superfluids, the SOC is generally absent and its effects have remained largely unexplored. Note that this issue is not limited to ultracold atoms since the Bose-Einstein condensation with SOC also exists for excitons in semiconductors [10,11]. There have been some theoretical works, where many interesting properties of spin-orbit coupled Bose-Einstein condensates (BECs) are explored [11–23]. For instance, it was shown in ref. [11] that SOC can lead to unconventional Bose-Einstein condensation with the breaking of timereversal symmetry. Later, a stripe phase that breaks rotational symmetry was found [12,13]. In this letter we concentrate on the superfluidity of the spin-orbit coupled BEC. (a)E-mail: [email protected] To put our study into perspective, we briefly review Landau’s theory for a superfluid without SOC. Consider such a superfluid flowing in a tube. With the Galilean transformation, Landau found that the excitation of this flowing superfluid is related to the excitation of a motionless superfluid as [1] εv(p) = ε0(p)+p ·v, (1) where ε0(p) is the excitation for a stationary superfluid and p is the momentum of the excitation. For phonon excitation ε0(p) = c|p|, the excitation εv(p) can be negative only when |v|> c. Therefore, the speed of sound c is the critical velocity beyond which the flowing superfluid loses its superfluidity and suffers viscosity. We switch to a different reference frame, where the superfluid is at rest while the tube is moving. It is apparent to many that these two reference frames are equivalent so that the superfluid will be dragged along only when the tube speed exceeds the speed of sound c. However, this equivalence is based on that the superfluid is invariant under the Galilean transformation. As SOC breaks the Galilean invariance of the system [24], we find that these two reference frames are no longer equivalent as shown in fig. 1: the critical speed for scenario (a) is different from the one for scenario (b). For easy reference, the critical speed for (a) is hereafter called the critical flowing speed and the one for (b) the critical dragging speed.