Bloch structure in a rotating BEC

We point out that the Bloch band picture is very suitable to describe a Bose-Einstein condensate (BEC) in a 1D ring. The ring geometry is periodic in the sense that some potential that breaks axisymmetry always has 2 pi periodicity. In this case, one finds that the rotation frequency (Omega) of the ring corresponds to the quasimomentum. As shown in Fig. 1, the energy levels certainly have the Bloch band structure.

Fig. 1: The first and second energy levels for the 1D ring system.

Fig. 2: Angular momentum corresponding to the levels shown in Fig. 1.

Now, we consider a quasi-2D BEC in a harmonic-plus-quartic potential with a rotating stirrer. When the stirring frequency (Omega) is larger than the frequency of the harmonic trap, this system can be approximated by the quasi-1D ring, because the quartic confinement and the centrifugal potential produces a Mexican-hat shaped potential, and the minimum of the potential traces a quasi-1D toroidal geometry. In fact, the structure of the energy levels of this system, shown in Fig. 3, is very similar to Fig. 2. Thus, we can say that the vortex nucleation occurs due to the Bragg reflection.

Fig. 3: Angular momenta of the two energy levels for a harmonic-plus-quartic potential rotating at frequency Omega.

Figure 3 suggests that if we prepare the nonvortex state, switch on the rotating stirrer at low frequency, and adiabatically increase the stirring frequency, we obtain the vortex state following the first Bloch band (red curve).

Fig. 4: Adiabatic vortex nucleation following the first Bloch band.

Interestingly, we can also nucleate vortices by decreasing Omega following the second Bloch band (blue curve).

Fig. 5: Adiabatic vortex nucleation following the second Bloch band.

We have considered so far the noninteracting atoms. If we change the strength of interaction g using the Feshbach resonance, the frequency Omega at which the Bragg reflection occurs is shifted. Using this shift, we can adiabatically nucleate vortices by changing the strength of interaction, which may be called "Feshbach-induced Bragg reflectoin."

Fig. 6: Adiabatic vortex nucleation with an increase in the interaction.

Fig. 7: Adiabatic vortex nucleation with a decrease in the interaction.

Another phenomenon peculiar to the Bloch structure is the formation of the gap soliton, in which the negative-mass dispersion counterbalances the repulsive interaction. The phenomenon analogous to the formation of the gap soliton occurs also in the present system.

When we prepare a nonvortex state, and increase the repulsive interaction at some fixed Omega, the point at which the Bragg reflection occurs is shifted as shown above, and vortices begin to enter the condensate (left image). However, this state is dynamically unstable. The instability is caused by the presence of an almost degenerate state that is dynamically stable. This stable state is a symmetry-broken localized state (right image), and can be interpreted as an analogue of the gap soliton.

Fig. 8: Left: A stationary but dynamically unstable state at the edge of the Brillouin zone. Right: A stationary and dynamically stable state for the same Omega and g.

Thus, when we increase the repulsive interaction and arrive at the Bragg reflection point, the condensate becomes unstable against the formation of the gap soliton. It is interesting to note that the localization occurs even with the repulsive interactions. We numerically simulated this process.
Symmetry breaking due to the emergence of the gap soliton: (GIF, 126KB) (WMV, 22KB)