Emergence of Bloch Bands in a Rotating Bose-Einstein Condensate

A rotating Bose-Einstein condensate is shown to exhibit a Bloch band structure even in the absence of a periodic potential. Vortices enter the condensate via Bragg reflection if the frequency of a rotating drive is adiabatically increased or decreased, or if the interaction is adiabatically changed at a constant rotating drive. A localized state analogous to a gap soliton in a periodic system is predicted to occur near the edge of the Brillouin zone.

Figure 1

(a) $E\equiv \mu +{\Omega }^2/4$ and (b) $⟨L⟩\equiv -i\int {\psi }_{1\mathrm{D}}^{*}{\partial }_{\theta }{\psi }_{1\mathrm{D}}d\theta$ of the first (red curves) and the second (blue curves) Bloch bands for the noninteracting (main panels) and interacting (right panels) BECs in a 1D ring described by Eq. (1) with $V(\theta )=0.1\cos 2\theta$. Since $V(\theta )$ has the twofold symmetry, there are two independent branches with even and odd values of $⟨L⟩$ [15], which are shown by the solid and dotted curves.

Figure 2

(a) Angular momentum $⟨L⟩$ versus rotation frequency $\Omega$ of the stationary states of Eq. (2) with $g=0$, $K=1$, and $\epsilon =0.02$. The images (1-1)–(1-5) and (2-1)–(2-5) correspond to the branches indicated by the red and blue curves. (b) Time evolutions with $g=0$, $K=1$, and an additional dissipation parameter $\lambda =0.03$, where $\Omega$ is changed linearly from 1.52 to 1.6 (red curve) and from 1.62 to 1.54 (blue curve) during $0\le t\le 600$. The initial state is the nonvortex ground state. The parameter $\epsilon$ is linearly ramped up from 0 to 0.04 during $0\le t\le 200$, kept at 0.04 during $200\le t\le 400$, and ramped down from 0.04 to 0 during $400\le t\le 600$ to suppress nonadiabatic disturbances. The size of the images is $5\times 5$ in units of $(\hslash /m{\omega }_{\perp }{)}^{1/2}$.

Figure 3

(a) The $g$ dependence of the first and second bands (left and right panels) with $K=1$ and $\epsilon =0.02$, where the ordinates are offset by $\pm g$ for clarity. The dynamically unstable regions are indicated by the red curves. The inset shows a localized stationary state—a gap soliton—formed near the first band at $g=0.8$ and $\Omega =1.55$. (b) Time evolutions with $\lambda =0.03$ and $K=1$, where $g$ is linearly ramped from $-1.2$ to 0.5 with $\Omega =1.58$ [red curve, corresponding to the red arrow in (a)], from 1.5 to $-0.3$ with $\Omega =1.56$ (blue curve, blue arrow), and from 0 to 1 with $\Omega =1.55$ (green curve, green arrow). The change in $\epsilon$ for the red and blue curves is the same as that in Fig. 2b. For the green curve, $\epsilon =0.04$ is maintained during $200\le t\le 600$ and a small perturbation to break the axisymmetry is added to the initial state.