Restoring quasireversibility with a single topological charge

We numerically study a rotating Bose-Einstein condensate placed transiently over the critical rotation frequency, i.e., in a regime where the rotation frequency is larger than the radial frequency of the confinement. We study the reversibility of this process depending on the strength of the interactions and the presence of vortices. We find that the reversibility is broken by the interactions in the absence of vortices but systematically quasirestored in the presence of a single vortex.

Figure 1

Density profile (left) and phase map (right) of the ground states for $\varepsilon =0,\mathrm{\Omega }=0.9$ and $\beta =2,5$, and 9.

Figure 2

Time evolution of the anisotropy $\varepsilon$ (top) and the rotation frequency $\mathrm{\Omega }$ (bottom). The initial state is in the stable zone (${\mathrm{\Omega }}_m=0.9$). The shaded area depicts the instability zone in the absence of interactions $\mathrm{\Omega }\left(t\right)\in [\sqrt{1-{\varepsilon }_{\max }};\sqrt{1+{\varepsilon }_{\max }}]$. To cross the instability zone, $\mathrm{\Omega }$ is either abruptly changed [scenario (b)] or smoothly ramped [scenario (a)] to ${\mathrm{\Omega }}_M$.

Figure 3

Relative energy difference between the initial and final states of the system $\mathrm{\Delta }E/E_0=(E_f-E_0)/E_0$ as a function of $2t_2$, applying scenario (b) to a noninteracting BEC ($\beta =0$) with ${\varepsilon }_{\max }=5\%$, and for various values of ${\mathrm{\Omega }}_m$ and ${\mathrm{\Omega }}_M$.

Figure 4

Relative energy difference between the initial and final states $\mathrm{\Delta }E/E_0$ as a function of $2{\omega }_0{t}_2$ applying the scenario (b) (with ${\varepsilon }_{\max }=5\%,{\mathrm{\Omega }}_m=0.9$, and ${\mathrm{\Omega }}_M=1.1$) to ground states without vortices for different interaction strengths $\beta$.

Figure 5

(a) Fidelity of the final state with respect to the initial one as a function of $2t_2$ applying scenario (b) (with ${\mathrm{\Omega }}_m=0.9$ and ${\mathrm{\Omega }}_M=1.1$) to a BEC with a single vortex ($\beta =5$) for various values of the anisotropic parameter ${\varepsilon }_{\max }$. (b, c) Evolution of the fidelity over time respectively at the first minimum and the first maximum of the final fidelity. (d) Relative energy difference between the initial and final states as a function of $2t_2$.

Figure 6

Period (top) and contrast $C_{st}$ (bottom) of the excess of energy $\mathrm{\Delta }E/E_0$ as a function of the interaction strength $\beta$ when applying scenario (b) (with ${\varepsilon }_{\max }=5\%,{\mathrm{\Omega }}_m=0.9$ and ${\mathrm{\Omega }}_M=1.1$) to different ground states with $N_v=0$ (circles), $N_v=1$ (diamonds), and $N_v=2$ (triangle) vortices. The square corresponds to the prediction of Sec. 4 in the noninteracting case and the lower dashed line to that in the Thomas-Fermi regime.

Figure 7

Amplitude $b_e$ and period $T_e$ of the excess of energy $\mathrm{\Delta }E/E_0$ with the maximum rotation frequency ${\mathrm{\Omega }}_M$ in the single-vortex case ($\beta =5$) and in the noninteracting case ($\beta =0$), following scenario (b) with ${\mathrm{\Omega }}_m=0.9$ and ${\varepsilon }_{\max }=5\%$. The vertical dashed line shows the upper limit of the instability zone $\sqrt{1+{\varepsilon }_{\max }}$ in the absence of interactions.

Figure 8

Relative energy difference $\mathrm{\Delta }E/E_0$ as a function of $2t_2$ in the noninteracting case ($\beta =0$), using scenario (a) (with ${\mathrm{\Omega }}_m=0.9$ and ${\mathrm{\Omega }}_M=1.1$) for various values of the anisotropic parameter ${\varepsilon }_{\max }$.

Figure 9

Relative energy difference as a function of $2t_2$ when applying scenario (a) to an interacting BEC without vortices: $\beta =2,{\varepsilon }_{\max }=5\%,{\mathrm{\Omega }}_m=0.9$, and ${\mathrm{\Omega }}_M=1.1$. The insets show the final density and phase profiles after the instability sweep at the local minimum at $2t_2=174$.

Figure 10

Relative energy difference as a function of $2t_2$ when applying scenario (a) to a BEC with a single vortex ($\beta =5$) for ${\mathrm{\Omega }}_m=0.9,{\mathrm{\Omega }}_M=1.1$, and ${\varepsilon }_{\max }=5\%$. The shaded area depicts the zone where the results are no longer reliable. The density (phase) profiles at the extrema denoted by the roman numbers are shown in the top (bottom) strips. Color scale from blue to red: amplitude varies from 0 to 0.12, and phase from $-\pi$ to $\pi$.