Fast control of topological vortex formation in Bose-Einstein condensates by counterdiabatic driving

Fast creation of a single vortex in BEC of alkali-metal atoms at a prescribed position and time is still challenging, even though various methods to create single and multiple vortices have been proposed and demonstrated. Topological vortex formation is advantageous in this respect over other methods in that the position and time of vortex formation is highly controllable. This method requires inversion of the bias magnetic field along the axis of the condensate, which leads to unwanted atom loss due to nonadiabatic transitions when the bias field crosses zero. It is the purpose of this paper to propose a scheme that enables a fast creation of a vortex in much shorter time than needed for adiabatic control time by introducing the counterdiabatic field to avert the atom loss. We further introduce a gauge transformation so that the required magnetic field is generated by manipulating the current of the Ioffe bars, which makes our proposal experimentally feasible.

Figure 1

Trajectories of $\mathit{B}$ (red line) and ${\langle \mathit{F}\rangle }_{\mathrm{W}}$ (red dashed curve) of a hyperfine spin with $\varphi =0$. The vertical axis is the $z$ axis, while the horizontal axis is the $x$ axis. The quadrupole field is made unphysically large compared to $B_z\left(0\right)$ for purposes of illustration, see text. Vectors at $t=0,T/2$, and $T$ are explicitly shown.

Figure 2

Time dependence of the ratio $|{\mathit{B}}_{\mathrm{CD}}|/|{\mathit{B}}_{\perp }|$, which is independent of $r$ and depends only on $t$. Parameters are ${log}_{10}(T/\tau )=1.4$ and $r_0/a_{\mathrm{HO}}=2$. For the parameters in the text, they amount to $T\sim 36\mu \mathrm{s}$ and $r_0\sim 1.83\mu \mathrm{m}$.

Figure 3

(a) Schematics of ${\mathit{B}}_{\perp }$ (red arrow) and ${\mathit{B}}_{\mathrm{CD}}$ (green arrow) in the $x-y$ plane. (b) ${\mathit{B}}_{\perp }, {\mathit{B}}_{\mathrm{CD}}$ and ${\mathit{B}}_{\perp }+{\mathit{B}}_{\mathrm{CD}}$ (blue arrow) for $x>0$ and $y=0$. ${\tilde{\mathit{B}}}_{\perp }$ (light blue arrow) is obtained by rotating ${\mathit{B}}_{\perp }+{\mathit{B}}_{\mathrm{CD}}$ by an angle ${\alpha }_{\mathrm{B}}$ so that it is parallel to ${\mathit{B}}_{\perp }$ (see Sec. 5). Figures are conceptual and coordinates and magnetic fields are in arbitrary units.

Figure 4

Fraction $N\left(t\right)/N\left(0\right)$ of atoms left in the trap a long time after a vortex is formed. Since $t\gg T$, the condensate is in the pure WFSS. $\circ$ shows the fraction without ${\mathit{B}}_{\mathrm{CD}}$, while other symbols show the fractions with ${\mathit{B}}_{\mathrm{CD}}$ for different $r_0$. Here $\tau \sim 1.43\mu \mathrm{s}$.

Figure 5

Ratio of $N\left(t\right)$ with the CDF to that without the CDF for $t\gg T$. The unit of time is $\tau \sim 1.43\mu \mathrm{s}$.

Figure 6

Snapshots of the amplitude of the order parameter $|{\mathrm{\Psi }}_{\mathrm{W}}{|}^2$ are displayed in the left (right) panel for the control without (with) the CDF with the same parameter set as in Fig. 2. The thin purple curves are $|{\mathrm{\Psi }}_{\mathrm{W}}{|}^2$ plotted sequentially at the same time interval for $T/2-\epsilon \le t\le T/2+\epsilon$ with $\epsilon =T/80$. The blue dashed curves and the red thick curves are $|{\mathrm{\Psi }}_{\mathrm{W}}{|}^2$ at $t=0$ and $T$, respectively. The inset is the profile of the condensate $f$ at $t=0$, which is obtained by solving Eq. (17) numerically. In the inset the units of $r$ and $f$ are $\mu \mathrm{m}$ and $\mu {\mathrm{m}}^{-3/2}$, respectively.

Figure 7

Instantaneous fraction of atoms $N_{\mathrm{W}}\left(t\right)/N\left(0\right)$ (red solid curve), $N_{\mathrm{N}}\left(t\right)/N\left(0\right)$ (green dashed curve), and $N_{\mathrm{S}}\left(t\right)/N\left(0\right)$ (blue dotted curve) with the CDF during the vortex formation for ${log}_{10}(T/\tau )=1.4$ and $r_0=2a_{\mathrm{HO}}$. The dashed black curve shows $N_{\mathrm{W}}\left(t\right)/N\left(0\right)$ without the CDF. Here $T\sim 36\mu \mathrm{s}$ as before.

Figure 8

(a) Fraction $N\left(t\right)/N\left(0\right)$ of atoms left in the trap a long time after a vortex is formed for $f\left(0\right)=20a_{\mathrm{HO}}^{-3/2}$ corresponding to ${\int \int }_{-\infty }^{\infty }{\left|\mathrm{\Psi }\right|}^{2}dxdy\sim 2.1\times {10}^4\mu {\mathrm{m}}^{-1}$. Since $t\gg T$, the condensate is in the pure WFSS. $\circ$ shows the fraction without ${\mathit{B}}_{\mathrm{CD}}$, while the other symbols show the fractions with ${\mathit{B}}_{\mathrm{CD}}$ for different $r_0$. (b) Ratio of $N(t\gg T)$ with the CDF to that without the CDF.

Figure 9

(a) $|{\tilde{\mathit{B}}}_{\perp }(\mathit{r},t)|/|{\mathit{B}}_{\perp }\left(\mathit{r}\right)|$ and (b) ${\tilde{B}}_z\left(t\right)/B_z\left(0\right)$ (solid red curve) and $B_z\left(t\right)/B_z\left(0\right)$ (dashed black line) for the same parameters as those used in Fig. 7 with $T\sim 36\mu \mathrm{s}$. Deviations from the original fields ${\mathit{B}}_{\perp }$ and $B_z\left(t\right)$ are prominent only when $t\sim T/2$.

Figure 10

(a) Trajectory of $\tilde{\mathit{B}}(\mathit{r},t)$ for $\varphi =0$ is shown in a solid red curve. The trajectory is in the $x-z$ plane. (b) Trajectories of $\tilde{\mathit{B}}(\mathit{r},t)$ (the solid red curve), $\overline{\mathit{B}}(\mathit{r},t)$ (the solid blue curve), ${\langle \mathit{F}\left(t\right)\rangle }_{\mathrm{W}}$ after the gauge transformation (broken red curve) and ${\langle \mathit{F}\left(t\right)\rangle }_{\mathrm{W}}$ before the gauge transformation (broken blue curve). $|{\mathit{B}}_{\perp }\left(0\right)|$ is taken unphysically large, as in Fig. 1, for purposes of illustration. The vertical black line is the axis of the sphere and is given as a guide.