Manipulation of vortices by localized impurities in Bose-Einstein condensates

We consider the manipulation of Bose-Einstein condensate vortices by optical potentials generated by focused laser beams. It is shown that for appropriate choices of the laser strength and width it is possible to successfully transport vortices to various positions inside the trap confining the condensate atoms. Furthermore, the full bifurcation structure of possible stationary single-charge vortex solutions in a harmonic potential with this type of impurity is elucidated. The case when a moving vortex is captured by a stationary laser beam is also studied, as well as the possibility of dragging the vortex by means of periodic optical lattices.

Figure 1

(Color online) Density plot showing a snapshot of the interaction of the vortex with a localized impurity in the absence of the harmonic trap (i.e., $\Omega =0$). The presence of the impurity (indicated by a white cross) induces a clockwise rotation of the vortex (narrower field depression) along a path depicted by the dark dots. The parameters are as follows: $\left(\mu ,\Omega ,s,V_{\text{Imp}}^{\left(0\right)},\epsilon \right)=\left(1,0,1,5,1\right)$. The colorbar shows the condensate density in adimensional units (i.e., in units of $\hslash {\omega }_z/g_{2\text{D}}$). Distances are given in units of the transverse harmonic oscillator length $a_z$.

Figure 2

(Color online) Phase diagram depicting the vortex pinning by the localized impurity of strength $V_{\text{imp}}^{\left(0\right)}$ (in units of $\hslash {\omega }_z$) and width $\epsilon$ (in units of transverse harmonic oscillator length $a_z$). Each curve represents a different pinning location at the indicated radii (i.e., distance from the center of the harmonic trap). Parameters are as follows: $\left(\mu ,\Omega ,s\right)=\left(1,0.065,1\right)$.

Figure 3

(Color online) The panel (a) is a one-dimensional slice (for $y_a=0$) of the solution surfaces (represented in the inset panel) as a function of the position of the impurity $\left(x_a,y_a\right)$, in units of $a_z$, for fixed $V_{\text{Imp}}^{\left(0\right)}=1$. The vertical axis corresponds to the $L^2$-norm squared of the solution (i.e., the number of atoms scaled by $\hslash {\omega }_z{a}_z^2/g_{2\text{D}}$). The dashed curves [yellow (lighter) surface] correspond to unstable solutions, while the solid curves [red (darker) surfaces] correspond to stable solutions. The critical value of the radius, $x_{a,\text{cr}}=3.72$ as well as the characteristic value $x_a=2$ and $x_a=0$ are represented by vertical lines. Panel (b) shows the squared bifurcating eigenvalues, ${\lambda }^2$ (in units of ${\omega }_z^2$), and along these branches ${\lambda }^2>0$ corresponds to an instability. Panel (c) shows the saddle-node bifurcation as a function of $V_{\text{Imp}}^{\left(0\right)}$ for $x_a=2$ fixed (there are lines at $V_{\text{Imp}}^{\left(\text{cr}\right)}=0.57$, $V_{\text{Imp}}^{\left(0\right)}=0.8$, and $V_{\text{Imp}}^{\left(0\right)}=1$). The solutions and spectra for each of the three branches, represented by circles (and letters) for each of the characteristic values of $x_a$ and $V_{\text{Imp}}^{\left(0\right)}$ are presented in Fig. 4. For these branches $\left(\mu ,s,\epsilon \right)=\left(1,1,0.5\right)$.

Figure 4

(Color online) The solutions (a,c, zoomed in to the region where the vortices live) and corresponding linearization spectra in a neighborhood of the origin (b,d), for the various parameter values indicated by the circles in Fig. 3. The depicted solutions correspond to cuts along $y=0$ of the two-dimensional density ${\left|\Psi \left(x,y\right)\right|}^2$ in units of $\hslash {\omega }_z/g_{2\text{D}}$. Solutions (a,b) and (c,d) correspond to $V_{\text{Imp}}^{\left(0\right)}=1$ and $V_{\text{Imp}}^{\left(0\right)}=0.8$, respectively (in units of $\hslash {\omega }_z$). [$\lambda$ and $x$ are again in units of ${\omega }_z$ and $a_z$, respectively.]

Figure 5

(Color online) Parameter regions for successful manipulation of a single vortex inside a harmonic trap. The area above each curve corresponds to the successful dragging region. Panel (b) depicts a zoomed region of the panel (a) where the asterisk and cross correspond, respectively, to the manipulation success and failure depicted in Fig. 6. These panels indicate that higher intensity beams and broader beams, can successfully drag vortices over shorter time scales than weaker, narrower beams. As usual, the units of $\left(\tau ,ϵ,V_{\text{imp}}^{\left(0\right)}\right)$ are $\left({\omega }_z^{-1},a_z,\hslash {\omega }_z\right)$.

Figure 6

Successful (top row) and failed (bottom row) vortex dragging cases by a moving laser impurity corresponding to the parameters depicted, respectively, by an asterisk and a cross in the bottom panel of Fig. 5. In both cases, the laser impurity, marked by a cross, with $\left(V_{\text{Imp}}^{\left(0\right)},\epsilon \right)=\left(5,1\right)$ is moved adiabatically from (0,0) to (5.43, $-5.43$) and the snapshots of the density are shown every $0.5t^{\ast }$. The top row corresponds to a successful manipulation for $\tau =18.5$ while the bottom row depicts a failed dragging for a slightly lower adiabaticity of $\tau =18$. Time is given in units of the transverse trapping frequency ${\omega }_z^{-1}$ and the field of view is $40a_z$ wide.

Figure 7

(Color online) Vortex dragging by an optical lattice potential as in Eq. (12). The central high intensity maximum of the optical lattice (depicted by a cross in the panels) is moved adiabatically from the initial position $\mathbf{\left(}{x}_a\left(0\right),y_a\left(0\right)\mathbf{\right)}=\left(0,0\right)$ to the final position (5.43, 5.43). The top row depicts the BEC density (the colorbar shows the density in adimensional units) while the bottom row depicts the phase of the condensate where the vortex position (“plus” symbol in the right column) can be clearly inferred from the $2\pi$ phase jump around its core. Observe how the vortex loses its guiding well and jumps to a neighboring well in the right column. The left, middle, and right columns correspond, respectively, to times $t=0$, $t=0.5t^{\ast }$, and $t=2.5t^{\ast }$. The remaining parameters are as follows: $\left(V_0,k,\tau ,{\theta }_x,{\theta }_y\right)=\left(1.4,0.3215,20,0,0\right)$. Distances are given in units of the transverse harmonic oscillator length $a_z$ and time in units of the inverse transverse trapping frequency ${\omega }_z^{-1}$.

Figure 8

(Color online) Capturing a precessing vortex by a stationary impurity for $\left(V_{\text{Imp}}^{\left(0\right)},\epsilon ,\Omega \right)=\left(5,1,0.065\right)$. The different paths correspond to isolated vortices that are released by adiabatically turning off a pinning impurity at the following off-center locations: (5,0) (thin black line), (6.5,0) (thick red line), and (8,0) (blue dashed line). The capturing impurity is located at ($-4$, 0). The first and third cases fail to produce capturing while the second case manages to capture the vortex. One interesting feature that we observed in the case of successful capture is that, before it gets captured, the vortex gets drawn into the impurity potential, but then almost gets knocked back out by the phonon radiation waves created from the capture which bounce around within the condensate. The axes are, again, given in units of $a_z$.