Dynamic and energetic stabilization of persistent currents in Bose-Einstein condensates

We study conditions under which vortices in a highly oblate harmonically trapped Bose-Einstein condensate (BEC) can be stabilized due to pinning by a blue-detuned Gaussian laser beam, with particular emphasis on the potentially destabilizing effects of laser beam positioning within the BEC. Our approach involves theoretical and numerical exploration of dynamically and energetically stable pinning of vortices with winding number up to $S=6$, in correspondence with experimental observations. Stable pinning is quantified theoretically via Bogoliubov-de Gennes excitation spectrum computations and confirmed via direct numerical simulations for a range of conditions similar to those of experimental observations. The theoretical and numerical results indicate that the pinned winding number, or equivalently the winding number of the superfluid current about the laser beam, decays as a laser beam of fixed intensity moves away from the BEC center. Our theoretical analysis helps explain previous experimental observations and helps define limits of stable vortex pinning for future experiments involving vortex manipulation by laser beams.

Figure 1

Evidence of superfluid current persistence and its decay. Black squares show the mean number of vortices formed and held in the presence of the laser barrier, observed in 10 runs for each of the hold times $t_h$ shown, with 3 s of hold time used to separate the vortices for determining the vortex number. Error bars represent statistical uncertainty. An exponential fit to the data (black curve) gives a $1/e$ lifetime of 31(4) s. Gray circles show the lifetime of vortices in the system under the same conditions, but with the pinning beam ramped down at the beginning of the hold time rather than at the end. An exponential fit (gray curve) gives the lifetime of free vortices to be 15(1) s, an indication that vortex and current lifetime decreases without the central barrier present. The BEC lifetime decreases with a $1/e$ lifetime of 24(3) s (not shown).

Figure 2

Experimental indication of relative drift between the harmonic trap center and pinning laser beam. Top panels: Each image of expanded BECs, acquired with a procedure described in the text, is an average of five images taken under identical conditions and for the hold times indicated. For $t_h=2$ s, a large fluid-free core indicating pinned vorticity is observed on the right side of the BEC center; this core is presumed to originate at the repeatable initial position of the pinning beam, prior to any drift. The darkness of this core in the BEC after image averaging indicates that this position is consistent from shot to shot. For longer hold times, a consistent drift in the core position relative to its initial position is observed. For $t_h=30$ s, the core is much less visible, and the relative position of the pinning beam and the harmonic trap is therefore much less repeatable from shot to shot. Bottom panel: Colored circles indicate the changes $\Delta x$ and $\Delta y$ of the position of the core in expanded images, relative to the mean core position at $t_h=2$ s, for the hold times shown in the centers of each circle (times given in seconds). The magnitude of statistical uncertainty is indicated with larger dashed circles about the center of each mean position. Position uncertainties for $t_h=30$ s, and later times are unusually large compared with earlier times and are thus not shown. As in the top row of images, data are obtained from averages over five experimental runs at each hold time given.

Figure 3

Maximal growth rate associated to energetically unstable index $\kappa$ for a vortex of charge $S=5$, with $w=7.42$ (equivalent to a $20\mu \mathrm{m}$ laser $1/e^2$ radius) and $N=2\times {10}^6$. Notice the threshold of dynamical stability associated with the vertical line in the top panel and the energetic stability indicated by the vertical line in the bottom panel (the energy spectra was computed on a finer discretization of $V_0$ in order to more precisely determine the threshold between the plotted data points).

Figure 4

Energetic stability threshold radii ($r_0^{cr}$) for various charge-$S$ currents pinned by the external laser. The (blue) circles depict the energetic stability obtained from the Bogoliubov-de Gennes analysis (the resolution in our numerics for $r_0^{cr}$ is $\pm 0.1$). The (red) squares depict experimental distances at which the indicated persistent current charges were observed. Error bars represent statistical uncertainty of the measurement results. All simulations are performed with ($N$,$V_0$, $w$, ${\omega }_r$, ${\omega }_z$, $a$, $m$) corresponding to dimensional parameters ($2\times {10}^6$, $146\hslash {\omega }_r$, $7.42\sqrt{\hslash /2m{\omega }_r}=20\mu \mathrm{m}$, $2\pi \times 8\mathrm{Hz}$, $2\pi \times 90\mathrm{Hz}$, $5.5\mu \mathrm{m}$, $87\mathrm{amu})$.

Figure 5

Snapshots of the density (top row) and phase (bottom row) of the evolution of a perturbed off-center $S=5$ state in the regime of energetic stability for the $S=4$ state ($w=7.42$, $N=2\times {10}^6$, and $V_0=144$). The dissipatively perturbed GPE followed here dynamically destabilizes the $S=5$ state by ejecting one vortex (see vortex inside the ellipse in the middle panels) and subsequently (and for the duration of the dynamical evolution) locks into the energetically stable $S=4$ state.