Figure 1

(Top) Density snapshots tracking the unpinning of a vortex due to global differential rotation, with $\Omega ={\Omega }_0+\dot{\Omega }t$, ${\Omega }_0=0.0$, and $\dot{\Omega }=-{10}^{-3}$. The pinning site is at (1.7,1.7) and $R=8.5$. The color plots of condensate density ${\left|\psi \right|}^2$ are snapshots taken at $\Omega =-0.660$ (top left), $-0.663$ (top center) and $-0.667$ (top right) for a pinning site of strength $V_i=2V_0$ ($V_0=60$). The range of densities represented in the color plots is ${0.8\left|\psi \right|}_{\max }^2$ to ${\left|\psi \right|}_{\max }^2$, to make low-amplitude sound waves visible. The black circles at the center of each snapshot indicate the region in which kinetic energy is calculated in Fig. 2. (Bottom) Total potential energy ${\int V\left|\psi \right|}^2{d}^{2}x$ versus $\Omega$ for pinning sites with strength $V_i=1.5V_0,2.0V_0$, and $2.5V_0$ (dotted, solid, and dashed curves), respectively.Reuse & Permissions

Figure 2

Kinetic energy ${\int \left|\nabla \psi \right|}^2{d}^{2}x$ in a unit disk centered at (0,0) (depicted in the top panels of Fig. 1), for pinning sites with strength $V_i=1.5V_0$, $2.0V_0$, and $2.5V_0$ (dotted, solid, and dashed curves, respectively) for the experiment described by Fig. 1. The inset zooms in on the region $-0.675\le \Omega \le -0.65$ (indicated by the vertical gray lines) for $V_i=2.0V_0$.Reuse & Permissions

Figure 3

(Top) Snapshots of the condensate density ${\left|\psi \right|}^2$ (color; dark/light represents low/high density) at times $t=0.005$, 0.780, 6.500, and 12.995 (left to right). A vortex initially pinned (with strength $2V_0$) at (1.7,1.7) is unpinned by an acoustic pulse launched by an impulsive spike in the potential of strength $4V_0$ at (0.0,0.0) at $t=0.5$. (Bottom) Kinetic energy $E_{\mathrm{kin}}$ integrated within the four unit disks in the top panels, which we label (clockwise from left) as regions A–D. Note that the vertical axis in the bottom panel is logarithmic. The vertical gray line at $t=5.49$ marks the time of the snapshot in Fig. 4 below. Simulation parameters: $R=12.5$, $\Omega =-0.65$, $V_{\mathrm{trap}}=200$, and $\gamma =0.02$.Reuse & Permissions

Figure 4

(Left) Cross sections of the condensate density ${\left|\psi \right|}^2$ taken vertically (solid curve) and horizontally (dotted curve) through a snapshot at $t=5.49$ from the simulation in Fig. 3. (Right) Contours of ${\left|\psi \right|}^2$ (color; dark/light represents low/high density).Reuse & Permissions

Figure 5

(Top) Snapshots of the condensate density ${\left|\psi \right|}^2$ (color; dark/light represents low/high density) at times $t=0.005$, 0.750, 2.495, and 4.985 (left to right, respectively). A vortex is that is initially pinned (with strength $1.97V_0$) in region B is unpinned by an impulsive acoustic pulse of strength $4V_0$ and launched in region A and then repins at a pinning site in region C ($V_i=6V_0$). The snapshots are taken before the initial sound pulse (left), as the wavefront from the sound pulse impacts on the pinned vortex (second from left), as the vortex travels between the two pinning sites (third from left), and once the vortex has repinned at C (right). (Bottom) Kinetic energy $E_{\mathrm{kin}}$ integrated within the four unit disks in the top panels, labeled regions A–D. Note that the vertical axis in the bottom panel is logarithmic. The vertical gray line at $t=3.29$ marks the time of the snapshot in Fig. 6 below. Simulation parameters: $R=12.5$, $\Omega =-0.65$, $V_{\mathrm{trap}}=200$, and $\gamma =0.02$. The gray curves correspond to a control experiment, in which the initial pulse (amplitude $2V_0$) is too weak to unpin the vortex.Reuse & Permissions

Figure 6

(Left) Cross sections of the condensate density ${\left|\psi \right|}^2$ taken vertically (solid curve) and horizontally (dotted curve) through a snapshot at $t=3.29$ from the simulation in Fig. 5. (Right) Contours of ${\left|\psi \right|}^2$ (color; dark/light represents low/high density).Reuse & Permissions

Figure 7

(Top) Snapshots of the condensate density ${\left|\psi \right|}^2$ (color; dark/light represents low/high density) at times $t=0.001$, 0.296, 1.849, 3.697. A vortex that is initially pinned (with strength $2.9V_0$) in region B is unpinned by an impulsive acoustic pulse of strength $4V_0$, launched from (0.0,1.7) and then repins at a pinning site in region C ($V_i=6.75V_0$). The sound waves emitted by the repinning vortex then strike and unpin a vortex pinned at D ($V_i=2.205V_0$). The snapshots are taken before the initial sound pulse (left), as the wavefront from the sound pulse strikes the pinned vortex (second from left), as the vortex travels between the two pinning sites (third from left), and once the vortex in region D has unpinned and is traveling to the left of the container (right). (Bottom) Kinetic energy $E_{\mathrm{kin}}$ integrated within the four unit disks in the top panels, labeled regions A–D. Note that the vertical axis in the bottom panel is logarithmic. The vertical gray line at $t=3.29$ marks the time of the snapshot in Fig. 8 below. Simulation parameters: $R=12.5$, $\Omega =-0.65$, $V_{\mathrm{trap}}=200$, and $\gamma =0.0025$.Reuse & Permissions

Figure 8

(Left) Cross section of the condensate density (shown in the contour plot at right) taken vertically (solid) and horizontally (dotted) through a snapshot at $t=3.51$ for the experiment shown in Fig. 7, when the bottom vortex has unpinned and is traveling away from region D. (Right) Contours of ${\left|\psi \right|}^2$ (color; dark/light represents low/high density).Reuse & Permissions

Figure 9

(Top) Total kinetic energy $E_{\mathrm{kin}}$ as a function of time $t$ for a condensate in a stationary container of radius 8.5. At $t=5.0$ a pulse of height $4V_0$ at (0.85,0.85) is generated. The curves show three different levels of dissipation [$\gamma =0.01$, 0.02, and 0.05 (solid, dotted, and dashed curves, respectively)]. The inset shows a contour plot of the condensate density at $t=10$. (Bottom) As for top but with an $11\times 11$ pinning grid [$V_i=0.5V_0$, $1.0V_0$, and $2.0V_0$ (gray, black, and light gray curves, respectively)]. The $0.5V_0$ and $2.0V_0$ curves have been shifted up and down, respectively, so that the initial value of $E_{\mathrm{kin}}$ agrees with that of the $1.0V_0$ curve.Reuse & Permissions

Figure 10

Snapshots of the condensate density in a container of radius $R=8.5$ containing an $11\times 11$ pinning grid, as a pinned vortex ($V_i=4.0V_0$), initially in region A, is dragged toward another, more weakly pinned, vortex ($V_i=1.5V_0$) in region B. The initial positions of the two vortices are indicated by black circles. The container, including the pinning grid, rotates with angular velocity $\Omega =-0.5$, in the opposite sense to the velocity field generated by the vortex. The top vortex is dragged with unit dimensionless speed directly down toward the bottom vortex, unpinning it when the separation is 2.0.Reuse & Permissions

Figure 11

Lattice energy, $E_{\mathrm{latt}}$ within a unit disk centered on the lower pinned vortex in Fig. 10, as a function of time (bottom axis) and vortex separation (top axis, $d_0$ is the initial vortex separation and $\Delta d$ is the distance traversed by the moving vortex). The five curves correspond to five different pinning strengths for the lower vortex (see legend for $V_i$ values). When the lower vortex unpins, $E_{\mathrm{latt}}$ drops sharply.Reuse & Permissions

Figure 12

Reciprocal of vortex separation $1/d_{\mathrm{unpin}}$ when the lower vortex in Figs. 10 and 11 unpins, as a function of the potential it unpins from $V_i$. The data points are overlaid by a linear regression, as predicted from Eq. (4), with slope 0.825.Reuse & Permissions

Figure 13

(Top) A series of snapshots of the top right quadrant of the condensate density in a container of radius $R=8.5$ with a lattice of $11\times 11$ pinning sites of strength $V_0$. Two pinned vortices are visible, as indicated by the solid (region A) and dashed (region B) circles (pinning strength $0.4V_0$ and $V_0$, respectively). Four additional vortices are pinned in the other three quadrants. At $t=0$ the angular velocity of the container is reduced from $\Omega =0.075$ to $\Omega =0.05$. The vortex in region A unpins at $t\approx 0.4$ and travels toward region B. When the intervortex separation is approximately 2.5 units, the vortex in region B unpins. (Bottom) Contributions to $E_{\mathrm{latt}}$ (in dimensionless units) from region A (solid, plotted against the left-hand vertical axis) and region B (dashed, plotted against the right-hand vertical axis). The sharp increase in each curve (at $t=0$ and $t=1.65$, respectively) marks the unpinning event.Reuse & Permissions

Figure 14

Total condensate angular momentum $\langle {\widehat{L}}_z\rangle$ as a function of container angular velocity $\Omega$ in the presence of zero ($V_i=0$, solid curve), weak ($V_i=0.13$, triple-dot-dashed curve), intermediate ($V_i=1.3$, dashed curve), and strong ($V_i=2.7$, dot-dashed curve) pinning. (Insets) Grayscale plots of condensate density at $t=90$ for $V_i=0$ to $V_i=2.7$ (left to right, top to bottom). The color table runs from dark (low density) to light (high density). Simulation parameters: $R=8.5$, $V_{\max }=200$, $N_c={10}^{-3}{I}_c$.Reuse & Permissions

Figure 15

Formation of a history-dependent core-corona structure in a vortex lattice that does not fill the condensate. (Top) Azimuthal velocity $v_{\phi }$ from simulation output (solid curve), rigid-body model ($v_{\phi }=\Omega r$, $\Omega =1.0$) (dotted curve), and “giant” vortex model ($v_{\phi }=\Omega R_c^2/r$) (dashed curve). (Bottom) Grayscale plot of azimuthal velocity (dark to light indicates low to high velocity); it increases smoothly with distance from the center, punctuated by vortices. Simulation parameters: $R=12.5$, $V_i=0.0$, $V_{\max }=200$.Reuse & Permissions

Figure 16

Angular momentum $\langle {\widehat{L}}_z\rangle$ as a function of time $t$ for spin-down experiments with pinning strengths $V_i=0.0$, 8.3, 16.6, and 33.3 (solid, dotted, dashed, and dot-dashed curves, respectively). Also shown is the $V_i=0.0$ case with no feedback (triple-dot-dashed). Overplotted are analytic predictions for the $V_i=0$ case with (solid gray) and without (triple-dot-dashed gray) feedback. The curves are rescaled vertically to intersect at $t=0$ to correct for the fact that the initial $\langle {\widehat{L}}_z\rangle$ varies with $V_i$ even when $\Omega$ is held constant, due to pinning hysteresis. The grayscale plots (top) show the condensate density at $t=1000$ ($V_i=0.0$ to $V_i=33.3$, left to right). (Inset) The initial state for $V_i=0$, showing the vortex-free corona described in the text (see Sec. 7b1). Simulation parameters: $R=12.5$, $V_{\max }=200$.Reuse & Permissions

Figure 17

Angular velocity $\Omega \left(t\right)$ as a function of time $t$ for spin-down experiments with pinning strengths $V_i=0.0$, 8.3, 16.6, and 33.3 (solid, dotted, dashed, and dot-dashed curves, respectively), corresponding to the angular-momentum curves in Fig. 16. Overplotted are analytic predictions for the $V_i=0.0$ case with feedback resulting from a condensate rotating as a rigid body (gray dot-dashed curve), feedback from a fluid with a vortex-free corona (gray dashed curve), and without feedback (dotted curve). Simulation parameters: $R=12.5$, $V_{\max }=200$.Reuse & Permissions

Figure 18

Angular velocity of the container $\Omega \left(t\right)$ as a function of time $t$ for the $V_i=16.6$ experiment described in Fig. 17. The insets zoom in on the local maxima at $t=439.5$ (bottom left) and $t=572.5$ (top right), revealing oscillations following vortex repinning. Detailed studies (see Sec. 4) associate the oscillations with acoustic radiation. Simulation parameters: $R=12.5$, $V_{\max }=200$.Reuse & Permissions