Finite-temperature vortex dynamics in Bose-Einstein condensates

We study the dynamics of a vortex in an atomic Bose-condensed gas at finite temperature within the Zaremba-Nikuni-Griffin formalism. In a harmonically trapped pancake-shaped condensate, an off-centered vortex is known to decay by spiraling out toward the edge of the condensate. We quantify the dependence of this decay on temperature, atomic collisions, and thermal cloud rotation. Near the trap center where the density varies slowly, we show that our numerical results agree with the predictions of the Hall-Vinen phenomenological friction force model used to describe quantized vorticity in superfluid systems. Our result thus clarifies the microscopic origin of the friction and provides an ab initio determination of its value.

Figure 1

(Color online) Three-dimensional density isosurface showing an off-centered vortex in a pancake-shaped condensate.

Figure 2

(Color online) Position of the vortex $\left(x_v,y_v\right)$ (in units of $a_{\perp }=\sqrt{\hslash /m{\omega }_{\perp }}$) as it spirals out of the condensate at $T=0.5T_c$ (top left, blue), $T=0.6T_c$ (top right, red), and $T=0.7T_c$ (bottom left, green) for a vortex initially located at $\left(1.3a_{\perp },0\right)$. At the bottom right, the radial position $r_v$ of the vortex is plotted versus ${\omega }_{\perp }t$ for each temperature. The trap parameters are ${\omega }_{\perp }=2\pi \times 129\text{Hz}$ and ${\omega }_z/{\omega }_{\perp }=\sqrt{8}$, with $N={10}^4$ ${}^{87}\text{R}\text{b}$ atoms.

Figure 3

(Color online) Density cross sections of the condensate (top row) and thermal cloud (bottom row) at $z=0$ for $T=0.7T_c$ and at the times (a) ${\omega }_{\perp }t=6$, (b) 12, (c) 18, and (d) 24. The colors range from brown or red (high density) to dark blue (low density), with different scales for the condensate and thermal cloud densities.

Figure 4

(Inset) Ln-log plot of the vortex radial position as a function of time for $T=0.5T_c$ (solid line). The dashed line is an exponential fit, $r_v\left(t\right)=r_0{e}^{\gamma t}$, to the data over $0\le {\omega }_{\perp }t\le 50$. The main figure plots the resulting values of $\gamma$ for an initial vortex position of $r_0\simeq 1.3$ (solid circles) and $r_0\simeq 0.65$ (open circles). For comparison, the solid and dashed lines plot the results of FS [37] and DLS [44], respectively.

Figure 5

(Color online) Precession angular frequency of the vortex, ${\omega }_v$, in units of the radial trap frequency ${\omega }_{\perp }$, as a function of the radial position of the vortex, for $T/T_c=0.5$ and $N={10}^4$. Results for two different initial vortex offset positions, $r_0=0.65a_{\perp }$ (squares) and $r_0=1.3a_{\perp }$ (circles), are illustrated. Note that near the center of the condensate ${\omega }_v$ changes little, but near the edge it changes rapidly.

Figure 6

(a) Precession angular frequency of the vortex, ${\omega }_v$, in units of the radial trap frequency ${\omega }_{\perp }$ as a function of reduced temperature $T/T_c$. (b) Friction coefficient $\alpha$ as a function of $T/T_c$ for a pancake trap geometry. The trap parameters are the same as in Fig. 2.

Figure 7

(a) Decay rate $\gamma$ and (b) precession frequency ${\omega }_v$, as a function of total number of atoms $N$, for a temperature $T/T_c=0.5$ and a pancake trap geometry.

Figure 8

Radial position of the vortex vs time for $T=0.6T_c$. The black line is the result with collisions, while the gray line is the result of a collisionless simulation as discussed in the text.

Figure 9

(Color online) Time evolution of vortex radial position for $T=0.7T_c$ and a rotating thermal cloud. The different curves represent varying thermal cloud rotation rates, with ${\Omega }_{\text{th}}=-0.2$ (black, top), ${\Omega }_{\text{th}}=0$ (red), ${\Omega }_{\text{th}}=0.2$ (green), and ${\Omega }_{\text{th}}=0.37$ (blue, bottom).

Figure 10

Values of the decay rate $b$ for different rotation rates of the thermal cloud, ${\Omega }_{\text{th}}$. The solid line plots the function $b=\gamma \left(1-{\Omega }_{\text{th}}/{\omega }_v\right)$ (where $\gamma$ and ${\omega }_v$ are the decay rate and the precession angular velocity, respectively, for a nonrotating thermal cloud), which is the expected dependence from Eq. (10).

Figure 11

(Color online). Vortex lattice decay at $T=0.7T_c$. The top row shows the time evolution (left to right) of a lattice with a vortex initially at the center; the bottom row shows the evolution of a lattice with the same number of vortices but having a ring configuration.