Observable vortex properties in finite-temperature Bose gases

We study the dynamics of vortices in finite temperature atomic Bose-Einstein condensates, focusing on decay rates, precession frequencies, and core brightness, motivated by a recent experiment [Science 329, 1182 (2010)] in which real-time dynamics of a single vortex was observed. Using the Zaremba, Nikuni, and Griffin (ZNG) formalism based on a dissipative Gross-Pitaevskii equation for the condensate coupled to a semiclassical Boltzmann equation for the thermal cloud, we find a rapid nonlinear increase of both the decay rate and precession frequency with increasing temperatures. The increase, which is dominated by the dynamical condensate-thermal coupling is also dependent on the intrinsic thermal cloud collisional dynamics; the precession frequency also varies with the initial radial coordinate. The integrated thermal cloud density in the vortex core is for the most part independent of the position of the vortex (except when it is near the condensate edge), with its value increasing with temperature. This could potentially be used as a variant to the method of Coddington et al. [Phys. Rev. A 70, 063607 (2004)] for experimentally determining the temperature.

Figure 1

3D isosurface plots of low condensate density (left, red) and high thermal cloud density (right, blue) for a cloud of $N_c=10000$ ${}^{87}$Rb atoms at a temperature of $0.5T_c$ for the trapping parameters as described in the text. Notice the tubular isosurface of the thermal cloud at a position corresponding to the vortex core in the condensate.

Figure 2

Two-dimensional (2D) densities, integrated along the $z$ direction for the condensate (left) and thermal cloud (right). Notice the effect of the vortex core on the thermal cloud. Trapping parameters are as in text for $T/T_c=0.5$, $N_c=10000$ ${}^{87}$Rb atoms.

Figure 3

Panels (a)–(c) top show $(x_v,y_v)$ trajectories of vortex initially at $r_0\simeq 0.4R_{\mathrm{TF}}$ at temperatures $T/T_c=0.3$ [53 nK (a), black], $0.5$ [89 nK (b), red], and $0.6$ [124 nK (c), blue]. In panels (a)–(c) bottom we plot for these temperatures the corresponding radial coordinate $r_v$ (solid lines) and the exponential fit $r_v=r_0{e}^{\gamma t}$ (dashed lines) with $\gamma =0.001,0.0026,\text{and}0.005$, respectively. These trajectories have been smoothed to remove any numerical “jitter” arising as a result of our tracking routine. Panel (d) shows the corresponding values of the decay rate $\gamma$ for vortices of variable initial position, $r_0={(x_0^2+y_0^2)}^{1/2}$, at these temperatures. The filled symbols indicate the position of the vortex subsequently analyzed in Fig. 4. Trapping parameters are as in text for $N_c=10000$ ${}^{87}$Rb atoms, $T_c$ varies between 175 and 205 nK for the range of temperatures shown. $R_{\mathrm{TF}}\simeq 5l_{\perp }$ with the maximum values of the thermal cloud density occurring at $\approx 0.85R_{\mathrm{TF}}$ in all cases.

Figure 4

Vortex decay dynamics as a function of temperature: values of vortex decay rate $\gamma$ (top) and corresponding vortex precession frequencies ${\omega }_v$ (bottom). Results for different levels of approximation are indicated by (i) black filled circles for all collision ($C_{12},C_{22}$) processes, (ii) open, magenta boxes for particle-transferring ($C_{12}$) collisions only, (iii) blue crosses for thermal-thermal ($C_{22}$) collisions only, (iv) open, green circles for no collisions, and (v) red stars for static thermal cloud approximation. An initial radial vortex offset of $r_0\simeq 0.26R_{\mathrm{TF}}$, as highlighted in Fig. 3, is used for all simulations. Clearly, all collision mechanisms contribute significantly to the decay. Trapping parameters as in text for $N_c=10000$ ${}^{87}$Rb atoms with $T_c$ varying between 175 and 280 nK for the results shown here.

Figure 5

Inset shows the frequencies of vortices with different initial positions $r_0$ in the range $0.1R_{\mathrm{TF}}$ to $0.8R_{\mathrm{TF}}$ (shown by filled circles of different colors) at a temperature of $T/T_c=0.6$. Main panel shows the corresponding averaged curve for this temperature $T/T_c=0.6$ (124 nK; blue filled circles and connecting line). For the remaining temperatures, from bottom to top with increasing temperature, $T/T_c=0$ (green diamonds), $0.3$ (53 nK, black triangles), and $0.5$ ($88.5$ nK, red squares), precession frequencies have been extracted as a function of radial coordinates. Trapping parameters are as in text for $N_c=10000$ ${}^{87}$Rb atoms, $R_{\mathrm{TF}}\approx 5l_{\perp }$ for these temperatures and $T_c$ varies between 175 and 205 nK for the results shown here.

Figure 6

Top panel shows precession frequencies of a vortex for increasing temperature from bottom to top, $T/T_c=0$ (green diamonds, dashed line), $0.2$ (39 nK, black triangles, dashed line), $0.4$ (78 nK, red squares, dashed line), $0.7$ (133 nK, blue circles, dashed line), and the respective analytically predicted frequencies from Eq. (3) (correspondingly colored solid lines). Note that the lines for $T/T_c=0$, 0.2, and 0.4 lie close together and as a result are difficult to identify individually in this figure. Bottom panel shows relative difference between the ZNG results and the GPE prediction at the vortex radial coordinate $0.25R_{\mathrm{TF}}$. For completeness we indicate the values of $N_c/N$ (top axis) in a nonlinear scale. The GPE results are obtained using the same number of condensate atoms as in the ZNG results. Trapping parameters as in text for $N_{\mathrm{TOT}}\approx 6\times {10}^5$ ${}^{87}$Rb atoms at $T_c=190$ nK with $R_{\mathrm{TF}}$ in the range $\approx 9l_{\perp }\text{to}9.5l_{\perp }$ for the range of temperatures indicated.

Figure 7

Decay rate (left) and precession frequency (right) of a vortex at a position $r_v=0.35R_{\mathrm{TF}}$ at temperatures $T/T_c=0.01$ (green diamonds), $0.2$ (39 nK, black triangles), $0.3$ (59 nK, cyan crosses), $0.4$ (78nK, red squares), $0.6$ (114 nK, magenta stars) and $0.7$ (133 nK, blue circles). Parameters are as in Fig. 6.

Figure 8

Local density plotted along the radial line in the $z=0$ plane (left) and density integrated along the $z$ direction (right) of the condensate (solid red line) and thermal cloud (dashed blue line). The dip in the condensate density occurs at the position of the vortex. Trapping parameters are as in text for $N_{\mathrm{TOT}}=10000$ ${}^{87}$Rb atoms at a temperature of $T/T_c=0.7$ where $T_c=177$ nK.

Figure 9

Vortex core brightness $\mathcal{B}$ for a vortex at position $r_v=0.2R_{\mathrm{TF}}$ as a function of temperature. Results for different levels of approximation are indicated by (i) black filled circles for all collision ($C_{12},C_{22}$) processes, (ii) open, magenta boxes for particle-transferring ($C_{12}$) collisions only, (iii) blue crosses for thermal-thermal ($C_{22}$) collisions only, (iv) open, green circles for no collisions, and (v) red stars for static thermal cloud approximation. Trapping parameters as in text for $N_{\mathrm{TOT}}=10000$${}^{87}$Rb atoms, $T_c=177$ nK. $R_{\mathrm{TF}}$ varies between $4.2l_{\perp }\text{to}4.9l_{\perp }$ for these results.

Figure 10

Top-left panel shows vortex trajectory $r_v=(x_v,y_v)$ for condensate containing a vortex with initial position $r_0=0.15R_{\mathrm{TF}}$ at $T/T_c=0.5$. Integrated projected condensate density $n_{2\mathrm{D}}^C({\mathbf{r}}_{\mathbf{v}},t)$ (top-right panel). Bottom panel shows integrated thermal cloud density $n_{2\mathrm{D}}^T({\mathbf{r}}_{\mathbf{v}},t)$ at the center of the vortex core, as a function of radial coordinate, for the temperatures $T/T_c=0.5$ (red squares), $0.6$ (magenta circles), and $0.7T_c$ (blue triangles). Parameters are as in Fig. 9.

Figure 11

Integrated thermal cloud density $n_{2\mathrm{D}}^T$ at the center of the vortex core as a function of temperature. Results for different levels of approximation are indicated by (i) black filled circles for all collision ($C_{12},C_{22}$) processes, (ii) open, magenta boxes for particle-transferring ($C_{12}$) collisions only, (iii) blue crosses for thermal-thermal ($C_{22}$) collisions only, (iv) open, green circles for no collisions, and (v) red stars for static thermal cloud approximation. Parameters are as in Fig. 9.