Persistent-current formation in a high-temperature Bose-Einstein condensate: An experimental test for classical-field theory

Experimental stirring of a toroidally trapped Bose-Einstein condensate at high temperature generates a disordered array of quantum vortices that decays via thermal dissipation to form a macroscopic persistent current [T. W. Neely et al., Phys. Rev. Lett. 111, 235301 (2013)]. We perform three-dimensional numerical simulations of the experimental sequence within the stochastic projected Gross-Pitaevskii equation using ab initio determined reservoir parameters. We find that both damping and noise are essential for describing the dynamics of the high-temperature Bose field. The theory gives a quantitative account of the formation of a persistent current, with no fitted parameters.

Figure 1

Schematic of the experimental sequence and simulation parameters [13]. (a) The initial state consists of a trapped Bose gas at $T\sim 0.9T_C$ in a cylindrically symmetric harmonic trap augmented with an optically induced Gaussian obstacle potential. The potential is constant in the $z$ direction and coincident with the symmetry axis of the harmonic trap. In situ absorption images of the experimental atom density are shown in the transverse (b) and axial (c) directions. (d) The center of the harmonic trap is induced to complete a single revolution around the $z$ axis. The obstacle beam executes one circular orbit centered at $[\overline{x}\left(t\right),\overline{y}\left(t\right)]=r_0(1-\cos \kappa t,\sin \kappa t)$, with $r_0=2.875$ $\mu \mathrm{m}$ and $\kappa =6\pi {\mathrm{s}}^{-1}$. This is modeled in the frame of the obstacle beam, with an incoherent (I) region coupled to a coherent (C) region described with the SPGPE (shown schematically, see text). (e) The dissipation has a significant effect during the experimental hold (time $t_h$ after the stir). Absorption images after significant hold times show either (f) many vortex cores after an additional radial expansion stage or (g) a large density minimum corresponding to a persistent current (without radial expansion).

Figure 2

Column densities and phase slices (through the $z=0$ plane) showing the classical-field dynamics of modeling the persistent-current formation experiment, with parameter set (a), $V_0=58\hslash \overline{\omega }$. The blue dashed circles show the boundary of the detection region for vortices that are labeled in the phase profile as positive (cyan plus) or negative (red circle) based on their circulation. Each image is $94\times 94$ $\mu \mathrm{m}$. Rows 1 and 2 show the SPGPE evolution for a single trajectory [44]. Rows 3 and 4 show the dGPE evolution.

Figure 3

Comparison of the total vorticity $\langle N_T\rangle$ [Eq. (3)] between SPGPE theory and experiment. In the main figure the SPGPE ensemble average is shown for $V_0=58\hslash \overline{\omega }$ [SPGPE (a), red thick curve] and $V_0=67\hslash \overline{\omega }$ [SPGPE (b), blue thick curve] and compared with experimental data (equilibrium). In both cases a stable persistent current is formed after $t\sim 3$ s. The dGPE simulations corresponding to $V_0=58\hslash \overline{\omega }$ [dGPE (a), red thin line] and $V_0=67\hslash \overline{\omega }$ [dGPE (b), blue thin line] develop a stable persistent current after $t\sim 25$ s. The inset shows the short-time dynamics. The average times at which there are no free vortices after the stir are shown by the dashed vertical lines at $t=\tau$ for the SPGPE and $t=\kappa$ for the dGPE, where the subscript denotes the parameter set (a) or (b). The shading for each SPGPE curve shows 1 standard deviation, where the lower bound for parameter set (a) and the upper bound for parameter set (b) are shown for clarity.