Dynamical thermalization and vortex formation in stirred two-dimensional Bose-Einstein condensates

We present a quantum-mechanical treatment of the mechanical stirring of Bose-Einstein condensates using classical field techniques. In our approach the condensate and excited modes are described using a Hamiltonian classical field method in which the atom number and (rotating frame) energy are strictly conserved. We simulate a $T=0$ quasi-two-dimensional condensate perturbed by a rotating anisotropic trapping potential. Vacuum fluctuations in the initial state provide an irreducible mechanism for breaking the initial symmetries of the condensate and seeding the subsequent dynamical instability. Highly turbulent motion develops and we quantify the emergence of a rotating thermal component that provides the dissipation necessary for the nucleation and motional damping of vortices in the condensate. Vortex lattice formation is not observed, rather the vortices assemble into a spatially disordered vortex liquid state. We discuss methods we have developed to identify the condensate in the presence of an irregular distribution of vortices, determine the thermodynamic parameters of the thermal component, and extract damping rates from the classical field trajectories.

Figure 1

(Color online) Classical field density at representative times during the evolution. Shown are (a) initial condition: laboratory frame ground state plus vacuum noise, (b) ejection of material during the dynamical instability, (c) nucleation of vortices at the interface between the condensate and the noncondensate material, (d) penetration of vortices into the condensate bulk, (e) nonequilibrium state with rapid vortex motion, and (f) equilibrium state. Vortices are indicated by +, and are shown only where the surrounding density of the fluid exceeds some threshold value. Parameters of the trajectory are given in the text.

Figure 2

(Color online) Fitting procedure for the thermodynamic parameters. Shown are the (time-averaged) radial density distribution times radius (solid line), the effective potential experienced by noncondensate atoms times radius (dashed line), and the fitted distribution of noncondensed atoms (dash-dot line). The dotted line indicates the classical turning point corresponding to energy cutoff $E_R$. The data shown corresponds to the period $t=3000–3010\mathrm{cyc.}$ [Fig. 1f].

Figure 3

(Color online) Evolution of the effective (a) temperature and (b) chemical potential of the noncondensed atoms, as measured by the semiclassical fitting procedure.

Figure 4

(Color online) Rotational response of the field. Shown are (a) vortex density, with dash-dot lines indicating ${\rho }_v∕{\rho }_v^c=0.7$ and ${\rho }_v∕{\rho }_v^c=1$ for reference, (b) classical (dashed line) and quantum moments of inertia, and (c) ratio of quantum and classical moments of inertia. Insets show the behavior of the same quantities over a longer time scale. Note that the initial $L_z$ is finite due to the initial vacuum occupation. For reference dotted lines in (a–c) separate phases of the evolution discussed in the text.

Figure 5

(Color online) Regions used for the evaluation of angular momenta and moments of inertia. Disc $\mathcal{A}$ lies entirely within the condensate, where the flow is (initially) irrotational. Annulus $\mathcal{B}$ (between the solid black line and the dashed line indicating the classical turning point of the condensate band) contains only noncondensed, normal fluid. The field density shown is that at $t=50\mathrm{cyc.}$

Figure 6

(Color online) (a) Azimuthally averaged power spectral density (PSD). Dashed lines indicate the cutoff energy, corresponding classical turning point, and (centrifugally dilated) trapping potential. (b) PSD traces at particular radii. The black (blue) and dark grey (red) lines correspond to radii $r=3.1894r_0$ and $r=11.9575r_0$, respectively. The light grey (cyan) line corresponds to the smallest measured radius. (c) Autocorrelation function (absolute value) obtained from the power spectrum. The dashed line indicates the classical turning point of the condensate band. (d) Autocorrelation magnitude at radii corresponding to Fig. 6b. Data corresponds to the period $t=9900–9910\mathrm{cyc.}$

Figure 7

(Color online) Effective chemical potential of the condensate and thermal cloud. The data displayed is obtained from 10 trap cycle long samples spaced at 100 trap cycle intervals. Inset: Data with samples spaced at 10 trap cycle intervals over the initial decay (rise) of ${\mu }_p\left({\mu }_{\mathrm{th}}\right)$.

Figure 8

(Color online) Plots of (a) the condensate band phase and (b) the corresponding correlation time as a function of radius, near time $t=100\mathrm{cyc.}$

Figure 9

(Color online) Mean vortex velocity measured from the classical field trajectory, revealing the rapid slowing of vortices during the initial stage of nucleation $(t\approx 0–120\mathrm{cyc.})$ and more gradual damping of the remaining collective excitations $(t\approx 100–1000\mathrm{cyc.})$. Inset: Logarithm of the mean vortex velocity (scaled by the minimum velocity measured ${\overline{v}}_{\min }$), and linear least-squares fit to the data over the damping period.

Figure 10

(Color online) Evolution of the field angular momentum for various initial chemical potentials.

Figure 11

(Color online) Equilibrium temperatures and chemical potentials reached as a function of initial chemical potential, as determined by the fitting procedure. Dotted and dash-dot lines indicate the predictions of Eqs. (30, 33), respectively. Numbers indicate times at which the measurements were made (thousands of trap cycles) in cases where numerical results were constrained by time. Points without numbers were measured at ${10}^4\mathrm{cyc.}$

Figure 12

(Color online) (a)–(c) Coordinate space densities of the classical fields with initial chemical potentials ${\mu }_i=5,10,18\hslash {\omega }_r$ at times $t=9900,9900,8800\mathrm{cyc.}$, respectively. (d)–(f) Simulated absorption images generated from the same densities.

Figure 13

(Color online) (a) Vortex motional damping for three representative choices of initial chemical potential ${\mu }_i$. (b) Mean vortex motion (averaged over the period $t=1900-2000\mathrm{cyc}$ as a function of initial chemical potential ${\mu }_i$.

Figure 14

(Color online) Field temperatures and chemical potentials reached as a function of cutoff. Dashed and dash-dot lines indicate the predictions of Eqs. (30, 33), respectively. Measurement times are indicated as in Fig. 11.

Figure 15

(Color online) Decay of the effective condensate chemical potential (as defined in Sec. 5C) for trajectories with initial chemical potential ${\mu }_i=20\hslash {\omega }_r$ and three different values of the energy cutoff $E_R$.