Quench dynamics of an ultracold two-dimensional Bose gas

We study the dynamics of a two-dimensional Bose gas after an instantaneous quench of an initially ultracold thermal atomic gas across the Berezinskii-Kosterlitz-Thouless phase transition, confirming via stochastic simulations that the system undergoes phase-ordering kinetics and fulfills the dynamical scaling hypothesis at late-time dynamics. Specifically, we find in that regime the vortex number decaying in time as $〈N_{\mathrm{v}}〉\propto t^{-1}$, consistent with a dynamical critical exponent $z\approx 2$ for both temperature and interaction quenches. Focusing on finite-size boxlike geometries, we demonstrate that such an observation is within current experimental reach.

Figure 1

Left: equilibrium profiles for the first-order correlation function $g^{\left(1\right)}\left(r\right)$ for different temperatures, in logarithmic scale (the colors are the same ones as in the inset and correspond to the same values of $T/T_{\mathrm{BKT}}^{\infty }$ from low, blue top line, to high, brown bottom line). The fitted algebraic (dotted lines) and exponential (dashed lines) functions are defined as in Eq. (9). The numerical box has size $L_x\times L_y=(50\times 50)\mu \mathrm{m}$, with periodic boundary conditions and contains $20000{}^{87}\mathrm{Rb}$ atoms. The averages are performed over $\mathcal{N}=100$ stochastic realizations for each temperature data set, and we used $\gamma =0.05$. Right: ${\chi }^2$ of the exponential and the algebraic fits (top) and their ratio (bottom) as a function of $T/T_{\mathrm{BKT}}^{\infty }$. The colored points correspond to the colored lines in the left panel; the critical region is reported as a shaded area.

Figure 2

(a) Exponent $\alpha$ of the algebraic fit as in Eq. (9), calculated for samples at thermal equilibrium at different values of $T/T_{\mathrm{BKT}}^{\infty }$ as in Fig. 1. The critical value ${\alpha }_c=0.25$ predicted by BKT theory in the thermodynamic limit is shown as a horizontal dashed green line. (b) Equilibrium quasicondensate fraction $n_{\mathrm{qc}}$ (green solid line and colored big dots), superfluid fraction $n_{\mathrm{sf}}$ (red small dots and solid line), and condensate fraction $n_0$ (blue small dots and dashed line) as a function of temperature. (c) Equilibrium Binder ratio $U$ as defined in Eq. (13) for the same samples. (d) Equilibrium vortex density computed by performing a short time average of the final values of $〈N_{\mathrm{v}}〉$. (e) Equilibrium order parameter $m$ as defined in Eq. (14), in comparison with the equilibrium value of $\sqrt{g_1\left(r\right)}$ computed at the edges of the system.

Figure 3

Phase distribution (a) before, (b) close to, and (c) far after the critical point in a single realization of an instantaneous temperature quench across the BKT transition in a $(200\times 200\mu {\mathrm{m}}^2)$ box with periodic boundary conditions. (d) Normalized correlation function $g^{\left(1\right)}/g_{\mathrm{eq}}^{\left(1\right)}$ as a function of $r$ at different times (inset). The value of $r$ where this quantity is equal to 0.2 (black dots) is used to define the correlation length $L\left(t\right)$, which is then used to plot $g^{\left(1\right)}/g_{\mathrm{eq}}^{\left(1\right)}$ as a function of the rescaled distance $r/L\left(t\right)$. (e) Temporal evolution of the number of vortices (blue, decreasing) and correlation length $L\left(t\right)$ (green, increasing) averaged over $\mathcal{N}=400$ stochastic realizations of the same temperature quench. (f) and (g) Same as in the (d) and (e), but for an instantaneous quench of the interaction parameter $\tilde{g}$ at fixed temperature. For both quench protocols we obtain a scaling exponent $\beta \approx -1$ within the gray-shaded regions. In all simulations we have used $\gamma =0.01$; for this choice of $\gamma$, the shaded region in (e) [(g)] corresponds to $1.9<t<4.8\mathrm{s}$ ($1.0<t<2.1\mathrm{s}$).

Figure 4

Decay of the number of vortices following an instantaneous quench of the interaction coupling constant $\tilde{g}$ for two different boxes with $L_x=L_y=40\mu \mathrm{m}$ (blue squares and solid line) and $100\mu \mathrm{m}$ (green dots and solid line), with hard wall boundary conditions, averaged over $\mathcal{N}=400$ stochastic realizations. Results with the parameters of the LKB-Paris and AMOP-Cambridge experimental setups are shown on the left and on the right, respectively. The scaling law $〈N_{\mathrm{v}}〉\approx t^{-1}$ is shown as a dashed line. Insets: plot in linear scale of the time evolution of the average number of vortices $〈N_{\mathrm{v}}〉$ (blue solid line) and $c$-field density $n_{\mathrm{c}-\mathrm{field}}$ (red dot-dashed line) in $\mu {\mathrm{m}}^{-2}$. The key is the same for both insets. In all simulations $\gamma =0.01$.