Kibble-Zurek Scaling and Spatial Statistics in Quenched Binary Bose Superfluids

Abstract

We study how gradually changing the chemical potential causes a two-dimensional binary Bose gas to condense from vacuum to finite density, resulting in either a mixed (miscible) or separated (immiscible) state depending on interaction strengths. In the immiscible case, random domains form, and their number, boundary length, and average size at the point of equilibration follow universal Kibble-Zurek (KZ) scaling with the cooling rate. These patterns continue to evolve in a self-similar way while maintaining KZ scaling. In the miscible regime, instead of domains, vortices appear, following the KZ scaling. The spatial distribution of domains and vortices is described by a Poisson point process with the KZ density. These findings highlight robust, universal features of how binary superfluids behave far from equilibrium, extending beyond the KZ theory.

Figures (14)

• Figure 1: Norm $\mathcal{N}(t)$ that represents the scaled number density of atoms is plotted during growth into (a) immiscible and (c) miscible phases for quench times $\tau_Q = [5, 10, 20, \dots, 100]$, arranged left to right for a single noise realization. The equilibration time $t_{\text{eq}}$ (black triangles) scales as $t_{\text{eq}} \propto \tau_Q^{0.48 \pm 0.002}$ for $g_{12} > g$ [inset of (b)] and $t_{\text{eq}} \propto \tau_Q^{0.48 \pm 0.002}$ for $g_{12} < g$ [inset of (d)], shown by red lines. (b,d) The early exponential growth ($t < t_{\text{eq}}$) collapses when time is scaled by $\tau_Q^{1/2}$, except for $\tau_Q = 5, 10$. Filled red markers on $\tau_Q = 50$ curves in (a,d) indicate times corresponding to density profiles in Fig. \ref{['density-growth']}.
• Figure 2: Condensate density evolution $|\psi_1|^2$ for $\tau_Q = 50$. Panels (a)–(c) show domain formation and evolution in the immiscible phase; (e)–(g) show vortex formation in the miscible phase, corresponding to red markers in Fig. \ref{['number-growth']}(a) and (c). Panel (d) shows the domain number $N_D$ as a function of $\tau_Q$ on a log scale, revealing KZM-like scaling: $N_{D_1} \propto \tau_Q^{-0.48 \pm 0.017}$, $N_{D_2} \propto \tau_Q^{-0.49 \pm 0.024}$. Panels (e)–(g) highlight the vortex core formation, with blue/red dots indicating positive/negative winding numbers. Panel (h) shows vortex number scaling: $N_{V_1} \propto \tau_Q^{-0.48 \pm 0.019}$, $N_{V_2} \propto \tau_Q^{-0.47 \pm 0.014}$ , averaged over 100 noise realizations. Error bars show standard deviations across these realizations. Grey shaded areas in (d) and (h) denote deviation from universal KZM behavior, indicating a smooth crossover between fast and slow quenches.
• Figure 3: (a) Average domain wall length $L$ at $t_{\text{eq}}$ as a function of the quench time $\tau_Q$ (log scale), averaged over $\mathcal{R} = 100$ runs, showing $L \propto \tau_Q^{-0.23 \pm 0.003}$. (b) Time evolution of $L$, rescaled by $\tau_Q^{1/4}$, for various $\tau_Q$; inset shows unscaled $L(t)$ from $t_{\text{eq}}$. (c) Mean domain area $A$ as a function of $\tau_Q$, showing KZM scaling: $A_1 \propto \tau_Q^{0.52 \pm 0.029}$, $A_2 \propto \tau_Q^{0.48 \pm 0.024}$. Error bars show standard deviations over 100 noise realizations.
• Figure 4: (a) Vortex distance distribution, regardless of the charge, for component $j=1$ at $t/t_{\text{eq}} = 0.930$ with $\tau_Q = 50$. The solid line corresponds to the disk line-picking formula with disk radius $R = 17.5$ and 25 bins. (b) The first nearest-neighbor spacing distribution of vortices for the same component and quench time, fitted with Eq. \ref{['kth-spacing']} with $k=1$ (solid line). Histograms are constructed with data from $\mathcal{R} = 400$ noise realizations.
• Figure 5: (a) Domain distance distribution for component $j=1$ at $t/t_{\text{eq}} = 0.930$ with $\tau_Q = 50$. The solid line corresponds to the disk line-picking formula with disk radius $R = 15.5$ and 25 bins. (b) Further, the first nearest-neighbor spacing distribution for mean domain positions follows Eq. \ref{['kth-spacing']} with $k=2$ (solid line). The distributions are computed at $t/t_{\text{eq}} = 0.930$ for $\tau_Q = 50$, using 25-bin histograms constructed from $\mathcal{R} = 400$ noise realizations. (Inset) Density profiles of both components at $t/t_{\text{eq}} = 0.930$, plotted together using a threshold of 50% of the maximum density (see SM), from a single noise realization.