Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Condensate via the Kibble-Zurek Mechanism

TL;DR

The study demonstrates spontaneous quantum turbulence during a finite-rate BEC transition in a 2D Bose gas using the SPGPE framework with a linear quench across the critical point. It shows that the newborn condensate hosts a vortex tangle whose density scales with the Kibble-Zurek correlation length $\hat{\xi}$, and that the incompressible energy spectrum exhibits Kolmogorov scaling $E_i(k) \propto k^{-5/3}$ within an inertial range set by $\hat{\xi}$; spectra collapse universally across different $\tau_Q$ when plotted against $k\hat{\xi}$, with $\hat{\xi} \propto \tau_Q^{\nu/(1+z\nu)}$ for mean-field exponents $\nu=1/2$, $z=2$, leading to $t_{eq} \propto \tau_Q^{1/2}$ and $E_i \propto \tau_Q^{-1}$. The compressible energy shows near-universal behavior across quenches, while low-temperature phonon emission from vortex annihilation introduces distinct scaling, captured by an ESS-based collapse; velocity structure functions display intermittency consistent with refined Kolmogorov (K62) predictions. Altogether, the work links nonequilibrium critical dynamics (KZM) with turbulent energy cascades in a newborn BEC, establishing nonequilibrium universality in SQT through both energy spectra and defect statistics.

Abstract

The Kibble-Zurek mechanism (KZM) predicts the spontaneous formation of topological defects in a continuous phase transition driven at a finite rate. We propose the generation of spontaneous quantum turbulence (SQT) via the KZM during Bose-Einstein condensation induced by a thermal quench. Using numerical simulations of the stochastic projected Gross-Pitaevskii equation in two spatial dimensions, we describe the formation of a newborn Bose-Einstein condensate proliferated by quantum vortices. We establish the nonequilibrium universality of SQT through the Kibble-Zurek and Kolmogorov scaling of the incompressible kinetic energy.

Figures (12)

• Figure 1: Typical condensate density and phase profiles at equilibration time, together with the vortex-number scaling. Panel (a) shows the condensate density $|\Psi_{\mathcal{C}}(\boldsymbol{r},t_{eq})|^{2}$ at equilibration time $t_{eq}$ following a quench of duration $\tau_{Q}=10$, while panel (b) displays the corresponding phase of $\Psi_{\mathcal{C}}$. Vortices with topological charge $w=+1$ are marked by red crosses and those with $w=-1$ by circles; no vortices with $\lvert w\rvert>1$ were observed. The main plot in panel (c) shows the vortex count $n_v$ at $t_{eq}$ versus quench time $\tau_Q$, demonstrating the Kibble-Zurek power-law scaling for $10\le\tau_Q\le1000$. The inset plots ${t}_{eq}$ against $\tau_Q$, where ${t}_{eq}$ is the equilibration time measured from the crossing of the critical point. Data are averaged over $\mathcal{R}=1000$ realizations, with error bars indicating one standard deviation.
• Figure 2: Incompressible and compressible kinetic energy spectra. Panels (a) and (b) show the incompressible kinetic energy spectrum $E_{i}(k)$ at the equilibration time following quenches with $\tau_{Q}=100$ and $\tau_{Q}=350$, respectively, each averaged over $\mathcal{R}=1000$ independent noise realizations. The corresponding fitted power-law exponents are $\alpha = 1.710 \pm 0.079$, $\beta = 3.025 \pm 0.030$ for (a), and $\alpha = 1.654 \pm 0.031$, $\beta = 3.036 \pm 0.047$ for (b). Panels (c) and (d) display the compressible spectrum $E_{c}(k)$ for the same quench protocols, with fit parameters $(\eta, \beta)=(0.992 \pm 0.008, 3.090 \pm 0.110)$ and $(1.002 \pm 0.008, 3.111 \pm 0.223)$ respectively. The vertical dashed lines mark $k = 2\pi / d_v$, where $d_v = 4 \xi_h$ gives a good estimate of the vortex diameter in our system, while the vertical dotted lines indicate $k = 2 \pi / l_v$, with $l_v$ being the mean nearest intervortex distance. Shaded error bands correspond to one standard deviation.
• Figure 3: Kibble-Zurek universality of the incompressible kinetic energy spectrum. Panel (a) shows $E_i \left( k \right)$ at equilibration time for various $\tau_{Q}$ values, each averaged over $\mathcal{R}=1000$ stochastic realizations. The dashed vertical line marks $k = 2\pi / d_v$ with the vortex diameter estimated to be $d_v = 4 \xi_h$. Panel (b) displays $\left\langle E_i \left( k \right) \right\rangle \tau_Q^{ 3/4 }$ as a function of the scaled momentum $k \tau_Q^{1/4} \propto k \hat{\xi}$. Shaded error bands indicate one standard deviation in (a) and 95% confidence interval in (b).
• Figure S1: Determination of the equilibration time. Panel (a) shows the time evolution of the $\mathcal{C}$-region number density (i.e., $N_{\mathcal{C}}(t)/L^2 = (1 / L^2) \int |\Psi_{\mathcal{C}}(\boldsymbol{r},t)|^2 \, d^{2}\boldsymbol{r}$) for quench times $\tau_{Q}$ from $10$ to $100$. The equilibration time $t_{eq}$ is determined by considering the second derivative of $N_{\mathcal{C}}/L^2$ and identifying the time interval $t_{1}\leq t\leq t_{2}$ such that $d^2 (N_{\mathcal{C}}/L^2) \left( t \right) / d t^2 \le - \Delta$ for $t_1 \le t \le t_2$. In particular, we set $\Delta$ to be $10\%$ of the maximum value of $d^2 (N_{\mathcal{C}}/L^2) \left( t \right) / d t^2$. This is shown for the case $\tau_{Q}=90$ in panel (b). Finally, the equilibration time is defined as $t_{eq}=t_2$.
• Figure S2: Condensate fraction at $t = t_{eq}$ and $t = \tau_Q$. The condensate fraction is shown at equilibration time $t_{eq}$ and at the end of the quench ($t = \tau_Q$). The values at $t_{eq}$ are averaged over $\mathcal{R}=1000$ stochastic realizations (same dataset as in the main text), while those at $t = \tau_Q$ are averaged over $\mathcal{R}=100$ realizations. Error bars denote 95% confidence interval.