Coupling-induced universal dynamics in bilayer two-dimensional Bose gases

TL;DR

This work demonstrates universal, diffusion-like phase-ordering dynamics in a bilayer 2D Bose gas after a rapid interlayer-coupling quench that explicitly breaks the relative-phase $U(1)$ symmetry. By measuring the two-point phase correlation $C(r,t)$ and vortex density $n_v(t)$ via matter-wave interferometry and supporting these observations with classical-field simulations, the authors identify self-similar coarsening characterized by $L_c(t)\sim t^{1/z}$ with $z=1.73(9)$ and a universal scaling collapse of $C(r,t)$ across different initial phase-space densities. The results reveal vortex–antivortex annihilation as the mechanism driving ordering and provide a robust benchmark for nonequilibrium effective field theories in coupled 2D systems, including the 2D XY and sine-Gordon frameworks. This coupling-quench platform opens avenues to explore universal nonequilibrium phenomena such as Kibble–Zurek scaling, light-cone dynamics, reverse-KZ processes, and Josephson effects in 2D quantum gases.

Abstract

The emergence of order in many-body systems and the associated self-similar dynamics governed by dynamical scaling laws is a hallmark of universality far from equilibrium. Measuring and classifying such nontrivial behavior for novel symmetry classes remains challenging. Here, we realize a well-controlled interlayer coupling quench in a tunable bilayer two-dimensional Bose gas, driving the system to an ordered phase. We observe robust self-similar dynamics and a universal critical exponent consistent with diffusion-like coarsening, driven by vortex and antivortex annihilation induced by the interlayer coupling. Our results extend the understanding of universal dynamics in many-body systems and provide a robust foundation for quantitative tests of nonequilibrium effective field theories.

Figures (10)

• Figure 1: Interlayer coupling quench in bilayer 2D quantum gases. (a) (Top) We quench the interlayer coupling $J$ from nearly zero to $34$ Hz, driving an initially decoupled bilayer with free vortices toward a phase-locked, coherent state. (Bottom) Relaxation dynamics are monitored by matter-wave interferometry, with local fluctuations of relative phase captured by optically pumping a thin slice (red sheet) before absorption imaging. A representative pre-quench interference image is shown, with the extracted relative phase profile plotted below. (b) The equilibrium phase diagram for the relative-phase mode, as a function of the interlayer coupling $J$ and phase-space density $\mathcal{D}$, obtained via renormalisation-group analysis mathey_phaselocking_2007rydow_observation_2025; the quench induced by the sudden increase in $J$ drives the system to a bilayer superfluid (BSF). The dashed line indicates the critical points. (c) Numerical simulation of the quench dynamics shows relaxation toward a nearly phase-locked state. Vortices (open circles) and antivortices (filled circles) decay over time through dynamical pairing and annihilation. The phase-locked domains (dark blue to dark red) grow progressively. (d) (Top) Typical single-shot interference image at different hold times after the quench show the evolution from an initial state containing vortices (left), identifiable by sharp phase discontinuities (see Fig. \ref{['fig:corr_vortex_analysis']}), to a phase-coherent state (right). (Bottom) Histogram of phase difference $\Delta\theta=\theta(x)-\theta(x^\prime)$ at fixed distance $\left|x-x^{\prime}\right|=5µ m$, showing suppression of phase fluctuations over time, obtained from 40 experimental runs.
• Figure 2: Dynamical phase-locking transition. (a) Probability density distribution of the relative phase, $P(\theta)$, at selected evolution times after the quench, obtained from experiments (histograms) and simulations (solid lines). (b) Time evolution of the normalized phase distribution, $\tilde{P}(\theta) = P(\theta)/\max(P(\theta))$, from experiments (left panel) and simulation (right panel), illustrating the emergence of phase locking. Each column represents a histogram of the phases (panel a) by colors. (c) Relative-phase order parameter $\langle\cos\theta\rangle$ plotted on a linear-log scale. Experimental data (dots) have error bars obtained via bootstrapping, while simulation results are shown as a solid line (mean) with a shaded region indicating uncertainty. The simulation exhibits a small, damped oscillation at a frequency on the order of the Josephson plasma frequency. Averaged interference patterns from over 40 experimental runs are shown for selected evolution times $t$ (insets).
• Figure 3: Build-up of phase coherence and the self-similar dynamics. (a) Phase correlation function $C(r, t)$ at various hold times after the quench, averaged over 50 realizations with error bars showing the standard error. The spatial decay is fitted with both algebraic and exponential decay models, revealing a transition to algebraic decay at a critical time $t_c$, as indicated by the crossover of reduced $\chi^2$ statistics of the fits (inset). The correlation function at $t_c$ is highlighted by the circular markers with a black outline, connected by black dashed line. We define a characteristic correlation length $L_c(t)$, serving as a proxy for the domain size, by the condition $C(L_c(t), t) = 0.75$; see text for details. (b) Correlation functions at different times beyond $t_c$ collapse onto a single curve, when plotted on a log-log scale. Inset: Power-law growth of the domain size $L_c(t)$, shown on a log-log plot. The black line is a fit to the form $L(t) = A\times t^{1/z_\mathrm{corr}}$. The solid portion of the line denotes the scaling window used for the fit. Error bars denote standard error.
• Figure 4: Universal scaling behavior. (a) Temporal evolution of $\eta(t)$, determined by fitting $C(r)$ with an algebraic model, for four different initial conditions, plotted on a log-log scale. Measurements (dots with error bars denoting fit uncertainties) are shown alongside simulations (represented by shaded regions indicating uncertainty). The horizontal dashed line marks the critical value $\eta_c$ for the onset of algebraic phase coherence. The solid lines are a guide to the eye, illustrating that $\eta$ follows a power-law behavior in the scaling window. (b) Rescaled time evolution of $\eta(t/t_{c})$, demonstrating universal dynamics (indicated by the black solid line). The horizontal error bars arise from the uncertainty in $t_{c}$. The inset shows the critical time $t_{c}$ for different initial conditions. (c) Time evolution of the vortex density $n_{v}(t)$ for different initial conditions on a log-log scale, with error bars indicating the standard error. For $\eta_\mathrm{f}= 0.04$ and $t > 150$ ms, no vortices were observed in the finite dataset obtained in this work. The inset is linear plot for a selected initial condition. (d) Rescaled vortex density $n_{v}(t/t_{c})$, showing universal dynamics. The solid line indicates a power-law behavior consistent with the scaling $t^{-2/1.73}$. The inset shows the fitted values of the dynamic critical exponent for different initial conditions; the dashed line indicates the mean value, and the shaded area represents the associated uncertainty. Open circles denote the exponent values extracted from simulations using the same fitting method.
• Figure 5: Determination of dynamic critical exponent. Histogram of bootstrapped estimates of the critical exponent $z$ obtained from power‑law fits $f(t) = A \times t^{-2/z}$ to the universal evolution of vortex density as a function of $t/t_c$ across all initial conditions. The blue solid line indicates the Gaussian fit to the observed probability density distribution and the black dash‑dotted line marks the mean value $z=1.73$.