Universal non-Gaussian order parameter statistics in 2D superfluids

TL;DR

This work experimentally confirms universal nonGaussian order-parameter statistics in 2D Bose gases across the BKT transition by measuring full distribution functions with matter-wave interferometry, revealing a low-temperature universal Gumbel form for the order parameter and extracting higher moments via spectra such as skewness, kurtosis, and Binder cumulants. It demonstrates a robust collapse of the normalized order-parameter distribution in the superfluid regime, validates finite-temperature XY model predictions after correcting for imaging noise, and identifies a precise critical phase-space density via Binder cumulant analysis. Extending to non-equilibrium, the study shows a universal scaling of higher-moment observables after a coherent splitting quench, with time-rescaled data collapsing onto a parameter-independent curve and described by real-time renormalization group theory, highlighting genuinely non-thermal fluctuations during vortex unbinding. Overall, the results establish interferometry as a powerful tool for accessing higher-order correlations in 2D quantum fluids and provide a framework for exploring prethermalization and non-equilibrium critical dynamics in low dimensions.

Abstract

Fluctuations are an intrinsic feature of many-body systems, and their full statistical distributions reveal a wealth of information about the underlying physics. Of particular interest are non-Gaussian, extreme-value statistics that arise when nontrivial correlations and criticality dominate over the central limit theorem. Strikingly, in two-dimensional (2D) quantum fluids, such effects have been predicted to manifest in the order parameter distribution in the Berezinskii-Kosterlitz-Thouless (BKT) superfluid phase, which approaches a universal extreme-value form in the low-temperature limit. Here, we measure the order parameter statistics of 2D Bose gases across the BKT critical point using matter-wave interferometry. This allows us to confirm the predicted convergence of the observed statistics to a universal Gumbel distribution at low temperatures, to the 0.1% level of the probability density. Furthermore, the intrinsic precision of the atom interferometer allows the robust extraction of higher-moment observables such as skewness and kurtosis; in particular, we report direct measurements of the Binder cumulant which allows us to precisely identify the onset of the phase transition. Extending this approach to the investigation of non-equilibrium systems, we probe vortex unbinding dynamics following a quench across the BKT critical point and identify parameter-independent scaling behaviour of higher moments.

Figures (10)

• Figure 1: Matter-wave interferometric probe of order parameter statistics in the 2D XY universality class. (a), 2D XY model spin configurations at various temperatures across the BKT transition. At low temperatures ($T/T_c =0.7$), the system is quasi-ordered with only spin-wave excitations. In the crossover region ($T/T_c =1.0$), vortex-antivortex pairs appear together with some unpaired vortices. In the normal phase ($T/T_c =1.3$), vortex excitations proliferate (red circles and squares, representing vortices and antivortices). (b), Numerically simulated order parameter distributions of the 2D XY model across the BKT critical point, exhibiting significant non-Gaussian features. The distribution in the superfluid regime has a universal Gumbel form. (c), Experimental protocol. Two independent but identical 2D Bose gases are prepared, whose phase fluctuations are illustrated with the colour map. Matter-wave interferometry turns relative-phase fluctuations into interference patterns, which are integrated over a 2D area. This gives a single realization of the interferometric contrast $C$, which is sampled over many experimental realizations to obtain its full distribution. (d), Experimental implementation. After time-of-flight expansion of the 2D Bose gases, interference patterns are detected by absorption imaging; integration along $y$ is realised intrinsically by absorption imaging with a controlled depth (red sheet indicates selective repumping Sunami2022), and along $x$ by image post-processing. The contrast is extracted from the integrated interference signal. (e), Examples of observed contrast statistics across the BKT transition, exhibiting non-Gaussian features. More than 1000 samples are obtained from over 200 experimental repeats.
• Figure 2: Convergence of order parameter statistics to universal Gumbel distribution. (a), Probability histograms of the experimental $C^2$ at various values of $\mathcal{D}$ compared to Monte Carlo (MC) simulations (dashed lines). (b), Empirical cumulative distributions (eCDF) and survival functions (eSF), each obtained from at least 900 contrast values, plotted on a logarithmic scale to highlight the collapse of the distributions onto a universal curve over several orders of magnitude. The shaded area indicates the 99 percent confidence interval for the highest $\mathcal{D}$ dataset. Inset shows the scaled sum-of-squared residuals between the eCDF of the low temperature MC simulation result and each experimental dataset. The shaded region indicates the identified universal regime. (c), Universal order parameter statistics in the low-temperature limit. Histogram of the experimental data within the universal regime ($14 \leq\mathcal{D} \leq 21$), containing over 7500 contrast values. The solid blue line is a kernel density estimate (KDE) and the shaded area indicates its 99 percent confidence interval, obtained via bootstrapping. The dashed lines are from Monte Carlo simulations for $14 \leq\mathcal{D} \leq 21$, analysed with (orange) and without (red) accounting for the systematic effect of imaging noise (Methods).
• Figure 3: Measurements of higher-moment observables associated with contrast FDFs across the BKT critical point. (a),(b), Skewness and kurtosis of the normalised distribution $p(\widetilde{C})$. The simulation data (red solid lines) is from moments obtained by Monte Carlo simulations of interference contrast statistics. (c), Binder cumulant measurement from the raw moments of full distribution functions of interference contrast. Experimental data (purple markers) and simulated data (red open markers) exhibit the characteristic step-like behaviour from 1 to 2 across the BKT critical point. The blue open markers show the result from direct integration of the individual fields from Monte Carlo simulations (the connecting lines are guide to the eye). The green shaded area marks the experimentally detected critical phase-space density from the complementary analysis of the two-point phase correlation functions. The yellow shaded area indicates the experimentally detected critical phase-space density from the analysis of Binder cumulants. The horizontal dashed line shows the critical Binder cumulant, as predicted from the finite-size scaling analysis (Supplementary Information).
• Figure 4: Probing out-of-equilibrium dynamics of 2D Bose gases with interference contrast full distribution measurements. (a), Schematic of the experimental protocol. A single layer 2D Bose gas is split coherently into two clouds. The decoupled gases are then evolved for variable amount of time before probing the system with a time-of-flight interferometer. (b), The coherent splitting quench shown on the phase diagram. (c), Probability histograms of the experimental $C^2$ at various times after the quench. (d), Real-time renormalization group flow diagram for the 2D XY model. The red arrow and the red line (for $x>0,\ y=0$) indicate the fixed points of the flow. The quench induces the equilibrium states in the superfluid phase to be driven into the normal phase (dotted blue trajectory), followed by dynamical vortex unbinding (solid blue trajectory).
• Figure 5: Out-of-equilibrium dynamics of higher-moment observables corresponding to the order parameter of 2D Bose gases following a quench across the BKT transition. (a)-(c), Temporal evolution of skewness, kurtosis and Binder cumulant as a function of time for three different initial conditions, indicated by crossover times $t_c$. (d)-(f), Universal dynamics of higher-moment observables in reduced time $t/t_c$. (g)-(i), Comparison of universal dynamics of higher-moment observables with equilibrium simulations through the mapping $f: \eta \rightarrow m_{eq}$ (blue dashed line).