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Universal dynamics in an isolated one-dimensional Bose gas far from equilibrium

Abstract

Understanding the behaviour of isolated quantum systems far from equilibrium and their equilibration is one of the most pressing problems in quantum many-body physics1,2. There is strong theoretical evidence that sufficiently far from equilibrium a wide variety of systems—including the early Universe after inflation3,4,5,6, quark–gluon matter generated in heavy-ion collisions7,8,9, and cold quantum gases4,10,11,12,13,14—exhibit universal scaling in time and space during their evolution, independent of their initial state or microscale properties. However, direct experimental evidence is lacking. Here we demonstrate universal scaling in the time-evolving momentum distribution of an isolated, far-from-equilibrium, one-dimensional Bose gas, which emerges from a three-dimensional ultracold Bose gas by means of a strong cooling quench. Within the scaling regime, the time evolution of the system at low momenta is described by a time-independent, universal function and a single scaling exponent. The non-equilibrium scaling describes the transport of an emergent conserved quantity towards low momenta, which eventually leads to the build-up of a quasi-condensate. Our results establish universal scaling dynamics in an isolated quantum many-body system, which is a crucial step towards characterizing time evolution far from equilibrium in terms of universality classes. Universality would open the possibility of using, for example, cold-atom set-ups at the lowest energies to simulate important aspects of the dynamics of currently inaccessible systems at the highest energies, such as those encountered in the inflationary early Universe.

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Fig. 1: Cooling quench and late-time evolution.
Fig. 2: Universal scaling dynamics.
Fig. 3: Scaling exponents.
Fig. 4: Universal scaling function and spatially averaged observables.

Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

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Acknowledgements

We thank J. Brand, L. Carr, M. Karl, P. Kevrekidis, P. Kunkel, D. Linnemann, A. N. Mikheev, B. Nowak, M. K. Oberthaler, J. M. Pawlowski, A. Piñeiro Orioli, M. Prüfer, W. Rohringer, C. M. Schmied, M. Schmidt, J. Schole and H. Strobel for discussions. We thank T. Berrada, S. van Frank, J.-F. Schaff and T. Schumm for help with the experiment during data collection. This work was supported by the SFB 1225 ‘ISOQUANT’ and grant number GA677/7,8 financed by the German Research Foundation (DFG) and Austrian Science Fund (FWF), the ERC advanced grant QuantumRelax, the Helmholtz Association (HA216/EMMI), the EU (FET-Proactive grant AQuS, project number 640800) and Heidelberg University (CQD). S.E. acknowledges partial support through the EPSRC project grant (EP/P00637X/1). J.S., J.B. and T.G. acknowledge the hospitality of the Erwin Schrödinger Institut in the framework of their thematic programme ‘Quantum Paths’.

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Nature thanks M. Kolodrubetz and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Contributions

S.E. performed the analysis, adapted the theory and wrote the paper, J.S. designed the experiment; R.B. conducted the experiment and initial data analysis. All authors contributed to interpreting the data and writing the manuscript.

Corresponding author

Correspondence to Jörg Schmiedmayer.

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Extended data figures and tables

Extended Data Fig. 1 Results of random-defect and quasi-condensate models.

The time evolution of the characteristic scales for the experimental data presented in Fig. 4a (initial condition 1) are shown. The resulting temperature T (blue) and defect density ns (red) are shown in the upper panel for the full time evolution. The defect width for the random-defect model is fixed to ξs = 0.087 μm, determined by the mean over the first 25 ms of the evolution. The defect density within the scaling region shows a power-law dependence consistent with the exponent β of the scaling evolution reported in the main text. For later times deviations occur, signalling the end of the scaling region. The quality of the model fit is depicted in the lower panel (black squares), where positive and negative values favour the random-defect and quasi-condensate models, respectively. The random-defect model is strongly preferred for the first roughly 100 ms, after which the system converges to a thermal quasi-condensate within about 400 ms. The absolute values of the reduced χ2 for the random-defect (RD) model are about 1 and 5 for early and late times, respectively; those for the quasi-condensate (QC) model are about 25 and 1.

Extended Data Fig. 2 Rescaling analysis for different initial conditions.

ac, Original (left) and rescaled (right) single-particle momentum distribution n(kt) for different initial conditions (ac correspond to initial conditions 1–3 in Fig. 4a). Each distribution is normalized by the time-dependent atom number N(t) and the time is encoded in the colour scale. The grey dashed vertical lines indicate the scaling regime in k. The scaling exponents α ≈ β and the deviation between them Δαβ = α − β are in excellent agreement with the mean values reported in the main text. We note that here we compare the data for the full experimental resolution in k. The distribution at the reference time t0 = 4.7 ms is given by the grey line; its width indicates the 95% confidence interval.

Extended Data Fig. 3 Likelihood function for different initial conditions.

ac, Two-dimensional likelihood functions (colour scales) and marginal-likelihood functions (top and right) for different initial conditions (ac correspond to initial conditions 1–3 in Fig. 4a). A clear peak at non-zero α ≈ β is visible for each realization, whereas the deviation between the two exponents is Δαβ = α − β ≈ 0. For scan 2 (b), a small condensate may have been present before the quench, which led to the larger extent of the likelihood function. Gaussian fits are in excellent agreement with the marginal-likelihood functions and determine the error of the scaling exponents reported in Extended Data Fig. 2.

Extended Data Fig. 4 Time evolution of scaling exponents for different initial conditions.

ac, Scaling exponents α ≈ β (blue) and deviation between the two exponents Δαβ = α − β (red) for different initial conditions (ac correspond to initial conditions 1–3 in Fig. 4a), determined from the likelihood function for each reference time t0, are in good agreement with the predicted mean (black solid and dashed lines). The error bars denote the standard deviation obtained from a Gaussian fit to the marginal-likelihood function at each reference time separately.

Extended Data Fig. 5 Spatially averaged observables for different initial conditions.

ac, Time evolution of the fraction of particles in the scaling region \(\bar{N}\propto {(t/{t}_{0})}^{{\Delta }_{\alpha \beta }}\) (red) and the mean kinetic energy per particle in the scaling region \({\bar{M}}_{2}\propto {(t/{t}_{0})}^{-2\beta }\) (blue) for different initial conditions (ac correspond to initial conditions 1–3 in Fig. 4a). Within the scaling region (grey-shaded areas), \(\bar{N}\) is approximately conserved. The solid black lines are the approximately conserved value and scaling solutions (5). The error bars indicate the 95% confidence interval.

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Erne, S., Bücker, R., Gasenzer, T. et al. Universal dynamics in an isolated one-dimensional Bose gas far from equilibrium. Nature 563, 225–229 (2018). https://doi.org/10.1038/s41586-018-0667-0

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Keywords

  • Bose Gas
  • Universal Dynamic Scaling
  • Scaling Exponent
  • Momentum Distribution
  • Cold Quantum Gases

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