Observation of universal dynamics in a spinor Bose gas far from equilibrium


Predicting the dynamics of quantum systems far from equilibrium represents one of the most challenging problems in theoretical many-body physics1,2. While the evolution of a many-body system is in general intractable in all its details, relevant observables can become insensitive to microscopic system parameters and initial conditions. This is the basis of the phenomenon of universality. Far from equilibrium, universality is identified through the scaling of the spatio-temporal evolution of the system, captured by universal exponents and functions. Theoretically, this has been studied in examples as different as the reheating process in inflationary Universe cosmology3,4, the dynamics of nuclear collision experiments described by quantum chromodynamics5,6, and the post-quench dynamics in dilute quantum gases in non-relativistic quantum field theory7,8,9,10,11. However, an experimental demonstration of such scaling evolution in space and time in a quantum many-body system has been lacking. Here we observe the emergence of universal dynamics by evaluating spatially resolved spin correlations in a quasi-one-dimensional spinor Bose–Einstein condensate12,13,14,15,16. For long evolution times we extract the scaling properties from the spatial correlations of the spin excitations. From this we find the dynamics to be governed by an emergent conserved quantity and the transport of spin excitations towards low momentum scales. Our results establish an important class of non-stationary systems whose dynamics is encoded in time-independent scaling exponents and functions, signalling the existence of non-thermal fixed points10,17,18. We confirm that the non-thermal scaling phenomenon involves no fine-tuning of parameters, by preparing different initial conditions and observing the same scaling behaviour. Our analogue quantum simulation approach provides the basis with which to reveal the underlying mechanisms and characteristics of non-thermal universality classes. One may use this universality to learn, from experiments with ultracold gases, about fundamental aspects of dynamics studied in cosmology and quantum chromodynamics.

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Fig. 1: Universal dynamics and experimental procedure.
Fig. 2: Scaling in space and time at a non-thermal fixed point.
Fig. 3: Characterization of the scaling regime.
Fig. 4: Robustness of universal dynamics at a non-thermal fixed point.

Data availability

The data presented in this paper are available from the corresponding author upon reasonable request.


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We thank D. M. Stamper-Kurn, J. Schmiedmayer, A. Piñeiro Orioli, M. Karl, J. M. Pawlowski and A. N. Mikheev for discussions. This work was supported by the Heidelberg Center for Quantum Dynamics, the European Commission FET-Proactive grant AQuS (project number 640800), the ERC Advanced Grant Horizon 2020 EntangleGen (project ID 694561) and the DFG Collaborative Research Center SFB1225 (ISOQUANT).

Reviewer information

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

Author information

The experimental concept was developed in discussion among all authors. M.P., P.K. and S.L. controlled the experimental apparatus. M.P., P.K., H.S., S.L. and M.K.O. discussed the measurement results and analysed the data. C.-M.S., J.B. and T.G. elaborated the theoretical framework. All authors contributed to the discussion of the results and the writing of the manuscript.

Correspondence to Maximilian Prüfer.

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Extended data figures and tables

Extended Data Fig. 1 Spin distributions for all evolution times.

a, The panels show the distributions of the transversal spin, Fx, measured at different evolution times as indicated. Initially, we find a narrow Gaussian distribution corresponding to the prepared coherent spin state. The excitations developing in the transversal spin lead to a double-peaked distribution within the interval of 2 s to 10 s. For long evolution times, t > 12 s, the distribution resembles a Gaussian, which is much broader than the initial distribution. b, The spin length and its root-mean-square fluctuation as a function of evolution time are extracted by a fit (see Methods). We find a slow decay of the spin length and nearly constant root-mean-square fluctuations in the scaling regime.

Extended Data Fig. 2 Build-up of transversal spin in momentum space.

Since the angular orientation θ cannot be extracted reliably for short evolution times, we choose to show the Fourier transform of the transversal spin for regimes 1–3 (see Fig. 1). The initial condition, all atoms prepared in mF = 0, is characterized by a flat distribution. There is a fast build-up of long-wavelength spin excitations by more than two orders of magnitude within the first second. This process is followed by a redistribution of momenta leading to the scaling form for times longer than 4 s.

Extended Data Fig. 3 Scaling of structure factor for all experimentally accessible length scales.

Same data as shown in Fig. 2. a, Unscaled data. b, Data rescaled with the scaling exponents reported in the main text. The rescaling does not apply for large momenta, k > 0.04 µm−1.

Source data

Source Data Fig. 2

Source Data Fig. 3

Source Data Fig. 4

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