Abstract
Hydrodynamics accurately describe relativistic heavy-ion collision experiments well before local thermal equilibrium is established1. This unexpectedly rapid onset of hydrodynamics—which takes place on the fastest available timescale—is called hydrodynamization2,3,4. It occurs when an interacting quantum system is quenched with an energy density that is much greater than its ground-state energy density5,6. During hydrodynamization, energy gets redistributed across very different energy scales. Hydrodynamization precedes local equilibration among momentum modes5, which is local prethermalization to a generalized Gibbs ensemble7,8 in nearly integrable systems or local thermalization in non-integrable systems9. Although many theories of quantum dynamics postulate local prethermalization10,11, the associated timescale has not been studied experimentally. Here we use an array of one-dimensional Bose gases to directly observe both hydrodynamization and local prethermalization. After we apply a Bragg scattering pulse, hydrodynamization is evident in the fast redistribution of energy among distant momentum modes, which occurs on timescales associated with the Bragg peak energies. Local prethermalization can be seen in the slower redistribution of occupation among nearby momentum modes. We find that the timescale for local prethermalization in our system is inversely proportional to the momenta involved. During hydrodynamization and local prethermalization, existing theories cannot quantitatively model our experiment. Exact theoretical calculations in the Tonks–Girardeau limit12 show qualitatively similar features.
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Data availability
The data for all the figures can be found at https://doi.org/10.7910/DVN/KFGNRH. Source data are provided with this paper.
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Acknowledgements
We acknowledge N. Malvania for preliminary experimental work related to this paper. This work was supported by NSF grants PHY-2012039 (D.S.W., Y.L. and Y.Z.), PHY-2012145 (Y.Z. and M.R.) and DMR-1653271 (S.G.). The computations were carried out at the Institute for Computational and Data Sciences at Penn State University.
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Y.L. carried out the experiments and experimental analysis. Y.Z. carried out the theoretical calculations. D.S.W. oversaw the experimental work. M.R. and D.S.W. oversaw the theoretical work. Y.L, Y.Z, S.G, M.R. and D.S.W were all involved in the analysis and interpretation of the results, and all contributed to writing the paper.
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Extended data figures and tables
Extended Data Fig. 1 Theoretical momentum distributions.
Calculations are done for a single 1D tube in the TG limit at zero temperature. We use the same trap and quench parameters as in the \({\bar{\gamma }}_{0}=2.3\) experiment, and choose the atom number in the tube to be N = 32 in order to match the experimental average energy density. These curves match the experimental evolution times in Fig. 1.
Extended Data Fig. 2 Experimental rapidity distributions for \({\bar{{\boldsymbol{\gamma }}}}_{{\bf{0}}}={\bf{2.3}}\) at a sequence of times after the Bragg scattering quench.
The central part of the rapidity distribution does not appreciably change over 1.5 ms, which is less than 10% of the trap oscillation period. The rapidities measurement for the side peaks are slightly distorted by the fact that the flat potential does not extend far enough for atoms moving that fast (after they have moved approximately 20 μm, they are accelerated slightly). Still, since the distortion is approximately the same for all times within the first approximately 0.5 ms, the fact that the measured distribution does not change implies that that part of the rapidity distribution also remains constant. After approximately1 ms, the side peaks start to be noticeably slowed as they climb up the potential of the Gaussian axial trap.
Extended Data Fig. 3 Theoretical rapidity distributions.
These simulations are done for a single 1D tube in the TG limit at zero temperature with same trap and quench parameters as in the \({\bar{\gamma }}_{0}=2.3\) experiment. The atom number in the tube is chosen to be N = 32 in order to match the experimental average energy density. These curves match the experimental evolution times of Extended Data Fig. 2.
Extended Data Fig. 4 Integrated experimental rapidity energy.
The curves are extracted from the rapidity profiles for \({\bar{\gamma }}_{0}=2.3\) integrating over different 0.2 ħk wide rapidity groups. The different colours denote different rapidity groups (as in Fig. 3), defined in the key. For the average rapidity \(\bar{\theta }=0.5\,\hbar k-1.3\,\hbar k\) curves, we have horizontally shifted the points for times longer than tev = 20 μs in order to better resolve the different rapidity group energies at the same time. The dashed lines show the average energy for each rapidity group. There is no detectable change in the energy of each rapidity group in the first approximately 300 μs, in stark contrast to the 3 dB-scale rapid changes in the energies associated with momentum groups in Fig. 3b.
Extended Data Fig. 5 Effect of finite temperature on momentum distributions and integrated kinetic energy.
a–c, Theoretical momentum distributions for a single 1D tube in the TG limit at 0 nK, 5 nK and 10 nK, respectively (see Methods). We use the same trap and quench parameters as in the \({\bar{\gamma }}_{0}=3.4\) experiment, and choose the atom number in the tube to be N = 29 to match the experimental average energy density. These temperatures are the highest we expect there to be in the experiment. One can see that the effect is mostly to smooth out the sharp peaks. Note also that similar smoothing also results from finite time-of-flight in the experiment [24]. d–f, Corresponding theoretical integrated kinetic energy at 0 nK, 5 nK and 10 nK, respectively. Each curve shows the time evolution of the integrated energy in a different 0.2 ħk0 wide momentum group. Curves in the same momentum group have very similar shapes at zero and finite temperatures. Since hydrodynamization involves a much larger energy scale than that associated with these temperatures, it is not surprising that temperature does not significantly affect the signatures of hydrodynamization. We conclude that the differences between the experimental and theoretical results in Fig. 3 are not a result of finite temperature effects.
Extended Data Fig. 6 The time evolution of the occupations (O) of different momentum groups within the central peak.
a, Experimental curves for \({\bar{\gamma }}_{0}=2.3\) are plotted on a log scale. Each curve is obtained by integrating the area of the normalized momentum distribution within the designated momentum range. Different colours denote the different momentum groups as shown in the legend. The last three momentum groups, 0.30 − 0.36 ħk, 0.36 − 0.42 ħk, and 0.42 − 0.48 ħk, have twice the integration range, so their occupations are divided by 2. b, Theoretical curves for a single 1D gas in the TG limit with the same average energy as in the experiment with \({\bar{\gamma }}_{0}=2.3\). In both the experiment and the theory, the occupation of higher momentum groups evolves faster than the lower ones, as expected for local prethermalization. The fact that each of these curves has a different shape makes it difficult to quantitatively compare time constants among them. The theoretical curves evolve more than twice as fast as the experimental curves, presumably reflecting the difference between infinite and finite g (see Eq. (1)). k = k0 for the experiment and k = 4k0 for the theory.
Extended Data Fig. 7 Time evolution of the FWHM.
a, The evolution of the FWHM of the central peak in the momentum distributions after the Bragg scattering quench for different coupling strengths. To a greater degree than for the p50(tev) curves of Fig. 3, these curves all have different shapes. b, The evolving momentum distribution of the central peak for \({\bar{\gamma }}_{0}=0.94\). These curves correspond to the first five points of the orange curve in a. The momentum distribution clearly evolves during the first 0.05 ms, even though the FWHM does not change. This illustrates that the FWHM is not a reliable marker of the evolution of these momentum distributions.
Extended Data Fig. 8 Relationships among pf, \({{\boldsymbol{\tau }}}_{{{\boldsymbol{p}}}_{{\boldsymbol{f}}}}\), and ϵ.
a, Experimental time constants associated with p40 (red circles), p50 (black squares), and p60 (blue triangles) as functions of \(1/k\sqrt{{\epsilon }}\). As in all the experiments presented in this paper, k is fixed at k0. The time constants are extracted from curves like those in Fig. 4a (see Methods). The dashed lines are least-squares linear fits; the intercepts are −2.5 ± 5.6, −5.8 ± 4.9, −6.0 ± 3.5 for p40, p50, and p60, respectively. The data are consistent with linear relationships between the time constant associated with each pf and the inverse of the square root of average energy per particle. For a given momentum distribution, the actual values of p40 and p60 span a range of ~ ± 30% around the steepest part of the distribution (at ~ p50). b, Experimental p50 vs \(\sqrt{{\epsilon }}\). The green triangles, magenta squares, and orange circles correspond to the initial (\({p}_{50}^{i}\)), middle (\({p}_{50}^{m}\)), and final (\({p}_{50}^{e}\)) values of p50 for each experimental condition, extracted from Fig. 4a. The dashed lines are least-squares linear fits; the intercepts are (3.5 ± 0.9) × 10−2, (1.7 ± 0.6) × 10−2, and (3.8 ± 4.4) × 10−3 for \({p}_{50}^{i}\), \({p}_{50}^{m}\), and \({p}_{50}^{e}\), respectively. The \({p}_{50}^{i}\) points at the lowest ϵ (highest \({\bar{\gamma }}_{0}\)) conditions are more likely to be affected by finite-size corrections to their momentum distributions. The data show a linear relationship between each measured value and \(\sqrt{{\epsilon }}\). c, Theoretical time constants obtained from the p40, p50, and p60 curves simulated with k = 4k0. The time constants are obtained from curves like those in Fig. 4b. The error bars are smaller than the marker size. The solid lines are least-squares linear fits; the intercepts are −0.065 ± 0.047, − 0.47 ± 0.13, and −0.67 ± 0.10 for p40, p50, and p60, respectively. d, Theoretical p50 vs \(\sqrt{{\epsilon }}\), simulated with k = 4k0. The intercepts are (14 ± 1.3) × 10−4, (2.7 ± 0.53) × 10−4, and (−19 ± 1.2) × 10−4, for \({p}_{50}^{i}\), \({p}_{50}^{m}\), and \({p}_{50}^{e}\), respectively. The momentum feature that is most clearly proportional to \(\sqrt{{\epsilon }}\) is \({p}_{50}^{m}\). Since that is the midpoint value of p50 during the evolution, it is likely to be close to \({\bar{p}}_{50}\), the effective momentum to which the p50 measurement is sensitive. Taken all together, this figure shows that, for p50, the time constants are proportional to \(1/\sqrt{{\epsilon }}\), which is in turn proportional to the characteristic momentum being measured. We have repeated the entire analysis for p40 and p60 and the conclusions are the same.
Extended Data Fig. 9 Effect of the average over 1D gases and of finite temperature on p50.
We plot theoretical simulations of p50 in the TG limit with the same trap and quench parameters as in the \({\bar{\gamma }}_{0}=3.4\) experiment. For the average over 1D gases, we use the Thomas-Fermi distribution with experimentally measured RTF = 23 μm and a total particle number of Ntot = 2.75 × 105 (see Methods). To simplify the calculations, we round the particle number in each tube in steps of 5. The circles show the results of ground state simulations for a single tube with N = 29 particles (which matches the experimental average energy density). The squares show the results of ground state simulations after averaging over all the 1D gases, as occurs in the experimental setup. The triangles (stars) show simulations for a single tube with N = 29 particles at a temperature of T = 5 nK (10 nK). We did the finite temperature simulations with a larger discretization a = 3.2 × 10−8 m (a = 4 × 10−8 m) due to numerical limitations (see Methods). The results show no significant changes in the time constant due to either the average over tubes or finite temperature.
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Le, Y., Zhang, Y., Gopalakrishnan, S. et al. Observation of hydrodynamization and local prethermalization in 1D Bose gases. Nature (2023). https://doi.org/10.1038/s41586-023-05979-9
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DOI: https://doi.org/10.1038/s41586-023-05979-9
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