The experiment of Le, Zhang, Gopalakrishnan, Rigol and Weiss (Nature 618, 494 (2023)) observed, in an array of one-dimensional Bose gases subjected to a strong Bragg pulse, both hydrodynamization (rapid redistribution of energy among distant momentum modes) and local prethermalization (slower relaxation of nearby momentum modes toward a local generalized Gibbs ensemble). It left three linked open questions: (i) no quantitatively accurate many-body dynamical theory exists at finite interaction strength for the complete post-quench evolution — only the Tonks–Girardeau limit is exactly computable and reproduces the data qualitatively; (ii) it is untested whether generalized hydrodynamics (GHD) describes the spatial evolution of the rapidity distribution during local prethermalization, i.e. before the local generalized Gibbs ensemble is established, as the authors conjectured; (iii) the origin of the quantitative discrepancies between Tonks–Girardeau theory and the finite-interaction experiment (prethermalization-time slope Cexp≈2.6 CthC_{\mathrm{exp}}\approx 2.6\,C_{\mathrm{th}}, about 2.5-times-larger hydrodynamization energy changes in theory, and more-than-twice-faster theoretical prethermalization) is not understood. The problem is open.

Background

Ultracold atoms tightly confined along two directions realize the Lieb–Liniger model, a gas of bosons with repulsive contact interactions of strength gg in one spatial dimension, governed by H=∫dz [(ℏ2/2m) ∂zΨ†∂zΨ+(g/2)Ψ†Ψ†ΨΨ]H=\int dz\,[(\hbar^2/2m)\,\partial_z\Psi^\dagger\partial_z\Psi+(g/2)\Psi^\dagger\Psi^\dagger\Psi\Psi]. The model is integrable: it has infinitely many conserved charges, its thermodynamics follows from the Bethe ansatz (an exact scattering-based solution), and its excitations are stable fermionic quasiparticles whose momenta are called rapidities. The dimensionless interaction strength γ=mg/(ℏ2n)\gamma=mg/(\hbar^2 n), with nn the 1D density, interpolates between weak coupling (γ≪1\gamma\ll 1) and the Tonks–Girardeau (TG) limit γ→∞\gamma\to\infty, where the bosons map to free spinless fermions and arbitrary out-of-equilibrium dynamics can be computed exactly. At finite γ\gamma, equilibrium properties remain exactly solvable, but genuinely out-of-equilibrium dynamics after a strong drive does not.

Integrability also changes relaxation. Instead of thermalizing, a locally perturbed integrable gas relaxes locally to a generalized Gibbs ensemble (GGE) — a maximum-entropy state fixed by all conserved charges, fully encoded by the rapidity distribution. On large space and time scales, the relaxation is described by generalized hydrodynamics (GHD), introduced by Castro-Alvaredo, Doyon and Yoshimura and by Bertini, Collura, De Nardis and Fagotti: an Euler-scale (i.e. ballistic, collisionless leading-order) equation, ∂tρ(θ,x,t)+∂x[veff(θ)ρ(θ,x,t)]=0\partial_t\rho(\theta,x,t)+\partial_x[v^{\mathrm{eff}}(\theta)\rho(\theta,x,t)]=0, for the space- and time-dependent rapidity distribution ρ\rho, where the effective velocity veffv^{\mathrm{eff}} is obtained from ρ\rho through dressing equations that encode the interactions between quasiparticles. GHD rests on the assumption that each mesoscopic cell is locally described by a GGE, and it has been quantitatively verified in 1D Bose gases for protocols whose initial state is a patchwork of stationary states — a domain wall on an atom chip (Schemmer et al.) and a vacuum–thermal junction in strongly interacting gases (Malvania et al.).

The term hydrodynamization comes from relativistic heavy-ion physics: the empirical fact that hydrodynamic descriptions of the quark-gluon plasma (the deconfined matter created in nucleus–nucleus collisions) work at times far shorter than local thermalization, on the fastest available timescale. Le, Zhang, Gopalakrishnan, Rigol and Weiss (Nature 618, 494 (2023)) realized a cold-atom analog. In their experiment, an array of 1D 87^{87}Rb gases with mean initial couplings γˉ0\bar\gamma_0 from 0.94 to 7.6 is driven by a short, strong Bragg pulse — a standing-wave potential V(z)∝cos⁡(2kz)V(z)\propto\cos(2kz) — which, in the Raman–Nath (short-pulse) regime, replicates the occupied momenta at shifts of ±2ℏk\pm 2\hbar k, ±4ℏk\pm 4\hbar k, ..., raising the energy density far above its initial value. Two distinct processes were measured, via momentum distributions obtained by time-of-flight imaging (the trap is released and the gas imaged after free expansion) and via rapidity distributions obtained from the 1D expansion of the gas in an approximately flat potential: hydrodynamization, a rapid redistribution of energy among distant momentum modes on the timescale 2π/ωhd2\pi/\omega_{\mathrm{hd}} with ωhd=ℏ(2k)2/m\omega_{\mathrm{hd}}=\hbar(2k)^2/m, with visible oscillations; and local prethermalization, a slower redistribution of occupation among nearby momentum modes toward the local GGE, with a timescale inversely proportional to the momentum involved. Exact TG simulations reproduce the phenomenology only qualitatively: energy changes during hydrodynamization are about 2.5 times greater in theory than in experiment, theoretical local prethermalization is more than twice as fast, and the measured local-prethermalization time, which scales as C/(kϵ)C/(k\sqrt{\epsilon}) with kk the Bragg wave number and ϵ\epsilon the average pre-quench energy per particle, has a proportionality constant that is larger in experiment than in theory, Cexp≈2.6 CthC_{\mathrm{exp}}\approx 2.6\,C_{\mathrm{th}}, which the authors attribute to the fixed finite-strength contact interactions (ruling out temperature and averaging-over-tubes effects).

The conceptual novelty is that a Bragg pulse creates coherent superpositions of momentum-shifted components — a state with no description in terms of a rapidity distribution — so the experiment probes GHD outside its established domain. The authors state that outside the hard-core limit no existing theory can calculate the complete dynamics during hydrodynamization and local prethermalization, and they conjecture that GHD still works during local prethermalization, i.e. that it describes the evolution of the rapidity distributions in space even though the local GGE is not yet satisfied — a relaxation of GHD's founding assumption analogous to hydrodynamics working after hydrodynamization but before local thermalization in the quark-gluon plasma, testable theoretically in the hard-core limit and experimentally with ultracold atoms.

Problem Statement

The problem is to establish a quantitative dynamical theory of the strongly Bragg-quenched one-dimensional Bose gas at finite interaction strength, and comprises three linked open questions, with the experimental parameters (mean initial couplings γˉ0≈0.94\bar\gamma_0\approx 0.94–7.6, the Bragg pulse and trap of the experiment) as the primary quantitative benchmark rather than a restriction on generality. Each sub-question has independent standing — any single one can be answered on its own — but the task is complete only when all three are answered:

(1) Construct a quantitatively accurate many-body dynamical theory at finite interaction strength γ\gamma that predicts the complete time-dependent evolution of the gas during both hydrodynamization and local prethermalization following a strong Bragg quench — the evolution of momentum distributions, of the spatially resolved rapidity distribution, and of mode-resolved energies — going beyond the Tonks–Girardeau limit, in which exact calculations exist but reproduce the experiment only qualitatively.

(2) Determine whether generalized hydrodynamics correctly describes the evolution of the spatially resolved rapidity distribution during the local prethermalization regime — that is, whether GHD applies even while the local generalized Gibbs ensemble is not yet locally established. This conjecture can be tested theoretically in the hard-core limit and experimentally with ultracold atoms. A confirmation would demonstrate a further relaxation of the required conditions for GHD, analogous to hydrodynamics working after hydrodynamization but before local thermalization in the quark-gluon plasma; a refutation would establish a sharp boundary of GHD's domain of validity.

(3) Explain the origin of the quantitative discrepancies between the Tonks–Girardeau-limit theory and the finite-interaction experiment — the local-prethermalization proportionality constant Cexp≈2.6 CthC_{\mathrm{exp}}\approx 2.6\,C_{\mathrm{th}} in the scaling τ=C/(kϵ)\tau=C/(k\sqrt{\epsilon}) of the local-prethermalization time (kk the Bragg wave number, ϵ\epsilon the average pre-quench energy per particle; measured experimentally versus its Tonks–Girardeau-theory value), the approximately 2.5-times-larger hydrodynamization energy changes in theory, and the more-than-twice-faster theoretical local prethermalization — including whether and how the fixed finite-strength contact interactions account for them.

Broader directions — extension of the phenomenology and scaling relations to non-integrable one-dimensional systems and higher dimensions, and the precise conditions under which the hydrodynamization and local-prethermalization timescales separate or merge — are outside the scope of this task.

References

  1. Y. Le, Y. Zhang, S. Gopalakrishnan, M. Rigol, and D. S. Weiss, Observation of hydrodynamization and local prethermalization in 1D Bose gases, Nature 618, 494 (2023), DOI 10.1038/s41586-023-05979-9, arXiv:2210.07318, https://arxiv.org/abs/2210.07318
  2. Y. Zhang, Y. Le, D. S. Weiss, and M. Rigol, Timescales and necessary conditions for hydrodynamization in one-dimensional Bose gases, Phys. Rev. A 111, 053306 (2025), DOI 10.1103/PhysRevA.111.053306, arXiv:2405.20376, https://arxiv.org/abs/2405.20376
  3. J. Leiberton and M. Rigol, Hydrodynamization in 1D Bose gases at nonzero temperature, arXiv:2608.16980 (2026), https://arxiv.org/abs/2608.16980
  4. L. Dubois, G. Thémèze, F. Nogrette, J. Dubail, and I. Bouchoule, Probing the Local Rapidity Distribution of a One-Dimensional Bose Gas, Phys. Rev. Lett. 133, 113402 (2024), DOI 10.1103/PhysRevLett.133.113402, arXiv:2312.15344, https://arxiv.org/abs/2312.15344
  5. L. Dubois, G. Thémèze, J. Dubail, and I. Bouchoule, Experimental investigation of a bipartite quench in a 1D Bose gas, SciPost Phys. 20, 008 (2026), DOI 10.21468/SciPostPhys.20.1.008, arXiv:2505.05839, https://arxiv.org/abs/2505.05839
  6. E. Granet, Wavelet representation of hardcore bosons, J. Stat. Mech. (2023) 123102, DOI 10.1088/1742-5468/ad082c, arXiv:2303.17494, https://arxiv.org/abs/2303.17494
  7. B. Doyon, S. Gopalakrishnan, F. Møller, J. Schmiedmayer, and R. Vasseur, Generalized Hydrodynamics: A Perspective, Phys. Rev. X 15, 010501 (2025), DOI 10.1103/PhysRevX.15.010501, arXiv:2311.03438, https://arxiv.org/abs/2311.03438
  8. M. L. Kerr and K. V. Kheruntsyan, The theory of generalised hydrodynamics for the one-dimensional Bose gas, AAPPS Bulletin 33, 25 (2023), DOI 10.1007/s43673-023-00095-2, arXiv:2304.04910, https://arxiv.org/abs/2304.04910
  9. Y. Le, Non-equilibrium experiments with nearly integrable 1D Bose gases, PhD thesis, The Pennsylvania State University (2024), INSPIRE record 2869248, https://inspirehep.net/literature/2869248, https://etda.libraries.psu.edu/catalog/22688yzl339
  10. N. Malvania, Y. Zhang, Y. Le, J. Dubail, M. Rigol, and D. S. Weiss, Generalized hydrodynamics in strongly interacting 1D Bose gases, Science 373, 1129 (2021), DOI 10.1126/science.abf0147, https://doi.org/10.1126/science.abf0147
  11. M. Schemmer, I. Bouchoule, B. Doyon, and J. Dubail, Generalized Hydrodynamics on an Atom Chip, Phys. Rev. Lett. 122, 090601 (2019), DOI 10.1103/PhysRevLett.122.090601, arXiv:1810.07170, https://arxiv.org/abs/1810.07170
  12. O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6, 041065 (2016), DOI 10.1103/PhysRevX.6.041065, arXiv:1605.07331, https://arxiv.org/abs/1605.07331
  13. B. Bertini, M. Collura, J. De Nardis, and M. Fagotti, Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents, Phys. Rev. Lett. 117, 207201 (2016), DOI 10.1103/PhysRevLett.117.207201, arXiv:1605.09790, https://arxiv.org/abs/1605.09790
  14. A. Takács, Y. Zhang, P. Calabrese, J. Dubail, M. Rigol, and S. Scopa, One-body correlations and momentum distributions of trapped one-dimensional Bose gases at finite temperature, Phys. Rev. A 111, 033317 (2025), DOI 10.1103/PhysRevA.111.033317, arXiv:2409.16929, https://arxiv.org/abs/2409.16929
  15. L. Biagetti, M. Łebek, M. Panfil, and J. De Nardis, Generalised BBGKY hierarchy for near-integrable dynamics, arXiv:2408.00593 (2024), https://arxiv.org/abs/2408.00593
  16. O. I. Pâţu, Nonequilibrium dynamics in one-dimensional strongly interacting two-component gases, Phys. Rev. A 108, 053304 (2023), DOI 10.1103/PhysRevA.108.053304, arXiv:2308.00486, https://arxiv.org/abs/2308.00486
  17. Y. Guo, S. Dhar, A. Yang, Z. Chen, H. Yao, M. Horvath, L. Ying, M. Landini, and H.-C. Nägerl, Observation of many-body dynamical localization, Science 389, eadn8625 (2025), DOI 10.1126/science.adn8625, arXiv:2312.13880, https://arxiv.org/abs/2312.13880
  18. M. Panfil and Z. Ristivojevic, Local Properties of the Rapidity Distribution in the Lieb-Liniger Model, Phys. Rev. Lett. 135, 020408 (2025), DOI 10.1103/hpfw-sfw1, arXiv:2410.12986, https://arxiv.org/abs/2410.12986