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arXiv:2410.08204 · Nature

A universal speed limit for spreading of coherence

Gevorg Martirosyan, Martin Gazo, Jiří Etrych, Simon M. Fischer, Sebastian J. Morris, C. J. Ho, Christoph Eigen, Zoran Hadzibabic

Published Nov 12, 2025• 6 citations
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Abstract

Abstract Discoveries of fundamental limits for the rates of physical processes, from the speed of light to the Lieb–Robinson bound for information propagation 1,2 , often lead to breakthroughs in the understanding of the underlying physics. Here we observe such a limit for a paradigmatic many-body phenomenon, the spreading of coherence during the formation of a weakly interacting Bose–Einstein condensate 3–18 . We study condensate formation in an isolated homogeneous atomic gas 19,20 that is initially far from equilibrium, in an incoherent low-energy state, and condenses as it relaxes towards equilibrium. Tuning the interatomic interactions that drive condensation, we show that the spreading of coherence through the system is initially slower for weaker interactions and faster for stronger ones, but always eventually reaches the same limit, at which the square of the coherence length grows at a universal rate given by the ratio of Planck’s constant and the particle mass, or, equivalently, by the quantum of velocity circulation associated with a quantum vortex. These observations are robust to changes in the initial state, the gas density, and the system size. Our results provide benchmarks for theories of universality far from equilibrium 21–34 , are relevant for quantum technologies that rely on large-scale coherence, and invite similar measurements in other systems.

DOI
10.1038/s41586-025-09735-z

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Abstract Discoveries of fundamental limits for the rates of physical processes, from the speed of light to the Lieb–Robinson bound for information propagation 1,2 , often lead to breakthroughs in the understanding of the underlying physics. Here we observe such a limit for a paradigmatic many-body phenomenon, the spreading of coherence during the formation of a weakly interacting Bose–Einstein condensate 3–18 . We study condensate formation in an isolated homogeneous atomic gas 19,20 that is initially far from equilibrium, in an incoherent low-energy state, and condenses as it relaxes towards equilibrium. Tuning the interatomic interactions that drive condensation, we show that the spreading of coherence through the system is initially slower for weaker interactions and faster for stronger ones, but always eventually reaches the same limit, at which the square of the coherence length grows at a universal rate given by the ratio of Planck’s constant and the particle mass, or, equivalently, by the quantum of velocity circulation associated with a quantum vortex. These observations are robust to changes in the initial state, the gas density, and the system size. Our results provide benchmarks for theories of universality far from equilibrium 21–34 , are relevant for quantum technologies that rely on large-scale coherence, and invite similar measurements in other systems.
Relaxation and condensation of an isolated low-energy Bose gas provide an ideal setting for the study of the universal features of far-from-equilibrium many-body dynamics and the emergence of long-range order. Conceptually, the emergence of such order involves two steps: the formation of local coherence, on a system-specific microscopic lengthscale, and the spreading of coherence, over lengthscales much larger than any microscopic scale. The latter is understood in terms of universal phase-ordering kinetics, or coarsening, characterized by an algebraic growth of the coherence length. Here, for a homogeneous Bose gas with tunable interactions, we show that the former also has a universal description, within the framework of weak wave turbulence (WWT). Specifically, the initial transport of particles to low momenta corresponds to an inverse turbulent cascade that is, in agreement with the WWT theory, characterized by a power-law momentum distribution, with exponent γ=2.4(1)γ= 2.4(1)γ=2.4(1), and transport times ∝(na)−2{\propto} (na)^{-2}∝(na)−2, where nnn is the gas density and aaa the sss-wave scattering length.
Coarsening of an isolated far-from-equilibrium quantum system is a paradigmatic many-body phenomenon, relevant from subnuclear to cosmological length scales and predicted to feature universal dynamic scaling. Here, we observed universal scaling in the coarsening of a homogeneous two-dimensional Bose gas, with exponents that match analytical predictions. For different initial states, we reveal universal scaling in the experimentally accessible finite-time dynamics by elucidating and accounting for the initial-state-dependent prescaling effects. The methods we introduce allow direct comparison between cold-atom experiments and nonequilibrium field theory and are applicable to any study of universality far from equilibrium.
Predicting universal transport coefficients in far-from-equilibrium quantum systems remains a fundamental challenge. A paradigmatic example is the non-thermal fixed point (NTFP) of isolated Bose gases, where coherence spreads as ℓ2(t)=Cℏt/m\ell^2(t) = C\hbar t/mℓ2(t)=Cℏt/m with a universal constant CCC. While the scaling exponent z=2z=2z=2 is well established, the amplitude CCC has remained elusive because the underlying particle cascade n(k)∼k−4n(k)\sim k^{-4}n(k)∼k−4 leads to a divergent kinetic energy, threatening the very existence of a constant speed limit. Here we resolve this paradox and present the first analytical, parameter-free prediction of a universal amplitude CCC. A deep interplay between symmetry and dissipation is uncovered. The emergent weak U(1) symmetry at the NTFP enforces a conserved total current, forcing the low-energy phase dynamics to obey a diffusive Langevin equation with noise entering as the divergence of a stochastic current. This structure, combined with dynamical decoherence of high-momentum modes, yields a universal power-law momentum distribution f~(v)∼(1+v2)−3\tilde{f}(v)\sim(1+v^2)^{-3}f~​(v)∼(1+v2)−3 (with v=kℓv=k\ellv=kℓ) that naturally regularizes the ultraviolet divergence. From this, a parameter-free geometric baseline C=3C=3C=3 is obtained, independent of microscopic details. The experimental value C=3.4(3)C=3.4(3)C=3.4(3) [Martirosyan et al., Nature 647, 608 (2025)] is then shown to be quantitatively consistent with universal logarithmic corrections arising from a marginally irrelevant coupling at the fixed point. A new paradigm is thus established for predicting transport coefficients in strongly correlated non-equilibrium systems: symmetry constraints determine the low-energy effective theory, dynamical decoherence provides a natural ultraviolet completion, and scaling analysis delivers testable predictions moving beyond scaling exponents to quantitative amplitude prediction.
Temporal coherence is a fundamental property of macroscopic quantum systems, such as lasers in optics and Bose-Einstein condensates in atomic gases and it is a crucial issue for interferometry applications with light or matter waves. Whereas the laser is an ``open'' quantum system, ultracold atomic gases are weakly coupled to the environment and may be considered as isolated. The coherence time of a condensate is then intrinsic to the system and its derivation is out of the frame of laser theory. Using quantum kinetic theory, we predict that the interaction with noncondensed modes gradually smears out the condensate phase, with a variance growing as At2+Bt+C\mathcal{A}{t}^{2}+\mathcal{B}t+\mathcal{C}At2+Bt+C at long times ttt, and we give a quantitative prediction for A\mathcal{A}A, B\mathcal{B}B, and C\mathcal{C}C. Whereas the coefficient A\mathcal{A}A vanishes for vanishing energy fluctuations in the initial state, the coefficients B\mathcal{B}B and C\mathcal{C}C are remarkably insensitive to these fluctuations. The coefficient B\mathcal{B}B describes a diffusive motion of the condensate phase that sets the ultimate limit to the condensate coherence time. We briefly discuss the possibility to observe the predicted phase spreading, also including the effect of particle losses.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2L^2L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01\textit{Full-text embargoed until: 2026-01-01}Full-text embargoed until: 2026-01-01
Lieb-Robinson and related bounds set an upper limit on the rate of spreading of information in non-relativistic quantum systems. Experimentally, they have been observed in the spreading of correlations in the Bose-Hubbard model after a quantum quench. Using a recently developed two particle irreducible (2PI) strong coupling approach to out-of-equilibrium dynamics in the Bose-Hubbard model we calculate both the group and phase velocities for the spreading of single-particle correlations in one, two and three dimensions as a function of interaction strength. Our results are in quantitative agreement with measurements of the velocities for the spreading of single particle correlations in both the one and two dimensional Bose-Hubbard model realized with ultra-cold atoms. They also agree with the claim that the phase velocity rather than the group velocity was observed in recent experiments in two dimensions. We demonstrate that there can be large differences between the phase and group velocities for the spreading of correlations and also explore the variation of the anisotropy in the velocity at which correlations spread across the phase diagram of the Bose-Hubbard model. Our results establish the 2PI strong coupling approach as a powerful tool to study out-of-equilibrium dynamics in the Bose-Hubbard model in dimensions greater than one.