Understanding the dynamics of far-from-equilibrium many-body systems, including the emergence of long-range order in such systems, is an outstanding problem in physics, relevant from subnuclear to cosmological length scales21,22,23,24,25,26,27,28,29,30,31,32,33,34. Conceptually, far-from-equilibrium relaxation and emergence of coherence have long been linked to decaying turbulence12,21,23,27, which often features self-similar scaling dynamics. More recently, within the framework of nonthermal fixed points (NTFPs)28, theorists have drawn analogies between such dynamics and the equilibrium properties of systems close to a continuous phase transition. Near the transition to an ordered state of matter, such as a superfluid or a ferromagnet, the system is scale-invariant and its salient properties do not depend on the microscopic details35. Analogously, in the NTFP theory, far-from-equilibrium systems, including the early universe undergoing reheating27, quark–gluon plasma in heavy-ion collisions36, quantum magnets37, and ultracold atomic gases38,39,40,41,42,43, generically show dynamic (spatiotemporal) scaling, with scaling exponents that could define far-from-equilibrium universality classes. Recently, far-from-equilibrium dynamic scaling was observed in several experiments with ultracold atoms, in both isolated (relaxing)44,45,46,47,48,49,50 and continuously driven51,52 systems.
Here we go beyond the elegant scaling properties of far-from-equilibrium relaxation44,45,46,47,48,49,50 to address the crucial question of how long it takes to establish long-range order. We study this problem for the paradigmatic macroscopically coherent state, the weakly interacting Bose–Einstein condensate, which is also a textbook example of a superfluid.
We study condensate formation in an isolated homogeneous Bose gas53, trapped in a cylindrical optical box19,20, such as sketched in Fig. 1a (Methods). The gas is prepared far from equilibrium and is initially incoherent but has very low energy and condenses as it relaxes towards equilibrium under the influence of interatomic interactions, characterized by the s-wave scattering length a. As illustrated in Fig. 1a, in a homogeneous system, the (global) condensate grows through coarsening25, the local spreading of coherence. This coarsening is quantified by the growth of the coherence length ℓ, over which the first-order correlation function g 1 (r) decays, and corresponds to narrowing of the momentum distribution n k (k) (in which k is the wavenumber), which is related to g 1 (r) by the Fourier transform.
Fig. 1: Universal coarsening of an isolated Bose gas. a, Real-space cartoon of coarsening. b, Momentum-space relaxation for different far-from-equilibrium initial states. Our initial states P 1,2,3 (left column) have different momentum distributions n k but the same energy, so the gas always relaxes towards the same equilibrium state. For P 1,2,3 , the system takes different times, t 1,2,3 , to evolve to the same n k shown in the middle column, but from this point onwards, it always evolves in the same way. The n k distributions are averages of at least 20 measurements. The red scale bar (top image) denotes 1 μm−1. c, Growth of the coherence length, ℓ (see text). Plotting ℓ2(t) reveals three stages of relaxation: (1) the non-universal initial dynamics; (2) the scaling regime in which ℓ2 grows linearly (dashed lines), as expected for the scaling exponent β = 1/2; and (3) the breakdown of scaling at long times owing to finite-size effects. The curves for P 1,2,3 are parallel, with the initial-state effects captured by the different time offsets t* (intercepts of the dashed lines). d, Dynamic scaling. In the scaling regime, the full low-k distributions for all three initial states (left panel) can be collapsed onto the same curve (right panel) according to equations (2) and (3) with β = 1/2 and t → t uni ≡ t − t*; t 0 = 60 ms is an arbitrary reference time. All error bars show standard errors of the measurements. a.u., arbitrary units. Full size image
Our experiments are performed with 39K atoms and we tune a using a Feshbach resonance, exploring coarsening for a = (50–400)a 0 , in which a 0 is the Bohr radius. Our cylindrical box has radius R = 21(2) μm, length L = 40(4) μm, and volume V = 55(12) × 103 μm3. Our gas density, n = 5.4(1.2) μm−3, corresponds (assuming ideal-gas thermodynamics) to the critical temperature for condensation T c = 127(19) nK. The kinetic energy per particle in our initial incoherent states is ε = k B × 20(2) nK (in which k B is the Boltzmann constant), corresponding to a large equilibrium condensed fraction η = 0.61(4) (see Extended Data Figs. 1 and 2). During coarsening, the total particle number, N ≈ 3 × 105, is essentially constant (see Methods) and the gas is always weakly interacting in the sense that na3 < 10−4. We measure n k by absorption imaging (along the z direction) after time-of-flight expansion, performing the inverse Abel transform on the line-of-sight integrated distributions; just for the images shown in Figs. 1b and 2a, we instead image only slices of the cloud54 corresponding to k z ≈ 0 (Methods). We normalize n k such that ∫n k 4πk2dk = N.
We first show that, although the short-time relaxation dynamics inevitably depend on the details of the initial state, the long-time relaxation does not (Fig. 1b). For this purpose, we engineer three different far-from-equilibrium states, starting with a quasi-pure condensate and using a time-varying force to perturb the cloud (see Extended Data Fig. 1). Our initial states P 1,2,3 have different n k (see left column) but the same ε. At time t = 0, the gas is non-interacting and we then initiate relaxation by switching a to 100a 0 . Starting from an initial state, n k (k) follows some trajectory (represented by the wavy coloured lines) in the space of functions with the same N and ε. The middle column shows that, still far from equilibrium, these trajectories converge to the same n k . The time that the gas takes to evolve to this n k depends on the initial state (t i=1,2,3 for P i=1,2,3 ) but further evolution from this n k is the same for all initial states; note that the state trajectories for P 1 and P 2 merge before merging with the P 3 trajectory, but we just show an n k for which all three have converged.
The long-time relaxation of a low-energy Bose fluid was theoretically studied in different frameworks. In ref. 23, this problem was studied for an incompressible superfluid, for which the spreading of coherence is associated with the decay of a random tangle of quantized vortices (variously known as the Kibble’s vortex tangle22, superfluid turbulence55 and Vinen turbulence26,33) and ℓ is set by the typical distance between the vortex lines. For ℓ ≫ ξ, in which ξ is the size of the vortex core, the prediction is that
$$\frac{{\rm{d}}{\ell }}{{\rm{d}}t}\propto \frac{{\rm{ln}}(A{\ell }/\xi )}{{\ell }},$$ (1)
in which A is a dimensionless constant. On the other hand, for coarsening of wave excitations, corresponding to a compressible-fluid flow, approximate kinetic equations give
$${\ell }(t)\propto {t}^{\beta },$$ (2)
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