Universal

Coarsening of an isolated far-from-equilibrium quantum system is a paradigmatic many-body phenomenon, relevant from subnuclear to cosmological lengthscales, and predicted to feature universal dynamic scaling. Here, we observe 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 are applicable to any quantitative study of universality far from equilibrium.

Understanding the dynamics of order formation in manybody systems is a long-standing problem in contexts including critical phenomena [1], driven systems [2], and active matter [3,4].Of particular interest are general ordering principles that are independent of the microscopic details of a system.A prime example of this is coarsening, the emergence of order characterized by a single lengthscale that grows in time [5,6].In systems that can exchange energy with the environment, coarsening is intuitively linked with cooling, most dramatically through a phase transition.However, the problem is more intricate if a far-from-equilibrium system is isolated and relaxes towards equilibrium only under the influence of internal interactions [7][8][9][10].In this case coarsening commonly involves bidirectional dynamics in momentum space, with most excitations flowing towards lower wavenumbers k, but the conserved energy flowing to high k.
In experiments with highly controllable Bose gases, selfsimilar dynamics akin to the NTFP predictions have been observed in a range of scenarios, including different dimensionalities and relaxation of both particle and spin distributions [24][25][26][27][28][29].However, many experimental and theoretical questions regarding the values of the scaling exponents remain open.A key challenge is that, for a generic initial state, universal scaling can be directly observed only for very long evolution times [10,[30][31][32][33], which are difficult to access experimentally.
Here, we study coarsening in a homogeneous twodimensional (2D) Bose gas [34,35], by engineering different far-from-equilibrium initial states and measuring the momentum distributions n k (k) by matter-wave focusing [36,37].Crucially, by elucidating and accounting for the non-universal effects of initial conditions, we reveal the universal long-time scaling in finite-range experimental data, introducing methods applicable to any quantitative study of universality far from equilibrium.We find that the low-k (IR) coarsening is characterized by the theoretically predicted dynamical exponent z = 2 [19,22] (see also [5,7,18,21,38]), and the form of the self-similarly evolving n k matches an analytical field-theory prediction [19], while the high-k (UV) energy dynamics corresponds to weak four-wave turbulence [39].

The experiment: bidirectional relaxation
We start with a quasi-pure interacting 2D condensate of 7×10 4 atoms of 39 K in the lowest hyperfine state, confined in a square box trap of size L = 50 µm [40].The interactions in the gas, characterized by the scattering length a, are tuneable via the magnetic Feshbach resonance at 402.7 G [41].To prepare our far-from-equilibrium initial states, we temporarily turn off 2. Non-universal dynamics, prescaling, and the universal-scaling exponents.(a) A system starting in a generic initial state takes some non-universal time t1 to join the scaling trajectory (red) associated with a non-thermal fixed point (NTFP).At t1, its state is the same as if it had always been on this trajectory and evolved for some t2 ̸ = t1.At t > t1, it exhibits scaling with respect to the universal-clock time tuni = t − t * , where t * = t1 − t2 can be positive or negative.(b) n0 ∝ (t − t * ) α shows 'prescaling', with a flowing exponent d ln(n0)/d ln(t) = α/(1 − t * /t) that asymptotically approaches α.(c) One can deduce α from finite-t data using the fact that n the interactions (a → 0) and shake the gas with a spatially uniform oscillating force F [see Fig. 1(a)].This destroys the condensate and, as previously studied in 3D [42,43], results in an isotropic highly nonthermal n k distribution.After preparing one of the three different initial states i1-i3 shown in Fig. 1(a), we stop the shaking, reinstate the interactions (a → 30 a 0 , where a 0 is the Bohr radius), and let the gas relax.For i1-i3, the per-particle energy E/k B is, respectively, 56(3) nK, 26(3) nK, and 15(3) nK; in all cases E is sufficiently low for a condensate to emerge during relaxation [44].
In Fig. 1(b) we show, for i2, the bidirectional dynamics of the full n k distribution (left panel) and the growth of n 0 = n k (k = 0), which characterizes the growth of the condensate (right panel).Here t = 0 corresponds to the time when the interactions are switched on and relaxation starts.

Dynamic scaling theory
According to the dynamic scaling hypothesis, the evolution of n k is given by where α and β are the scaling exponents, d is the system dimensionality, f is a dimensionless scaling function, and ℓ 0 is the characteristic lengthscale at an arbitrary reference time t 0 .For bidirectional dynamics, this scaling should apply separately in the IR and the UV, with different α, β, and f .
For the UV dynamics in our gas, the prediction based on the theory of weak four-wave turbulence [39] [46], with α/β = 4 reflecting energy-conserving transport.In this case, the theory does not predict the scaling function f .
Crucially, for a far-from-equilibrium system to display universal scaling dynamics, it must first 'forget' its specific initial state (but not yet approach equilibrium).

Stages of relaxation: from non-universal to universal
Starting with the IR dynamics, specifically the growth of n 0 , for which Eq. ( 1) reduces to  (t uni /t 0 ) β k (µm -1 ) in Fig. 2 we outline the different stages of relaxation and show how to extract α from finite-t data.For a generic initial state, it first takes some non-universal time t 1 for n k to acquire the scaling form f .At t 1 the system joins the scaling trajectory shown in red in Fig. 2(a), which follows Eq. ( 1).However, importantly, if the system had always followed Eq. ( 1), starting with n 0 = 0 at t = 0, it would have arrived to the same point on the scaling trajectory at some t 2 that in general differs from t 1 .Defining t * = t 1 − t 2 and the 'universal-clock time' t uni = t − t * , time-invariance of the Hamiltonian dictates that from thereon n 0 ∝ t α uni = (t − t * ) α [47].The system then exhibits 'prescaling' [31][32][33], with the flowing exponent d ln(n 0 )/d ln(t) = α/(1 − t * /t) [see Fig. 2(b)].Hence, only for t ≫ t * the physical evolution is given by Eq. (1).In essence, by t 1 the system forgets what its initial state was, but the memory that it had not always been on the scaling trajectory fades slowly.
In practice, the regime t ≫ |t * | may be inaccessible; the required t may be prohibitively long, or a (finite-size) system may reach equilibrium before t ≫ |t * |.However, one can deduce α from the finite-t prescaling data (t > t 1 , but not necessarily t ≫ |t * |), as we show in Fig. 2(c) for two different t * values: plotting n 1/α ′ 0 (t) for various α ′ gives a linear plot (with intercept t * ) only for α ′ = α.
In Fig. 2(d) we show the results of such analysis for our data taken with initial states i1-i3.The predicted α = 1 gives three parallel straight lines, with the differences between the initial states fully captured by the intercepts t * .In the inset we show that requiring the linearity of n 1/α 0 for i1-i3 separately gives consistent α values.
For comparison, n 0 (t) plots in Fig. 2(e) show different prescaling for each initial state (each t * value), and the powerlaw fits (black) give slopes that vary between 0.5 and 1.6.These non-universal prescaling exponents, approximated by the finite-range fits, also depend on how one defines t = 0; this choice is arbitrary, because any state during evolution can be treated as the initial one for further evolution.In contrast, our extraction of α is independent of this choice, since any arbitrary shift of t is simply absorbed by t * (see [37]).

Universal coarsening
We now turn to the dynamics of the full IR distributions, leveraging the fact that we have deduced the non-universal t * values for i1-i3 from the n 0 data.In Fig. 3(a) we show five n k curves, for different initial states and evolution times, which illustrate that distributions corresponding to the same t uni = t − t * are the same, irrespective of the initial conditions.
Note that universality requires only the exponents α, β, and κ to be the same for different initial states, while we observe that the IR dynamics for i1-i3, measured with respect to t uni , are essentially identical, even though the three states have very different energies [48].This 'superuniversality' is restricted to states that have the same value of the transport-conserving quantity, which in the IR is the atom number.

UV dynamics
Finally, we discuss the complementary UV dynamics, for which the ideas from Fig. 2 also apply.For an arbitrary initial n k , there is no reason for t 1 or t * to be the same as in the IR, and in the UV there is no fixed k at which n k ∝ (t − t * ) α for t > t 1 .However, the peaks of our energy spectra, ε(k) ∝ k 3 n k , provide a characteristic wavenumber k ε (t) [see Fig. 4(a)], which we extract without presuming the shape of n k .Then Eq. ( 1) gives the asymptotic behavior k ε ∝ t −β , and more generally for t > t 1 one expects: One can thus use the ideas from Fig. 2(c) to determine β and t * .In Fig. 4 we show our results for initial states i1 and i2; for i3, with the lowest total energy, our high-k signal is too weak for a (with t * intercepts) for the predicted β = −1/6, and that treating β as a free parameter gives consistent values, while Fig. 4(b) shows the scaling of the UV distributions according to Eq. ( 1) with t → t uni , α = −2/3, β = −1/6, and no free parameters.Note that the two k 6 ε lines are not parallel, and the two n k sets do not collapse onto the same universal curve, because here the transport-conserved quantity (the energy) is not the same for the two initial states.

Conclusions and outlook
Our experiments provide a comprehensive picture of universal coarsening in a 2D Bose gas dominated by wave excitations, in parameter-free agreement with theoretical predictions.More broadly, we establish methods for direct quantitative comparisons of experiments and analytical field theories of far-from-equilibrium quantum phenomena.Specifically, we show how to extract the theoretically relevant asymptotic longtime evolution from the experimentally accessible finite-time dynamics.In the future, similar studies with vortex-rich initial states [49,50] could reveal dynamic scaling with anomalous exponents [18,51], while studies in gases with a supersolid ground state [52] could reveal a fascinating interplay of coarsening and pattern-formation dynamics.It would also be interesting to extend our work to (miscible) two-component gases, with one component acting as a bath for the other, which would allow studies of the effects of a tuneable openness of the system [21].

Measurements of momentum distributions n k
We use matter-wave focusing [36] to measure n k (k).Following the relaxation time t, we turn off the 2D trap and interactions (a → 0), and turn on harmonic confinement with a tuneable isotropic in-plane frequency f 0 , created by two crossed laser beams.After a quarter of the trap period, Generally, a larger f 0 minimizes the effects of the focusingtrap anharmonicity and gives better signal-to-noise ratio, making it more suitable for measurements of the high-k (UV) part of the distribution, while a smaller f 0 gives better k-space resolution, which is favorable for measuring the low-k (IR) distribution.We use f 0 = 20 Hz (40 Hz) to study the IR (UV) dynamics.As a compromise, we use f 0 = 30 Hz to observe the full n k distributions shown in the left panel of Fig. 1(b).

Insensitivity to laboratory clock shifts
In our experiments t = 0 is naturally defined by the moment we turn on the interactions and the relaxation starts, while in some other protocols the preparation of a far-from-equilibrium state and the onset of relaxation may not be as unambiguously separated.However, accurate determination of the onset of the (initially non-universal) relaxation is neither sufficient nor necessary for the correct measurement of the universal scaling exponents.Even for the 'correct' t = 0, in general t * ̸ = 0 and for non-infinite t one observes prescaling.On the other hand, in our analysis the correct determination of the relevant t uni and the universal scaling exponents is independent of how t = 0 is assigned.
We illustrate this in Fig. S1 for two shifts of the laboratoryclock time, t → t ′ = t+δt, with arbitrary δt = ±30 ms.These shifts lead to dramatic changes in the prescaling exponents observed in the same data [Fig.S1(a)], but in our analysis they are simply absorbed by the deduced t * values, so that the correct t uni and the universal scaling exponent are recovered [Fig.S1(b)].
Additional panels for Fig.

FIG. 1 .
FIG. 1. Bidirectional relaxation in a box-trapped 2D Bose gas.(a) Starting with a quasi-pure condensate in a box of size L = 50 µm, we temporarily turn off the interatomic interactions and drive the gas with an oscillating force F to engineer different far-from-equilibrium momentum distributions i1-i3.(b) After we stop the driving and turn on interactions (at t = 0), the gas exhibits bidirectional relaxation, shown here for i2: most particles flow to the IR (low-k modes), while the net energy flow is to the UV (high-k).The right panel shows n0(t) = n k (k = 0, t), which characterizes the growth of the condensate; for t ≳ 150 ms the observed growth of n0 is limited by our k-space resolution.All measurements are repeated about 30 times and the error bars show standard error of the mean.
FIG.2.Non-universal dynamics, prescaling, and the universal-scaling exponents.(a) A system starting in a generic initial state takes some non-universal time t1 to join the scaling trajectory (red) associated with a non-thermal fixed point (NTFP).At t1, its state is the same as if it had always been on this trajectory and evolved for some t2 ̸ = t1.At t > t1, it exhibits scaling with respect to the universal-clock time tuni = t − t * , where t * = t1 − t2 can be positive or negative.(b) n0 ∝ (t − t * ) α shows 'prescaling', with a flowing exponent d ln(n0)/d ln(t) = α/(1 − t * /t) that asymptotically approaches α.(c) One can deduce α from finite-t data using the fact that n 1/α ′ 0 (t) is linear only for α ′ = α.(d) Analysis of our i1-i3 data.The main panel shows that the predicted α = 1 gives straight lines that differ just in the intercepts t * , and the inset shows χ 2 (α ′ ) for linear fits of n 1/α ′ 0 .The gray symbols indicate early times excluded from the analysis, and the shading in the inset indicates 1/α ′ = 1.00 ± 0.15.(e) Log-log plots of n0(t) show different prescaling for each initial state.The converging colored lines, with slopes 1 for t ≫ |t * |, are the same as the straight ones in (d), while the naïve power-law fits shown in black give slopes that vary between 0.5 and 1.6.

FIG. 3 .
FIG. 3. Universal coarsening.(a) For i1-i3, the IR distributions after the same evolution time t (here 65 ms) are different, but n k at the same universal-clock time tuni = t − t * (here ≈ 60 ms) are the same; the t * values for i1-i3 are those independently determined in Fig. 2(d).(b) The n k curves for all three initial states collapse onto a universal curve (right panel) according to Eq. (1) with t → tuni, α = 1, and β = 1/2, without any free parameters; t0 = 80 ms is an arbitrary reference time.The solid black line shows a fit to the collapsed data with A/(1 + (k/k0) 3 ).
1/(4f 0 ), when the spatial atom distribution reflects n k at the start of focusing, we transfer a variable fraction of atoms from the |F = 1, m F = 1⟩ state to |F = 2, m F = 2⟩, and measure the density distribution by absorption imaging on the cycling |F = 2, m F = 2⟩ → |F ′ = 3, m F ′ = 3⟩ transition.Varying the transferred fraction between 2 and 100% allows us to measure n k values over four orders of magnitude.
FIG. S1.Effects of arbitrary shifts of the laboratory time, t → t ′ = t + δt, illustrated for our IR data.(a) Shifting by δt = ±30 ms results in different prescaling exponents; here slopes are fitted (solid lines) for t ′ > 10 ms.(b) In our analysis method, δt is simply absorbed by a shift in t * , so the correct tuni and α are recovered.The dashed lines show the same n0 ∝ tuni curves in (a) and (b).

Fig. 3 (
Fig.3(b)shows the simultaneous collapse of n k curves for initial states i1-i3, when scaled according to Eq. (1) with t → t uni , α = 1, and β = 1/2.In Fig.S2we separately show the scaling of n k for each initial state.

) 2 FIG
FIG. S2.Data from Fig. 3(b) shown separately for the three initial states i1-i3.In the right panels the curves are collapsed according to Eq. (1) with t → tuni, α = 1, and β = 1/2, and the solid lines are the same as the solid line in the right panel of Fig. 3(b).