Non-equilibrium many-body states can persist for a very long time if, for example, a system is integrable, the transition to the lower free-energy state is inhibited by an energy barrier, or the target equilibrium state is continuously evolving owing to dissipation. Ultracold atomic gases offer excellent possibilities for fundamental studies of non-equilibrium phenomena3,4,5,6,7,8,9,10,11,12,13,14 and have been used to create counter-intuitive states such as repulsively bound atom pairs5 and Mott insulators with attractive inter-particle interactions12.

Our superheated Bose gas is reminiscent of superheated distilled water, which remains liquid above 100 °C. Specifically, as the temperature characterizing the average energy per particle and the populations of the excited states rises above Tc, the cloud remains in the partially condensed phase, which in true equilibrium should exist only below Tc. However, there are also important differences. Boiling of water is a first-order phase transition and is stalled in clean samples by the absence of nucleation centres. In that case the transition is inhibited by an energy barrier. For a second-order phase transition such a barrier does not exist and the superheating we observe is a purely dynamical non-equilibrium effect, which arises because different properties of the system evolve at different rates. In this respect our gas also bears resemblance to the long-lived non-equilibrium spin structures observed in spinor condensates6,9, pre-thermalized states in quenched one-dimensional Bose gases14 and supercritical superfluids predicted to occur in quenched two-dimensional gases15. In all of those cases, however, non-equilibrium states are observed owing to the system’s slow approach to true equilibrium. Here, the system actually evolves away from equilibrium.

In Fig. 1 we summarize the basic idea of our experiments and the key concepts needed to understand them. In an equilibrium gas, a BEC is present only if T<Tc, where Tc depends on the total particle number N, or equivalently if the chemical potential μ>μc. In a standard experiment, after a BEC is produced, it gradually decays because T rises, owing to technical heating, and/or Tc decreases, because N decays through various inelastic processes. As T/Tc increases, elastic collisions redistribute the atoms between the thermal and condensed components, aiming to ensure the equilibrium particle distribution. The BEC atom number, N0, can therefore decay in two ways: by direct inelastic loss, and through elastic transfer of atoms into the thermal component. Here we reduce the rate of the elastic particle transfer by tuning the strength of inter-particle interactions, characterized by the s-wave scattering length a. This protects the BEC deep into the superheated regime, where N0>0 even though T>Tc.

Figure 1: Creating and understanding a superheated Bose-condensed gas.
figure 1

a, In equilibrium, a BEC is present if T<Tc or equivalently μ>μc (here, β = 1/(kBT)). The arrow indicates the cooling trajectory along which a BEC is produced. The insets show measured momentum distributions, with the condensed component indicated in red. b, Two-component picture. The thermal and condensed components have chemical potentials μ′ and μ0, and inelastic decay rates Γ′ and Γ0, respectively. The net flow of particles between the two components, κ, depends on μ′, μ0 and the scattering rate a2. In equilibrium μ′ = μ0 = μeq. The Landau damping of the collective modes in the BEC has a rate . c, Time sequence of the experiment. Reducing a after preparing a BEC reduces the coupling between the two components and extends the condensate lifetime. In the superheated regime μ′<μeq<μc but μ0>μc.

We can understand our observations within the two-fluid picture outlined in Fig. 1b. Here we treat the thermal and condensed components as two coupled subsystems with atom numbers N′ and N0, chemical potentials μ′ and μ0, and instantaneous per-particle inelastic decay rates Γ′ and Γ0, respectively. In equilibrium μ′ = μ0; note that μ0 is defined only if N0>0, so μ0>μc.

The two components are coupled in two ways, both dependent on the scattering length a. First, the local kinetic thermal equilibrium between the collective excitations in the BEC (phonons) and the thermal bath is ensured by Landau damping, the rate of which is (refs 16, 17). Second, the global phase equilibrium (that is, the equilibrium condensed fraction N0/N) is ensured by the elastic scattering with a rate a2. Crucially, owing to the different scalings witha, we find a large parameter space where the two components can be considered to be in kinetic equilibrium while the system is not in global phase equilibrium. In other words, the two components are at the same temperature, but have different chemical potentials.

In our optically trapped 39K gas18, we control a by an external magnetic field tuned close to a Feshbach resonance at 402 G (ref. 19), the dominant source of Γ′ and Γ0 is spontaneous scattering of photons from the trapping laser beams, and Γ0 has an additional contribution from three-body recombination.

The key steps in our experimental sequence are summarized in Fig. 1c. We start by preparing a partially condensed gas in the |F,mF〉 = |1,1〉 hyperfine ground state by evaporative cooling at a = 135a0, where a0 is the Bohr radius18. We then reduce a (over 50 ms) and follow the subsequent evolution of the cloud, probing the atomic momentum distribution by absorption imaging in time-of-flight expansion. Reducing a (at constant N0) initially reduces μ0 below μ′ (ref. 13), but subsequently μ0 decays slower.

In Fig. 2 we quantitatively contrast the equilibrium evolution of a cloud at a = 83a0 and the non-equilibrium evolution at 5a0. In both cases we start at time t = 0 (Fig. 1c) with N0≈2×104 and N≈2×105 at T≈160 nK. In both cases Tc decreases at a similar rate owing to similar N decay. At 5a0, the temperature rises faster owing to less effective evaporative cooling at a fixed optical trap depth.

Figure 2: Equilibrium versus non-equilibrium BEC decay.
figure 2

a, At a = 83a0 the cloud is always in quasi-static equilibrium. The measured N0 is in excellent agreement with the predicted N0eq and vanishes when Teq = Tc; the three separately calculated chemical potentials, μ0, μ′ and μeq, all agree with each other. The dotted green line marks the equilibrium critical time, tc, and the dashed red lines show the experimental bounds on the time when the BEC actually vanishes. b, At 5a0, the BEC persists in the superheated regime (Teq>Tc) for .

Whether the gas is in equilibrium or not, it can always be characterized by two extensive variables, the total particle number N and energy E. We measure these quantities by direct summation of the momentum distribution and its second moment. To measure N0 we count the atoms within the central peak rising above the smooth thermal distribution.

From the measured N(t) alone we calculate the equilibrium Tc(t) (ref. 2). From N(t) and E(t) we calculate the equilibrium intensive thermodynamic variables μeq(t) and Teq(t), and the equilibrium number of condensed atoms, N0eq(t) (refs 2, 20, 21); in these calculations N0eq>0 if and only if Teq<Tc. For comparison, we also directly fit a temperature Tf to the wings of the momentum distribution. In addition, supposing only that the two components are separately in equilibrium, from the measured N0 and N′ we calculate μ0 and μ′. (For theoretical details see Supplementary Information.)

At 83a0 (Fig. 2a) we find excellent agreement between the measured N0 and the N0eq predicted without any free parameters. The BEC vanishes exactly at the equilibrium critical time tc (dotted green line), at which Teq = Tc. Note that the dashed red lines show the experimental bounds on the time when the BEC vanishes. The separately calculated μ0, μ′ and μeq are all consistent and we have also checked that the fitted Tf coincides with the calculated Teq. All this gives us full confidence in our equilibrium calculations.

At 5a0 (Fig. 2b) we observe strikingly different behaviour. The BEC now survives much longer than it would in true equilibrium; . We also see that μ0 and μ′ diverge from each other for t>tc, so the system is moving away from the global phase equilibrium rather than towards it. The observed superheating can thus not be understood as just a transient effect. (Note that μ0μc is always very small owing to weak interactions.)

At 5a0 the gas is not in global phase equilibrium, but it is still a good approximation to view its two components as two equilibrium subsystems at the same (kinetic) temperature, as in Fig. 1b. We have checked that the momentum distribution in the non-condensed component is still fitted well by a thermal distribution, with Tf always within 10% of the calculated Teq(N,E) (see Supplementary Information). For the BEC, in a weakly interacting gas the equilibrium relation μ0(N0) relies on the macroscopic occupation of a single quantum state1, rather than on global equilibrium. Moreover, even for the lowest-energy collective modes we estimate the Landau damping time to be <1 s (refs 16, 17; see Supplementary Information), that is, much shorter than the characteristic timescale of our experiments. Thus, although this distribution is not directly measurable, we expect the distribution of collective excitations in the BEC to be characterized by a temperature T0 that is also close to TeqTf.

These conclusions hold for any a1a0 (see Supplementary Information). Exactly at a = 0 our theoretical picture does break down, because the Landau damping rate vanishes and the BEC has no equilibrium features; the two components are simply completely decoupled. Bearing this small caveat in mind, from here on we refer to T0TfTeq simply as the temperature of the system T.

A way to directly see that the gas is superheated is to suddenly increase the coupling of the BEC to the thermal bath. In Fig. 3 we show the results of two experimental series in which a is quenched (within 10 ms) from 3a0 to 62a0 at different times in the superheated regime. The filled (open) symbols show N0 measured before (after) the quench. The small Γ0 is almost unaffected by the change in a, and the sudden N0 decay is due to the increase in κ (Fig. 1b). For reference, the green line shows the calculated N0eq at 3a0 and orange shading indicates the superheated regime. As shown in the inset of Fig. 3, we have checked that an interaction quench at t<tc (that is, when T<Tc and N0eq>0) does not kill the BEC.

Figure 3: Quenching the superheated Bose-condensed gas.
figure 3

Filled symbols show the evolution of N0 at a = 3a0, the green solid line shows N0eq and orange shading indicates the superheated regime. Open symbols show the rapid decay of the BEC after it is strongly coupled to the thermal bath by an interaction quench to a = 62a0 at time tq. We show two experimental series in which tq = 20 s (black) and 30 s (red). Inset: an interaction quench at a time when T<Tc and N0eq>0 does not kill the BEC. (Note that here we make an even stronger quench, to 100a0.)

As shown in Fig. 4, we have explored the limits of superheating for a range of interaction strengths, including small negative values of a. For a<0, a BEC is stable against collapse only for N0<−C/a, with C≈2×104a0 for our trap parameters22,23,24. However, after N0 drops below this critical value, at small |a| it decays slowly.

Figure 4: Limits of superheating.
figure 4

a, The highest temperature at which we observe a BEC, , scaled to the equilibrium Tc. Close to a = 0 the BEC survives up to ≈1.5Tc. The red line shows the results of our numerical calculations, with the shaded area indicating the theoretical uncertainty. Experimental error bars are statistical. The point at 62a0 is fixed to unity by the absolute atom number calibration2,27. b, Temporal phase diagram. For each value of a we plot the equilibrium tc (green points) and the time at which the BEC actually vanishes (red points). The errors correspond to dashed lines in Fig. 2 and the uncertainty in tc is indicated by the scatter of points. Solid curves are spline fits to the data. For a≈0 the BEC survives in the superheated regime for a whole minute. Inset: numerically calculated phase diagram, with data overlaid.

In Fig. 4a we plot the highest temperature at which we still observe a BEC, , scaled to the equilibrium Tc at the same N. For a→0, the BEC survives up to T≈1.5Tc. (For comparison, this is analogous to superheated water at 280 °C.)

In Fig. 4b we reconstruct the temporal phase diagram of our non-equilibrium gas. Here, a horizontal cut through the graph corresponds to a time series such as shown in Fig. 2. For each a, we plot the measured (red points) and the equilibrium tc (green points). The solid curves are spline fits to the data. The width of the orange-shaded region corresponds to the time that the BEC survives in the superheated regime. For a≈0 this region spans a whole minute.

The phase diagram in Fig. 4b is measured by always starting with N0≈2×104. In general, non-equilibrium behaviour can strongly depend on the initial conditions. However, we find that is essentially constant (within experimental errors) for initial N0 in the range (1–5)×104. The primary reason for this is that the three-body contribution to Γ0 grows with N0; this leads to self-stabilization of the condensed atom number on timescales much shorter than .

We theoretically reproduce our non-equilibrium observations using a two-component model directly corresponding to Fig. 1b. Starting with the measured initial N0, we numerically simulate the evolution of a BEC coupled to a thermal bath characterized by μ′(t). To do this we calculate Γ0 from our experimental parameters, and for κ we use the form25

Here γela2 is the elastic collision rate and A is a dimensionless coefficient. The largest uncertainty in our calculations comes from the theoretical uncertainty in A≈1–10 (ref. 26). (For details see Supplementary Information.)

In Fig. 4a we show the calculated . The red line corresponds to A = 3 and the shaded area to the range A = 1–10. The calculation generally captures our experimental observations well. With A = 3 we obtain quantitative agreement with the data, except exactly at a = 0, where the model is not valid. In the inset of Fig. 4b we show the calculated temporal phase diagram, with A = 3, together with the experimental data. Again the general features of the diagram are captured well for .

The success of our calculations supports a conceptually simple way to think about dynamical non-equilibrium effects near a continuous phase transition. Extending the BEC lifetime by tuning interactions could also have practical benefits for precision measurements and quantum information processing.