Abstract
Our understanding of various states of matter usually relies on the assumption of thermodynamic equilibrium. However, the transitions between different phases of matter can be strongly affected by nonequilibrium phenomena. Here we demonstrate and explain an example of nonequilibrium stalling of a continuous, secondorder phase transition. We create a superheated atomic Bose gas, in which a Bose–Einstein condensate (BEC) persists above the equilibrium critical temperature^{1,2}, T_{c}, if its coupling to the surrounding thermal bath is reduced by tuning interatomic interactions. For vanishing interactions the BEC persists in the superheated regime for a minute. However, if strong interactions are suddenly turned on, it rapidly boils away. Our observations can be understood within a twofluid picture, treating the condensed and thermal components of the gas as separate equilibrium systems with a tunable intercomponent coupling. We experimentally reconstruct a nonequilibrium phase diagram of our gas, and theoretically reproduce its main features.
Main
Nonequilibrium manybody states can persist for a very long time if, for example, a system is integrable, the transition to the lower freeenergy 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 nonequilibrium phenomena^{3,4,5,6,7,8,9,10,11,12,13,14} and have been used to create counterintuitive states such as repulsively bound atom pairs^{5} and Mott insulators with attractive interparticle interactions^{12}.
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 T_{c}, the cloud remains in the partially condensed phase, which in true equilibrium should exist only below T_{c}. However, there are also important differences. Boiling of water is a firstorder 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 secondorder phase transition such a barrier does not exist and the superheating we observe is a purely dynamical nonequilibrium effect, which arises because different properties of the system evolve at different rates. In this respect our gas also bears resemblance to the longlived nonequilibrium spin structures observed in spinor condensates^{6,9}, prethermalized states in quenched onedimensional Bose gases^{14} and supercritical superfluids predicted to occur in quenched twodimensional gases^{15}. In all of those cases, however, nonequilibrium 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<T_{c}, where T_{c} depends on the total particle number N, or equivalently if the chemical potential μ>μ_{c}. In a standard experiment, after a BEC is produced, it gradually decays because T rises, owing to technical heating, and/or T_{c} decreases, because N decays through various inelastic processes. As T/T_{c} increases, elastic collisions redistribute the atoms between the thermal and condensed components, aiming to ensure the equilibrium particle distribution. The BEC atom number, N_{0}, 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 interparticle interactions, characterized by the swave scattering length a. This protects the BEC deep into the superheated regime, where N_{0}>0 even though T>T_{c}.
We can understand our observations within the twofluid picture outlined in Fig. 1b. Here we treat the thermal and condensed components as two coupled subsystems with atom numbers N′ and N_{0}, chemical potentials μ′ and μ_{0}, and instantaneous perparticle inelastic decay rates Γ′ and Γ_{0}, respectively. In equilibrium μ′ = μ_{0}; note that μ_{0} is defined only if N_{0}>0, so μ_{0}>μ_{c}.
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 N_{0}/N) is ensured by the elastic scattering with a rate a^{2}. 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 ^{39}K gas^{18}, 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 Γ_{0} is spontaneous scattering of photons from the trapping laser beams, and Γ_{0} has an additional contribution from threebody recombination.
The key steps in our experimental sequence are summarized in Fig. 1c. We start by preparing a partially condensed gas in the F,m_{F}〉 = 1,1〉 hyperfine ground state by evaporative cooling at a = 135a_{0}, where a_{0} is the Bohr radius^{18}. We then reduce a (over 50 ms) and follow the subsequent evolution of the cloud, probing the atomic momentum distribution by absorption imaging in timeofflight expansion. Reducing a (at constant N_{0}) initially reduces μ_{0} below μ′ (ref. 13), but subsequently μ_{0} decays slower.
In Fig. 2 we quantitatively contrast the equilibrium evolution of a cloud at a = 83a_{0} and the nonequilibrium evolution at 5a_{0}. In both cases we start at time t = 0 (Fig. 1c) with N_{0}≈2×10^{4} and N≈2×10^{5} at T≈160 nK. In both cases T_{c} decreases at a similar rate owing to similar N decay. At 5a_{0}, the temperature rises faster owing to less effective evaporative cooling at a fixed optical trap depth.
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 N_{0} we count the atoms within the central peak rising above the smooth thermal distribution.
From the measured N(t) alone we calculate the equilibrium T_{c}(t) (ref. 2). From N(t) and E(t) we calculate the equilibrium intensive thermodynamic variables μ^{eq}(t) and T^{eq}(t), and the equilibrium number of condensed atoms, N_{0}^{eq}(t) (refs 2, 20, 21); in these calculations N_{0}^{eq}>0 if and only if T^{eq}<T_{c}. For comparison, we also directly fit a temperature T^{f} to the wings of the momentum distribution. In addition, supposing only that the two components are separately in equilibrium, from the measured N_{0} and N′ we calculate μ_{0} and μ′. (For theoretical details see Supplementary Information.)
At 83a_{0} (Fig. 2a) we find excellent agreement between the measured N_{0} and the N_{0}^{eq} predicted without any free parameters. The BEC vanishes exactly at the equilibrium critical time t_{c} (dotted green line), at which T^{eq} = T_{c}. Note that the dashed red lines show the experimental bounds on the time when the BEC vanishes. The separately calculated μ_{0}, μ′ and μ^{eq} are all consistent and we have also checked that the fitted T^{f} coincides with the calculated T^{eq}. All this gives us full confidence in our equilibrium calculations.
At 5a_{0} (Fig. 2b) we observe strikingly different behaviour. The BEC now survives much longer than it would in true equilibrium; . We also see that μ_{0} and μ′ diverge from each other for t>t_{c}, 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 μ_{0}–μ_{c} is always very small owing to weak interactions.)
At 5a_{0} 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 noncondensed component is still fitted well by a thermal distribution, with T^{f} always within 10% of the calculated T^{eq}(N,E) (see Supplementary Information). For the BEC, in a weakly interacting gas the equilibrium relation μ_{0}(N_{0}) relies on the macroscopic occupation of a single quantum state^{1}, rather than on global equilibrium. Moreover, even for the lowestenergy 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 T_{0} that is also close to T^{eq}≈T^{f}.
These conclusions hold for any a≳1a_{0} (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 T_{0}≈T^{f}≈T^{eq} 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 3a_{0} to 62a_{0} at different times in the superheated regime. The filled (open) symbols show N_{0} measured before (after) the quench. The small Γ_{0} is almost unaffected by the change in a, and the sudden N_{0} decay is due to the increase in κ (Fig. 1b). For reference, the green line shows the calculated N_{0}^{eq} at 3a_{0} and orange shading indicates the superheated regime. As shown in the inset of Fig. 3, we have checked that an interaction quench at t<t_{c} (that is, when T<T_{c} and N_{0}^{eq}>0) does not kill the BEC.
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 N_{0}<−C/a, with C≈2×10^{4}a_{0} for our trap parameters^{22,23,24}. However, after N_{0} drops below this critical value, at small a it decays slowly.
In Fig. 4a we plot the highest temperature at which we still observe a BEC, , scaled to the equilibrium T_{c} at the same N. For a→0, the BEC survives up to T≈1.5T_{c}. (For comparison, this is analogous to superheated water at 280 °C.)
In Fig. 4b we reconstruct the temporal phase diagram of our nonequilibrium 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 t_{c} (green points). The solid curves are spline fits to the data. The width of the orangeshaded 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 N_{0}≈2×10^{4}. In general, nonequilibrium behaviour can strongly depend on the initial conditions. However, we find that is essentially constant (within experimental errors) for initial N_{0} in the range (1–5)×10^{4}. The primary reason for this is that the threebody contribution to Γ_{0} grows with N_{0}; this leads to selfstabilization of the condensed atom number on timescales much shorter than .
We theoretically reproduce our nonequilibrium observations using a twocomponent model directly corresponding to Fig. 1b. Starting with the measured initial N_{0}, we numerically simulate the evolution of a BEC coupled to a thermal bath characterized by μ′(t). To do this we calculate Γ_{0} from our experimental parameters, and for κ we use the form^{25}
Here γ_{el}a^{2} 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 nonequilibrium 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.
Change history
28 March 2013
In the version of this Letter originally published online, in Fig. 3, the orange shading indicating the superheated region should have extended to the righthand edge of the figure. This error has now been corrected in all versions of the Letter.
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Acknowledgements
We thank S. Beattie and S. Moulder for experimental assistance. This work was supported by EPSRC (Grant No. EP/K003615/1), the Royal Society, AFOSR, ARO and DARPA OLE.
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Gaunt, A., Fletcher, R., Smith, R. et al. A superheated Bosecondensed gas. Nature Phys 9, 271–274 (2013). https://doi.org/10.1038/nphys2587
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DOI: https://doi.org/10.1038/nphys2587
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