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Condensed-matter physics

Relaxation after a tight squeeze

Are the rules that determine relaxation to equilibrium the same in the classical and quantum worlds? Recent experiments supported the idea that they are — but an investigation with ultracold atoms now contradicts that.

Our everyday experiences lead us to expect things to seek equilibrium quickly when perturbed from rest. Cold milk poured into hot coffee, for example, soon distributes itself evenly, equalizing the temperature throughout the liquid. This kind of system, which over time samples every state available to it and so can relax to a true equilibrium state, is called ergodic. But unlike the milk-in-coffee example, not all systems that are governed by classical mechanics are ergodic, and it is difficult to push a system that is not naturally ergodic into an ergodic, equilibrium-seeking state. On page 324 of this issue1, Hofferberth and colleagues report experiments that seem to indicate that a similar operation is much easier in the quantum world.

In classical statistical mechanics, concepts of ergodicity and thermodynamic irreversibility were originally addressed by Ludwig Boltzmann. His celebrated 'H-theorem' of 1872 states that an ideal gas will reach equilibrium starting from an arbitrary initial state. But even in classical mechanics, equilibration and the equalization of temperature are not guaranteed. One class of many-body systems, described by so-called integrable models, follows very different, non-ergodic dynamics, which does not allow relaxation to true equilibrium. Such systems have an infinite number of conservation laws, which prevents the sampling of the entire phase space, meaning that true equilibrium cannot be attained.

In fact, according to a well-founded hypothesis of classical dynamics, the Kolmogorov–Arnold–Moser (KAM) theorem, a classical system does not even have to be exactly integrable to be non-ergodic; being nearly integrable is enough. For example, coupled nonlinear oscillators in a linear array do not equilibrate (share their energy equally), even though this system is integrable only when the distance between the oscillators tends to zero2. Theorists continue to argue whether the KAM theorem has an equivalent in quantum dynamics, but others have set out to address the question experimentally. Their chosen proving-ground is the dynamics of one particular near-integrable quantum system — a condensate of cold atoms caught in a one-dimensional trap.

Seeking equilibrium — cold milk in hot coffee. Credit: PHOTOLIBRARY.COM

Just last year, the momentum distribution of a condensate of atoms confined to a one-dimensional tube and then kicked out of equilibrium was measured3. It did not re-equilibrate for a very long time — instead, individual atoms passed through each other repeatedly without ever coming to rest. This 'quantum Newton's cradle' was interpreted as a signature of non-ergodic dynamics in a near-integrable system, exactly as seen in the classical case.

Hofferberth et al.1 investigated a similar system brought out of equilibrium in a different way, and came to precisely the opposite conclusion. They took a single condensate of ultracold rubidium atoms, caught in a one-dimensional magnetic trap, and split it along its length into two identical halves. After holding the atomic clouds separate for a certain time, the authors released them and observed the fringes formed as the matter waves of each condensate interfered. Repeating these experiments many times, Hofferberth et al.1 noted that the position of interference fringes becomes more random with increasing 'hold time'.

The explanation for this observation is that, after splitting, the initially identical phases of the two condensates evolve independently. The relative phase between the two condensates becomes scrambled as the atoms within each condensate interact, and this loss of coherence expresses itself in a randomization of the interference fringes. From a classical point of view, this argument might seem surprising: identical systems prepared in the same state should surely evolve in the same way. But with a quantum system, identical measurements on identical wavefunctions can give different results.

More precisely in this case, the wavefunction describing the relative phase between the two condensates is in a 'squeezed' state just after splitting, because there is a very small uncertainty in the relative phase. But as time progresses after the separation, the wavefunction spreads out to form a wider probabilistic distribution. In any individual measurement, the relative phase will have a non-zero value somewhere within the spread of values covered by the wavefunction, even though its average value after many measurements remains zero.

This kind of analysis was first applied to the single-wave mode that describes a large, three-dimensional condensate4,5, where it predicted a 'diffusion' of phase over time that was indeed observed6. In one-dimensional condensates, rigorous theorems forbid the existence of long-range order, and the multi-mode character of phase diffusion must be considered7. This analysis was extended by suggesting that degrees of freedom other than the relative phase — in particular, the summed phase of both condensates — might act as a heat reservoir8, resulting in a characteristic 'subexponential' decay of the coherence with hold time. This theoretically predicted signature of a system relaxing to thermal equilibrium is exactly that reported by Hofferberth and colleagues1, suggesting that the dynamics they observe are indeed ergodic.

The idea of modelling a subset of the degrees of freedom by an effective heat bath is common to many areas of physics. But in the particular case of atoms in a one-dimensional trap this is hardly an obvious assumption, because such a system is very nearly integrable. It would be exactly integrable, were it not for the shallow trapping potential along the axis of the condensate. A common assumption is that as the potential varies little on the scale of interatomic distances, equilibration should still be prevented. Whereas the previous result3 supported this point of view, Hofferberth et al. seem to present a different answer.

What caused this difference? Perhaps it comes from the initial states or the degree of radial confinement, which was much greater in the earlier experiment3, and which affects the accuracy with which the system can be described by an integrable model. Perhaps it originates in the temperature of the initial state, which was much higher relative to the chemical potential of the system in the new experiment1. This experiment might also have been subject to a weak potential caused by low levels of magnetic-field disorder. It is a fundamental open question exactly what kind or magnitude of perturbations of an integrable quantum model are needed to effectively restore ergodicity, and to allow a system to reach equilibrium. A resolution of these two experimental results1,3 would provide a strong boost to our understanding of these fundamental questions.

Meanwhile, a practical lesson to be learnt from the experiments of Hofferberth and colleagues1 is that the multi-mode character of quantum dynamics in low-dimensional systems might put fundamental limits on applications of cold-atom interferometers for metrology and spectroscopy, as it would accelerate the scrambling of interference fringes. But the worth of their investigation lies, above all, in showing how systems of ultracold atoms can address fundamental questions in the non-equilibrium quantum dynamics of many interacting particles. The results will provoke heated debate, with implications beyond cold-atom research.

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Altman, E., Demler, E. Relaxation after a tight squeeze. Nature 449, 296–297 (2007). https://doi.org/10.1038/449296a

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