Old(-ish) stuff: quenches and coarsening

Quenches are a generic term for when you take an initial density matrix (say the thermal density matrix of Hamiltonian H) and time-evolve it with a Hamiltonian H' that's very different from H. A classical quench is typically a thermal quench -- you cool a system rapidly across a phase transition, and watch the domain walls grow. A good review of the old stuff is:

Theory of Phase-Ordering Kinetics
A.J. Bray
cond-mat/9501089 (no figures)
Adv. Phys. 43, 357 (1994)

There are three basic take-away messages -- (1) the dynamic scaling hypothesis, which states that all the time-dependence of correlation functions, response functions, etc. depends on a single scale, which is set (in some models, at least) by the average bubble size. (2) The bubble growth laws depend on whether the order parameter is locally conserved (e.g., binary fluid at separation/mixing) or not (e.g., magnets), as well as on the dimensionality of the order parameter (so, e.g., domain walls grow differently from vortices and hedgehogs). (3) Within any bubble, the system behaves like an ordered state with the equilibrium order parameter at the final temperature: therefore, response functions tend to have a two-time/two-length behavior, depending on whether they're probing the short-time respones, which is basically equilibrium, or the long-time response, which depends on the motion of the domain walls.

There are variants on this problem that involve, e.g., quenching from/to a critical point (so that the initial state has long-range correlations), or quenching across a Kosterlitz-Thouless transition, or quenching in the presence of disorder.

Zero-temperature quenches (microcanonical)

Here the initial state is a wavefunction that is very far from being an eigenfunction of the Hamiltonian it evolves under. (Usually it's the ground state of a totally different Hamiltonian but that's unnecessary.) Most of the experimental work in this field has involved quenching across the superfluid-to-Mott transition in optical lattices; it is at present unclear whether the systems ever approach a locally equilibrium state.

Here are some papers that don't necessarily agree with each other about thermalization:

Oscillating superfluidity of bosons in optical lattices
E. Altman and A. Auerbach
PRL 89, 250404 (2002)

Altman and Auerbach argue that quenching from the Mott to the superfluid phase will lead to macroscopic oscillations of the order parameter (which eventually die out because of quasiparticle collisions), and that underdamped relaxation should be observable. I'm not sure that underdamped relaxation has been seen experimentally for this particular case, but it was observed experimentally for a quench from the SF to the Mott state.

Collapse and revival of the matter wave field of a Bose–Einstein condensate
Markus Greiner, Olaf Mandel, Theodor W. Hänsch & Immanuel Bloch
Nature 419, 51 (2002)

This famous experiment demonstrated the periodic collapse and revival of Bragg peaks (indicative, usually, of phase coherence) when a BEC was quenched into the Mott state. This is an instance of underdamped relaxation -- trap-induced anharmonicities, etc., eventually kill phase coherence. Interestingly, the deeper the quench, the slower the system is to relax.

A quantum Newton's cradle
T. Kinoshita, T. Wenger, and D. Weiss
Nature 440, 900 (2006)

In this experiment, a 1D Bose gas -- bosons confined in an array of very narrow tubes -- was started off in a far-from-equilibrium condition, with half the atoms flying off to the left and the other half flying off to the right, and allowed to time-evolve. The momentum distribution didn't seem to approach a thermal distribution on the experimental timescales; this was interpreted as a consequence of the integrability of the system. (Roughly, an integrable system has a huge number of conserved quantities, and therefore retains much more information about its initial conditions than a normal system does.) This is a stronger result than underdamped relaxation; the experiment showed no tendency towards relaxation at all.

Quench dynamics and nonequilibrium phase diagram of the Bose-Hubbard model
C. Kollath, A. Lauchli, and E. Altman
PRL 98, 180601 (2007)

Kollath et al. found, rather surprisingly, that thermalization doesn't necessarily happen even in nonintegrable models. They simulated a quench from the SF to the Mott phase, using exact diagonalization techniques as well as a density-matrix renormalization group algorithm. The result is that if one quenches from the SF deep into the Mott, the system doesn't seem to thermalize. They explained their results with an argument that goes something like this -- thermalization occurs as a consequence of the decay of double occupancies; however, a double occupancy has energy U (on-site repulsion) and has to decay into quasi-particles that have energy of order J (roughly the bandwidth of the lower Mott band). If U >> J then this can only happen at very high orders in perturbation theory, so it takes forever.

One could interpret this result as being about the observation, rather than the occurrence, of thermalization -- from a practical point of view, a system that takes very long to thermalize looks like one that doesn't thermalize.

Thermalization and its mechanism for generic isolated quantum systems
M. Rigol, V. Dunjko, and M. Olshanii
Nature 452, 854-858 (arxiv version)

This paper presents the "eigenstate thermalization" hypothesis. The hypothesis, as I understand it (which is not very well), states that the expectation values of physically interesting observables are essentially constant for eigenstates that are near each other in energy. There is some numerical reason to believe that this is true, but as far as I know there hasn't been much analytic work. Rigol et al. also suggest a generalization of this hypothesis to integrable systems; the upshot is the idea of a generalized Gibbs ensemble, which introduces "chemical potentials" (i.e. Lagrange multipliers) corresponding to all the conserved quantities.

[... to be continued]