Session B32: Invited Session: Thermalization and Prethermalization in Isolated Systems after a Quantum Quench

How does an isolated quantum system relax?

One of the biggest challenges in probing non-equilibrium dynamics of many-body quantum systems is that there is no general approach to characterize the resulting quantum states. Interference experiments give access to the phase of the order parameter. The full distribution functions of the interference amplitude, and the full phase correlation functions allow us to study the relaxation dynamics in one-dimensional quantum systems. Starting form a coherently split 1d quantum gas, the initial coherence slowly decays. Due to the approximate conserved quantities in our nearly integrable system, this relaxation leads to a pre-thermalized state [1], which is characterized by thermal like distribution functions but exhibits an effective temperature much lower than the kinetic temperature of the initial system. A detailed study of the correlation functions reveals that these thermal-like properties emerge locally in their final form and propagate through the system in a light-cone-like evolution [2]. Furthermore we demonstrate that the pre- thermalized state is connected to a Generalized Gibbs Ensemble and show the pathways for further relaxation towards thermal equilibrium. \\[4pt] [1] M. Gring et al., Science \textbf{337, }1318 (2012); \\[0pt] [2] T. Langen et al. Nature Physics \textbf{9}, 640-643 (2013).   

Universal dynamics of a degenerate unitary Bose gas

It has long been thought that one can not study a degenerate Bose gas with fully resonant (unitary) interactions because the gas is unstable to three-body recombination. We find empirically instead that after a Bose-Einstein condensate has been tuned from a weakly interacting state to a fully unitary gas at the peak of a Feshbach resonance, it survives for a time long enough to permit the characterization momentum-population dynamics. In particular, a high momentum tail forms and comes to a quasi-steady state in perhaps 100 microseconds, while the sample continues to survive and indeed remains degenerate for considerably longer. We show that the shape- and time-dependence of the momentum distribution scale in a universal way with sample density. This work was done in collaboration with Phil Makotyn, Deborah Jin, Cathy Klauss and David Goldberger.   

Undephasing the generalized Gibbs ensemble

Understanding the long time limit of closed quantum many-body systems prepared in some initial non-equilibrium pure state has attracted a lot of interest in recent years. One central question is whether such a system will thermalize or not. In integrable systems the long time limit leads generically to a generalized Gibbs ensemble (GGE) description [1]. For example for the one dimensional transverse field Ising model one can prove that asymptotically all local observables can be calculated in the GGE [2]. In my talk I will show how one can approximately reverse the time arrow of this dynamics using a spin echo-like local Hamiltonian. In this sense the time evolved system never forgets that it is in a pure state and remembers the initial values of local observables like the longitudinal and transverse magnetization. The time evolution can be thought of as dephasing leading to a GGE, which can be undone with this spin echo-like setup. This and related kinds of echo dynamics will be demonstrated for the transverse field Ising model and other integrable models. \\[4pt] [1] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007). \newline [2] M. Fagotti and F. Essler, Phys. Rev. B 87, 245107 (2013).   

Equilibration and coarsening in the quantum O(N) model at infinite N

The quantum O(N) model in the infinite-N limit is a paradigm for symmetry breaking. In this talk, I will investigate the physics of this model out of equilibrium, specifically its response to global quenches starting in the disordered phase. In the infinite-N limit, I will show that not only does the model not lead to equilibration on account of an infinite number of conserved quantities, it also does not relax to a generalized Gibbs ensemble (GGE) consistent with these conserved quantities. Instead, an infinite number of new conservation laws emerge at late times and the system relaxes to an emergent GGE consistent with these. Nevertheless, the late-time states following quenches bear strong signatures of the equilibrium phase diagram. Notably, we find that the model exhibits coarsening to a nonequilibrium critical state only in dimensions d $>$ 2, that is, if the equilibrium phase diagram contains an ordered phase at nonzero temperatures.   

Fluctuation-dissipation relations in isolated quantum systems after a quench

In recent years, there has been increasing interest in understanding under which conditions observables in isolated quantum systems far from equilibrium relax to the predictions of traditional statistical ensembles. Despite being guided by unitary dynamics, this has been shown to occur in nonintegrable systems [1,2] and has been understood within the eigenstate thermalization hypothesis [1-4]. In integrable systems, on the other hand, observables have been found to relax to nonthermal values, which instead can be described by generalized Gibbs ensembles (GGEs) [5]. In this talk, we review some of the early results on this topic and examine whether standard fluctuation dissipation relations apply after relaxation following a quantum quench. We focus on the dynamics of trapped hard-core bosons in one-dimensional lattices with dipolar interactions, as realized in recent experiments with ultracold gases in optical lattices, whose strength is changed during the quench. We consider both nonintegrable and integrable regimes and discuss how, at integrability, the results after relaxation depend on the properties of the initial state selected [6].\\[4pt] [1] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).\\[0pt] [2] M. Rigol, Phys. Rev. Lett. 103, 100403 (2009).\\[0pt] [3] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).\\[0pt] [4] M. Srednicki, Phys. Rev. E 50, 888 (1994).\\[0pt] [5] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007).\\[0pt] [6] E. Khatami, G. Pupillo, M. Srednicki, and M. Rigol, Phys. Rev. Lett. 111, 050403 (2013).