Local Versus Global Equilibration near the Bosonic Mott-Insulator–Superfluid Transition

Abstract

We study the timescales for adiabaticity of trapped cold bosons subject to a time-varying lattice potential using a dynamic Gutzwiller mean-field theory. We explain apparently contradictory experimental observations by demonstrating a clear separation of timescales for local dynamics (∼ ms) and global mass redistribution (∼ 1s). We provide a simple explanation for the short and fast timescales, finding that while density/energy transport is dominated by low energy phonons, particle-hole excitations set the adiabaticity time for fast ramps. We show how mass transport shuts off within Mott domains, leading to a chemical potential gradient that fails to equilibrate on experimental timescales.Introduction.-A wide range of experiments have forced us to confront questions of dynamics in strongly correlated systems. These include studies of high density nuclear matter at the Relativistic Heavy Ion Collider (RHIC) [1], transport through metal-insulator interfaces [2], and femtosecond spectroscopy [3] of quantum dots after sudden changes in gate voltages [4]. This is a conceptually rich area where computation is difficult, and where it is hard to devise experiments which are straightforward to analyze. Experiments in cold atoms are beginning to play an important role in this areathey have started providing a framework for understanding the non-equilibrium dynamics of strongly correlated materials [5][6][7][8][9][10]. In cold gas experiments, not only is the Hamiltonian known, but one can dynamically tune between weak and strong interactions, readily producing highly non-equilibrium situations that allow one to explore both linear and nonlinear responses, the reestablishment of equilibrium, and the generation of topological defects during rapid quenches [11]. The latter physics is relevant to astrophysical models of the early universe. Here we theoretically explore the timescales governing local and global transport of bosons in optical lattices, the prototypical example of strongly correlated cold atom physics.Adding further interest to this area, initial experiments [5][6][7] probing the dynamics of bosons in optical lattices have found adiabaticity/relaxation timescales that differ by two orders of magnitude. The shortest of these timescales was particularly noteworthy, as it was an order of magnitude smaller than the inverse of the single particle tunneling energy, t ∼ 0.1J −1 [6]. How can the system adjust itself on a timescale which is short compared to the tunneling time? Conversely, experiments on a nearly identical system at Chicago [5], found that the global density profile didn't attain it's equilibrium value even on times t ∼ 10J −1 ! Here we resolve this contradiction by demonstrating a separation of timescales for global transport and local equilibration, and show that the timescale for adiabiaticity is largely set by the gap towards particle-hole excitations in the strongly correlated superfluid.

Equilibration and prethermalization in the Bose-Hubbard and Fermi-Hubbard models

“…Apart from some general statements concerning the relaxation of a quantum system towards equilibrium [29][30][31], quantum quenches have been studied by using certain approximations, such as Bogoliubov-type approxi-mations or strong-coupling perturbation theory [41][42][43][44], the Gutzwiller approximation [45], or related (semi) classical methods [46][47][48], as well as (truncated) exact diagonalization [25]. However, these approximations are only reliable in certain regions of parameter space.…”

Linear ramps of interaction in the fermionic Hubbard model

“…Recent works address this issue and look for the optimal ramping protocol which produces the minimal heating 29,30 . Other investigations on the slow quench dynamics in trapped cold gases address the issue of equilibration of local and global quantities 31,32 . Finally, we note that while those questions mainly address the dynamics during the ramp, there are interesting issues as well that concern the evolution of the system once the ramp is over, namely for times t > τ .…”

Rigorous mean-field dynamics of lattice bosons: Quenches from the Mott insulator

“…This approximation can be shown to be exact on fully connected lattices [8] and in the limit of infinite dimensions [13,14] regardless of the strength of the interaction, which means that the method is fully non-perturbative. Gutzwiller mean-field (a) E-mail: m.snoek@uva.nl theory offers a straightforward extension towards out-ofequilibrium situations [15], which has already been applied to many problems: creation of a molecular condensate [15], dynamics of the superfluid-insulator phase transition [16], transport in inhomogeneous systems [17], collapse and revival oscillations [18], atom lasers [19], ramp-up dynamics of the optical lattice [20,21], Bragg scattering [22], and trap dynamics [23]. So far this method has been justified by semi-classical arguments, but no rigorous derivation is available, making it questionable in which circumstances this approximation can be trusted.…”