Session U40: Non-equilibrium Cold Atom Systems

Emergence of an effective thermal correlation length in the course of prethermalization

Understanding non-equilibrium processes in many-body quantum systems is an important unsolved problem in many areas of physics. Here, we study the relaxation dynamics of a coherently split one-dimensional Bose gas by measuring the full probability distribution functions of matter-wave interference. After splitting, the system rapidly relaxes to a thermal-like quasi-steady state retaining partial information about the initial conditions. We observe this state to be independent on the initial temperature before splitting and associate the relaxation dynamics with prethermalization. Observing the system on different length scales allows us to probe the dynamics of excitations on different energy scales, revealing two distinct length-scale dependent regimes of relaxation. We measure the crossover length-scale separating these two regimes and identify it with the prethermalized phase-correlation length of the system. Our work provides a direct vizualization of prethermalization and multimode dynamics in a one-dimensional many-body quantum system.   

Quasi-universal transient behavior of a nonequilibrium Mott insulator driven by an electric field

We use a self-consistent strong-coupling expansion for the self-energy (perturbation theory in the hopping) to describe the nonequilibrium dynamics of strongly correlated lattice fermions. We study the three-dimensional homogeneous Fermi-Hubbard model driven by an external electric field showing that the damping of the ensuing Bloch oscillations depends on the direction of the field, and that for a broad range of field strengths, a long-lived transient prethermalized state emerges. This long-lived transient regime implies that thermal equilibrium may be out of reach of the time scales accessible in present cold atom experiments, but shows that an interesting new quasi-universal transient state exists in nonequilibrium governed by a thermalized kinetic energy, but not a thermalized potential energy. In addition, when the field strength is equal in magnitude to the interaction between atoms, the system undergoes a rapid thermalization, characterized by a different quasi-universal behavior of the current and spectral function for different values of the hopping.   

Quench Dynamics in the Presence of a Bath

Feshbach resonance are now widely used to tune the interaction strength in cold atoms. This allows one to experimentally study the out-of-equilibrium dynamics of a quench associated with instantaneously changing the strength of the interactions between fermionic and bosonic atoms. Previous theoretical studies based on standard time dependent Bogoliubov or BCS theory (for bosons and fermions) have not included the presence of a thermal bath. This bath is essential for ultimate equilibration. In this talk we show how to include the bath following a Leggett-Caldeira type approach. We point out some of the important differences in the quench dynamics between bosonic and fermionic superfluids.   

Many-body analysis of a quasi-disordered integrable lattice system after a quench

It has been recently argued that the transition between a delocalized and a localized regime in a quasi-disordered integrable lattice system affects the dynamics and description of one-body observables after relaxation following a quench [1]. Specifically, the generalized Gibbs ensemble description was found to be applicable in the delocalized phase, but to break down in the localized phase. Here we present a many-body analysis of those quenches. We discuss how the expectation values of one-body observables in the many-body eigenstates behave in both regimes, and provide a microscopic understanding of the results in Ref. [1]. \\ Ref. [1]: C. Gramsch and M. Rigol, Phys. Rev. A (in press); arXiv:1206.3570.   

When Is a Bath a Bath? Relaxation Dynamics and Thermalization in a Fermionic Chain

We study thermalization in a one-dimensional quantum system consisting of a non-interacting fermionic chain with each site of the chain coupled to an additional bath site. Using a time-dependent density matrix renormalization group algorithm we investigate the time evolution of observables in the chain after a quantum quench. For a weakly interacting bath and low densities we show that the dynamics can be quantitatively described by a system of coupled equations of motion. For higher densities our numerical results show equilibration for local observables and a thermalization to the canonical ensemble independent of the initial state. In particular, we find a Fermi momentum distribution in the chain in equilibrium in spite of the seemingly oversimplified bath in our model.   

Quench dynamics in the one-dimensional sine-Gordon model: Quantum kinetic equation approach

We study dynamics after a quantum quench in the one-dimensional sine-Gordon model in its gapless phase. We construct the Dyson equation to leading (quadratic) order in the cosine potential and show that the resulting quantum kinetic equation is atypical in that it involves multi-particle scattering processes. We also show that using an effective action, which generates the Dyson equation by a variational principle, the conserved stress-momentum tensor can be constructed. We solve the dynamics numerically by making a quasi-classical approximation that makes the quantum kinetic equation local in time while retaining the multi-particle nature of the scattering processes. We find that the boson distribution function reaches a steady-state characterized by an effective temperature in the long-wavelength limit. We present an analytic argument for the value of the effective temperature and the time-scales to reach this steady-state.   

Thermalization in isolated quantum many-body systems and dependence on initial states

We study the viability of thermalization in isolated quantum many-body systems described by one-dimensional Heisenberg spin-1/2 models. We show that the onset of thermal equilibrium depends on the interplay between initial states, observables and regimes. Our numerical studies are based on the spectrum analysis of the system and on its long-time evolution after a quench.   

Initial-state dependence of the quench dynamics in integrable quantum systems at finite temperature

We study properties of isolated integrable quantum systems after a sudden quench starting from thermal states. We show that, even if the system is initially in equilibrium at finite temperature, the diagonal entropy after a quench remains a fraction of the entropy in the generalized ensembles introduced to describe integrable systems after relaxation. The latter is also, in general, different from the entropy in thermal equilibrium. Furthermore, we examine the difference between the distribution of conserved quantities in the thermal and generalized ensembles after a quench and show that they are also, in general, different from each other. This explains why these systems fail to thermalize in the usual sense. A finite size scaling analysis is presented for each quantity, which allows us making predictions for thermodynamically large lattice sizes.   

Non-Equilbrium Behavior and Thermalization in 1D Bose Gases

Using a new numerical renormalization group based on exploiting an underlying exactly solvable nonrelativistic theory, we study the equilibrium properties and out-of-equilibrium dynamics of interacting many-body quantum systems. Focusing on the example of the Lieb-Liniger model we study quantum quenches with a focus on protocols in which the gas is released from a parabolic trap. Our method allows one not only to accurately describe the equilibrium state of the gas in the trap, but also to track the post-quench dynamics all the way to infinite time. Exploiting integrability, we are also able to exhibit a general protocol for the explicit construction of the generalized Gibbs ensemble, which is a candidate to govern the equilibriation of the trapped gas after its release. This construction does not rely on the underlying Hamiltonian being quadratic and works for arbitrary initial conditions. By comparing the predictions of equilibration from this ensemble against the long time dynamics observed in our method, we find that it is considerably more accurate than the effective grand canonical ensemble. See J.S. Caux and R. M. Konik, PRL 109, 175301 (2012).   

How Long Does it Take for a Non-Equilibrium System to Reach a Quasi-Thermal State?

We study the relaxation of an interacting system driven out of equilibrium by a constant electric field using Non-Equilibrium Dynamical Mean Field Theory. We use on the one hand a DMFT method which solves the steady state problem directly in frequency space, and on the other hand, a DMFT method that follows the transient time evolution of the system on the Keldysh contour. The system is described by the Falicov Kimball model which we follow across the metal - insulator transition. We find that the retarded Green's function quickly approaches that of the steady state while the lesser Green's function and, as a result the distribution function, slowly approach that of a steady state with an increased temperature due to the additional energy transferred to the system by the electric field. Analyses of this type can help understand the results of some experiments involving ultracold atomic gases.   

Probing thermalization and dephasing using the Kibble-Zurek mechanism

The Kibble-Zurek mechanism was introduced to describe defect creation after ramping through critical points. Recent work has extended this concept to a full non-equilibrium scaling theory, described by the same low-energy critical exponents as in equilibrium. In this talk, I will discuss applying Kibble-Zurek analysis and its extensions to probe open questions in non-equilibrium dynamics, specifically working to understand thermalization or -- in the case of integrable systems -- dephasing to a generalized Gibbs ensemble. The major advantage of investigating these questions within the Kibble-Zurek scaling regime is that the results are universal in the renormalization group sense, i.e., insensitive to microscopic details that often confound analyses of thermalization. I will describe both analytical and numerical (TEBD) approaches to address the problem, with an emphasis on understanding the long-time behavior after a slow ramps and small quenches.   

Dynamics of Large Quantum Systems: Equilibration, Thermalization and Interactions

The question of how/whether large quantum systems equilibrate and/or thermalize when prepared in an out-of-equilibrium state has been of enormous interest given recent experimental progress. We address this question in fermionic [1,2] and bosonic [3] systems, by following the dynamics of the full density matrix. We particularly study the case of two large-twin systems connected by a weak link (a quantum impurity), and we show that the total system equilibrates and thermalizes when the weak link is susceptible to incoherent and inelastic processes. We thus provide an experimentally feasible prescription for equilibrating and thermalizing large finite quantum systems. Our calculations are based on extending methods originally developed to treat subsystem dynamics (such as impurity), namely, the quantum Langevin equation method, the well known fermionic trace formula, and an iterative path integral approach. We also explore the role of interactions. While the fermionic system [1,2] shares many common features with the bosonic analog [3], we will describe certain crucial differences that arise as a result of different statistics.\\[4pt] [1] M. Kulkarni, K. L. Tiwari, D. Segal, arXiv:1206.2408\\[0pt] [2] M. Kulkarni, K. L. Tiwari, D. Segal, arXiv:1208.5725\\[0pt] [3] M. Kulkarni and D. Segal (in preparation)   

Quench Dynamics of the Interacting Bose Gas in one Dimension

We obtain an exact expression for the time evolution of the interacting Bose gas following a quench from a generic initial state using the Yudson representation for integrable systems. We study the evolution of the density and noise correlation for a small number of bosons and their asymptotic for any number. We show that for any value of the coupling, as long as it is repulsive, the system asymptotes towards a strongly repulsive gas, while for any value of an attractive coupling long time behavior is dominated by the maximal bound state. This occurs independently of the initial state and can be viewed as an emerging ``dynamic universality''.   

Quantum Quenches of Ultracold Atoms in the Presence of Synthetic Gauge Fields

Motivated by the experimental realization of synthetic gauge fields for ultracold atoms in optical lattices, we consider quantum quenches in such gauge field backgrounds. We show that the density dynamics following sudden anisotropic quenches can be used as a probe of equilibrium mass currents of atoms. We show, using diverse examples of Bose superfluids and normal Fermi fluids, that bulk equilibrium currents produced by the background gauge fields can be uncovered using this method. Such quenches are also shown to provide an effective route to probing the edge currents in topological states such as quantum Hall or quantum spin Hall insulators.   

Thermalization Processes in Quantum Mechanics

In quantum mechanics, the emergence of thermalization processes from unitary evolution has remained one of the greatest challenges. The two outstanding theories of this issue by Srednicki and Tasaki cannot address the concepts of temperature, heat, and work. Here, we present a theory using multiple quenches to examine the thermalization processes to advance thermodynamics concepts. To perform multiple quenches, one can vary one single control parameter ($\lambda )$ in a series of time evolutions, which create a set of density operators. The average of these density operators results into a diagonal operator with probability distribution function that can describe the emerging ensembles. Measuring probability distribution functions of key physical observables, temperature, equal to the derivative of energy with respect to entropy, can be easily measured. Therefore, simulations via multiple quenches can mimic dynamics in open quantum systems with much cheaper computational cost. They allow (1) tuning of temperature and entropy via $\lambda $, (2) measuring work distribution functions from distributions of a reaction coordinate, and (3) computing free-energy changes via Jarzynski's Equality. We hope that this approach can provide a new foundation and open up new directions for studying control of quantum systems.