Session X4: Focus Session: Quantum Quench Dynamics in Cold Atom Systems

Integrability versus Thermalizability in Isolated Quantum Systems

The purpose of this presentation is to assess the status of our understanding of the transition from integrability to thermalizability in isolated quantum systems. In Classical Mechanics, the boundary stripe between the two is relatively sharp: its integrability edge is marked by the appearance of finite Lyapunov's exponents that further converge to a unique value when the ergodicity edge is reached. Classical ergodicity is a universal property: if a system is ergodic, then every observable attains its microcanonical value in the infinite time average over the trajectory. On the contrary, in Quantum Mechanics, Lyapunov's exponents are always zero. Furthermore, since quantum dynamics necessarily invokes coherent superpositions of eigenstates of different energy, projectors to the eigenstates become more relevant; those in turn never thermalize. All of the above indicates that in quantum many-body systems, (a) the integrability-thermalizability transition is smooth, and (b) the degree of thermalizability is not absolute like in classical mechanics, but it is relative to the class of observables of interest. In accordance with these observations, we propose a concrete measure of the degree of quantum thermalizability, consistent with the expected empirical manifestations of it. As a practical application of this measure, we devise a unified recipe for choosing an optimal set of conserved quantities to govern the after-relaxation values of observables, in both integrable quantum systems and in quantum systems in between integrable and thermalizable.   

Chaos and statistical relaxation in quantum systems of interacting particles

Recent experimental progresses in the studies of quantum systems of interacting particles with optical lattices have triggered the interest in basic problems of many-body physics. One of the issues that has been widely discussed in the literature is the onset of thermalization in an isolated quantum system caused by interparticle interactions. A prerequisite for thermalization is the statistical relaxation of the system to some kind of equilibrium and its viability has been associated with the onset of quantum chaos. We propose a method to study the transition to chaos in isolated quantum many-body systems, which is based on the concept of delocalization of eigenstates in the energy shell. We show that although the fluctuations of energy levels and delocalization measures in integrable and non-integrable systems differ, global properties of the eigenstates may be quite similar, provided the interaction between particles exceeds some critical value. In this case the quench dynamics can be described analytically, demonstrating the universal statistical relaxation of the systems irrespectively of whether they are chaotic or not.   

Non-adiabatic ramps in quantum many-particle systems

A change of system parameter can be neither truly instantaneous nor truly adiabatic in real life. For several quantum many-particle systems, I will consider non-equilibrium dynamics induced by finite-rate ramps. The ramp rate extrapolates between an instantaneous quench and an adiabatic sweep. I will characterize the deviation from adiabaticity through the excess energy or ``heating'' of the system. For cold-atom systems in a harmonic trapping potential, I will show that the non-adiabatic heating in finite-time ramps has universal features common to a wide range of systems.   

Entropy of Isolated Quantum Systems after a Quench

A diagonal entropy, which depends only on the diagonal elements of the system's density matrix in the energy representation, has been argued to be the proper definition of thermodynamic entropy in out-of-equilibrium quantum systems. We study this quantity after an interaction quench in lattice hardcore bosons and spinless fermions, and after a local chemical potential quench in a system of hard-core bosons in a superlattice potential. The former systems have a chaotic regime, where the diagonal entropy approaches the equilibrium microcanonical entropy, coinciding with the onset of thermalization. The latter system is integrable. We show that its diagonal entropy is additive and different from the entropy of a generalized Gibbs ensemble, which has been introduced to account for the effects of conserved quantities at integrability [1]. \\[4pt] [1] Lea F. Santos, Anatoli Polkovnikov, and Marcos Rigol, Phys. Rev. Lett. 107, 040601 (2011).   

Mode coupling induced dissipative and thermal effects at long times after a quantum quench

An interaction quench in a Luttinger liquid can drive it into an athermal steady state. We analyze the effects on such an out of equilibrium state of a mode coupling term due to a periodic potential. Employing a perturbative renormalization group approach we show that even when the periodic potential is an irrelevant perturbation in equilibrium, it has important consequences on the athermal steady state as it generates a temperature as well as a dissipation and hence a finite life-time for the bosonic modes.   

Random phase approximation study of one-dimensional fermions after a quantum quench

The effect of interactions on a system of fermions that are in a nonequilibrium steady state due to a quantum quench is studied employing the random phase approximation. As a result of the quench, the distribution function of the fermions is greatly broadened. This gives rise to an enhanced particle-hole spectrum and overdamped collective modes for attractive interactions between fermions. On the other hand, for repulsive interactions, an undamped mode above the particle-hole continuum survives. The sensitivity of the result to the nature of the nonequilibrium steady state is explored by also considering a quench that produces a current-carrying steady state.   

Ballistic expansion of interacting fermions in one-dimensional optical lattices

In most quantum quenches, no net particle currents arise. Access to studying transport properties can be gained by letting a two-component Fermi gas that is originally confined by the presence of a trapping potential expand into an empty optical lattice. In recent experiments, this situation was addressed in 2D and 3D optical lattices [1]. We focus on the 1D case in which an exact numerical simulation of the time-evolution is possible by means of the DMRG method. Concretely, we study the expansion in the 1D Hubbard model with repulsive interactions, driven by quenching the trapping potential to zero, and we concentrate on the most direct experimental observable, namely density profiles [2]. In the strict 1D case, we identify conditions for which the expansion is ballistic, characterized by an increase of the cloud's radius that is linear in time. This behavior is found whenever initial densities are smaller or equal to one, both for the expansion from box and harmonic traps. We make quantitative predictions for the expansion velocity as a function of onsite repulsion and initial density that can be probed in experiments. \\[4pt] [1] Schneider et al., arXiv:1005.3545\\[0pt] [2] Langer et al., arXiv:1109.4364   

Quenching across a quantum critical point: dependence of scaling laws on spatial periodicity

We study the quenching dynamics of a quantum many-body system in one dimension described by a Hamiltonian having spatial periodicity. Specifically, we consider a spin-1/2 $XX$ chain subject to a periodically varying magnetic field in the $\hat z$ direction or, equivalently, a tight-binding model of spinless fermions having a periodic local chemical potential. If the strength of the magnetic field (or chemical potential) is varied slowly in time at a rate $1/\tau$ so as to take the system across a quantum critical point, we find that the density of excitations thereby produced scales as a power of $1/\tau$.Remarkably, the power depends on the spatial periodicity of the field and deviates from the $1/\sqrt{\tau}$ scaling that is ubiquitous to a range of systems. This behavior is analyzed by mapping the slow quenching problem to a collection of fermionic two-level systems, labeled by the lattice momentum $k$, for which the effective Hamiltonians vary as a power of the time close to the quantum critical point. For a magnetic field described by multiple periodicities, the power depends on the smallest period for very large values of $\tau$. Finally, we find that if there are interactions between the fermions, the power varies continuously with the interaction strength.   

Kibble-Zurek Scaling: Universality and scaling

Near a critical point, the equilibrium relaxation time of a system diverges and any change of control/thermodynamic parameters leads to non-equilibrium behavior. The Kibble-Zurek (KZ) problem is to determine the dynamical evolution of the system parametrically close to its critical point when the change is parametrically slow. We formulate the KZ problem as a scaling limit and compute its universal content analytically (critical exponents+scaling functions) in a few classical and quantum models. We also use gauge-gravity duality to compute KZ response functions in more exotic critical theories.   

Universal Quantum Dynamics of the Transverse-Field Ising Model

The one-dimensional transverse field Ising model is a prototypical example of a quantum phase transition. While its equilibrium scaling has been known for more than half a century, we discuss the non-equilibrium quantum dynamics as the system is swept slowly through the critical point (a Kibble-Zurek ramp). Scaling is well understood for Kibble-Zurek ramps that end at the quantum critical point or deep in the ferromagnetic regime. We solve for the full finite-size scaling forms of excess heat and spin-spin correlation function for an arbitrary point along the ramp. We also confirm the postulated universality of the dynamic scaling forms by numerically simulating Mott insulating bosons in a tilted potential, an experimentally realizable model in the same universality class [Simon et. al., Nature 472, 372 (2011)]. Our numerics indicate that the time-scales necessary to see non-equilibrium scaling should already be within the reach of experiment.   

Self-consistent theory of instabilities in the spin-1 Bose gas

We discuss instabilities of a spin-1 Bose condensate using a Hartree-Fock-Bogoliubov approximation to account for the interactions between the unstable modes. There is a close analogy to the ``S-theory'' that describes parametric excitation of magnons in solid state systems. We particularly emphasize the pair-breaking effect of phase fluctuations in the parent condensate and their role in inhibiting the instability.   

Decay of classical quasiperiodic state and emergence of prethermalization in quenched Fermi-Pasta-Ulam system

We will discuss the relaxation of the Fermi Pasta Ulam system in the presence of quantum fluctuations. In order to make comparisons with the classical relaxation, we strongly excite a single normal mode, while the rest of the modes are initially in the quantum ground state. We confine ourselves to the quasiperiodic regime where the classical system never thermalizes. We show that the short time dynamics of the quantum problem are very different from classical evolution, with the quantum zero point energy playing a key role. The short time dynamics can be viewed as an enhancement of zero point energy, parametrically driven by the classical degrees of freedom. This introduces nontrivial off-diagonal correlations in the low momentum sector and dampens the classical oscillations eventually leading to both dephasing and decay, and we identify the time scales associated with these processes. Eventually the system reaches a nontrivial very long lived quasistationary regime where off-diagonal correlations disappear and the energy remains mostly localized in the low q sector while the high q sector relaxes to a uniform effective temperature. In this regime, correlations are very well described by a generalized Gibbs ensemble.   

Fisher zeroes and non-analytic real time evolution for quenches in the transverse field Ising model

We study quenches of the magnetic field in the transverse field Ising model. For quenches across the quantum critical point, the boundary partition function in the complex temperature-time-plane shows lines of Fisher zeroes that intersect the time axis, indicating non-analytic real time evolution in the thermodynamic limit (analogous to well-known thermodynamic phase transitions). We obtain exact analytical results for these dynamic transitions and show that the dynamic behavior cannot be obtained from a naive analytic continuation of the thermal equilibrium partition function: Real time evolution across this quantum critical point generates a new non-equilibrium energy scale. We argue that this behavior is expected to be generic for interaction quenches across quantum critical points in other models as well.