Lea F. Santos

Quantum chaos and thermalization in isolated systems of interacting particles

This review is devoted to the problem of thermalization in a small isolated conglomerate of interacting constituents. A variety of physically important systems of intensive current interest belong to this category: complex atoms, molecules (including biological molecules), nuclei, small devices of condensed matter and quantum optics on nano- and micro-scale, cold atoms in optical lattices, ion traps. Physical implementations of quantum computers, where there are many interacting qubits, also fall into this group. Statistical regularities come into play through inter-particle interactions, which have two fundamental components: mean field, that along with external conditions, forms the regular component of the dynamics, and residual interactions responsible for the complex structure of the actual stationary states. At sufficiently high level density, the stationary states become exceedingly complicated superpositions of simple quasiparticle excitations. At this stage, regularities typical of quantum chaos emerge and bring in signatures of thermalization. We describe all the stages and the results of the processes leading to thermalization, using analytical and massive numerical examples for realistic atomic, nuclear, and spin systems, as well as for models with random parameters. The structure of stationary states, strength functions of simple configurations, and concepts of entropy and temperature in application to isolated mesoscopic systems are discussed in detail. We conclude with a schematic discussion of the time evolution of such systems to equilibrium.

Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization.

By means of full exact diagonalization, we study level statistics and the structure of the eigenvectors of one-dimensional gapless bosonic and fermionic systems across the transition from integrability to quantum chaos. These systems are integrable in the presence of only nearest-neighbor terms, whereas the addition of next-nearest-neighbor hopping and interaction may lead to the onset of chaos. We show that the strength of the next-nearest-neighbor terms required to observe clear signatures of nonintegrability is inversely proportional to the system size. Interestingly, the transition to chaos is also seen to depend on particle statistics, with bosons responding first to the integrability breaking terms. In addition, we discuss the use of delocalization measures as main indicators for the crossover from integrability to chaos and the consequent viability of quantum thermalization in isolated systems.

Entropy of Isolated Quantum Systems after a Quench

A diagonal entropy, which depends only on the diagonal elements of the systems density matrix in the energy representation, has been recently introduced as the proper definition of thermodynamic entropy in out-of-equilibrium quantum systems. We study this quantity after an interaction quench in lattice hard-core 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 becomes equivalent to 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.

Localization and the effects of symmetries in the thermalization properties of one-dimensional quantum systems

We study how the proximity to an integrable point or to localization as one approaches the atomic limit, as well as the mixing of symmetries in the chaotic domain, may affect the onset of thermalization in finite one-dimensional systems. We consider systems of hard-core bosons at half-filling with nearest-neighbor hopping and interaction, and next-nearest-neighbor interaction. The latter breaks integrability and induces a ground-state superfluid to insulator transition. By full exact diagonalization, we study chaos indicators and few-body observables. We show that when different symmetry sectors are mixed, chaos indicators associated with the eigenvectors, contrary to those related to the eigenvalues, capture the onset of chaos. The results for the complexity of the eigenvectors and for the expectation values of few-body observables confirm the validity of the eigenstate thermalization hypothesis in the chaotic regime, and therefore the occurrence of thermalization. We also study the properties of the off-diagonal matrix elements of few-body observables in relation to the transition from integrability to chaos and from chaos to localization.

Onset of chaos and relaxation in isolated systems of interacting spins: Energy shell approach

We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.

Enhanced Convergence and Robust Performance of Randomized Dynamical Decoupling

We demonstrate the advantages of randomization in coherent quantum dynamical control. For systems which are either time-varying or require decoupling cycles involving a large number of operations, we find that simple randomized protocols offer superior convergence and stability as compared to deterministic counterparts. In addition, we show how randomization may allow us to outperform purely deterministic schemes at long times, including combinatorial and concatenated methods. General criteria for optimally interpolating between deterministic and stochastic design are proposed and illustrated in explicit decoupling scenarios relevant to quantum information storage.

Quantum chaos and thermalization in gapped systems

We investigate the onset of thermalization and quantum chaos in finite one-dimensional gapped systems of hard-core bosons. Integrability in these systems is broken by next-nearest-neighbor repulsive interactions, which also generate a superfluid to insulator transition. By employing full exact diagonalization, we study chaos indicators and few-body observables. We show that with increasing system size, chaotic behavior is seen over a broader range of parameters and, in particular, deeper into the insulating phase. Concomitantly, we observe that, as the system size increases, the eigenstate thermalization hypothesis extends its range of validity inside the insulating phase and is accompanied by the thermalization of the system.

Coherent Control of Quantum Dynamics with Sequences of Unitary Phase-Kick Pulses

Coherent-optical-control schemes exploit the coherence of laser pulses to change the phases of interfering dynamical pathways and manipulate dynamical processes. These active control methods are closely related to dynamical decoupling techniques, popularized in the field of quantum information. Inspired by nuclear magnetic resonance spectroscopy, dynamical decoupling methods apply sequences of unitary operations to modify the interference phenomena responsible for the system dynamics thus also belonging to the general class of coherent-control techniques. This article reviews related developments in the fields of coherent optical control and dynamical decoupling, emphasizing the control of tunneling and decoherence in general model systems. Considering recent experimental breakthroughs in the demonstration of active control of a variety of systems, we anticipate that the reviewed coherent-control scenarios and dynamical-decoupling methods should raise significant experimental interest.

General features of the relaxation dynamics of interacting quantum systems

We study numerically and analytically isolated interacting quantum systems that are taken out of equilibrium instantaneously (quenched). The probability of finding the initial state in time, the so-called fidelity, decays fastest for systems described by full random matrices, where simultaneous many-body interactions are implied. In the realm of realistic systems with two-body interactions, the dynamics is slower and depends on the interplay between the initial state and the Hamiltonian characterizing the system. The fastest fidelity decay in this case is Gaussian and can persist until saturation. A simple general picture, in which the fidelity plays a central role, is also achieved for the short-time dynamics of few-body observables. It holds for initial states that are eigenstates of the observables. We also discuss the need to reassess analytical expressions that were previously proposed to describe the evolution of the Shannon entropy. Our analyses are mainly developed for initial states that can be prepared in experiments with cold atoms in optical lattices.

Quantum chaos, delocalization, and entanglement in disordered Heisenberg models

We investigate disordered one- and two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication of the basis dependence of state delocalization indicators (such as the number of principal components) is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spin- and fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.