Quenched dynamics of classical isolated systems: the spherical spin model with two-body random interactions or the Neumann integrable model

We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body interactions drawn from a Gaussian probability distribution. In the statistical physics framework, the potential energy is of the so-called $p=2$ spherical disordered kind. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable system. We take initial conditions in thermal equilibrium and we subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian. We identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We obtain the {\it global} dynamical observables with numerical and analytic methods and we show that, in most cases, they are out of thermal equilibrium. We note, however, that for shallow quenches from the condensed phase the dynamics are close to (though not at) thermal equilibrium. Surprisingly enough, for a particular relation between parameters the global observables comply Gibbs-Boltzmann equilibrium. We next set the analysis of the system with finite number of degrees of freedom in terms of $N$ non-linearly coupled modes. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the $N-1$ integrals of motion and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs-Boltzmann behaviour of global observables. We elaborate on the role played by these constants of motion in the post-quench dynamics and we briefly discuss the possible description of the asymptotic dynamics in terms of a Generalised Gibbs Ensemble

Quenched dynamics of classical isolated systems: The spherical spin model with two-body random interactions or the Neumann integrable model

We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body random interactions. In the statistical physics framework, the potential energy is of the so-called p = 2 kind, closely linked to the scalar field theory. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable model. We take initial conditions from the Boltzmann equilibrium measure at a temperature that can be above or below the static phase transition, typical of a disordered (paramagnetic) or of an ordered (disguised ferromagnetic) equilibrium phase. We subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian, obtained from an instantaneous global rescaling of the elements in the interaction random matrix. In the limit of infinitely many degrees of freedom, , we identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We next set the analysis of the system with finite number of degrees of freedom in terms of N non-linearly coupled modes. We argue that in the limit the modes decouple at long times. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the N - 1 integrals of motion, notably, their scaling with N, and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs-Boltzmann behaviour of the global observables. We elaborate on the role played by these constants of motion after the quench and we briefly discuss the possible description of the asymptotic dynamics in terms of a generalised Gibbs ensemble.Fil: Cugliandolo, Leticia Fernanda. Université Pierre et Marie Curie; Francia. Université Pierre et Marie Curie. Laboratoire de Physique Théorique et Hautes Energies; FranciaFil: Lozano, Gustavo Sergio. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Nessi, Emilio Nicolás. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Picco, Marcos Fernando. Université Pierre et Marie Curie. Laboratoire de Physique Théorique et Hautes Energies; FranciaFil: Tartaglia, Alessandro. Université Pierre et Marie Curie. Laboratoire de Physique Théorique et Hautes Energies; Franci

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble and discuss its connection with ideas of prethermalization in weakly interacting systems.This work was supported by the Army Research Office [grant number W911NF1410540] (L.D., A.P, and M.R.), the U.S.-Israel Binational Science Foundation [grant number 2010318] (Y.K. and A.P.), the Israel Science Foundation [grant number 1156/13] (Y.K.), the National Science Foundation [grant numbers DMR-1506340 (A.P.)and PHY-1318303 (M.R.)], the Air Force Office of Scientific Research [grant number FA9550-13-1-0039] (A.P.), and the Office of Naval Research [grant number N000141410540] (M.R.). The computations were performed in the Institute for CyberScience at Penn State. (W911NF1410540 - Army Research Office; 2010318 - U.S.-Israel Binational Science Foundation; 1156/13 - Israel Science Foundation; DMR-1506340 - National Science Foundation; PHY-1318303 - National Science Foundation; FA9550-13-1-0039 - Air Force Office of Scientific Research; N000141410540 - Office of Naval Research)Accepted manuscrip

Quantum Quenches in the Luttinger model and its close relatives

A number of results on quantum quenches in the Luttinger and related models are surveyed with emphasis on post-quench correlations. For the Luttinger model and initial gaussian states, we discuss both sudden and smooth quenches of the interaction and the emergence of a steady state described by a generalized Gibbs ensemble. Comparisons between analytics and numerics, and the question of universality or lack thereof are also discussed. The relevance of the theoretical results to current and future experiments in the fields of ultracold atomic gases and mesoscopic systems of electrons is also briefly touched upon. Wherever possible, our approach is pedagogical and self-contained. This work is dedicated to the memory of our colleague Alejandro Muramatsu.Comment: 51+epsilon pages. Review article for or special issue of JSTAT on non-equilibrium dynamics in integrable systems, Feedback is welcom

Collisional shifts in optical-lattice atom clocks

We theoretically study the effects of elastic collisions on the determination of frequency standards via Ramsey fringe spectroscopy in optical-lattice atom clocks. Interparticle interactions of bosonic atoms in multiply-occupied lattice sites can cause a linear frequency shift, as well as generate asymmetric Ramsey fringe patterns and reduce fringe visibility due to interparticle entanglement. We propose a method of reducing these collisional effects in an optical lattice by introducing a phase difference of $\pi$ between the Ramsey driving fields in adjacent sites. This configuration suppresses site to site hopping due to interference of two tunneling pathways, without degrading fringe visibility. Consequently, the probability of double occupancy is reduced, leading to cancellation of collisional shifts.Comment: 15 pages, 11 figure

A minimal mathematical model of nonphotochemical quenching of chlorophyll fluorescence

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