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An exact formalism for quench dynamics

Deepak Iyer, Huijie Guan, Natan Andrei

Published Apr 2, 2013• 1 citation
OpenAlex

Abstract

We describe a formulation for studying the quench dynamics of integrable systems generalizing an approach by Yudson. We study the evolution of the Lieb-Liniger model, a gas of interacting bosons moving on the continuous infinite line and interacting via a short range potential. The formalism allows us to quench the system from any initial state. We find that for any value of repulsive coupling independently of the initial state the system asymptotes towards a strongly repulsive gas, while for any value of attractive coupling, the system forms a maximal bound state that dominates at longer times. In either case the system equilibrates but does not thermalize. We compare this to quenches in a Bose-Hubbard lattice and show that there, initial states determine long-time dynamics independent of the sign of the coupling.

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Exact formalism for the quench dynamics of integrable models

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We describe a formulation for studying the quench dynamics of integrable systems generalizing an approach by Yudson. We study the evolution of the Lieb-Liniger model, a gas of interacting bosons moving on the continuous infinite line and interacting via a short-range potential. The formalism allows us to quench the system from any initial state. We find that for any value of repulsive coupling independently of the initial state the system asymptotes towards a strongly repulsive gas, while for any value of attractive coupling, the system forms a maximal bound state that dominates at longer times. In either case the system equilibrates but does not thermalize. We compare this to quenches in a Bose-Hubbard lattice and show that there initial states determine long-time dynamics independent of the sign of the coupling.

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This chapter discusses aspects of the physics revealed by quantum quenches and presents a formalism for studying the quench dynamics of integrable systems. This formalism is applied to the Lieb–Liniger model of a gas of N interacting bosons moving on the continuous infinite line while interacting via a short-range potential. The quench is carried out from several initial states and for any number of particles, and the evolution of the density and the noise correlations is computed. In the long-time limit, the system dilutes and, for any value of repulsive coupling, independently of the initial state, it asymptotes toward a strongly repulsive gas, while for any value of attractive coupling, it forms a maximal bound state that dominates at longer times. In either case, the system equilibrates but does not thermalize, an effect that is consistent with prethermalization. These results can be confronted with experiments.

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We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.

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Márton Kormos, Aditya Shashi, Yang-Zhi Chou +1 more

We obtain exact results on interaction quenches in the 1D Bose gas described by the integrable Lieb-Liniger model. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble and a description within the generalized Gibbs ensemble (GGE) is needed. For a non-interacting initial state and arbitrary final interactions, we find that the presence of infinitely many conserved charges generates a non-analytic behavior in the equilibrated density of quasimomenta. This manifests itself in a dynamically generated Friedel-like oscillation of the non-local correlation functions with interaction dependent oscillation momenta. We also exactly evaluate local correlations and the generalized chemical potentials within GGE.

arXiv:1204.3889•Apr 17, 2012•4 citations
We describe a formulation for studying the quench dynamics of integrable systems generalizing an approach by Yudson. We study the evolution of the Lieb-Liniger model, a gas of interacting bosons moving on the continuous infinite line and interacting via a short range potential. The formalism allows us to quench the system from any initial state. We find that for any value of repulsive coupling independently of the initial state the system asymptotes towards a strongly repulsive gas, while for any value of attractive coupling, the system forms a maximal bound state that dominates at longer times. In either case the system equilibrates but does not thermalize. We compare this to quenches in a Bose-Hubbard lattice and show that there, initial states determine long-time dynamics independent of the sign of the coupling.
We describe a formulation for studying the quench dynamics of integrable systems generalizing an approach by Yudson. We study the evolution of the Lieb-Liniger model, a gas of interacting bosons moving on the continuous infinite line and interacting via a short-range potential. The formalism allows us to quench the system from any initial state. We find that for any value of repulsive coupling independently of the initial state the system asymptotes towards a strongly repulsive gas, while for any value of attractive coupling, the system forms a maximal bound state that dominates at longer times. In either case the system equilibrates but does not thermalize. We compare this to quenches in a Bose-Hubbard lattice and show that there initial states determine long-time dynamics independent of the sign of the coupling.
This chapter discusses aspects of the physics revealed by quantum quenches and presents a formalism for studying the quench dynamics of integrable systems. This formalism is applied to the Lieb–Liniger model of a gas of N interacting bosons moving on the continuous infinite line while interacting via a short-range potential. The quench is carried out from several initial states and for any number of particles, and the evolution of the density and the noise correlations is computed. In the long-time limit, the system dilutes and, for any value of repulsive coupling, independently of the initial state, it asymptotes toward a strongly repulsive gas, while for any value of attractive coupling, it forms a maximal bound state that dominates at longer times. In either case, the system equilibrates but does not thermalize, an effect that is consistent with prethermalization. These results can be confronted with experiments.
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 time evolution of the density and noise correlation for a small number of bosons and their asymptotic behavior 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 the 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."
This thesis studies the quench dynamics of strongly correlated quantum systems described by one dimensional integrable Hamiltonians. We develop the Yudson approach for such systems in terms of contour integrals allowing the expansion of arbitrary states in terms of the Bethe Ansatz eigenstates which in turn provides the means to calculate the time evolution of such observables as densities or noise correlations from any initial states. As a motivation to the present work, I present the state of art ultracold atom techniques and fundamental questions related to the quench dynamics. Then various Bethe ansatz solutions are discussed. This is followed by an introduction to the Yudson approach where advantages and difficulties are listed. As applications of the Yudson approach, I studied the nonequilibrium dynamics of the Lieb-Liniger model,(bosonic) Gaudin-Yang model, quenched from a Mott state with a super fluid Hamiltonian. Integration contours are specified for various models and different interactions. It is shown that the Yudson approach incorporates all free states and bound states into the contour, separating them apart sheds light on the validity of the String hypothesis. The result is affirmative for Lieb-Linger model, but not for the Gaudin-Yang model. Our calculations for the density and correlation shows that for interacting system, if the pre-quench state has negligible overlap among the particles, the system retains this feature after the quench. Particularly, normalized noise function c(z; -z) at the origin shows different stages. Shortly after the quench, the sign are different for attractive and repulsive interaction. Then they both quickly approaches the value where possibility to find both particles equals zero. Then the value increases gradually for attractive models, while c(0; 0) remains small for repulsive systems.
We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.
We obtain exact results on interaction quenches in the 1D Bose gas described by the integrable Lieb-Liniger model. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble and a description within the generalized Gibbs ensemble (GGE) is needed. For a non-interacting initial state and arbitrary final interactions, we find that the presence of infinitely many conserved charges generates a non-analytic behavior in the equilibrated density of quasimomenta. This manifests itself in a dynamically generated Friedel-like oscillation of the non-local correlation functions with interaction dependent oscillation momenta. We also exactly evaluate local correlations and the generalized chemical potentials within GGE.