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arXiv:2304.02314 · Nuclear Physics B

Quantum quenches from an excited state

Gesualdo Delfino A, Marianna Sorba

Published Aug 1, 2023• 5 citations
arXivHTMLPDF

Abstract

Determining the role of initial conditions in the late time evolution is a key issue for the theory of nonequilibrium dynamics of isolated quantum systems. Here we extend the theory of quantum quenches to the case in which before the quench the system is in an excited state. In particular, we show perturbatively in the size of the quench (and for arbitrarily strong interactions among the quasiparticles) that persistent oscillations of one-point functions require the presence of a one-quasiparticle contribution to the nonequilibrium state, as originally shown in [J. Phys. A 47 (2014) 402001] for the quenches from the ground state. Also in the present case, we argue that the results generically have nonperturbative implications. Oscillations staying undamped within the accessible time interval, far beyond the perturbative time scale, are nowadays observed in numerical simulations.

DOI
10.1016/j.nuclphysb.2023.116312

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Determining the role of initial conditions in the late time evolution is a key issue for the theory of nonequilibrium dynamics of isolated quantum systems. Here we extend the theory of quantum quenches to the case in which before the quench the system is in an excited state. In particular, we show perturbatively in the size of the quench (and for arbitrarily strong interactions among the quasiparticles) that persistent oscillations of one-point functions require the presence of a one-quasiparticle contribution to the nonequilibrium state, as originally shown in [J. Phys. A 47 (2014) 402001] for the quenches from the ground state. Also in the present case, we argue that the results generically have nonperturbative implications. Oscillations staying undamped within the accessible time interval, far beyond the perturbative time scale, are nowadays observed in numerical simulations.
We previously showed that a quantum quench in a one-dimensional translation invariant system produces undamped oscillations of a local observable when the post-quench state includes a single-quasiparticle mode and the observable couples to that mode [J. Phys. A 47 (2014) 402001]. Here we consider quenches that break initial translation invariance. Focusing on quenches performed only on an interval of the whole system, we analytically determine the time evolution of local observables, which occurs inside a truncated light cone spreading away from the quenched interval as time increases. If the quench excites a single-quasiparticle mode, oscillations with the frequency of the quasiparticle mass stay undamped until a time increasing with the length of the quenched interval, before eventually decaying as t−1/2. The translation invariant case with no damping is recovered as the length of the interval goes to infinity.
Abstract The theory of quantum quenches in near-critical one-dimensional systems formulated in Delfino (2014 J. Phys. A: Math. Theor . 402001) yields analytic predictions for the dynamics, unveils a qualitative difference between non-interacting and interacting systems, with undamped oscillations of one-point functions occurring only in the latter case, and explains the presence and role of different time scales. Here we examine additional aspects, determining in particular the relaxation value of one-point functions for small quenches. For a class of quenches we relate this value to the scaling dimensions of the operators. We argue that the E 8 spectrum of the Ising chain can be more accessible through a quench than at equilibrium, while for a quench of the plane anisotropy in the XYZ chain we obtain that the one-point function of the quench operator switches from damped to undamped oscillations at Δ = 1 / 2 .
We extend the theory of quantum quenches to the case of d -dimensional homogeneous systems with long range interactions. This is achieved treating the long range interactions as switched on by the quench and performing the derivation within the basis of asymptotic states of the short range interacting pre-quench theory. In this way we analytically determine the post-quench state and the one-point functions of local observables such as the order parameter. One implication is that, as in the short range case, some oscillations induced by the quench remain undamped at large times under conditions specified by the theory. This explains, in particular, why such undamped oscillations have been numerically observed also in presence of long range interactions.
The theory of quantum quenches in translationally invariant isolated systems near quantum critical points is formulated from first principles. The theory generally explains the role of interactions among the quasi-particle excitations, the quantum mechanism at the origin of light cone spreading of correlations, and the presence of different time scales in the dynamics. In one dimension the theory clarifies the role of integrability and predicts that persistent oscillations of the order parameter arise depending on the coupling subjected to the quench. The oscillation frequencies correspond to the masses of the quasi-particle excitations. The predictions of the theory are confirmed by independent numerical studies. References: [1] G. Delfino, J. Phys. A: Math. Theor. 47 (2014) 402001. [2] G. Delfino and J. Viti, J. Phys. A: Math. Theor. 50 (2017) 084004. [3] G. Delfino, Phys. Rev. E 97, 062138 (2018).
We obtain analytical results for the time evolution of local observables in systems undergoing quantum quenches in d spatial dimensions. For homogeneous systems we show that oscillations undamped in time occur when the state produced by the quench includes single-quasiparticle modes and the observable couples to those modes. In particular, a quench of the transverse field within the ferromagnetic phase of the Ising model produces undamped oscillations of the order parameter when d>1. For the more general case in which the quench is performed only in a subregion of the whole d-dimensional space occupied by the system, the time evolution occurs inside a light cone spreading away from the boundary of the quenched region as time increases. The additional condition for undamped oscillations is that the volume of the quenched region is extensive in all dimensions.
The eigenstate thermalization hypothesis provides one picture of thermalization in a quantum system by looking at individual eigenstates. However, it is also important to consider how local observables reach equilibrium values dynamically. Quench protocol is one of the settings to study such questions. A recent numerical study [Ba\~nuls, Cirac, and Hastings, Phys. Rev. Lett. 106, 050405 (2007)] of a nonintegrable quantum Ising model with longitudinal field under such a quench setting found different behaviors for different initial quantum states. One particular case called the ``weak-thermalization'' regime showed apparently persistent oscillations of some observables. Here we provide an explanation of such oscillations. We note that the corresponding initial state has low energy density relative to the ground state of the model. We then use perturbation theory near the ground state and identify the oscillation frequency as essentially a quasiparticle gap. With this quasiparticle picture, we can then address the long-time behavior of the oscillations. Upon making additional approximations which intuitively should only make thermalization weaker, we argue that the oscillations nevertheless decay in the long-time limit. As part of our arguments, we also consider a quench from a BEC to a hard-core boson model in one dimension. We find that the expectation value of a single-boson creation operator oscillates but decays exponentially in time, while a pair-boson creation operator has oscillations with a ${t}^{\ensuremath{-}3/2}$ decay in time. We also study dependence of the decay time on the density of bosons in the low-density regime and use this to estimate decay time for oscillations in the original spin model.