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Exact nonequilibrium dynamics of a class of initial states in one-dimensional two-component integrable quantum gases

Neil J. Robinson, Jean-Sébastien Caux, Robert Konik

Published Feb 17, 2016• 3 citations
OpenAlex

Abstract

We present the numerically exact time-evolution of a class of initialstates in integrable one-dimensional two-component quantum gases. Thisspecial class of states, formed from a simple superposition ofeigenstates, contains a well-localized particle of one species and abackground gas containing a density-depletion (hole) in the vicinity ofthis particle, looking much like an exciton. The special structure ofthe initial states means that we can compute the time-evolution in anumerically exact manner for large numbers N= 100-1000 of interactingparticles, comparable with those studied in experiments on cold atomicgases. The initial state can be pictured as a linear superposition ofspin wave excitations, and has significant overlap with simple spin flipexcitations, which have been studied in existing experimental setups. Westudy both fermionic and bosonic quantum gases and highlight somedifferences between the two cases. In both cases, the initiallywell-localized exciton dissolves, with particle-hole excitationsspreading through the gas within a light cone. The behavior of the lightcone on varying the interaction strength and density of the gas can beunderstood from existing exact results for the spin wave mass in thesesystems. Within the light cone of the Bose gas, we see evidence ofadditional excitations about (finite momentum) roton-like minima in thespin wave dispersion.

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Exact methods in the analysis of the non-equilibrium dynamics of integrable models: application to the study of correlation functions for non-equilibrium 1D Bose gas

Vladimir Gritsev, Timofei Rostunov, Eugene Demler

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Nonequilibrium dynamics of a one-dimensional Bose gas via the coordinate Bethe ansatz

Jan C. Zill

This thesis is concerned with non-equilibrium phenomena in interacting quantum many-body systems. Specifically, we investigate the time-evolution and relaxation dynamics of Bose gases in a highly restricted geometry, which constrains the dynamics to one spatial dimension. This leads to a description of the system in terms of a simple model Hamiltonian which permits exact many-body quantum mechanical solutions due to its integrability. In the first part of this thesis, a computational method is developed to obtain experimentally relevant correlation functions in the framework of the coordinate Bethe ansatz. We employ this method to compute exact ground-state correlation functions of the Lieb-Liniger gas for up to seven particles covering the whole regime of repulsive interactions. We also investigate the dynamics of the system after an instantaneous change of the interaction strength. This quantum quench deposits large amounts of energy that cannot be dissipated due to the system being closed, and so the dynamics far from equilibrium are probed. We prepare the system in two different initial states, and quench to the same final interaction strength. The latter is determined in such a way that the added energy due to the quench is the same for both scenarios. Conventional statistical mechanics predicts the same relaxed state, but due to the integrability of the system all considered correlation functions of the relaxed states differ from each other and also from the thermal ones. We then investigate the dynamics and relaxed state for a quench from zero to repulsive interactions in more detail, focussing on the mechanism of relaxation and the involved time-scales. We find that local correlation functions relax on time-scales determined by the interaction strength, in contrast to non-local correlation functions, whose relaxation time-scale is proportional to the system size. Next, we employ the same methodology to study the one-dimensional Bose gas with attractive interactions. In this case many-body bound states are permissible solutions of the Lieb-Liniger model. We compare exact ground-state correlation functions of up to seven particles to their corresponding mean-field solution. The latter displays a quantum phase transition at a critical interaction strength, marking the transition from a uniform-density state to a localized bright soliton. Our exact results agree remarkably well with the corresponding mean-field solution past the critical point. We also investigate the dynamics following an interaction strength quench, starting again from the non-interacting ground state. Bound states strongly influence correlation functions for all post-quench interaction strengths, and local correlation functions are largely increased compared to their initial value. In the last part of this thesis, we investigate the behavior of the one-dimensional Bose gas under periodic driving of the interaction strength. To this end, we extend the coordinate Bethe-ansatz formalism by employing Floquet theory to obtain solutions for the full time- dependent Hamiltonian. This realizes an extension to non-integrable systems in a way that allows controlled breaking of integrability. We compare the dynamics of the system following an interaction quench to a fixed final interaction strength to that of a system with periodically modulated post-quench interaction strength. For fast driving, the system evolves according to the time-averaged Hamiltonian and correlation functions are nearly identical to those of the undriven system. The response of the system qualitatively changes at resonances of the time-averaged Hamiltonian, where we observe energy absorption and a marked change in correlation functions. However, the system does not absorb energy indefinitely.

•Jan 30, 2017•0 citations

Exact quantum dynamics of yrast states in the finite 1D Bose gas

Eriko Kaminishi, Jun Sato, Tetsuo Deguchi

We demonstrate that the quantum dynamics of yrast states in the one-dimensional (1D) Bose gas gives an illustrative example to equilibration of an isolated quantum many-body system. We first formulate the energy spectrum of yrast states in terms of the dressed energy by applying the method of finite-size corrections. We then review the exact time evolution of quantum states constructed from yrast states shown by the Bethe ansatz. In time evolution the density profile of an initially localized quantum state constructed from yrast states collapses into a flat profile in the case of a large particle number such as N = 1000, while recurrence of the localized state occurs in the case of a small particle number such as N = 20. We suggest that the dynamical relaxation behavior for the large N case is consistent with the viewpoint of typicality for generic quantum states: the expectation values of local operators evaluated in most of quantum states are very close to those of the micro-canonical ensemble.

arXiv:1401.4262•Journal of Physics Conference Series•Apr 9, 2014•3 citations
We present the numerically exact time-evolution of a class of initialstates in integrable one-dimensional two-component quantum gases. Thisspecial class of states, formed from a simple superposition ofeigenstates, contains a well-localized particle of one species and abackground gas containing a density-depletion (hole) in the vicinity ofthis particle, looking much like an exciton. The special structure ofthe initial states means that we can compute the time-evolution in anumerically exact manner for large numbers N= 100-1000 of interactingparticles, comparable with those studied in experiments on cold atomicgases. The initial state can be pictured as a linear superposition ofspin wave excitations, and has significant overlap with simple spin flipexcitations, which have been studied in existing experimental setups. Westudy both fermionic and bosonic quantum gases and highlight somedifferences between the two cases. In both cases, the initiallywell-localized exciton dissolves, with particle-hole excitationsspreading through the gas within a light cone. The behavior of the lightcone on varying the interaction strength and density of the gas can beunderstood from existing exact results for the spin wave mass in thesesystems. Within the light cone of the Bose gas, we see evidence ofadditional excitations about (finite momentum) roton-like minima in thespin wave dispersion.
Abstract We consider the non-equilibrium dynamics of two-component one dimensional quantum gases in the limit of extreme population imbalance where the minority species has but a single particle. We consider the situation where the gas is prepared in a state with a single spatially localized exciton: the single particle of the minority species is spatially localized while the density of the majority species in the vicinity of the minority particle sees a depression. Remarkably, we are able to consider cases where the gas contains on the order of N = 100 particles, comparable to that studied in experiments on cold atomic gases. We are able to do so by exploiting the integrability of the gas together with the observation that the excitonic state can be constructed through a simple superposition of exact eigenstates of the gas. The number of states in this superposition, rather than being exponentially large in the number of particles, scales linearly with N . We study the evolution of such spatially localized states in both strongly interacting Bose and Fermi gases. The behavior of the light cones when the interaction strength and density of the gas is varied can be understood from exact results for the spin excitation spectrum in these systems. We argue that the light cone in both cases exhibits scaling collapse. However unique to the Bose gas, we show that the presence of gapped finite-momentum roton-like excitations provide the Bose gas dynamics with secondary light cones.
In this paper we study nonequilibrium dynamics of one dimensional Bose gas from the general perspective of dynamics of integrable systems. After outlining and critically reviewing methods based on inverse scattering transform, intertwining operators, q-deformed objects, and extended dynamical conformal symmetry, we focus on the form-factor based approach. Motivated by possible applications in nonlinear quantum optics and experiments with ultracold atoms, we concentrate on the regime of strong repulsive interactions. We consider dynamical evolution starting from two initial states: a condensate of particles in a state with zero momentum and a condensate of particles in a gaussian wavepacket in real space. Combining the form-factor approach with the method of intertwining operator we develop a numerical procedure which allows explicit summation over intermediate states and analysis of the time evolution of non-local density-density correlation functions. In both cases we observe a tendency toward formation of crystal-like correlations at intermediate time scales.
Through an exact method, we numerically solve the time evolution of the density profile for an initially localized state in the one-dimensional bosons with repulsive short-range interactions. We show that a localized state with a density notch is constructed by superposing one-hole excitations. The initial density profile overlaps the plot of the squared amplitude of a dark soliton in the weak coupling regime. We observe the localized state collapsing into a flat profile in equilibrium for a large number of particles such as N=1000. The relaxation time increases as the coupling constant decreases, which suggests the existence of off-diagonal long-range order. We show a recurrence phenomenon for a small number of particles such as N=20.
In this paper we study nonequilibrium dynamics of one dimensional Bose gas from the general perspective of dynamics of integrable systems. After outlining and critically reviewing methods based on inverse scattering transform, intertwining operators, q-deformed objects, and extended dynamical conformal symmetry, we focus on the form-factor based approach. Motivated by possible applications in nonlinear quantum optics and experiments with ultracold atoms, we concentrate on the regime of strong repulsive interactions. We consider dynamical evolution starting from two initial states: a condensate of particles in a state with zero momentum and a condensate of particles in a gaussian wavepacket in real space. Combining the form-factor approach with the method of intertwining operator we develop a numerical procedure which allows explicit summation over intermediate states and analysis of the time evolution of non-local density-density correlation functions. In both cases we observe a tendency toward formation of crystal-like correlations at intermediate time scales.
This thesis is concerned with non-equilibrium phenomena in interacting quantum many-body systems. Specifically, we investigate the time-evolution and relaxation dynamics of Bose gases in a highly restricted geometry, which constrains the dynamics to one spatial dimension. This leads to a description of the system in terms of a simple model Hamiltonian which permits exact many-body quantum mechanical solutions due to its integrability. In the first part of this thesis, a computational method is developed to obtain experimentally relevant correlation functions in the framework of the coordinate Bethe ansatz. We employ this method to compute exact ground-state correlation functions of the Lieb-Liniger gas for up to seven particles covering the whole regime of repulsive interactions. We also investigate the dynamics of the system after an instantaneous change of the interaction strength. This quantum quench deposits large amounts of energy that cannot be dissipated due to the system being closed, and so the dynamics far from equilibrium are probed. We prepare the system in two different initial states, and quench to the same final interaction strength. The latter is determined in such a way that the added energy due to the quench is the same for both scenarios. Conventional statistical mechanics predicts the same relaxed state, but due to the integrability of the system all considered correlation functions of the relaxed states differ from each other and also from the thermal ones. We then investigate the dynamics and relaxed state for a quench from zero to repulsive interactions in more detail, focussing on the mechanism of relaxation and the involved time-scales. We find that local correlation functions relax on time-scales determined by the interaction strength, in contrast to non-local correlation functions, whose relaxation time-scale is proportional to the system size. Next, we employ the same methodology to study the one-dimensional Bose gas with attractive interactions. In this case many-body bound states are permissible solutions of the Lieb-Liniger model. We compare exact ground-state correlation functions of up to seven particles to their corresponding mean-field solution. The latter displays a quantum phase transition at a critical interaction strength, marking the transition from a uniform-density state to a localized bright soliton. Our exact results agree remarkably well with the corresponding mean-field solution past the critical point. We also investigate the dynamics following an interaction strength quench, starting again from the non-interacting ground state. Bound states strongly influence correlation functions for all post-quench interaction strengths, and local correlation functions are largely increased compared to their initial value. In the last part of this thesis, we investigate the behavior of the one-dimensional Bose gas under periodic driving of the interaction strength. To this end, we extend the coordinate Bethe-ansatz formalism by employing Floquet theory to obtain solutions for the full time- dependent Hamiltonian. This realizes an extension to non-integrable systems in a way that allows controlled breaking of integrability. We compare the dynamics of the system following an interaction quench to a fixed final interaction strength to that of a system with periodically modulated post-quench interaction strength. For fast driving, the system evolves according to the time-averaged Hamiltonian and correlation functions are nearly identical to those of the undriven system. The response of the system qualitatively changes at resonances of the time-averaged Hamiltonian, where we observe energy absorption and a marked change in correlation functions. However, the system does not absorb energy indefinitely.
We demonstrate that the quantum dynamics of yrast states in the one-dimensional (1D) Bose gas gives an illustrative example to equilibration of an isolated quantum many-body system. We first formulate the energy spectrum of yrast states in terms of the dressed energy by applying the method of finite-size corrections. We then review the exact time evolution of quantum states constructed from yrast states shown by the Bethe ansatz. In time evolution the density profile of an initially localized quantum state constructed from yrast states collapses into a flat profile in the case of a large particle number such as N = 1000, while recurrence of the localized state occurs in the case of a small particle number such as N = 20. We suggest that the dynamical relaxation behavior for the large N case is consistent with the viewpoint of typicality for generic quantum states: the expectation values of local operators evaluated in most of quantum states are very close to those of the micro-canonical ensemble.