Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace

Hara, Takashi; Tasaki, Hal

doi:10.1088/1367-2630/17/4/045002

Cited by 89 publications

(126 citation statements)

References 33 publications

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“…(1)] than the measure of all t ∈ ½0; T. Altogether, we thus can conclude that, for the overwhelming majority of U's, the difference hAi ρðtÞ − hAi¯ρ remains below the resolution limit δA for the vast majority of times t contained in any sufficiently large time interval ½0; T. The same conclusion carries over to our actual Hamiltonian H and observable A, given their eigenbases are related by a typical transformation U as discussed above. To establish quantitative bounds for T is a subject of considerable current interest [8,13,[29][30][31][32] but goes beyond our present scope. The salient point is that (5) holds independently of the initial condition ρð0Þ.…”

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confidence: 99%

Generalization of von Neumann’s Approach to Thermalization

Reimann

2015

Phys. Rev. Lett.

108

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Thermalization of isolated many-body systems is demonstrated by generalizing an approach originally due to von Neumann: For arbitrary initial states with a macroscopically well-defined energy, quantum mechanical expectation values become indistinguishable from the corresponding microcanonical expectation values for the overwhelming majority of all sufficiently late times. As in von Neumann's work, the eigenvectors of the Hamiltonian and of the considered observable are required to not exhibit any specially tailored (untypical) orientation relative to each other. But all of von Neumann's further assumptions about the admitted observables are abandoned.

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confidence: 99%

Generalization of von Neumann’s Approach to Thermalization

Reimann

2015

Phys. Rev. Lett.

108

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“…Finally, we noted that fast thermalization on timescales of order the inverse temperature seems to be generic for broad classes of quantum states and Hamiltonians [10,11]. If, as is sometimes conjectured, all QFTs have gravitational duals, this result might be understood in terms of horizon formation: the timescale for gravitational collapse is of order the light crossing time of the resulting black hole, which is roughly the inverse temperature.…”

Section: Resultsmentioning

confidence: 66%

“…Explicit demonstrations of fast quantum thermalization have been obtained for broad classes initial states, on timescales of order the inverse temperature (i.e., average energy per mode) [10,11]. Indeed the conceptual challenge is to understand the conditions which lead to the more familiar slow (semi-classical) thermalization.…”

Section: Quantum Thermalization and Properties Of Typical Pure Statesmentioning

confidence: 99%

Entanglement and fast quantum thermalization in heavy ion collisions

Hsu

2016

Mod. Phys. Lett. A

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Let A be subsystem of a larger system A ∪ B, and ψ be a typical state from the subspace of the Hilbert space HAB satisfying an energy constraint. Then ρA(ψ) = TrB|ψ ψ| is nearly thermal. We discuss how this observation is related to fast thermalization of the central region (≈ A) in heavy ion collisions, where B represents other degrees of freedom (soft modes, hard jets, collinear particles) outside of A. Entanglement between the modes in A and B plays a central role: the entanglement entropy SA increases rapidly in the collision. In gauge-gravity duality, SA is related to the area of extremal surfaces in the bulk, which can be studied using gravitational duals.The usual particle scattering description of thermalization in heavy ion collisions (HIC) considers individual scattering events for each degree of freedom (i.e., parton). These scattering events are implicitly assumed to be incoherent -largely independent of each other. In the standard kinetic theory picture, multiple scatterings of each parton, one after another, are required to bring their distribution to the high entropy thermal configuration. By the usual reasoning, a timescale larger than 1 fm is required in order for these multiple, successive scatterings to occur.However, nucleus by nucleus scattering in HIC is a case of coherent scattering of two objects, each with many internal degrees of freedom. We can, in principle, describe the evolution of the wave functional ψ of the entire two nucleus system from asymptotic separation at early times to some specific post-collision moment in time, such as τ ≈ 1 fm after the initial overlap of the two nuclei. The wave functional then describes a superposition of each possible set of interactions or scatterings of individual partons in the system. The bulk or macroscopic properties of the remnant post-collision system are determined by an average over this complex superposition. Only a fraction of the total number of degrees of freedom are in the central region of the collision. The density matrix describing its properties is obtained through a trace over the remainder of the complex state.Recent results in the foundations of quantum statistical mechanics suggest that the system can thermalize through the spread of entanglement (i.e., superposition) much faster than expected from the usual (incoherent) kinetic theory. That is, the system can evolve from a very atypical state (two heavy nuclei, each with large boost) to a more typical one (entangled superposition state) over a short timescale.Below, we discuss fast thermalization in HIC (i.e., on timescales of τ ≈ 1 fm or less) in relation to the complex nature of the entangled superposition state, and the resulting growth of entanglement entropy. To be precise, what requires explanation is not full thermalization of the central region A, but rather the weaker condition of isotropization so that a hydrodynamic description becomes approximately valid. At minimum, the stress tensor of the matter in A must assume a diagonal form in its rest frame. For an overvie...

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“…The former is important since equilibration on very long time-scales would mean we can never actually observe it. General bounds on thermalization times, based on very few generic assumptions, are available [32][33][34][35][36][37], but they are far off from those observed in numerical calculations and actual experiments, see Ref. [28] and references therein.…”

Section: Quantum Thermalizationmentioning

confidence: 99%

“…The center-of-mass motion then decouples, and the z motion is free. The relative Hamiltonian then readŝ 37) where r = r 2 − r 1 is the relative coordinate between the two atoms and the transverse…”

Section: Interactions In Ultra-cold Gasesmentioning

confidence: 99%

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

Zill¹

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This thesis is concerned with non-equilibrium phenomena in interacting quantum manybody 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 LiebLiniger 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 uniformdensity 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...

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Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace

Cited by 89 publications

References 33 publications

Generalization of von Neumann’s Approach to Thermalization

Generalization of von Neumann’s Approach to Thermalization

Entanglement and fast quantum thermalization in heavy ion collisions

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

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