Convergence to equilibrium under a random Hamiltonian

We analyze the thermalization properties and the validity of the Eigenstate Thermalization Hypothesis in a generic class of quantum Hamiltonians where the quench parameter explicitly breaks a Z 2 symmetry. Natural realizations of such systems are given by random matrices expressed in a block form where the terms responsible for the quench dynamics are the offdiagonal blocks. Our analysis examines both dense and sparse random matrix realizations of the Hamiltonians and the observables. Sparse random matrices may be associated with local quantum Hamiltonians and they show a different spread of the observables on the energy eigenstates with respect to the dense ones. In particular, the numerical data seems to support the existence of rare states, i.e. states where the observables take expectation values which are different compared to the typical ones sampled by the micro-canonical distribution. In the case of sparse random matrices we also extract the finite size behavior of two different time scales associated with the thermalization process.

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“…In contrast, the time scale for dense matrices was recently argued to go as the inverse of the volume [41].…”

Section: Time Scales Of Thermalizationmentioning

confidence: 99%

“…It is therefore of interest to give a precise estimate of the time needed for thermalization. There have been different approaches which have tried to clarify this question [39,40,41,42]. In Fig.…”

Section: Time Scales Of Thermalizationmentioning

confidence: 99%

Quench dynamics in randomly generated extended quantum models

et al. 2012

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“…[340] for an early review and Refs. [341,342,343,344] for further results. Note that, in each subensemble E sub , the expectation values of the observables of S + M may evolve according to (11.11) on the time lapse τ sub ; however, they remain constant for the full ensemble sinceD(t) has already reached its stationary value: When the subensembles E k of some decomposition (11.4) of E are put back together, the time dependences issued from (11.11) compensate one another.…”

Section: Subensemble Relaxation Of the Pointer Alonementioning

confidence: 99%

Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

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“…Since the mid2000s there has been increasing research activities in the field of "equilibration" and "thermalization" with respect to closed quantum systems, although the latter mechanisms are traditionally associated with stochastic processes. Most of these research activities have focused on the remarkable fact that after some, possibly very long, time [3][4][5], the behavior of many observables is very well be practically indistinguishable from standard phenomenological equilibrium behavior, despite the fact that the Schrödinger equation does not feature any attractive fixed point. Some of these attempts follow concepts of pure state quantum statistical mechanics [6,7], typicality [8][9][10], or eigenstate thermalization hypothesis [9,11].…”

Section: Introductionmentioning

confidence: 99%

Numerical evidence for approximate consistency and Markovianity of some quantum histories in a class of finite closed spin systems

Schmidtke

Gemmer

2016

Phys. Rev. E

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Closed quantum systems obey the Schrödinger equation whereas nonequilibrium behavior of many systems is routinely described in terms of classical, Markovian stochastic processes. Evidently, there are fundamental differences between those two types of behavior. We discuss the conditions under which the unitary dynamics may be mapped onto pertinent classical stochastic processes. This is first principally addressed based on the notions of "consistency" and "Markovianity." Numerical data are presented that show that the above conditions are to good approximation fulfilled for Heisenberg-type spin models comprising 12-20 spins. The accuracy to which these conditions are met increases with system size.

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Convergence to equilibrium under a random Hamiltonian

Cited by 66 publications

References 48 publications

Quench dynamics in randomly generated extended quantum models

Quench dynamics in randomly generated extended quantum models

Understanding quantum measurement from the solution of dynamical models

Numerical evidence for approximate consistency and Markovianity of some quantum histories in a class of finite closed spin systems

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