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arXiv:1910.05982

Quantum equilibration of a model system Porphine

Guillermo Albareda, Arnau Riera, Miguel González, Josep María Bofill, Iberio de P. R Moreira, Rosendo Valero, Ivano Tavernelli

Published Oct 14, 2019• 0 citations
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Abstract

There is a renewed interest in the derivation of statistical mechanics from the dynamics of closed quantum systems. A central part of this program is to understand how far-from-equilibrium closed quantum system can behave as if relaxing to a stable equilibrium. Equilibration dynamics has been traditionally studied with a focus on the so-called quenches of large-scale many-body systems. Alternatively, we consider here the equilibration of a molecular model system describing the double proton transfer reaction in porphine. Using numerical simulations, we show that equilibration in this context indeed takes place and does so very rapidly ($\sim \!\! 200$fs) for initial states induced by pump-dump laser pulse control with energies well above the synchronous tunneling barrier.

DOI
10.48550/arxiv.1910.05982

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There is a renewed interest in the derivation of statistical mechanics from the dynamics of closed quantum systems. A central part of this program is to understand how far-from-equilibrium closed quantum system can behave as if relaxing to a stable equilibrium. Equilibration dynamics has been traditionally studied with a focus on the so-called quenches of large-scale many-body systems. Alternatively, we consider here the equilibration of a molecular model system describing the double proton transfer reaction in porphine. Using numerical simulations, we show that equilibration in this context indeed takes place and does so very rapidly (∼ ⁣ ⁣200\sim \!\! 200∼200fs) for initial states induced by pump-dump laser pulse control with energies well above the synchronous tunneling barrier.
There is a renewed interest in the derivation of statistical mechanics from the dynamics of closed quantum systems. A central part of this program is to understand how closed quantum systems, i.e., in the absence of a thermal bath, initialized far-from-equilibrium can share a dynamics that is typical to the relaxation towards thermal equilibrium. Equilibration dynamics has been traditionally studied with a focus on the so-called quenches of large-scale many-body systems. We consider here the equilibration of a two-dimensional molecular model system describing the double proton transfer reaction in porphine. Using numerical simulations, we show that equilibration indeed takes place very rapidly (∼200 fs) for initial states induced by pump-dump laser pulse control with energies well above the synchronous barrier. The resulting equilibration state is characterized by a strong delocalization of the probability density of the protons that can be explained, mechanistically, as the result of (i) an initial state consisting of a large superposition of vibrational states, and (ii) the presence of a very effective dephasing mechanism.
One of the main questions of research on quantum many-body systems following unitary out of equilibrium dynamics is to find out how local expectation values equilibrate in time. For non-interacting models, this question is rather well understood. However, the best known bounds for general quantum systems are vastly crude, scaling unfavorable with the system size. Nevertheless, empirical and numerical evidence suggests that for generic interacting many-body systems, generic local observables, and sufficiently well-behaved states, the equilibration time does not depend strongly on the system size, but only the precision with which this occurs does. In this discussion paper, we aim at giving very simple and plausible arguments for why this happens. While our discussion does not yield rigorous results about equilibration time scales, we believe that it helps to clarify the essential underlying mechanisms, the intuition and important figure of merits behind equilibration. We then connect our arguments to common assumptions and numerical results in the field of equilibration and thermalization of closed quantum systems, such as the eigenstate thermalization hypothesis as well as rigorous results on interacting quantum many-body systems. Finally, we complement our discussions with numerical results - both in the case of examples and counter-examples of equilibrating systems.
We study the equilibration dynamics of closed finite quantum systems and address the question of the time needed for the system to equilibrate. In particular, we focus on the scaling of the equilibration time Teq{T}_{\mathrm{eq}}Teq​ with the system size LLL. For clean systems, we give general arguments predicting Teq=O(L0){T}_{\mathrm{eq}}=O({L}^{0})Teq​=O(L0) for clustering initial states, while for small quenches around a critical point we find ${T}_{\mathrm{eq}}=O({L}^{\ensuremath{\zeta}})$ where $\ensuremath{\zeta}$ is the dynamical critical exponent. We then analyze noisy systems where exponentially large time scales are known to exist. Specifically, we consider the tight-binding model with diagonal impurities and give numerical evidence that in this case ${T}_{\mathrm{eq}}\ensuremath{\sim}B{e}^{C{L}^{\ensuremath{\psi}}}$ where $B,C,\phantom{\rule{0.16em}{0ex}}\ensuremath{\psi}$ are observable-dependent constants. Finally, we consider another noisy system whose evolution dynamics is randomly sampled from a circular unitary ensemble. Here, we are able to prove analytically that Teq=O(1){T}_{\mathrm{eq}}=O(1)Teq​=O(1), thus showing that noise alone is not sufficient for slow equilibration dynamics.
Closed quantum systems evolve unitarily and therefore cannot converge in a strong sense to an equilibrium state starting out from a generic pure state. Nevertheless for large system size one observes temporal typicality. Namely, for the overwhelming majority of the time instants, the statistics of observables is practically indistinguishable from an effective equilibrium one. In this paper we consider the Loschmidt echo (LE) to study this sort of unitary equilibration after a quench. We draw several conclusions on general grounds and on the basis of an exactly solvable example of a quasifree system. In particular we focus on the whole probability distribution of observing a given value of the LE after waiting a long time. Depending on the interplay between the initial state and the quench Hamiltonian, we find different regimes reflecting different equilibration dynamics. When the perturbation is small and the system is away from criticality the probability distribution is Gaussian. However close to criticality the distribution function approaches a double-peaked, universal form.
ABSTRACT In the last decade, the rapid evolution of ultrafast laser spectroscopy has provided new impetus to the branch of chemistry dealing with tunneling processes involving electrons and light atoms. The possibility of measuring the time required by a particle to cross a barrier opens new perspectives for understanding the kinetics and dynamics of tunneling processes. In this study, a model is developed to estimate the tunneling time in tautomerization reactions based on proton transfer through a classically forbidden barrier. The proposed approach uses the asymptotic values of the wavefunction to compute the dwell time, which is the average time spent by the proton inside the potential barrier. The theory is formulated for both static barriers, which are good approximations for isolated molecules, and dynamic barriers to account for the effects of the chemical bath. The model performance is evaluated by two numerical simulations, in which a proton is transferred via tunneling in a double‐well potential; the tunneling barrier is approximated by a static and dynamical rectangular potential. A possible extension of the theory to open systems is briefly discussed within the framework of the Caldeira–Leggett model.
Over the last decade new experimental tools and theoretical concepts are providing new insights into collective nonequilibrium behavior of quantum systems. The exquisite control provided by laser trapping and cooling techniques allows us to observe the behavior of condensed bose and degenerate Fermi gases under nonequilibrium drive or after quenches' in which a Hamiltonian parameter is suddenly or slowly changed. On the solid state front, high intensity short-time pulses and fast (femtosecond) probes allow solids to be put into highly excited states and probed before relaxation and dissipation occur. Experimental developments are matched by progress in theoretical techniques ranging from exact solutions of strongly interacting nonequilibrium models to new approaches to nonequilibrium numerics. The summer school Strongly interacting quantum systems out of equilibrium' held at the Les Houches School of Physics as its XCIX session was designed to summarize this progress, lay out the open questions and define directions for future work. This books collects the lecture notes of the main courses given in this summer school.