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Physical Review Letters

Computational Study of Many-Dimensional Quantum Energy Flow: From Action Diffusion to Localization

Sarah Ann Schofield, Peter Guy Wolynes, Róbert E. Wyatt

Published May 8, 1995• 36 citations
Publisher

Abstract

Many-dimensional quantum calculations of energy flow are presented, revealing dynamical scaling in agreement with theoretical predictions. A model system of coupled anharmonic oscillators which undergoes a localization transition as a function of coupling strength was studied. For strong coupling, action diffusion is clearly seen. For near critical coupling, the average survival probability varies inversely with time but shows no simple relationship to the average extent of the spread of the wave function in action space.

DOI
10.1103/physrevlett.74.3720

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Many-dimensional quantum calculations of energy flow are presented, revealing dynamical scaling in agreement with theoretical predictions. A model system of coupled anharmonic oscillators which undergoes a localization transition as a function of coupling strength was studied. For strong coupling, action diffusion is clearly seen. For near critical coupling, the average survival probability varies inversely with time but shows no simple relationship to the average extent of the spread of the wave function in action space.
We present an analysis of quantum energy flow and localization in molecules from the point of view of a scaling approach. This scaling approach is based on earlier scaling theories of Anderson localization in disordered metals. The picture provides a simple general framework for describing both energy flow and the localization of eigenstates. This framework can be applied to molecules whose Hamiltonians are described by local random matrices, in which transitions between states nearby in quantum space occur more easily than those between distant states. In this theory the central quantity of interest is the survival probability. It is the squared overlap of a wave function at a later time with its initial condition. Its long time behavior varies depending on whether the eigenstates are localized or not. In the scaling description the survival probability approaches its long time limit varying inversely with time raised to some power. Different power laws are valid in different time regimes, depending on the degree of localization and the size of the molecule’s phase space. Near the localization transition we find that the survival probability decays as t−1. This decay is much slower than when energy flows easily, for a large number of dimensions participating in the energy flow.
Vibrational energy flow in molecules, like the dynamics of other many dimensional finite systems, involves quantum transport across a dense network of near resonant states. For molecules in their electronic ground state, the network is ordinarily provided by anharmonic vibrational Fermi resonances. Surface crossing between different electronic states provides another route to chaotic motion and energy redistribution. We show that nonadiabatic coupling between electronic energy surfaces facilitates vibrational energy flow, and conversely, anharmonic vibrational couplings facilitate nonadiabatic electronic state mixing. A generalization of the Logan-Wolynes theory of quantum energy flow in many-dimensional Fermi resonance systems to the two-surface case gives a phase diagram describing the boundary between localized quantum dynamics and global energy flow. We explore these predictions and test them using a model inspired by the problem of electronic excitation energy transfer in the photosynthetic reaction center. Using an explicit numerical solution of the time dependent Schrödinger equation for this ten-dimensional model, we find quite good agreement with the expectations from the approximate analytical theory.
Criteria for ergodicity and rates of energy flow in a quantum mechanical system of NNN coupled anharmonic oscillators where NNN is large are determined at energies near the ground state of the system. High-order resonances are important for the transition at large NNN. The role of numerous virtual transitions, ``vibrational superexchange,'' in global transport is examined both for typical parts of the state space and special states often interrogated experimentally.
We review a theory that predicts the onset of thermalization in a quantum mechanical coupled non-linear oscillator system, which models the vibrational degrees of freedom of a molecule. A system of N non-linear oscillators perturbed by cubic anharmonic interactions exhibits a many-body localization (MBL) transition in the vibrational state space (VSS) of the molecule. This transition can occur at rather high energy in a sizable molecule because the density of states coupled by cubic anharmonic terms scales as N3, in marked contrast to the total density of states, which scales as exp(aN), where a is a constant. The emergence of a MBL transition in the VSS is seen by analysis of a random matrix ensemble that captures the locality of coupling in the VSS, referred to as local random matrix theory (LRMT). Upon introducing higher order anharmonicity, the location of the MBL transition of even a sizable molecule, such as an organic molecule with tens of atoms, still lies at an energy that may exceed the energy to surmount a barrier to reaction, such as a barrier to conformational change. Illustrative calculations are provided, and some recent work on the influence of thermalization on thermal conduction in molecular junctions is also discussed.
Several theoretical methods for the computation of quantum dynamical quantities are formulated, implemented, and applied, in the overall context of overcoming the exponential scaling property of quantum mechanics. The convergence of the approximate forward-backward semiclassical dynamics (FBSD) method has been accelerated by working with only the imaginary part of the real-time position and momentum time correlation functions. Simple manipulations allow the computationally favorable limit of a single factorization (“bead”) of the Boltzmann operator to yield greatly improved results. Numerical examples for a one-dimensional anharmonic oscillator and Lennard-Jones fluids are presented. Bohm's trajectory formulation of quantum mechanics is an exact mapping of the time-dependent Schrödinger equation onto classical fluid dynamics, plus one extra term responsible for all non-classical effects, the dreaded quantum potential. A computational procedure has been introduced to stably solve Bohm's hydrodynamic equations of motion in the forward-backward quantum dynamics (FBQD) formulation of time correlation functions, and applied to one-dimensional bound systems at zero and low temperature. A key step in the formulation of this procedure was the use of Hamilton's law of varying action, a generalization of Hamilton's principle of stationary action applicable to initial-value problems. This has resulted in the first reported synthetic computation of non-stationary quantum trajectories for a fully anharmonic oscillator. Success in one dimension has laid the groundwork for future extension to a rigorous quantum-classical scheme, which is briefly discussed. Other unpublished methods for obtaining time correlation functions and propagators in quantum dynamics are also discussed.
Quantum dynamics on quasiperiodic geometries has recently gathered significant attention in ultracold atom experiments where nontrivial localized phases have been observed. One such quasiperiodic model is the so-called Fibonacci model. In this tight-binding model, noninteracting particles are subject to on-site energies generated by a Fibonacci sequence. This is known to induce critical states, with a continuously varying dynamical exponent, leading to anomalous transport. In this work, we investigate whether anomalous diffusion present in the noninteracting system survives in the presence of interactions and establish connections to a possible transition towards a localized phase. We investigate the dynamics of the interacting Fibonacci model by studying real-time spread of density-density correlations at infinite temperature using the dynamical typicality approach. We also corroborate our findings by calculating the participation entropy in configuration space and investigating the expectation value of local observables in the diagonal ensemble.