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Molecular Physics

Memory propagator matrix for long-time dissipative charge transfer dynamics

Roberto Lambert, Nancy Makri

Published Jul 13, 2012• 51 citations
Publisher

Abstract

Iterative path integral methods provide a numerically exact approach to the dynamics of a quantum system coupled to a dissipative bath. These methods involve step-by-step propagation of a generalized multi-time reduced density matrix with a propagator that includes influence functional interactions that span the bath-induced memory length. Low-frequency baths lead to long memory that can span many time steps, necessitating the use of filtering procedures that select and store only statistically significant path segments, avoiding storage of the entire sparse propagator. The present paper further enhances the iterative path integral methodology through a deterministic procedure that identifies at the start of the calculation all memory-length path segments whose weights exceed a chosen threshold and builds the propagator matrix for use in all propagation steps. A binning procedure allows efficient pairing of path segments required in the construction of the propagator. Additional savings are achieved via a simple criterion for assessing the adequacy of a chosen path selection threshold prior to propagation. Application to a tight-binding model for charge transfer in extended systems reveals the important role of bath-induced memory on the dynamics of these systems.

DOI
10.1080/00268976.2012.700408

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Iterative path integral methods provide a numerically exact approach to the dynamics of a quantum system coupled to a dissipative bath. These methods involve step-by-step propagation of a generalized multi-time reduced density matrix with a propagator that includes influence functional interactions that span the bath-induced memory length. Low-frequency baths lead to long memory that can span many time steps, necessitating the use of filtering procedures that select and store only statistically significant path segments, avoiding storage of the entire sparse propagator. The present paper further enhances the iterative path integral methodology through a deterministic procedure that identifies at the start of the calculation all memory-length path segments whose weights exceed a chosen threshold and builds the propagator matrix for use in all propagation steps. A binning procedure allows efficient pairing of path segments required in the construction of the propagator. Additional savings are achieved via a simple criterion for assessing the adequacy of a chosen path selection threshold prior to propagation. Application to a tight-binding model for charge transfer in extended systems reveals the important role of bath-induced memory on the dynamics of these systems.
An on-the-fly filtered propagator functional path integral scheme is introduced as an efficient way of calculating the iterative dynamics of complex condensed systems. Time evolution of the reduced density matrix of a dissipative quantum system is evaluated iteratively by filtering the negligible propagator elements at each propagation step. This on-the-fly filtering along with the finiteness of the bath memory dramatically reduces the configuration space to be integrated without losing numerical accuracy of the results. The required computational storage space for the configuration and the weight of the survived path segments increases linearly with the bath memory length. Up to the bath memory time, it is found that a strikingly small fraction of the configurations survives the on-the-fly filtering process and the number of surviving configurations increases algebraically with the propagation time. At times longer than the bath memory time, the number of surviving configurations required for numerically accurate results is essentially saturated and is less than 0.1% of the total number of the configurations. This new scheme is extremely useful for the problems of low-frequency solvents for which the bath memory spans many time steps of the propagation.
The reduced density matrix of a system coupled to a harmonic dissipative bath can be obtained accurately using an iterative path integral methodology that involves a propagator tensor, whose rank is roughly equal to the number of time steps that span the bath-induced memory. The present work shows that even in situations where the bath is characterized by high-frequency modes (which necessitate a small time step) and slow oscillators (which give rise to very long memory), efficient propagation that takes into account the memory on all time scales is feasible by renormalizing the propagator matrix to include successively larger-length influence functional correlations. At its crudest level, the renormalized propagator is an ordinary matrix, thus the computational demands of the method are very modest, similar to those for propagating the bare system. Yet, test applications on dissipative two-level systems show that the method produces accurate results in demanding situations. By including memory on successively increasing time scales, the method is capable of capturing nonexponential kinetics induced by sluggish solvents.
The discretized path integral expression for the reduced density matrix (RDM) of a system interacting with a dissipative harmonic bath is fully entangled because of influence functional terms that couple the variables at different time points. The iterative decomposition of the path integral, which exploits the finite length of influence functional memory, involves a tensor propagator whose size grows exponentially with the memory length. The present Communication disentangles the path integral by recursively spreading the temporal entanglement over longer path segments, while decreasing its contribution. Eventually, the entangled term becomes sufficiently small and may be neglected, leading to iterative propagation of the RDM through simple multiplication of matrices whose size is equal to that of the bare system. It is found that the temporal entanglement length is practically equal to the bath-induced memory length. The small matrix decomposition of the path integral (SMatPI) is stable and very efficient, extending the applicability of numerically exact real-time path integral methods to multi-state systems.
The iterative decomposition of the blip-summed path integral [N. Makri, J. Chem. Phys. 141, 134117 (2014)] is described. The starting point is the expression of the reduced density matrix for a quantum system interacting with a harmonic dissipative bath in the form of a forward-backward path sum, where the effects of the bath enter through the Feynman-Vernon influence functional. The path sum is evaluated iteratively in time by propagating an array that stores blip configurations within the memory interval. Convergence with respect to the number of blips and the memory length yields numerically exact results which are free of statistical error. In situations of strongly dissipative, sluggish baths, the algorithm leads to a dramatic reduction of computational effort in comparison with iterative path integral methods that do not implement the blip decomposition. This gain in efficiency arises from (i) the rapid convergence of the blip series and (ii) circumventing the explicit enumeration of between-blip path segments, whose number grows exponentially with the memory length. Application to an asymmetric dissipative two-level system illustrates the rapid convergence of the algorithm even when the bath memory is extremely long.
Recent progress in numerical methods for evaluating the real-time path integral in dissipative harmonic environments is reviewed. Quasi-adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi-adiabatic propagator path integral is evaluated by efficient system-specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunneling dynamics in the condensed phase are described.
A small matrix decomposition of the path integral expression (SMatPI) that yields the reduced density matrix of a system interacting with a dissipative harmonic bath is obtained by recursively spreading the entangled influence functional terms over longer time intervals while simultaneously decreasing their magnitude until these terms become negligible. This allows summation over the path integral variables one by one through multiplication of small matrices with dimension equal to that of the bare system. The theoretical framework of the decomposition is described using a diagrammatic approach. Analytical and numerical calculations show that the necessary time length for the temporal entanglement to become negligible is practically the same as the bath-induced memory. The properties and structure of the propagator matrices are discussed, and applications to multistate systems are presented.