Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Physical Chemistry Letters

Parsing the Influence Functional: Harmonic Bath Mapping and Anharmonic Small Matrix Path Integral

Nancy Makri

Published Apr 19, 2024• 10 citations
Publisher

Abstract

The influence functional (IF) encodes all of the information required for calculating the dynamical properties of a system in contact with its environment. A direct and simple procedure is introduced for extracting from a few numerical evaluations of the IF, without computing time correlation functions or evaluating integrals, the parameters required for path integral calculations, either within or beyond the harmonic mapping, and for assessing the accuracy of the harmonic bath approximation. In addition, the small matrix decomposition of the path integral (SMatPI) is extended to anharmonic environments and the required matrices are constructed directly from the IF.

DOI
10.1021/acs.jpclett.4c00908

Related papers

Two-Tier Modular Anharmonic Small Matrix Path Integral with Composite Spin-Boson Baths

Nancy Makri

The anharmonic small matrix path integral (anh-SMatPI) algorithm is adapted to composite environments, where each anharmonic bath unit couples to the system and to its own harmonic bath. The propagation matrices are obtained from numerical evaluation of the composite influence functional, which is based on separate iterative path integral calculations for each bath module. The numerically exact nature of the algorithm is demonstrated on a two-level system (TLS) coupled to dissipative TLS baths of one, two, and 50 units.

Journal of Chemical Theory and Computation•Sep 20, 2024•6 citations

Small Matrix Path Integral for System-Bath Dynamics

Nancy Makri

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.

Journal of Chemical Theory and Computation•Jun 15, 2020•95 citations

Small Matrix Path Integral for Driven Dissipative Dynamics

Nancy Makri

The small matrix decomposition of the quasi-adiabatic propagator path integral (SMatPI) for a system coupled to a harmonic bath, which accounts for multitime memory correlations in the influence functional without the use of tensors, is extended to include a time-dependent term that drives the system. In the case of a periodic field, the algorithm requires the construction of SMatPI matrices initialized over a short time interval. The SMatPI algorithm circumvents the large array storage of tensor-based iterative path integral decompositions and, in the case of a periodic field, also eliminates the demanding tensor multiplication at each time step, leading to dramatic savings which allow the fully quantum mechanical treatment of multistate systems and long-memory environments.

The Journal of Physical Chemistry A•Nov 23, 2021•29 citations

The Linear Response Approximation and Its Lowest Order Corrections: An Influence Functional Approach

Nancy Makri

A cumulant expansion of the influence functional entering the path integral representation of the reduced density matrix is presented. Truncation of the expansion at the quadratic terms allows mapping of a many-body environment onto an effective harmonic model with the same force autocorrelation function, recovering the conventional linear response limit. It is shown that the latter becomes exact in the thermodynamic limit if the bath consists of independent modes coupled to the system via coefficients that scale appropriately with the size of the bath. Low-order correction terms to the linear response approximation are found to be given by multitime correlation functions of the force exerted by the bath on the system. Within a classical approximation, such correction terms can be obtained from molecular dynamics calculations and incorporated in the path integral, which can be evaluated by means of iterative procedures. Numerical calculations on an intrinsically nonlinear bath of two-level systems confirm the above behaviors and indicate that the deviations from the linear response approximation are more significant at low temperatures where quantum mechanical effects play an important role to the bath dynamics.

The Journal of Physical Chemistry B•Mar 27, 1999•250 citations

Small matrix disentanglement of the path integral: Overcoming the exponential tensor scaling with memory length

Nancy Makri

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 Journal of Chemical Physics•Jan 27, 2020•101 citations

Iterative evaluation of the path integral for a system coupled to an anharmonic bath

Nancy Makri

An iterative algorithm is presented for evaluating the path integral expression for the reduced density matrix of a quantum system interacting with an anharmonic dissipative bath whose influence functional is obtained via numerical methods. The method allows calculation of the reduced density matrix over very long time periods.

The Journal of Chemical Physics•Oct 8, 1999•60 citations
The influence functional (IF) encodes all of the information required for calculating the dynamical properties of a system in contact with its environment. A direct and simple procedure is introduced for extracting from a few numerical evaluations of the IF, without computing time correlation functions or evaluating integrals, the parameters required for path integral calculations, either within or beyond the harmonic mapping, and for assessing the accuracy of the harmonic bath approximation. In addition, the small matrix decomposition of the path integral (SMatPI) is extended to anharmonic environments and the required matrices are constructed directly from the IF.
The anharmonic small matrix path integral (anh-SMatPI) algorithm is adapted to composite environments, where each anharmonic bath unit couples to the system and to its own harmonic bath. The propagation matrices are obtained from numerical evaluation of the composite influence functional, which is based on separate iterative path integral calculations for each bath module. The numerically exact nature of the algorithm is demonstrated on a two-level system (TLS) coupled to dissipative TLS baths of one, two, and 50 units.
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.
The small matrix decomposition of the quasi-adiabatic propagator path integral (SMatPI) for a system coupled to a harmonic bath, which accounts for multitime memory correlations in the influence functional without the use of tensors, is extended to include a time-dependent term that drives the system. In the case of a periodic field, the algorithm requires the construction of SMatPI matrices initialized over a short time interval. The SMatPI algorithm circumvents the large array storage of tensor-based iterative path integral decompositions and, in the case of a periodic field, also eliminates the demanding tensor multiplication at each time step, leading to dramatic savings which allow the fully quantum mechanical treatment of multistate systems and long-memory environments.
A cumulant expansion of the influence functional entering the path integral representation of the reduced density matrix is presented. Truncation of the expansion at the quadratic terms allows mapping of a many-body environment onto an effective harmonic model with the same force autocorrelation function, recovering the conventional linear response limit. It is shown that the latter becomes exact in the thermodynamic limit if the bath consists of independent modes coupled to the system via coefficients that scale appropriately with the size of the bath. Low-order correction terms to the linear response approximation are found to be given by multitime correlation functions of the force exerted by the bath on the system. Within a classical approximation, such correction terms can be obtained from molecular dynamics calculations and incorporated in the path integral, which can be evaluated by means of iterative procedures. Numerical calculations on an intrinsically nonlinear bath of two-level systems confirm the above behaviors and indicate that the deviations from the linear response approximation are more significant at low temperatures where quantum mechanical effects play an important role to the bath dynamics.
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.