# Low-Temperature Quantum Fokker-Planck and Smoluchowski Equations and Their Extension to Multistate Systems

## TL;DR

This paper develops quantum Fokker-Planck and Smoluchowski equations with low-temperature corrections for simulating coupled electronic and vibrational dynamics in open quantum systems, extending to multi-state cases.

## Contribution

It introduces rigorous quantum equations with non-Markovian low-temperature corrections and extends them to multi-state systems, enabling efficient non-adiabatic dynamics simulations.

## Key findings

- Numerical validation for single-state Brownian systems.
- Application to multi-state double-well systems comparing with other methods.
- Quantum low-temperature corrections are crucial for accurate transient spectra.

## Abstract

Simulating electron-nucleus coupled dynamics poses a non-trivial challenge and an important problem in the investigation of ultrafast processes involving coupled electronic and vibrational dynamics. Because irreversibility of the system dynamics results from thermal activation and dissipation caused by the environment, in dynamical studies, it is necessary to include heat bath degrees of freedom in the total system. When the system dynamics involves high-energy electronic transitions, the environment is regarded to be in a low-temperature regime and we must treat it quantum mechanically. In this paper, we present rigorous and versatile approaches for investigating the dynamics of open systems with coupled electronic and vibrational degrees of freedom within a fully quantum mechanical framework. These approaches are based on a quantum Fokker-Planck equation and a quantum Smoluchowski equation employing a heat bath with an Ohmic spectral density, with non-Markovian low-temperature correction terms, and extensions of these equations to the case of multi-state systems. The accuracy of these equations was numerically examined for a single-state Brownian system, while their applicability was examined for multi-state double-well systems by comparing their results with those of the fewest-switch surface hopping and Ehrenfest methods with a classical Markovian Langevin force. Comparison of the transient absorption spectra obtained using these methods clearly reveals the importance of the quantum low-temperature correction terms. These equations allow us to treat non-adiabatic dynamics in an efficient way, while maintaining numerical accuracy. The C++ source codes that we developed, which allow for the treatment of the phase and coordinate space dynamics with any single-state or multi-state potential forms, are provided as Supporting Information.