# Extension of the Nakajima-Zwanzig approach to multitime correlation functions of open systems

## TL;DR

This paper extends the Nakajima-Zwanzig approach to multitime correlation functions in open quantum systems, enabling combined short- and long-time simulations through memory kernels and stochastic unraveling.

## Contribution

The authors develop a method to compute multitime correlation functions using memory kernels and stochastic unraveling, bridging short- and long-time dynamics in open quantum systems.

## Key findings

- Validated approach with 2D-spectra simulations for donor-acceptor models
- Compared results with reduced hierarchy equations of motion
- Applicable to periodically driven two-level systems in equilibrium

## Abstract

We extend the Nakajima-Zwanzig projection operator technique to the determination of multitime correlation functions of open quantum systems. The correlation functions are expressed in terms of certain multitime homogeneous and inhomogeneous memory kernels for which suitable equations of motion are derived. We show that under the condition of finite memory times these equations can be used to determine the memory kernels by employing an exact stochastic unraveling of the full system-environment dynamics. The approach thus allows to combine exact stochastic methods, feasible for short times, with long-time master equation simulations. The applicability of the method is demonstrated by numerical simulations of 2D-spectra for a donor-acceptor model, and by comparison of the results with those obtained from the reduced hierarchy equations of motion. We further show that the formalism is also applicable to the time evolution of a periodically driven two-level system initially in equilibrium with its environment.