# Non-Markovian Dynamics of Quantum Open Systems Embedded in a Hybrid Environment

## TL;DR

This paper introduces an exact stochastic Schrödinger equation for open quantum systems interacting with a hybrid environment of bosonic and fermionic particles, enabling detailed analysis of their non-Markovian dynamics.

## Contribution

It develops a novel stochastic approach using classical Gaussian and non-commutative fermionic noises to model hybrid quantum baths, providing a new tool for studying complex open quantum systems.

## Key findings

- Derived a stochastic Schrödinger equation for hybrid environments.
- Validated the approach with three illustrative examples.
- Revealed unique features of fermionic and bosonic baths.

## Abstract

Quantum systems of interest are typically coupled to several quantum channels (more generally environments). In this paper, we develop an exact stochastic Schr\"{o}dinger equation for an open quantum system coupled to a hybrid environment containing both bosonic and fermionic particles. Such a stochastic differential equation may be obtained directly from a microscopic model through employing a classical complex Gaussian noise and a non-commutative fermionic noise to simulate the hybrid bath. As an immediate application of our developed stochastic approach, we show that the evolution of the reduced density matrix can be derived by taking the average over both the bosonic noise and the fermionic noise. Three specific examples are given in this paper to illustrate that the hybrid quantum trajectory is fully consistent with the standard quantum mechanics. Our examples also shed new light on the special features exhibited by the fermionic bath and bosnoic bath.