# Quantum Impurity Models coupled to Markovian and Non Markovian Baths

## TL;DR

This paper introduces a new method for analyzing quantum impurity models with both Markovian and non-Markovian baths, enabling the calculation of the impurity's reduced density matrix evolution in complex environments.

## Contribution

The authors develop an exact real-time hybridization expansion and a Dyson equation approach with Non-Crossing-Approximation for impurity models coupled to diverse baths, including Markovian dissipation.

## Key findings

- Successfully applied to a fermionic impurity with Markovian pump, losses, and dephasing.
- Provides a framework for stochastic sampling via diagrammatic Monte Carlo.
- Extends existing methods to include non-linear couplings and Markovian environments.

## Abstract

We develop a method to study quantum impurity models, small interacting quantum systems linearly coupled to an environment, in presence of an additional Markovian quantum bath, with a generic non-linear coupling to the impurity. We aim at computing the evolution operator of the reduced density matrix of the impurity, obtained after tracing out all the environmental degrees of freedom. First, we derive an exact real-time hybridization expansion for this quantity, which generalizes the result obtained in absence of the additional Markovian dissipation, and which could be amenable to stochastic sampling through diagrammatic Monte Carlo. Then, we obtain a Dyson equation for this quantity and we evaluate its self-energy with a resummation technique known as the Non-Crossing-Approximation. We apply this novel approach to a simple fermionic impurity coupled to a zero temperature fermionic bath and in presence of Markovian pump, losses and dephasing.