Quantum trajectories for a system interacting with environment in N-photon state

TL;DR

This paper develops a stochastic master equation for a quantum system interacting with an environment in an N-photon state, providing analytical solutions, quantum trajectories, and a diagrammatic technique for understanding the output photon statistics.

Contribution

It introduces a novel stochastic master equation framework for systems coupled with N-photon states and offers a simple diagrammatic method for analyzing the solutions.

Findings

Derived the stochastic master equation for N-photon environment interactions.

Provided analytical formulas for output photon statistics.

Developed a diagrammatic technique with Feynman rules for solution analysis.

Abstract

We derive stochastic master equation for a quantum system interacting with an environment prepared in a continuous-mode $N$-photon state. To determine the conditional evolution of the quantum system depending on continuous in time measurements of the output field the model of repeated interactions and measurements is applied. The environment is defined as an infinite chain of harmonic oscillators which do not interact between themselves and they are prepared initially in an entangled state being a discrete analogue of a continuous-mode $N$-photon state. We provide not only the quantum trajectories but also the analytical formulae for the whole statistics of the output photons and the solution to the master equation. The solution in the continuous case is represented by a simple diagrammatic technique with very transparent "Feynmann rules". This technique considerably simplifies the structure of the solution and enables one to find physical interpretation for the solution in terms of a few elementary processes.