Open systems dynamics: Simulating master equations in the computer

TL;DR

This paper provides a comprehensive guide to coding and simulating quantum master equations, especially in dissipative systems, using Matlab, with potential adaptation to other programming environments.

Contribution

It introduces practical methods for simulating master equations and finding steady states, tailored for quantum optics applications, based on the author's tested approaches over four years.

Findings

Effective Matlab code for master equation simulation

Methods applicable to various quantum optical systems

Guidelines for steady state and time evolution calculations

Abstract

Master equations are probably the most fundamental equations for anyone working in quantum optics in the presence of dissipation. In this context it is then incredibly useful to have efficient ways of coding and simulating such equations in the computer, and in this notes I try to introduce in a comprehensive way how do I do so, focusing on Matlab, but making it general enough so that it can be directly translated to any other language or software of choice. I inherited most of my methods from Juan Jos\'e Garc\'ia-Ripoll (whose numerical abilities I cannot praise enough), changing them here and there to accommodate them to the way my (fairly limited) numerical brain works, and to connect them as much as possible to how I understand the theory behind them. At present, the notes focus on how to code master equations and find their steady state, but I hope soon I will be able to update them with time evolution methods, including how to deal with time-dependent master equations. During the last 4 years I've tested these methods in various different contexts, including circuit quantum electrodynamics, the laser problem, optical parametric oscillators, and optomechanical systems. Comments and (constructive) criticism are greatly welcome, and will be properly credited and acknowledged.