A simple derivation of the Lindblad equation

TL;DR

This paper provides an accessible derivation of the Lindblad equation for non-unitary quantum evolution, focusing on a simple system-bath interaction at zero temperature, suitable for undergraduate students.

Contribution

It offers a straightforward derivation of the Lindblad equation from the Born-Markov approximation in a simple quantum system, illustrating the emergence of Lindbladian dynamics.

Findings

Lindblad equation derived from basic assumptions

Markov approximation is essential for Lindbladian form

Simplified model clarifies environment's effect on quantum systems

Abstract

We present a derivation of the Lindblad equation - an important tool for the treatment of non-unitary evolutions - that is accessible to undergraduate students in physics or mathematics with a basic background on quantum mechanics. We consider a specific case, corresponding to a very simple situation, where a primary system interacts with a bath of harmonic oscillators at zero temperature, with an interaction Hamiltonian that resembles the Jaynes-Cummings format. We start with the Born-Markov equation and, tracing out the bath degrees of freedom, we obtain an equation in the Lindblad form. The specific situation is very instructive, for it makes it easy to realize that the Lindblads represent the effect on the main system caused by the interaction with the bath, and that the Markov approximation is a fundamental condition for the emergence of the Lindbladian operator. The formal derivation of the Lindblad equation for a more general case requires the use of quantum dynamical semi-groups and broader considerations regarding the environment and temperature than we have considered in the particular case treated here.