Minimalistic analytical approach to non-Markovian open quantum systems

TL;DR

This paper introduces a minimalistic analytical method using simple projection operators to solve the dynamics of finite-dimensional non-Markovian open quantum systems, providing exact short-time solutions and recursive long-time dynamics.

Contribution

It presents a novel, simplified analytical approach that yields exact solutions for non-Markovian quantum system dynamics using one-dimensional projection operators.

Findings

Analytical solutions reproduce short-time dynamics accurately.

Method applies recursively for arbitrary time intervals.

Number of relevant degrees of freedom is (n-1)(n+2)/2 for an n-dimensional system.

Abstract

The dynamics of finite dimension open quantum systems is studied with the help of the simplest possible form of projection operators, namely the ones which project only onto one dimensional subspaces. The simplicity of the action of the projection operators always leads to an analytical solution of the dynamical master equation, even in the non-Markovian case, in any perturbative order. The analytical solution correctly reproduces the short-time dynamics, and can be used to recursively recover the dynamics for an arbitrary time interval with arbitrary precision. The necessary number of relevant degrees of freedom to completely characterise an open quantum system is $(n-1)(n+2)/2,$ where $n$ is the dimension of the Hilbert space of the open system. The method is illustrated by two examples, the relaxation of a qubit in a thermal bath and the dynamics of two interacting qubits in a common environment.