Dynamics of Open Quantum Systems II, Markovian Approximation

Marco Merkli

doi:10.22331/q-2022-01-03-616

Abstract

A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperatureT&#x003E;0. We show that for fixed, small values of the coupling constant&#x03BB;, the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics isO(|&#x03BB;|1/4)for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracyO(|&#x03BB;|1/4)for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as1/t4or faster, which is a significant improvement relative to the previously required exponential decay.

Highlights

1.1 MotivationThe fundamental evolution equation of quantum theory is the Schrödinger equation d i ψ(t) = Hψ(t), dt (1.1)where ψ(t) is a pure state in a Hilbert space H and H is the Hamiltonian, the operator representing the energy observable. (We choose units in which we have effectively= 1.) Equivalently, density matrices ρ(t), which are mixtures of pure states, evolve according to the von-Neumann equation d i ρ(t) = [H, ρ(t)]. dt (1.2)The state of a part, or subsystem, of a whole system, is obtained by ‘tracing out’ the degrees of freedom of the complement
What is the evolution equation for ρS(t)? Generically, when S and R interact, which is expressed by the fact that H contains an interaction operator, the dynamics ρS(t) is very complicated, as it contains all information about the dynamics of R and the influence of R on S
The dynamics of an open quantum systems is usually presented in form of the evolution t → ρ(t) of the density matrix

Summary

Motivation

The state ωR,β is a linear functional on the observable algebra P of polynomials in creation and annihilation operators, or the Weyl algebra, and it can be represented by a density matrix ρR,β (having rank one), albeit in a Hilbert space different from F. Said differently, (1.18) shows that the markovian approximation is guaranteed to hold for t → ∞ only if at the same time, one takes λ → 0 with a bounded value for λ2t For this reason, (1.18) is sometimes called the weak-coupling (or van Hove) limit. Our current work uses the results of [30] to prove the validity of the markovian approximation

Main result

Example

Purification of the initial state

Liouville operator

We have