Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rotating-Wave and Secular Approximations for Open Quantum Systems

Daniel Burgarth, Paolo Facchi, Giovanni Gramegna, Kazuya Yuasa

Abstract

We derive a nonperturbative bound on the distance between evolutions of open quantum systems described by time-dependent generators. We show how this result can be employed to provide an explicit upper bound on the error of the rotating-wave approximation in the presence of dissipation and decoherence. We apply the derived bound to the strong-coupling limit in open quantum systems and to the secular approximation used to obtain a master equation from the Redfield equation.

Rotating-Wave and Secular Approximations for Open Quantum Systems

Abstract

We derive a nonperturbative bound on the distance between evolutions of open quantum systems described by time-dependent generators. We show how this result can be employed to provide an explicit upper bound on the error of the rotating-wave approximation in the presence of dissipation and decoherence. We apply the derived bound to the strong-coupling limit in open quantum systems and to the secular approximation used to obtain a master equation from the Redfield equation.

Paper Structure

This paper contains 10 sections, 13 theorems, 156 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of the Results and Outline of the Paper
  3. Preliminaries on Evolution Operators
  4. Integration-by-Part Lemma
  5. Main Result: Constant Strong Generator
  6. Examples
  7. Secular Approximation in the Redfield Equation
  8. Conclusions
  9. Diamond Norm
  10. Useful Properties of Superoperators

Key Result

Proposition 1

Let $t\in[0,T]\mapsto \mathcal{L}(t)$ be a continuous family of bounded generators of contraction semigroups. Then, the time propagator generated by $\mathcal{L}(t)$ according to eq:evolution is a contraction, i.e. $\|\Lambda(t,s)\|\le1$ for $0\le s\le t\le T$.

Figures (2)

  • Figure 1: Comparison of the exact diamond distances computed numerically and the bounds \ref{['eq:boundEx1uniform']} and \ref{['eq:boundEx2uniform']} obtained for Examples \ref{['ex:2']} (Left) and \ref{['ex:3']} (Right).
  • Figure 2: Comparison of the exact diamond distances computed numerically and the corresponding bounds. In the left panel the distance \ref{['eq:boundEx3uniform']} between the true evolution and the approximate one projected on the peripheral subspace is shown, while in the right panel the distance \ref{['eq:boundEx3Puniform']} to the approximate evolution projected on the peripheral subspace is shown.

Theorems & Definitions (32)

  • Proposition 1: Ref. reed1975ii, Theorem X.70. See also Refs. kato1953integrationavron2012adiabatic
  • Remark 1
  • Proposition 2: Ref. chruscinski2022dynamical, Corollary 7
  • Lemma 3
  • proof
  • Lemma 4: Integration-by-part lemma
  • proof
  • Theorem 5
  • proof
  • Remark 2: Effective generator
  • ...and 22 more