Tutorial: projector approach to master equations for open quantum systems

C. Gonzalez-Ballestero

doi:10.22331/q-2024-08-29-1454

Tutorial: projector approach to master equations for open quantum systems

C. Gonzalez-Ballestero

Institute for Theoretical Physics, Vienna University of Technology (TU Wien), 1040 Vienna, Austria

Published:	2024-08-29, volume 8, page 1454
Eprint:	arXiv:2305.19704v4
Doi:	https://doi.org/10.22331/q-2024-08-29-1454
Citation:	Quantum 8, 1454 (2024).

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Abstract

Most quantum theorists are familiar with different ways of describing the effective quantum dynamics of a system coupled to external degrees of freedom, such as the Born-Markov master equation or the adiabatic elimination. Understanding the deep connection between these -- sometimes apparently unrelated -- methods can be a powerful tool, allowing us to derive effective dynamics in unconventional systems or regimes. This tutorial aims at providing quantum theorists across multiple fields (e.g., quantum and atom optics, optomechanics, or hybrid quantum systems) with a self-contained practical toolbox to derive effective quantum dynamics, applicable to systems ranging from $N$-level emitters to mechanical resonators. First, we summarize the projector approach to open quantum systems and the derivation of the fundamental Nakajima-Zwanzig equation. Then, we show how three common effective equations, namely the Brownian master equation, the Born-Markov master equation, and the adiabatic elimination used in atom and molecular optics, can be derived from different perturbative expansions of the Nakajima-Zwanzig equation. We also solve in detail four specific examples using this formalism, namely a harmonic oscillator subject to displacement noise, the effective equations of a mechanical resonator cooled by an optical cavity, the Purcell effect for a qubit coupled to an optical cavity, and the adiabatic elimination in a Lambda system.

Featured image: Schematic depiction of the projector approach: the time evolution of a system density matrix in the full Hilbert space (blue curve) is projected by a projector P onto a subspace of interest (blue surface). As shown below, this subspace typically corresponds to product states between the reduced density matrix of a chosen set of degrees of freedom – the "system" S – and a suitably chosen density matrix for the remaining ones – the "bath" B. The dynamical equation within this subspace describes the evolution of the system (orange curve) influenced by the bath. This general and exact equation (the Nakajima-Zwanzig equation) can be perturbatively expanded to recover various conventional methods to describe reduced dynamics, e.g., the Born-Markov equation for a system coupled to a continuum (upper box, the continuum is represented by the blue arrows), the adiabatic elimination of levels in an N-level system (middle box), or the cooling of a mechanical resonator coupled to an optical cavity (lower box).

► BibTeX data

@article{GonzalezBallestero2024tutorialprojector,
  doi = {10.22331/q-2024-08-29-1454},
  url = {https://doi.org/10.22331/q-2024-08-29-1454},
  title = {Tutorial: projector approach to master equations for open quantum systems},
  author = {Gonzalez-Ballestero, C.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1454},
  month = aug,
  year = {2024}
}

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