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arXiv:2303.16449 · PRX Quantum

Quantum Master Equations: Tips and Tricks for Quantum Optics, Quantum Computing, and Beyond

Francesco Campaioli, Jared H. Cole, Harini Hapuarachchi

Published Jun 10, 2024• 89 citations
arXivHTMLPDF

Abstract

Quantum master equations are an invaluable tool to model the dynamics of a plethora of microscopic systems, ranging from quantum optics and quantum information processing to energy and charge transport, electronic and nuclear spin resonance, photochemistry, and more. This tutorial offers a concise and pedagogical introduction to quantum master equations, accessible to a broad, cross-disciplinary audience. The reader is guided through the basics of quantum dynamics with hands-on examples that increase in complexity. The tutorial covers essential methods such as the use of the Lindblad master equation, Redfield relaxation, and Floquet theory, as well as techniques such as Suzuki-Trotter expansion and numerical approaches for sparse solvers. These methods are illustrated with code snippets implemented in and other languages, which can be used as a starting point for generalization and more sophisticated implementations. Published by the American Physical Society 2024

DOI
10.1103/prxquantum.5.020202

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The theory of open quantum systems is one of the most essential tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) master equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field.
The Lindblad master equation is a foundational tool for modeling the dynamics of open quantum systems. As its use has extended far beyond its original domain, the boundaries of its validity have grown opaque. In particular, the rise of new research areas including open quantum many body systems, non-equilibrium condensed matter, and the possibility to test its limits in driven-open quantum simulators, call for a critical revision of its regimes of applicability. In this pedagogical review, we re-examine the folklore surrounding its three standard approximations (Born, Markov, and Rotating Wave Approximation), as we build our narrative by employing a series of examples and case studies accessible to any reader with a solid background on the fundamentals of quantum mechanics. As a synthesis of our work, we offer a checklist that contrasts common lore with refined expectations, offering a practical guideline for assessing the breakdown of the Lindblad framework in the problem at hand.
Abstract This chapter introduces the fundamentals of the description of the quantum dynamics of open systems in terms of quantum master equations, together with its most important applications. Special emphasis is laid on the theory of completely positive quantum dynamical semigroups, which leads to the concept of a quantum Markov process. It discusses the relaxation to equilibrium and the multi-time structure of quantum Markov processes, as well as their irreversible nature which is characterized by an appropriate entropy functional. Microscopic derivations for various quantum master equations are presented, such as the quantum optical master equation and the master equation for quantum Brownian motion. As a further application, the master equation describing continuous measurements is derived and used to study the quantum Zeno effect. The chapter also contains a treatment of non-linear, mean field quantum master equations together with applications to laser theory and super-radiance.
Understanding system-environment correlations in open quantum systems is vital for various quantum information and technology applications. However, these correlations are often overlooked or hidden in derivations of open-quantum-system master equations, especially when applying the Born approximation. To address this issue, given a microscopic model, we demonstrate how to retain system-environment correlation within commonly used master equations, such as the Markovian Lindblad, Redfield, second-order time convolutionless, second-order Nakajima-Zwanzig, and second-order universal Lindblad-like equations. We show that each master equation corresponds to a particular approximation on the system-environment correlation operator. In particular, our analysis exposes the form of the hidden system-environment correlation in the Markovian Lindblad equation derived using the Born approximation. We also identify that the processes leading to the Redfield equation yield an inaccurate initial-time system-environment correlation approximation. By fixing this problem, we propose a corrected Redfield equation with an improved prediction for early stages of the time evolution. We further illustrate our results in two examples, which imply that the second-order universal Lindblad-like equation captures correlation more accurately than the other standard master equations. Published by the American Physical Society 2024
Motivated by the need to study nonequilibrium evolutions of many-electron systems at the atomistic ab initio level, as they occur in modern devices and applications, we developed a quantum dynamics approach bridging master equations and surface hopping (SH). The Lindblad master equation (LME) allows us to propagate efficiently ensembles of particles, while SH provides nonperturbative evaluation of transition rates that evolve in time and depend explicitly on nuclear geometry. We implemented the LME-SH technique within real-time time-dependent density functional theory using global flux SH, and we demonstrated its efficiency and utility by modeling metallic films, in which charge-phonon dynamics was studied experimentally and showed an unexpectedly strong dependence on adhesion layers. The LME-SH approach provides a general framework for modeling efficiently quantum dynamics in a broad range of complex many-electron condensed-matter and nanoscale systems.