Martino Stefanini, Aleksandra A. Ziółkowska, Dmitry Budker, Ulrich Poschinger, Ferdinand Schmidt‐Kaler, Antoine Browaeys, Ataç İmamoğlu, Darrick E. Chang, Jamir Marino
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.
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.
Away from equilibrium, the properties of open quantum systems depend on the details of their environment. A microscopic derivation of a master equation (ME) is therefore crucial. Of particular interest are Lindblad-type equations, not only because they provide the most general class of Markovian MEs, but also since they are the starting point for efficient quantum trajectory simulations. Lindblad-type MEs are commonly derived from the Born-Markov-Redfield equation via a rotating-wave approximation (RWA). However the RWA is valid only for ultraweak system-bath coupling and often fails to accurately describe nonequilibrium processes. Here we derive an alternative Lindbladian approximation to the Redfield equation, which does not rely on ultraweak system-bath coupling. Applying it to an extended Hubbard model coupled to Ohmic baths, we show that, especially away from equilibrium, it provides a good approximation in large parameter regimes where the RWA fails.
Recent advances in quantum technologies and related experiments have created a need for highly accurate, versatile, and computationally efficient simulation techniques for the dynamics of open quantum systems. Long-lived correlation effects (non-Markovianity), system-environment hybridization, and the necessity for accuracy beyond the Born-Markov approximation form particular challenges. Approaches to meet these challenges have been introduced, originating from different fields, such as hierarchical equations of motion, Lindblad-pseudomode formulas, chain-mapping approaches, quantum Brownian motion master equations, stochastic unravelings, and refined quantum master equations. This diversity, while indicative of the field's relevance, has inadvertently led to a fragmentation that hinders cohesive advances and their effective cross-community application to current problems for complex systems. How are different approaches related to each other? What are their strengths and limitations? Here we give a systematic overview and concise discussion addressing these questions. We make use of a unified framework which very conveniently allows to link different schemes and, this way, may also catalyze further progress. In line with the state of the art, this framework is formulated not in a fully reduced space of the system but in an extended state space which in a minimal fashion includes effective reservoir modes. This in turn offers a comprehensive understanding of existing methods, elucidating their physical interpretations, interconnections, and applicability.
We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of noninteracting particles (bosons or fermions) on an arbitrary lattice of N sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the N=2 case, we show that RQME gives results which agree with exact analytical results for steady-state properties and with exact numerics for time-dependent properties over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in nonequilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed-form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED, and far-from-equilibrium quantum dot experiments.
This text is a short introduction to the physics of driven-dissipative many-body systems, focusing on a few selected topics. Beyond its more historical interest in the study of atomic physics and quantum optics, presently the modeling and studying of dissipative phenomena in open quantum systems are pivotal to understanding quantum hardware platforms. Although the lack of a thermodynamic potential for these out-of-equilibrium open systems makes it theoretically challenging to investigate their physics, at the same time it allows one to go beyond the thermodynamic paradigms and investigate new and exotic phenomena. We will focus on one of the simplest, yet most effective, descrip-tions of open quantum systems, namely the (Gorini-Kossakowski-Sudarshan-) Lindblad master equation. This phenomenological approach has been devel-oped to describe quantum systems that weakly interact (Born approximation) with a memoryless (Markovian) surrounding environment. However, it is often a remarkably simple and predictive tool to characterize the essential features of an open quantum system. Although many of the results derived below will ap-ply to any quantum system, we will focus in particular on bosonic/spin systems.
The Lindblad theory of open quantum systems has been successfully applied in various fields ranging from quantum optics, condensed matter physics, and quantum thermodynamics to quantum chemistry. Nevertheless, there are situations where we can compare its predictions to those of exact methods or numerically exact results and find discrepancies between them. One example is the case where a single harmonic oscillator is coupled to a thermal bath of independent harmonic oscillators. In this case, for certain parameters, the exact master equation describes equilibration of the system to a thermal state with a temperature different from the one predicted by the Lindblad master equation. The Lindblad theory, based on a weak coupling between the system and environment as well as the Born-Markov approximation, predicts that the system will evolve to a thermal state whose temperature is equal to the initial temperature of the bath. This result is a consequence of the Born approximation, which supposes that throughout the evolution, the bath only fluctuates around its initial equilibrium state. In this master thesis we study the validity of the Born approximation as well as its consistency with other approximations. By starting from a Hamiltonian for the total system we show that, if the the Born approximation is not invoked, the dynamics of the system is determined by a hierarchy of equations for matrices which can be used to reconstruct the reduced density matrix of the system. This hierarchy approach is fundamentally different from the standard approach, which describes the system dynamics with a weak-coupling master equation, because the hierarchy does not restrict the amount of entanglement possible between the system and the environment. Furthermore, we show how invoking the Born approximation reduces to the usual result and how a generalised ansatz, which we refer to as a generalised Born approximation, reduces the hierarchy to a master equation where the bath temperature is an additional degree of freedom. Finally, we analyse the conservation of the energy of the total system and its relationship to the Markov and secular approximations, and discuss whether imposing it as a condition can fix the time-dependent temperature.
We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovian master equation for the reduced density matrix, incorporating secular effects and independent of the initial preparation. The kernel is determined by a systematic diagrammatic expansion in powers of the interaction. We consider the Born approximation for the kernel. Applying a condensed-matter pole or, equivalently, a quasiparticle-type approximation, equivalent to the usual assumption of a timescale separation, we derive a master equation of the Markov type. Furthermore, imposing the rotating-wave approximation, we obtain a Markov master equation of the Lindblad form. To illustrate the method, we consider the standard example of a single qubit coupled to a thermal heat bath.