In a previous work we developed a field theoretical approach to open quantum systems using condensed matter methods. In the Born approximation we derived the Redfield equation on the basis of a multi-oscillator bath, a Dyson equation, a diagrammatic expansion and a quasi-particle approximation. In addition applying a rotating wave approximation we obtained the Lindblad equation describing a proper quantum map. The issue regarding the additional rotating wave approximation was left as an open problem.
The present work addresses the open problem and presents new results. We identify a discrepancy in the popular and standard Redfield equation. The discrepancy is associated with the well-known fact that the Redfield equation does not represent a proper quantum map. The discrepancy is related to the diagrammatic expansion and a consistency requirement in the quasi-particle approximation. The explicit resolution of this discrepancy is obtained by imposing energy conservation on the Born level. As a result we obtain formal equivalence between the energy-conserving Redfield equation and the Lindblad equation without invoking the rotating wave approximation. We provide a detailed mapping of the field theoretical approach to the standard microscopic derivation in the theory of open quantum systems.
In a previous work we developed a field theoretical approach to open quantum systems using condensed matter methods. In the Born approximation we derived the Redfield equation on the basis of a multi-oscillator bath, a Dyson equation, a diagrammatic expansion and a quasi-particle approximation. In addition applying a rotating wave approximation we obtained the Lindblad equation describing a proper quantum map. The issue regarding the additional rotating wave approximation was left as an open problem.
The present work addresses the open problem and presents new results. We identify a discrepancy in the popular and standard Redfield equation. The discrepancy is associated with the well-known fact that the Redfield equation does not represent a proper quantum map. The discrepancy is related to the diagrammatic expansion and a consistency requirement in the quasi-particle approximation. The explicit resolution of this discrepancy is obtained by imposing energy conservation on the Born level. As a result we obtain formal equivalence between the energy-conserving Redfield equation and the Lindblad equation without invoking the rotating wave approximation. We provide a detailed mapping of the field theoretical approach to the standard microscopic derivation in the theory of open quantum systems.
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.
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.
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.
Abstract Quantum master equations are widely used to describe the dynamics of open quantum systems. All these different master equations rely on specific approximations that may or may not be justified. Starting from a microscopic model, applying the standard Born and Markov approximations results in the Redfield equation that does not guarantee to preserve positivity. The latter is typically achieved by additionally applying the secular approximation resulting in a quantum master equation in Lindblad form. There are other ways to obtain an equation in Lindblad form, one of which is the recently proposed universal Lindblad equation (ULE). It has been shown that it is in the same equivalence class of approximations as the Redfield master equation although avoiding the heuristic secular approximation (Nathan and Rudner 2020 Phys. Rev. B 102 115109). In this work, we prove that the solutions of the master equation obtained by applying the secular approximation are also obtained by an approximation of the same order as the one performed to obtain the Redfield equation. We hereby provide a mathematical justification for the secular approximation. We show that the result of applying the secular approximation is obtained naturally by applying a self-consistency argument. This shows that the resulting master equation is also in the same equivalence class of approximations as the Redfield master equation and the ULE. We furthermore give numerical considerations about the accuracy of the secular master equation compared to the ULE.
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
In this paper we derive the Lindblad and Redfield forms with and without secular approximation from the Born-Markov master equation for open quantum systems. The spectral correlation tensor of bath (the Fourier transform of the bath correlation function) and then the coefficients in the two forms of the master equation are reevaluated according to the scheme in Ref.[Phys. Rev. A 99, 022118 (2019)]. They are complex numbers rather than the real numbers getting from traditional simplified methods. The dynamics of two models [one is an open three-level quantum system model, and the other is the model of phycoerythrin 545 (PE545) of modeling a photosynthesis reaction center] are studied by using the obtained equations. The non-secular Lindblad and Redfield equations with the complex coefficients predict almost the same dynamical results from the Born-Markov master equation. However, the results obtained from the traditional Lindblad and Redfield equations deviate the actual dynamics of the open quantum systems.