We investigate the long-time behavior of quantum Markovian dynamics generated by time-dependent Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) master equations. We introduce a notion of weak relaxation and derive sufficient conditions guaranteeing asymptotic independence from the initial state. Our results provide a quantitative extension of the Spohn–Frigerio theorem to time-dependent generators, yielding explicit contraction bounds in terms of the instantaneous steady state and the jump operators. For a class of microscopically derived master equations, we further obtain a graph-theoretic characterization of the aforementioned conditions that directly links the structure of the jump operators to the relaxation properties. The general theory is illustrated by applications to driven finite-level systems, including a detailed three-level example, and is extended to a non-Markovian setting by means of time-local master equations that become of GKLS form at long times. These findings pave the way for the development of a more general theory of relaxation beyond the Markovian case.
We investigate the long-time behavior of quantum Markovian dynamics generated by time-dependent Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) master equations. We introduce a notion of weak relaxation and derive sufficient conditions guaranteeing asymptotic independence from the initial state. Our results provide a quantitative extension of the Spohn–Frigerio theorem to time-dependent generators, yielding explicit contraction bounds in terms of the instantaneous steady state and the jump operators. For a class of microscopically derived master equations, we further obtain a graph-theoretic characterization of the aforementioned conditions that directly links the structure of the jump operators to the relaxation properties. The general theory is illustrated by applications to driven finite-level systems, including a detailed three-level example, and is extended to a non-Markovian setting by means of time-local master equations that become of GKLS form at long times. These findings pave the way for the development of a more general theory of relaxation beyond the Markovian case.
The theory of open quantum systems represents one of the major developments of quantum mechanics over the last 50 years. Its central object of study is constituted by quantum systems that cannot be regarded as fully isolated, but instead interact with additional degrees of freedom, giving rise to irreversible phenomena such as dissipation and decoherence. The tools provided by this framework are nowadays of paramount importance across many active areas of research, including quantum computation, thermodynamics, metrology and biology. In this thesis, we investigate three main topics within the field. The first contribution concerns the microscopic derivation of the evolution equations, socalled quantum master equations, for open systems which are also subject to external driving fields. Standard techniques developed for time-independent Hamiltonians encounter several difficulties in the time-dependent case, largely restricting their applicability to specific instances, such as periodic or adiabatic drivings. In contrast, a core result of the derivation presented in the thesis is that, when the environment is sufficiently Markovian (essentially, memoryless), the non-adiabatic contributions to the driving field can be neglected in a controlled fashion, yielding an accurate master equation with time-dependent generator, capable of correctly reproducing the open dynamics even for strong drivings. This result is further supported by benchmarks against numerically exact simulations. The second contribution addresses the long-time behavior of general types of (Markovian) quantum master equations, including those of the form introduced above. While this problem has been extensively analyzed in the time-independent setting, virtually no results are known for time-dependent generators. In particular, we establish the foundations of a general theory of relaxation, that is, convergence towards an asymptotic trajectory in the long-time limit, independently of the initial state. The main results include a theorem guaranteeing sufficient conditions for relaxation and the derivation of a rate function governing the convergence to the asymptotic trajectory. The analysis is further extended to a simple non-Markovian case. Moreover, we treat in detail the specific master equation derived in the first part, connecting the aforementioned relaxation property directly to the underlying microscopic dynamics. The third contribution investigates chirality-induced spin selectivity (CISS) effects from the perspective of open quantum systems. Although the majority of theoretical studies aiming to model CISS are based on a coherent description of the spin dynamics, such effects have been observed also for electrons subject to incoherent transport across molecular networks. In this regard, we provide a rigorous ab initio derivation of the microscopic Hamiltonian describing the electronic transport, incorporating electron spin effects and contributions stemming from the nuclear vibrations. Within this result, we critically revise a master equation approach aiming to explain the origin of CISS effects observed in photosynthetic complexes. Taken together, these contributions offer fresh insights and original results on the interplay between driving fields and dissipation, relaxation phenomena, and electronic spin transport in the presence of incoherent dynamics. More broadly, these results provide new theoretical tools with several potential applications in quantum technologies, from modeling of control pulses in noisy environments to quantum reservoir engineering, and contribute to a deeper understanding of the role of quantum effects in biological systems.
We derive sufficient criteria that guarantee convergence of open quantum system dynamics to a unique long-time state independent of the initial conditions. We refer to this behavior as complete relaxation. Focusing on systems described by Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equations on finite-dimensional Hilbert spaces, our approach uses only elementary mathematical arguments. For autonomous systems, we recover three previously derived relaxation criteria, which can be expressed as algebraic conditions on the Lindblad operators and the Hamiltonian entering the GKSL equation. For periodically driven systems, we obtain a new criterion involving a similar algebraic condition and a local continuity requirement on the time-dependent GKSL generator. This result implies an earlier criterion for complete relaxation formulated in Ref. [J. Phys. A 52, 43LT01 (2019)]. We also discuss implications of our approach for periodically modulated classical master equations, as well as extensions to driven systems without a periodicity constraint.
We present a rigorous and comprehensive classification of the asymptotic behavior of time-dependent Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equations under the assumption of Hermitian jump operators. Our results apply to a broad class of GKSL equations whose time dependence is assumed to be recurrent, including time-independent, periodic, quasiperiodic, and certain classes of random time dependence. Our main contributions are twofold: first, we establish a criterion for the uniqueness of steady states. The criterion is formulated in terms of the algebra generated by the GKSL generators and provides a necessary and sufficient condition when the generators are analytic functions of time. We demonstrate the utility of our criterion through prototypical examples, including quantum many-body spin chains. Second, we extend the concept of strong symmetry for time-dependent GKSL equations by introducing two distinct forms, strong symmetry in the Schrödinger picture and that in the interaction picture, and completely classify the asymptotic dynamics with them. More concretely, we rigorously uncover that the strong symmetry in the interaction picture is responsible for non-trivial time-dependent steady states, such as coherent oscillations, whereas that in the Schrödinger picture controls the existence of time-independent steady states. This classification not only encompasses established mechanisms underlying non-trivial oscillatory steady states, such as strong dynamical symmetry and Floquet dynamical symmetry, but also reveals symmetry-predicted, time-dependent asymptotic dynamics in a novel class of open quantum systems. Our framework thus provides a rigorous foundation for controlling dissipative quantum systems in a time-dependent manner.
A well established approach to describe open quantum system dynamics is the celebrated Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. In this article we aim to shed light on semigroup dynamics that are not of this form. Such dynamics may appear in real physical systems describing asymptotic long-time dynamics. Thus, they correspond to semigroup dynamics applicable to a restricted set of initial states only. As a way of motivating our studies, first we review known results about the single-mode Gaussian Quantum Brownian motion channel and corresponding master equations. Next we discuss a conceptually simpler family of qubit semigroup generators and identify parameter regions of completely positive, merely positive, and non-positive dynamics. We then focus on the latter case to which we will refer to as restricted-positive, as the semigroup generator may well be physically meaningful when applied to a restricted set of admissible initial states.
Stochastic unravelings provide a useful way to represent open quantum system dynamics in terms of pure state realizations, and have been widely studied both from a fundamental and from a computational point of view. They were initially formulated for Markovian dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad master equation. However, due to recent technological and experimental development, most physical relevant dynamics present temporal correlations beyond the Markov approximation. Such correlations cause decay rates to turn temporarily negative, thus requiring the generalization of stochastic unravelings from Markovian to non-Markovian scenarios. Indeed, many unraveling techniques have been introduced in this regime, and a comprehensive review of the different jump methods is currently missing. In this work, we provide an overview of widely used quantum jump unraveling techniques for non-Markovian systems and also discuss them in terms of their numerical efficiency, divisibility requirements, Hilbert space extension, and measurement interpretation.
In 1976, Gorini, Kossakowski, Sudarshan and Lindblad independently discovered a general form of master equations for an open quantum Markovian dynamics. In honor of all the authors, the equation is nowadays called the GKLS master equation. In this paper, we show universal constraints on the relaxation times valid for any d-level GKLS master equations, which is a generalization of the well-known constraints for 2-level systems. Specifically, we show that any relaxation rate, the inverse-relaxation time, is not greater than half of the sum of all relaxation rates. Since the relaxation times are measurable in experiments, our constraints provide a direct experimental test for the validity of the GKLS master equations, and hence for the conditions of the complete positivity and Markovianity.