Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account non-secular effects, but, nevertheless, has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form (in particular, preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The non-secular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account non-secular effects, but, nevertheless, has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form (in particular, preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The non-secular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account nonsecular effects, but nevertheless has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan form (in particular, it preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The nonsecular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.
This preprint presents a self-contained and mathematically explicit reconstruction of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation starting from exact microscopic unitary dynamics through the Nakajima-Zwanzig projection operator formalism. Beginning with the exact non-local generalized master equation, we systematically analyze the hierarchy of approximations required for the emergence of a Markovian quantum dynamical semigroup. The Born (weak-coupling), Markov (time-localization), and secular approximations are derived and interpreted as successive information-reduction steps, each associated with clearly identifiable mathematical assumptions and physical consequences. Particular attention is devoted to the role of secularization in transforming the Redfield generator into a completely positive GKSL structure. To provide a concrete worked example, the formalism is applied to a damped two-level system coupled to a bosonic thermal reservoir, where the Redfield and GKSL equations are derived explicitly and compared at the microscopic level. Beyond the derivation itself, the work develops a systematic map of validity limits and failure regimes. The breakdown of the Born, Markov, secular, factorization, and stationarity assumptions is analyzed in terms of environmental correlation times, spectral-density structure, non-Markovian memory effects, information backflow, and loss of semigroup behavior. The resulting framework clarifies the precise operational domain in which GKSL dynamics can be regarded as a physically reliable effective description and provides a rigorous foundation for applications involving entropy-production theorems, quantum thermodynamics, and open-system information theory.
Time-local quantum master equations that describe open quantum systems beyond the limit of ultraweak system-bath coupling are often not of Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form. Prominent examples are the Redfield equation approximating general open quantum systems and the Hu-Paz-Zhang equation precisely describing a damped harmonic oscillator. In this study, we demonstrate that not only the former but also the latter can be brought to pseudo-Lindblad form. This form features a dissipator resembling that of a GKSL equation, with the distinction that some of the terms have negative weights. Moreover, we systematically investigate transformations that leave the dissipator of pseudo-Lindblad equations unchanged, while changing the relative weight between its positive and negative terms. These can be used to minimize the weights of the negative terms, which is optimal both for the convergence of a recently developed quantum-trajectory unraveling of pseudo-Lindblad equations and the truncation of the negative terms to obtain a GKSL equation.
We develop a framework that allows us to describe the dynamics of the total state of an open quantum system and its bosonic environment in the usual Born (weak coupling) and Markov approximation. By shifting the whole time-dependence into an unnormalized s-operator of the open system, the full dynamics is captured by an s-master equation of similar structure than the well-known Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation for the reduced dynamics. By varying the ordering parameter s (0 ≤ s ≤ 1) we obtain the partial Husimi representation (s = 0) and the partial Glauber-Sudarshan representation (s = 1) for the dynamics of the total state. For the reduced density operator the GKSL master equation can be derived easily. The case of s = 1/2, leading to a partial Wigner representation, is helpful to study the overlap of states in the total Hilbert space of system and environment.
A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to a GKSL master equation for the full density matrix.
We present a systematic thermodynamic analysis of nonsecular GKLS master equations. When some energy transitions in the system are comparable with the dissipative rate, the secular approximation is not valid, and the positivity of the dynamics must be ensured via other approximations. The dynamics then presents features reminiscent from strong coupling to the environment: e.g. the equilibrium state with an environment at thermal equilibrium is not a Gibbs state of the local system Hamiltonian and the dynamics couples energy coherences and populations. Nonetheless, we build a unified, consistent thermodynamic framework for those dynamics. We focus on two approximations beyond the full secular approximation (the local and the more accurate geometric-arithmetic approximations), and then employ systematic perturbation theory on microscopic expressions for the first and second law of thermodynamics which relate irreversibility to the increase of system-environment correlations. We show that, in spite of the weak system-bath coupling, the system-bath interaction energy participates to the energy balance, and is directly linked to the Lamb-shift Hamiltonian. We compare this microscopic entropy production with the definition based on the contractivity of the reduced system dynamics (Spohn inequality). We show that, unlike for secular master equations, the two entropy production rates differ because of the presence of non-vanishing stationary coherences in the energy eigenbasis. However, in the case of a single thermal bath, the difference is purely transient, and no work can be cyclically extracted from the steady-state despite its non-Gibbs form. Finally, we illustrate our results with simple examples, clarifying and completing the thermodynamic picture of Markovian dynamics in the quantum regime.