Thermodynamics entails a set of mathematical conditions on quantum Markovian dynamics.In particular, strict energy conservation between the system and environment implies that the dissipative dynamical map commutes with the map of the system's unitary evolution.Employing spectral analysis, we prove the general form of the ensuing master equation.The obtained structure extends thermodynamical considerations to dynamical processes.Comparing this form with master equations obtained from microscopic derivations allows validating their compatibility with thermodynamics.It predicts that coherence is not generated spontaneously under steady-state transport.Moreover, for a bipartite system-environment it singles out the global master equation as the thermodynamically compatible choice for nondriven systems, as well as supplying insight into the validity of the secular approximation.
Thermodynamics entails a set of mathematical conditions on quantum Markovian dynamics.In particular, strict energy conservation between the system and environment implies that the dissipative dynamical map commutes with the map of the system's unitary evolution.Employing spectral analysis, we prove the general form of the ensuing master equation.The obtained structure extends thermodynamical considerations to dynamical processes.Comparing this form with master equations obtained from microscopic derivations allows validating their compatibility with thermodynamics.It predicts that coherence is not generated spontaneously under steady-state transport.Moreover, for a bipartite system-environment it singles out the global master equation as the thermodynamically compatible choice for nondriven systems, as well as supplying insight into the validity of the secular approximation.
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.
Abstract Markovian master equations provide a versatile tool for describing open quantum systems when memory effects of the environment may be neglected. As these equations are of an approximate nature, they often do not respect the laws of thermodynamics when no secular approximation is performed in their derivation. Here we introduce a Markovian master equation that is thermodynamically consistent and provides an accurate description whenever memory effects can be neglected. The thermodynamic consistency is obtained through a rescaled Hamiltonian for the thermodynamic bookkeeping, exploiting the fact that a Markovian description implies a limited resolution for heat. Our results enable a thermodynamically consistent description of a variety of systems where the secular approximation breaks down.
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.
The Bloch equation that set the foundation for open quantum systems, was conceived by pure physical reasoning. Since then, the Lindblad (GKLS) form of a quantum master equation, its most general mathematical representation, became an established staple in the open quantum systems toolbox. It allows to describe a multitude of quantum phenomena, however its universality comes at a cost - without additional constraints, the resultant dynamics are not necessarily thermodynamically consistent, and the equation itself lacks an intuitive interpretation. We present a mathematically equivalent form of the Lindblad master equation under a single constraint of strict energy conservation. The "elemental Bloch" equation separates the system dynamics into its elemental parts, making an explicit distinction between thermal mixing, dephasing, and energy relaxation, and thus reinstating the physical intuition in the equation. We derive the equation for a many-level system by accounting for all relevant transitions between pairs of levels. Finally, the formalism is illustrated by calculating the fixed point of the dynamics and exploring the conditions for canonical invariance in quantum systems.
Thermodynamics imposes fundamental constraints on the evolution of quantum systems. These constraints and their dynamical consequences have been formulated within distinct paradigms, including axiomatic approaches based on quantum master equations, microscopic descriptions of open-system dynamics, and operational formulations rooted in resource theories. While each perspective has yielded important insights into thermodynamically consistent quantum dynamics, their precise relationship has remained unresolved. Here we establish the exact equivalence of these three paradigms in the Markovian regime. We prove that thermal Lindbladians satisfying Markovianity, time-translation symmetry, and quantum detailed balance are precisely those admitting a microscopic realisation as an energy-conserving thermal collision model and, equivalently, those generating Markovian thermal operations. This unifies the existing approaches to Markovian quantum thermodynamics and identifies its dynamical underpinnings. We further provide an explicit microscopic protocol for simulating thermal Markovian processes with controlled finite-time simulation errors. We illustrate its applicability by providing faithful thermal collision-model implementations of a qubit thermalising in a bosonic environment and of a three-level autonomous thermal machine. In the latter case, the protocol gives rise to a finite-stroke thermal engine that not only reproduces the continuous-time dynamics but also its steady-state thermodynamic performance. As a whole, these results establish a unified foundation for Markovian quantum thermodynamics, showing that its axiomatic, microscopic, and operational formulations are exactly equivalent and providing a universal protocol for implementing thermal processes and machines.
Abstract The Bloch equation which first inspired the field of open quantum systems, was conceived by pure physical reasoning. Since then, the Lindblad (GKLS) form of a quantum master equation, its most general mathematical representation, became an established staple in the open quantum systems toolbox. It allows the description of a multitude of quantum phenomena, however its universality comes at a cost—without additional constraints, the resultant dynamics are not necessarily thermodynamically consistent, and the equation itself lacks an intuitive interpretation. We present a mathematically equivalent form of the Lindblad master equation under a single constraint of strict energy conservation, which reinstates physical intuition, separating the system dynamics into its elemental parts: thermal mixing, dephasing, and energy relaxation. The analytical capabilities of the formalism are illustrated by calculating the fixed point of the dynamics and exploring the conditions for canonical invariance in quantum systems.