Abstract Optical Bloch equations (OBE) describe the coherent exchange of energy between a quantum bit (qubit) and a quasi-resonant driving field in the presence of a thermal bath. Despite it being an ubiquitous process in quantum technologies, a sound thermodynamic analysis is still missing. We hereby provide such an analysis, by deriving the relevant framework from first principles. We start from a complete microscopic description of the qubit-bath system where definitions of heat, work and entropy production are unambiguous. We trace out the bath and coarse-grain the resulting expressions in time, using a methodology similar to the derivation of the dynamical master equation, to derive closed expressions for the first and second law in terms of system properties. Long coarse graining times yield the Floquet Master equation and its already known thermodynamic description. Short coarse-graining times yield instead the OBE and a novel thermodynamic framework which explicitly depends on quantum coherences in the qubit’s energy basis which produce quantum signatures in the heat and entropy production flows. This allows us to characterize a genuinely quantum non-equilibrium situation, where the coherences created by the driving field are continuously erased by the bath. Our findings can be readily extended to larger open quantum systems. They carry the seeds for future thermodynamic analyses of quantum gates and the design of quantum engines in the strong coherent driving regime.
Abstract Optical Bloch equations (OBE) describe the coherent exchange of energy between a quantum bit (qubit) and a quasi-resonant driving field in the presence of a thermal bath. Despite it being an ubiquitous process in quantum technologies, a sound thermodynamic analysis is still missing. We hereby provide such an analysis, by deriving the relevant framework from first principles. We start from a complete microscopic description of the qubit-bath system where definitions of heat, work and entropy production are unambiguous. We trace out the bath and coarse-grain the resulting expressions in time, using a methodology similar to the derivation of the dynamical master equation, to derive closed expressions for the first and second law in terms of system properties. Long coarse graining times yield the Floquet Master equation and its already known thermodynamic description. Short coarse-graining times yield instead the OBE and a novel thermodynamic framework which explicitly depends on quantum coherences in the qubit’s energy basis which produce quantum signatures in the heat and entropy production flows. This allows us to characterize a genuinely quantum non-equilibrium situation, where the coherences created by the driving field are continuously erased by the bath. Our findings can be readily extended to larger open quantum systems. They carry the seeds for future thermodynamic analyses of quantum gates and the design of quantum engines in the strong coherent driving regime.
Optical Bloch Equations (OBEs) are canonical equations of quantum open systems, that describe the dynamics of a classically driven atom coupled to a thermal bath. Their thermodynamics is highly relevant to establish fundamental energetic bounds of key quantum processes. A consistent framework is available in the regime where drives and baths can be treated classically, i.e. remain insensitive to the coupling with the atom. This regime, however, is not adapted to explore minimal energy costs, nor to measure atom-induced energy variations inside drives and baths – a key ability to directly measure and optimize work and heat exchanges. This calls for a new framework where the atomic back-action on drives and baths would be accounted for. Here we build such a framework suitable to analyze the situation where the atom, the drive and the bath form a joint autonomous system, the drive and the bath being parts of the same electromagnetic field. Our approach captures atom-field correlations at fundamental timescales, as well as the atomic back-action on the field. This allows us to define work-like (heat-like) flows as energy flows stemming from effective unitary dynamics induced by one system on the other (non-unitary dynamics induced by correlations). Time-integrated work-like and heat-like flows are shown to be directly measurable in the field, as changes of energy locked in the mean field and fluctuations, respectively. Our approach differs from standard open analyses by identifying an additional unitary contribution in the atom’s dynamics, the self-drive, and its energetic counterpart, the self-work, which yields a tighter expression of the second law. We quantitatively relate this tightening to the extra-knowledge about the field state as compared with usual treatments of the atom as an open system. Our autonomous framework deepens the current understanding of thermodynamics in the quantum regime and its potential for energy management at quantum scales. Its predictions can be probed in state-of-the-art quantum hardware, such as superconducting and photonic circuits.
A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.
In quantum thermodynamics, the decomposition of energy exchanges into heat and work remains an open problem beyond weak-coupling and slow-driving regimes. Recent formulations have shown that quantum coherence introduces additional energy contributions whose thermodynamic interpretation is still under debate, raising fundamental questions about the structure of the quantum first law. In this work, we investigate this problem through a time-dependent perturbative framework applied to the first law of quantum thermodynamics. By expanding the thermodynamic quantities up to second order, we derive explicit perturbative corrections for work, heat, and coherence contributions. Our results show that the coherence term can be consistently decomposed into coherent heat and coherent work, demonstrating that quantum coherence does not require the introduction of an independent energetic contribution beyond heat and work. The formalism resolves inconsistencies associated with previous formulations of the quantum first law, including the interpretation of coherence contributions and their connection with entropy fluxes. At second order, the perturbative corrections become directly connected to transition rates governed by Fermi's golden rule, establishing a bridge between microscopic quantum transitions and macroscopic thermodynamic quantities. These results provide a physically transparent framework to investigate coherence-driven thermodynamic processes and offer new perspectives for the analysis of driven quantum systems and nonequilibrium quantum technologies.
We study the quantum thermodynamics of a coherent macroscopic electromagnetic field (laser) coupled to a two-level system (qubit) near resonance, from weak- to strong-driving regimes. This combined system is, in turn, weakly coupled to a thermal radiation field and can be described by an autonomous quantum master equation. We show that the laser acts as an autonomous work source and that, in the macroscopic limit, the work produced is independent of the phase of the laser. Using the dressed qubit approach, we show that the variation of energy in the laser is not the work transferred to the dressed qubit, which is instead obtained from the ``dressed laser''---a coherent superposition of the laser and the qubit. Using a two-point measurement technique with counting fields, we obtain the full counting statistics for the work of the laser and dressed laser, and show that they satisfy the Crooks fluctuation theorems. We then use these theorems as criteria to investigate the thermodynamic consistency of quantum master equations, first in the autonomous setup for the combined system, then in the nonautonomous setup for the quantum system where the coherent field is eliminated and effectively described by a time-dependent external field. Treating the laser as an external field is known to yield expressions for the work which are in contradiction with quantum thermodynamics predictions in the strong-driving regime. We show that these inconsistencies stem from a confusion between the laser and dressed laser, and show how to correct them. We also derive a generalized Bloch master equation, thermodynamically consistent across all driving regimes, from which the Bloch and Floquet master equations can be obtained using additional approximations (of which we also examine the consistency).
We report an alternate formulation of the quantum master equation (QME) to describe the dynamics of a quantum system weakly coupled to a heat bath, in the presence of weak external driving. A key feature of this approach is the introduction of an explicit Hamiltonian to model the thermal fluctuations in the heat bath. We show that the resulting time coarse-grained dynamical equation for the quantum system has dissipators with a natural regulator, which emerges from an ensemble average over the fluctuations. Importantly, such regularized dissipators arise from the second-order contributions of both the external drive as well as the system-environment coupling. We show that the second-order drive terms, regularized to time-scales set by the fluctuations, result in dynamic drive-induced frequency shifts as well as drive-dependent relaxation phenomena. Considering the specific case of an ensemble of two-level systems, subjected to a linearly polarized external drive, we derive the modified Bloch Equations with such drive-dependent shift and damping terms. The resulting drive-induced frequency shifts converge to the known forms of dynamic frequency shifts, such as the Bloch-Siegert shift or the dynamic Stark shift, in appropriate limits. The Kramers-Kronig pairs of these frequency shifts - manifest as drive-dependent damping terms in the modified Bloch Equations and help explain the Redfield limit of free-induction-decay (FID) rates as well as the non-Bloch decay of Rabi oscillations in isotropic medium. Our method predicts correct orders of magnitudes of non-Bloch decay rates in isotropic medium as well as their observed temperature dependence. The QME reported here, correctly describes all known aspects of the driven-dissipative dynamics up to second-order of an open quantum system with appropriate thermal signatures and as such is expected to provide deeper insights into the study of quantum information processing on real systems.
Thermodynamics is a phenomenological theory describing the energy conversion of work and heat. At its origins thermodynamics was developed in order to understand and improve heat engines. In conventional thermodynamics, however, only such processes are completely describable which are slow enough to keep the system of interest in an equilibrium state with its thermal environment at all times. On the contrary, all real physical processes are accompanied by non-equilibrium phenomena. These are mathematically described with the help of the irreversible entropy production, i.e. the dissipated work divided by the temperature of the bath. The second law of thermodynamics merely states that the irreversible entropy production is on average always positive. The modern trend of miniaturization, however, leads to smaller and smaller devices. On short length scales thermal noise as well as quantum fluctuations become important. Thus, usual thermodynamic quantities as work and heat acquire stochastic nature. Moreover, in the quantum regime a completely new theory had to be invented, since classical notions of work and heat are no longer valid. The present dissertation contributes to this prevailing field by the derivation of analytical expressions for the entropy production in open and closed quantum system far from thermal equilibrium. The theoretical treatment is motivated by an experimental point of view. The obtained results were derived from the reduced dynamics of the quantum mechanical system, which may be coupled to a thermal bath. To this end, it was dealt with methods and approaches of statistical physics, conventional thermodynamics, quantum information theory and the theory of open quantum systems. In this dissertation the most important physical systems are treated. First, an isolated system is considered before later on a heat bath is included. The limit of weak as well as of strong coupling are analyzed in detail. For isolated systems it turns out that a geometric approach is able to capture the thermodynamic properties, completely. To this end, the irreversible entropy production is sharply estimated from below by the angle between the real non-equilibrium state and the corresponding equilibrium one. Further, it is shown that the Heisenberg uncertainty relation for energy and time can be formulated more precisely in terms of that angle. As experimental system cold ion traps are introduced, in which the theoretical predictions can be verified. For weakly coupled systems, then, the irreversible entropy production is identified and a fluctuation theorem is derived by merely thermodynamic arguments. As experimental system we, again, propose the cold ion traps, where the heat bath can be simulated with the help of LASER light. In the limit of strong coupling a semiclassical description of the reduced quantum system is discussed. Again, an analytically exact expression for the entropy production and the corresponding fluctuation theorem is derived. A physical system, in which strong coupling to the environment can be analyzed, is given by Josephson junctions. This dissertation discusses the entropy production for isolated, weakly coupled, and overdamped quantum systems. For these physical regimes analytical relation are derived and experimental systems proposed, with which the predictions can be verified.