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.
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.
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.
Traditional quantum thermodynamic frameworks associate work to energy exchanges induced by unitary transformations generated by external controls, and heat to energy exchanges induced by bath interaction. Recently, a framework was introduced aiming at extending the thermodynamic formalism to genuine quantum settings, also referred to as autonomous quantum systems: free from external controls, only quantum systems interacting with each other. In this paper, we apply such a thermodynamic framework to common experimental situations of interacting quantum systems. In situations where traditional frameworks detect only heat exchanges, the recent autonomous thermodynamic framework points at work exchanges based on two mechanisms: population inversion and coherence generation / consumption. Such mechanisms are well known in the literature for being related to work expenditure and extraction, in particular in relation with ergotropy, which emphasizes the relevance of the autonomous framework and the limitations of traditional ones. Furthermore, the autonomous framework also identifies a genuine non-unitary mechanism of work exchange related to athermality. %, also pointed out as a resource for work extraction. Finally, in the semi-classical limit, the autonomous framework identifies all energy exchanges as pure work, but distinguishes between local work and interaction energy.
Our results show that the autonomous framework provides a refined analysis of work exchange mechanisms in the quantum realm and serves as a consistent approach to analyze thermodynamic processes in realistic quantum devices.
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).
During the last decades, there have been many theoretical and experimental advances both in the extension of thermodynamics to comprise microscopic systems out-of-equilibrium and in the understanding of quantum mechanics.Along with the state-of-the-art capability of controlling fragile quantum systems in a wide variety of physical platforms, this context has paved the way for the current strategic efforts to develop a thermodynamic theory of quantum systems.In this sense, the research field coined as quantum thermodynamics (QT) already plays a key role in the design and development of future quantum-based technologies.More specifically, QT aims both to apply the usual thermodynamic concepts and notions to describe arbitrary non-equilibrium quantum systems and to understand the emergence of classical thermodynamic behaviour from the underlying fundamentally quantum dynamics.However, despite all current progress, there is still no consolidated formalism for a general thermodynamic description of fully autonomous quantum objects.Besides, the lack of consensus on some central aspects, such as the definitions of quantum counterparts of thermodynamic quantities, is particularly notorious.In this thesis, we focus on the energetic analysis within autonomous quantum systems.To this aim, we propose a novel and general formalism for a dynamic description of the energy exchanges between interacting subsystems.From the Schmidt decomposition approach, we identify effective Hamiltonians as the representative operators for characterizing the local internal energies, whose expectation values satisfy the usual thermodynamic notion of energy additivity.In contrast to the currently used methodologies, such procedure treats the subsystems with equal footing and do not rely on any sort of approximations and additional hypotheses, e.g., semi-classical description, weak-coupling regime, strict energy conservation and Markovian dynamics.In short, our proposal contributes to the development of QT by providing a new formalism that does not suffer from the usual restrictive shortcomings and establishes a new and exact route for defining other general thermodynamic quantities to the quantum regime.
During the last decades, there have been many theoretical and experimental advances both in the extension of thermodynamics to comprise microscopic systems out-of-equilibrium and in the understanding of quantum mechanics. Along with the state-of-the-art capability of controlling fragile quantum systems in a wide variety of physical platforms, this context has paved the way for the current strategic efforts to develop a thermodynamic theory of quantum systems. In this sense, the research field coined as quantum thermodynamics (QT) already plays a key role in the design and development of future quantum-based technologies. More specifically, QT aims both to apply the usual thermodynamic concepts and notions to describe arbitrary non-equilibrium quantum systems and to understand the emergence of classical thermodynamic behaviour from the underlying fundamentally quantum dynamics. However, despite all current progress, there is still no consolidated formalism for a general thermodynamic description of fully autonomous quantum objects. Besides, the lack of consensus on some central aspects, such as the definitions of quantum counterparts of thermodynamic quantities, is particularly notorious. In this thesis, we focus on the energetic analysis within autonomous quantum systems. To this aim, we propose a novel and general formalism for a dynamic description of the energy exchanges between interacting subsystems. From the Schmidt decomposition approach, we identify effective Hamiltonians as the representative operators for characterizing the local internal energies, whose expectation values satisfy the usual thermodynamic notion of energy additivity. In contrast to the currently used methodologies, such procedure treats the subsystems with equal footing and do not rely on any sort of approximations and additional hypotheses, e.g., semi-classical description, weak-coupling regime, strict energy conservation and Markovian dynamics. In short, our proposal contributes to the development of QT by providing a new formalism that does not suffer from the usual restrictive shortcomings and establishes a new and exact route for defining other general thermodynamic quantities to the quantum regime.
Quantum-Induced Stochastic Dynamics arises from the coupling between a classical system and a quantum environment. Unlike standard thermal reservoirs, this environment acts as a dynamic bath, capable of simultaneously exchanging heat and performing work. We formulate a thermodynamic framework for this semi-classical regime, defining heat, work, and entropy production. We derive a modified Second Law that accounts for non-equilibrium quantum features, such as squeezing. The framework is exemplified by an optomechanical setup, where we characterize the thermodynamics of the non-stationary noise induced by the cavity field.