We use a canonical quantization procedure to set up a quantum Fokker-Planck-Kramers equation that accounts for quantum dissipation in a thermal environment. The dissipation term is chosen to ensure that the thermodynamic equilibrium is described by the Gibbs state. An expression for the quantum entropy production that properly describes quantum systems in a nonequilibrium stationary state is also provided. The time-dependent solution is given for a quantum harmonic oscillator in contact with a heat bath. We also obtain the stationary solution for a system of two coupled harmonic oscillators in contact with reservoirs at distinct temperatures, from which we obtain the entropy production and the quantum thermal conductance.
We use a canonical quantization procedure to set up a quantum Fokker-Planck-Kramers equation that accounts for quantum dissipation in a thermal environment. The dissipation term is chosen to ensure that the thermodynamic equilibrium is described by the Gibbs state. An expression for the quantum entropy production that properly describes quantum systems in a nonequilibrium stationary state is also provided. The time-dependent solution is given for a quantum harmonic oscillator in contact with a heat bath. We also obtain the stationary solution for a system of two coupled harmonic oscillators in contact with reservoirs at distinct temperatures, from which we obtain the entropy production and the quantum thermal conductance.
Abstract The theory of open quantum systems plays a fundamental role in several scientific and technological disciplines, from quantum computing and information science to molecular electronics and quantum thermodynamics. Despite its widespread relevance, a rigorous formulation of quantum dissipation in conjunction with thermal noise remains a topic of active research. In this work, we establish a formal correspondence between classical stochastic thermodynamics, in particular the Fokker–Planck and Klein–Kramers equations, and the quantum master equation. Building on prior studies of multiplicative noise in classical stochastic differential equations, we demonstrate that thermal noise at the quantum level manifests as a multidimensional geometric stochastic process. By applying canonical quantization, we introduce a novel Hermitian dissipation operator that serves as a quantum analogue of classical viscous friction. This operator allows for a well-defined expression of heat exchange between a system and its environment, enabling the formulation of an alternative quantum equipartition theorem. Our framework ensures a precise energy balance that aligns with the first law of thermodynamics and an entropy balance consistent with the second law. The theoretical formalism is applied to two prototypical quantum systems, the harmonic oscillator and a particle in an infinite potential well, for which it provides new insights into nonequilibrium thermodynamics at the quantum scale. Our results advance the understanding of dissipation in quantum systems and establish a foundation for future studies on stochastic thermodynamics in the quantum domain.
The transport properties of a bosonic chain have been calculated by placing the ends of the chain in contact with thermal and particle reservoirs at different temperatures and chemical potentials. The contact with the reservoirs is described by the use of a quantum Fokker-Planck-Kramers equation, which is a canonical quantization of the classical Fokker-Planck-Kramers equation. From the quantum equation we obtain equations for the covariances of the creation and annihilation boson operators and solve them in the stationary state for small interactions. From the covariances we determine the Onsager coefficients and in particular the conductance, which was found to be finite for any chain size leading to an infinite conductivity and the absence of Fourier's law.
Abstract We propose a Langevin equation to describe the quantum Brownian motion of bounded particles based on a distinctive formulation concerning both the fluctuation and dissipation forces. The fluctuation force is similar to that employed in the classical case. It is a white noise with a variance proportional to the temperature. The dissipation force is not restricted to be proportional to the velocity and is determined in a way as to guarantee that the stationary state is given by a density operator of the Gibbs canonical type. To this end we derived an equation that gives the time evolution of the density operator, which turns out to be a quantum Fokker–Planck–Kramers equation. The approach is applied to the harmonic oscillator in which case the dissipation force is found to be non Hermitian and proportional to the velocity and position.
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.
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.
In this thesis we develop an approach to nonequilibrium quantum-statistical mechanics which is based on the consideration of the thermodynamic entropy as being a quantum observable with an associated hermitian operator and with its own equation of motion, from which the rate of entropy production can be studied. The relationship of this quantum observable---the expectation value of which is proved to obey the laws of thermodynamics and is thus called the thermodynamic entropy---with heat dissipation in quantum many-body systems is investigated in detail. After showing how the classical theory of nonequilibrium thermodynamics is obtained from our entropy-production formalism in the limit of very weakly coupled subsystems of a larger isolated quantum system, the solution of the equation of motion for the thermodynamic entropy operator is formally obtained and applied to the case of electrons interacting with phonons and being driven by an external electric field, arriving at an explicit expression for the Joule heat without any a priori consideration of the rate of change of the energy of the system. Finally, the formalism is applied to solve a puzzle introduced with the most basic model for atom-light interactions: the Jaynes-Cummings model. Without using the correct thermodynamic arguments implied by our entropy-production theory, this model leads to the conclusion that a stream of two-level atoms sent to a cavity filled with monochromatic photons---in the so called one-atom maser configuration---thermalize, under off-resonant conditions, to a temperature different from that of the photons, casting doubts on the validity of the principle of conservation of energy.