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Communications Physics

Generalized Clausius inequalities in a nonequilibrium cold-atom system

Daniel Mayer, Eric Lutz, Artur Widera

Published Apr 5, 2023• 7 citations
Publisher

Abstract

Abstract Thermodynamic inequalities, such as the Clausius inequality, characterize the direction of nonequilibrium processes. However, the latter result presupposes a system coupled to a heat bath that drives it to a thermal state. Far from equilibrium, the Clausius inequality can be generalized using information-theoretic quantities. For initially isolated systems that are moved from an equilibrium state by a dissipative heat exchange, the generalized Clausius inequality is predicted to be reversed. We here experimentally investigate the nonequilibrium thermodynamics of an initially isolated dilute gas of ultracold Cesium atoms that can be either thermalized or pushed out of equilibrium by means of laser cooling techniques. We determine in both cases the phase-space dynamics by tracing the evolution with position-resolved fluorescence imaging, from which we evaluate all relevant thermodynamic quantities. We confirm the validity of the generalized Clausius inequality for the first process and of the reversed generalized Clausius inequality for the second transformation.

DOI
10.1038/s42005-023-01175-3

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Abstract Thermodynamic inequalities, such as the Clausius inequality, characterize the direction of nonequilibrium processes. However, the latter result presupposes a system coupled to a heat bath that drives it to a thermal state. Far from equilibrium, the Clausius inequality can be generalized using information-theoretic quantities. For initially isolated systems that are moved from an equilibrium state by a dissipative heat exchange, the generalized Clausius inequality is predicted to be reversed. We here experimentally investigate the nonequilibrium thermodynamics of an initially isolated dilute gas of ultracold Cesium atoms that can be either thermalized or pushed out of equilibrium by means of laser cooling techniques. We determine in both cases the phase-space dynamics by tracing the evolution with position-resolved fluorescence imaging, from which we evaluate all relevant thermodynamic quantities. We confirm the validity of the generalized Clausius inequality for the first process and of the reversed generalized Clausius inequality for the second transformation.
The second law of thermodynamics is a fundamental law of Nature. It is almost universally associated with the Clausius inequality that lower bounds a change in entropy by the ratio of supplied heat and temperature. However, this result presupposes that a system is in contact with a heat bath that drives it to a thermal state. For isolated systems that are moved from an initial equilibrium state by a dissipative heat exchange, the Clausius inequality has been predicted to be reversed. We here experimentally investigate the nonequilibrium thermodynamics of an isolated dilute gas of ultracold Cesium atoms that can be either thermalized or pushed out of equilibrium by means of laser cooling techniques. We determine in both cases the phase-space dynamics by tracing the evolution of the gas with position-resolved fluorescence imaging, from which we evaluate all relevant thermodynamic quantities. Our results confirm the validity of the usual Clausius inequality for the first process and of the reversed Clausius inequality for the second transformation.
This paper has been withdrawn by the author. For the reason, see the bottom paragraph of this abstract. By generalizing Tasaki's work on the second law of thermodynamics for an adiabatic process between two equilibrium states of a macroscopic quantum compound system, we obtain an extension of the second law to a transient adiabatic process that takes a macroscopic quantum compound system consisting of a system of interest and two heat reservoirs from an initial equilibrium state to a final non-equilibrium state, where the system of interest is in a steady heat conduction state. For the system of interest, we define its generalized entropy so that it becomes an experimentally measurable quantity and exactly satisfies an extension of the Clausius relation. This paper has been withdrawn by the author because the generalized entropy presented in this article violates the second law when the system in a steady heat conduction state is detached from the two reservoirs and then attached to a reservoir at the temperature T^bar defined in Sec.1.2 so that it will reach an equilibrium state at T^bar. In this irreversible process, the total entropy for the system and the reservoir remains constant and thus violates the second law.
Abstract We consider a situation where an N-level system (NLS) is coupled to a heat bath without being necessarily thermalized. For this situation, we derive general Jarzynski-type equations and conclude that heat and entropy is flowing from the hot bath to the cold NLS and, vice versa, from the hot NLS to the cold bath. The Clausius relation between increase of entropy and transfer of heat divided by a suitable temperature assumes the form of two inequalities which have already been considered in the literature. Our approach is illustrated by an analytical example.
Abstract The entropy produced when a quantum system at thermal equilibrium is driven to a non-equilibrium state can be considered as having components from coherence generation and from a population mismatch. The entropy component which is related to coherence represents a truly quantum effect and can be calculated using two different decompositions, based on either the relative entropy of coherence or on the thermal state corresponding to the Hamiltonian dephased in the eigen basis of the actual state. However, the decomposition based on the relative entropy was unable to quantify highly coherent processes or processes occurring in the low temperature regime. We experimentally explored a general driving scheme (a non-equilibrium process) on a nuclear magnetic resonance quantum processor, and were able to isolate the contribution of coherence to irreversible entropy generation. Furthermore, we numerically explored an infinitesimal driving process to distinguish between high and low temperature limits. The thermal equilibrium temperature and the final energy gap were varied to establish the relationship among temperature, energy level spacing, entropy production and coherence generation. Our results indicate that the contribution to entropy from coherence is higher in the low temperature regime. We also verified a generalized Clausius inequality, which affirms that the irreversible entropy production is lower-bounded.
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.