Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0601006 · New Journal of Physics

Irreversible performance of a quantum harmonic heat engine

Yair Rezek, Ronnie Kosloff

Published May 31, 2006• 326 citations
arXivHTMLPDF

Abstract

The unavoidable irreversible loss of power in a heat engine is found to be of quantum origin. Following thermodynamic tradition, a model quantum heat engine operating in an Otto cycle is analysed, where the working medium is composed of an ensemble of harmonic oscillators and changes in volume correspond to changes in the curvature of the potential well. Equations of motion for quantum observables are derived for the complete cycle of operation. These observables are sufficient to determine the state of the system and with it all thermodynamical variables. Once the external controls are set, the engine settles to a limit cycle. Conditions for optimal work, power and entropy production are derived. At high temperatures and quasistatic operating conditions, the efficiency at maximum power coincides with the endoreversible result . The optimal compression ratio varies from in the quasistatic limit where the irreversibility is dominated by heat conductance to in the sudden limit when the irreversibility is dominated by friction. When the engine deviates from adiabatic conditions, the performance is subject to friction. The origin of this friction can be traced to the noncommutability of the kinetic and potential energy of the working medium.

DOI
10.1088/1367-2630/8/5/083

Related papers

Irreversible Performance of a Quantum Harmonic Heat Engine

Rezek, Y, Kosloff, R

The unavoidable irreversible losses of power in a heat engine are found to be of quantum origin. Following thermodynamic tradition a model quantum heat engine operating by the Otto cycle is analyzed. The working medium of the model is composed of an ensemble of harmonic oscillators. A link is established between the quantum observables and thermodynamical variables based on the concept of canonical invariance. These quantum variables are sufficient to determine the state of the system and with it all thermodynamical variables. Conditions for optimal work, power and entropy production show that maximum power is a compromise between the quasistatic limit of adiabatic following on the compression and expansion branches and a sudden limit of very short time allocation to these branches. At high temperatures and quasistatic operating conditions the efficiency at maximum power coincides with the endoreversible result. The optimal compression ratio varies from the square root of the temperature ratio in the quasistatic limit where their reversibility is dominated by heat conductance to the temperature ratio to the power of 1/4 in the sudden limit when the irreversibility is dominated by friction. When the engine deviates from adiabatic conditions the performance is subject to friction. The origin of this friction can be traced to the noncommutability of the kinetic and potential energy of the working medium.

•Jan 2, 2006•172 citations

Finite-power performance of quantum heat engines in linear response

Qin Liu, Jizhou He, Yong-li Ma +1 more

We investigate the finite-power performance of quantum heat engines working in the linear response regime where the temperature gradient is small. The engine cycles with working substances of ideal harmonic systems consist of two heat transfer and two adiabatic processes, such as the Carnot cycle, Otto cycle, and Brayton cycle. By analyzing the optimal protocol under maximum power we derive the explicitly analytic expression for the irreversible entropy production, which becomes the low dissipation form in the long duration limit. Assuming the engine to be endoreversible, we derive the universal expression for the efficiency at maximum power, which agrees well with that obtained from the phenomenological heat transfer laws holding in the classical thermodynamics. Through appropriate identification of the thermodynamic fluxes and forces that a linear relation connects, we find that the quantum engines under consideration are tightly coupled, and the universality of efficiency at maximum power is confirmed at the linear order in the temperature gradient.

Physical review. E•Jul 8, 2019•10 citations

Performance analysis of an irreversible quantum heat engine working with harmonic oscillators

Bihong Lin, Jincan Chen

The cycle model of a regenerative quantum heat engine working with many noninteracting harmonic oscillators is established. The cycle consists of two isothermal and two constant-frequency processes. The performance of the cycle is investigated, based on the quantum master equation and semigroup approach. The inherent regenerative losses in the two constant-frequency processes are calculated. The expressions of several important performance parameters such as the efficiency, power output, and rate of the entropy production are derived for several interesting cases. Especially, the optimal performance of the cycle in high-temperature limit is discussed in detail. The maximum power output and the corresponding parameters are calculated. The optimal region of the efficiency and the optimal ranges of the temperatures of the working substance in the two isothermal processes are determined.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 11, 2003•138 citations

The Quantum Harmonic Otto Cycle

Ronnie Kosloff, Yair Rezek

The quantum Otto cycle serves as a bridge between the macroscopic world of heat engines and the quantum regime of thermal devices composed from a single element. We compile recent studies of the quantum Otto cycle with a harmonic oscillator as a working medium. This model has the advantage that it is analytically trackable. In addition, an experimental realization has been achieved, employing a single ion in a harmonic trap. The review is embedded in the field of quantum thermodynamics and quantum open systems. The basic principles of the theory are explained by a specific example illuminating the basic definitions of work and heat. The relation between quantum observables and the state of the system is emphasized. The dynamical description of the cycle is based on a completely positive map formulated as a propagator for each stroke of the engine. Explicit solutions for these propagators are described on a vector space of quantum thermodynamical observables. These solutions which employ different assumptions and techniques are compared. The tradeoff between power and efficiency is the focal point of finite-time-thermodynamics. The dynamical model enables the study of finite time cycles limiting time on the adiabatic and the thermalization times. Explicit finite time solutions are found which are frictionless (meaning that no coherence is generated), and are also known as shortcuts to adiabaticity.The transition from frictionless to sudden adiabats is characterized by a non-hermitian degeneracy in the propagator. In addition, the influence of noise on the control is illustrated. These results are used to close the cycles either as engines or as refrigerators. The properties of the limit cycle are described. Methods to optimize the power by controlling the thermalization time are also introduced. At high temperatures, the Novikov–Curzon–Ahlborn efficiency at maximum power is obtained. The sudden limit of the engine which allows finite power at zero cycle time is shown. The refrigerator cycle is described within the frictionless limit, with emphasis on the cooling rate when the cold bath temperature approaches zero.

arXiv:1612.03582•Entropy•Mar 23, 2017•373 citations

Performance analysis of a spin quantum heat engine cycle with internal friction

Jianhui Wang, Jizhou He, Yong Xin

An irreversible cycle model of a quantum heat engine with internal friction is established, which is composed of two adiabatic and two isomagnetic field processes. The working substance of the cycle consists of an ensemble of many non-interacting spin-1/2 systems. Based on a quantum master equation and semi-group approach, the general performance characteristics of the heat engine are investigated. The general expressions for several important parameters, such as the efficiency, power output, and rate of the entropy production, are derived. The performance of the cycle is optimized with respect to the temperatures of the working substance. By numerical solutions, the maximum power output and the corresponding parameters are calculated. The optimal regions of efficiency, temperatures of the working substance, and cycle period are determined. Moreover, the performance of the heat engine in the frictionless case is obtained, which is different from that in the friction case. Finally, the results obtained are generalized to the performance optimization of the heat engine working with spin-J systems.

Physica Scripta•Jan 17, 2007•52 citations

Thermodynamical analysis of a quantum heat engine based on harmonic oscillators

Andrea Roberto Insinga, Bjarne Andresen, Peter Salamon

Many models of heat engines have been studied with the tools of finite-time thermodynamics and an ensemble of independent quantum systems as the working fluid. Because of their convenient analytical properties, harmonic oscillators are the most frequently used example of a quantum system. We analyze different thermodynamical aspects with the final aim of the optimization of the performance of the engine in terms of the mechanical power provided during a finite-time Otto cycle. The heat exchange mechanism between the working fluid and the thermal reservoirs is provided by the Lindblad formalism. We describe an analytical method to find the limit cycle and give conditions for a stable limit cycle to exist. We explore the power production landscape as the duration of the four branches of the cycle are varied for short times, intermediate times, and special frictionless times. For short times we find a periodic structure with atolls of purely dissipative operation surrounding islands of divergent behavior where, rather than tending to a limit cycle, the working fluid accumulates more and more energy. For frictionless times the periodic structure is gone and we come very close to the global optimal operation. The global optimum is found and interestingly comes with a particular value of the cycle time.

Physical review. E•Jul 15, 2016•51 citations
The unavoidable irreversible loss of power in a heat engine is found to be of quantum origin. Following thermodynamic tradition, a model quantum heat engine operating in an Otto cycle is analysed, where the working medium is composed of an ensemble of harmonic oscillators and changes in volume correspond to changes in the curvature of the potential well. Equations of motion for quantum observables are derived for the complete cycle of operation. These observables are sufficient to determine the state of the system and with it all thermodynamical variables. Once the external controls are set, the engine settles to a limit cycle. Conditions for optimal work, power and entropy production are derived. At high temperatures and quasistatic operating conditions, the efficiency at maximum power coincides with the endoreversible result . The optimal compression ratio varies from in the quasistatic limit where the irreversibility is dominated by heat conductance to in the sudden limit when the irreversibility is dominated by friction. When the engine deviates from adiabatic conditions, the performance is subject to friction. The origin of this friction can be traced to the noncommutability of the kinetic and potential energy of the working medium.
The unavoidable irreversible losses of power in a heat engine are found to be of quantum origin. Following thermodynamic tradition a model quantum heat engine operating by the Otto cycle is analyzed. The working medium of the model is composed of an ensemble of harmonic oscillators. A link is established between the quantum observables and thermodynamical variables based on the concept of canonical invariance. These quantum variables are sufficient to determine the state of the system and with it all thermodynamical variables. Conditions for optimal work, power and entropy production show that maximum power is a compromise between the quasistatic limit of adiabatic following on the compression and expansion branches and a sudden limit of very short time allocation to these branches. At high temperatures and quasistatic operating conditions the efficiency at maximum power coincides with the endoreversible result. The optimal compression ratio varies from the square root of the temperature ratio in the quasistatic limit where their reversibility is dominated by heat conductance to the temperature ratio to the power of 1/4 in the sudden limit when the irreversibility is dominated by friction. When the engine deviates from adiabatic conditions the performance is subject to friction. The origin of this friction can be traced to the noncommutability of the kinetic and potential energy of the working medium.
We investigate the finite-power performance of quantum heat engines working in the linear response regime where the temperature gradient is small. The engine cycles with working substances of ideal harmonic systems consist of two heat transfer and two adiabatic processes, such as the Carnot cycle, Otto cycle, and Brayton cycle. By analyzing the optimal protocol under maximum power we derive the explicitly analytic expression for the irreversible entropy production, which becomes the low dissipation form in the long duration limit. Assuming the engine to be endoreversible, we derive the universal expression for the efficiency at maximum power, which agrees well with that obtained from the phenomenological heat transfer laws holding in the classical thermodynamics. Through appropriate identification of the thermodynamic fluxes and forces that a linear relation connects, we find that the quantum engines under consideration are tightly coupled, and the universality of efficiency at maximum power is confirmed at the linear order in the temperature gradient.
The cycle model of a regenerative quantum heat engine working with many noninteracting harmonic oscillators is established. The cycle consists of two isothermal and two constant-frequency processes. The performance of the cycle is investigated, based on the quantum master equation and semigroup approach. The inherent regenerative losses in the two constant-frequency processes are calculated. The expressions of several important performance parameters such as the efficiency, power output, and rate of the entropy production are derived for several interesting cases. Especially, the optimal performance of the cycle in high-temperature limit is discussed in detail. The maximum power output and the corresponding parameters are calculated. The optimal region of the efficiency and the optimal ranges of the temperatures of the working substance in the two isothermal processes are determined.
The quantum Otto cycle serves as a bridge between the macroscopic world of heat engines and the quantum regime of thermal devices composed from a single element. We compile recent studies of the quantum Otto cycle with a harmonic oscillator as a working medium. This model has the advantage that it is analytically trackable. In addition, an experimental realization has been achieved, employing a single ion in a harmonic trap. The review is embedded in the field of quantum thermodynamics and quantum open systems. The basic principles of the theory are explained by a specific example illuminating the basic definitions of work and heat. The relation between quantum observables and the state of the system is emphasized. The dynamical description of the cycle is based on a completely positive map formulated as a propagator for each stroke of the engine. Explicit solutions for these propagators are described on a vector space of quantum thermodynamical observables. These solutions which employ different assumptions and techniques are compared. The tradeoff between power and efficiency is the focal point of finite-time-thermodynamics. The dynamical model enables the study of finite time cycles limiting time on the adiabatic and the thermalization times. Explicit finite time solutions are found which are frictionless (meaning that no coherence is generated), and are also known as shortcuts to adiabaticity.The transition from frictionless to sudden adiabats is characterized by a non-hermitian degeneracy in the propagator. In addition, the influence of noise on the control is illustrated. These results are used to close the cycles either as engines or as refrigerators. The properties of the limit cycle are described. Methods to optimize the power by controlling the thermalization time are also introduced. At high temperatures, the Novikov–Curzon–Ahlborn efficiency at maximum power is obtained. The sudden limit of the engine which allows finite power at zero cycle time is shown. The refrigerator cycle is described within the frictionless limit, with emphasis on the cooling rate when the cold bath temperature approaches zero.
An irreversible cycle model of a quantum heat engine with internal friction is established, which is composed of two adiabatic and two isomagnetic field processes. The working substance of the cycle consists of an ensemble of many non-interacting spin-1/2 systems. Based on a quantum master equation and semi-group approach, the general performance characteristics of the heat engine are investigated. The general expressions for several important parameters, such as the efficiency, power output, and rate of the entropy production, are derived. The performance of the cycle is optimized with respect to the temperatures of the working substance. By numerical solutions, the maximum power output and the corresponding parameters are calculated. The optimal regions of efficiency, temperatures of the working substance, and cycle period are determined. Moreover, the performance of the heat engine in the frictionless case is obtained, which is different from that in the friction case. Finally, the results obtained are generalized to the performance optimization of the heat engine working with spin-J systems.
Many models of heat engines have been studied with the tools of finite-time thermodynamics and an ensemble of independent quantum systems as the working fluid. Because of their convenient analytical properties, harmonic oscillators are the most frequently used example of a quantum system. We analyze different thermodynamical aspects with the final aim of the optimization of the performance of the engine in terms of the mechanical power provided during a finite-time Otto cycle. The heat exchange mechanism between the working fluid and the thermal reservoirs is provided by the Lindblad formalism. We describe an analytical method to find the limit cycle and give conditions for a stable limit cycle to exist. We explore the power production landscape as the duration of the four branches of the cycle are varied for short times, intermediate times, and special frictionless times. For short times we find a periodic structure with atolls of purely dissipative operation surrounding islands of divergent behavior where, rather than tending to a limit cycle, the working fluid accumulates more and more energy. For frictionless times the periodic structure is gone and we come very close to the global optimal operation. The global optimum is found and interestingly comes with a particular value of the cycle time.