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arXiv:quant-ph/0303046 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum four-stroke heat engine: Thermodynamic observables in a model with intrinsic friction

Tova Feldmann, Ronnie Kosloff

Published Jul 3, 2003• 225 citations
arXivHTMLPDF

Abstract

The fundamentals of a quantum heat engine are derived from first principles. The study is based on the equation of motion of a minimum set of operators, which is then used to define the state of the system. The relation between the quantum framework and the thermodynamical observables is examined. A four-stroke heat engine model with a coupled two-level system as a working fluid is used to explore the fundamental relations. In the model used, the internal Hamiltonian does not commute with the external control field, which defines the two adiabatic branches. Heat is transferred to the working fluid by coupling to hot and cold reservoirs under constant field values. Explicit quantum equations of motion for the relevant observables are derived on all branches. The dynamics on the heat transfer constant field branches is solved in closed form. On the adiabats, a general numerical solution is used and compared with a particular analytic solution. These solutions are combined to construct the cycle of operation. The engine is then analyzed in terms of the frequency-entropy and entropy-temperature graphs. The irreversible nature of the engine is the result of finite heat transfer rates and frictionlike behavior due to noncommutability of the internal and external Hamiltonians.

DOI
10.1103/physreve.68.016101

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The fundamentals of a quantum heat engine are derived from first principles. The study is based on the equation of motion of a minimum set of operators, which is then used to define the state of the system. The relation between the quantum framework and the thermodynamical observables is examined. A four-stroke heat engine model with a coupled two-level system as a working fluid is used to explore the fundamental relations. In the model used, the internal Hamiltonian does not commute with the external control field, which defines the two adiabatic branches. Heat is transferred to the working fluid by coupling to hot and cold reservoirs under constant field values. Explicit quantum equations of motion for the relevant observables are derived on all branches. The dynamics on the heat transfer constant field branches is solved in closed form. On the adiabats, a general numerical solution is used and compared with a particular analytic solution. These solutions are combined to construct the cycle of operation. The engine is then analyzed in terms of the frequency-entropy and entropy-temperature graphs. The irreversible nature of the engine is the result of finite heat transfer rates and frictionlike behavior due to noncommutability of the internal and external Hamiltonians.
The finite-rate performance of a quantum heat engine, constructed from a three-level amplifier, is analyzed. Consistent definitions of thermodynamical quantities in terms of quantum observables are postulated. The performance is analyzed in steady state, where the operation of the amplifier only influences the surroundings. Quantum master equations describe the irreversible dynamics induced by the coupling of the working medium to the reservoirs. It is shown that the standard assumption of field-independent dissipation is inconcistent with thermodynamics. Field-dependent relaxation equations, based upon the semigroup approach, and consistent with thermodynamics, are formulated. These equations are valid if the time scale of the external field is slow compared to that associated with the bath fluctuations. The steady-state values of the thermodynamical quantities are evaluated. The power is found to have maxima as a function of important controls, such as the field amplitude, frequency, and the coupling with the baths. The existence and locations of these maxima differ from those obtained in the standard treatment, where the dissipation is field independent. The irreversible nature of engine operation is due to the finite rate of heat transfer and a genuine ``quantum-friction'' loss term due to dephasing.
After starting a reciprocating heat engine it eventually settles to a stable mode of operation. A first principle quantum heat engine also approaches this stable limit cycle. The studied engine is based on a working medium consisting of an ensemble of quantum systems composed of two coupled spins. A four-stroke cycle of operation is studied, with two isochore branches where heat is transferred from the hot/cold baths and two adiabats where work is exchanged. The dynamics is generated by a completely positive map. It has been shown that the performance of this model resembles an engine with intrinsic friction. The quantum conditional entropy is employed to prove the monotonic approach to a limit cycle. Other convex measures such as the quantum distance display the same monotonic approach. The equations of motion of the engine are solved for the different branches and are combined to a global propagator that relates the state of the engine in the beginning of the cycle to the state after one period of operation of the cycle. The eigenvalues of the propagator define the rate of relaxation toward the limit cycle. A longitudinal and transverse mode of approach to the limit cycle is thus identified. The entropy balance is used to explore the necessary conditions which lead to a stable limit cycle. The phenomena of friction can be identified with a zero change in the von Neumann entropy of the working medium.
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 manifestations of the three laws of thermodynamics are explored in a model of an irreversible quantum heat engine. The engine is composed of a three-level system simultaneously coupled to hot and cold heat baths, and driven by an oscillating external field. General quantum heat baths are considered, which are weakly coupled to the three-level system. The work reservoir is modeled by a classical electro-magnetic field of arbitrary intensity, which is driving the three-level system. The first law of thermodynamics is related to the rate of change of energy obtained from the quantum master equation in the Heisenberg picture. The fluxes of the thermodynamic heat and work are then directly related to the expectation values of quantum observables. An analysis of the standard quantum master equation for the amplifier, first introduced by Lamb, is shown to be thermodynamically inconsistent when strong driving fields are used. A generalized master equation is rigorously derived, starting from the underlying quantum dynamics, which includes relaxation terms that explicitly depend upon the field. For weak fields the generalized master equation reduces to the standard equation. In very intense fields the amplifier splits into two heat engines. One engine accelerates as the field intensifies, while the other slows down and eventually switches direction to become a heat pump. The relative weight of the slower engine increases with the field intensity, leading to a maximum in power as a function of the field intensity. The amplifier is shown to go through four ‘‘phases’’ as the driving field is intensified, throughout all of which the second law of thermodynamics is generally satisfied. One phase corresponds to a ‘‘refrigeration window’’ which allows for the extraction of heat out of a cold bath of temperatures down to the absolute zero. This window disappears at absolute zero, which is conjectured to be a dynamical manifestation of the third law of thermodynamics.
The minimal-coupling quantum heat engine is a thermal machine consisting of an explicit energy storage system, heat baths, and a working body, which alternatively couples to subsystems through discrete strokes --- energy-conserving two-body quantum operations. Within this paradigm, we present a general framework of quantum thermodynamics, where a work extraction process is fundamentally limited by a flow of non-passive energy (ergotropy), while energy dissipation is expressed through a flow of passive energy. It turns out that small dimensionality of the working body and a restriction only to two-body operations make the engine fundamentally irreversible. Our main result is finding the optimal efficiency and work production per cycle within the whole class of irreversible minimal-coupling engines composed of three strokes and with the two-level working body, where we take into account all possible quantum correlations between the working body and the battery. One of the key new tools is the introduced ``control-marginal state" --- one which acts only on a working body Hilbert space, but encapsulates all features regarding work extraction of the total working body-battery system. In addition, we propose a generalization of the many-stroke engine, and we analyze efficiency vs extracted work trade-offs, as well as work fluctuations after many cycles of the running of the engine.