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arXiv:quant-ph/0411018 · Physical Review E

Work extraction in the spin-boson model

ARMEN E. ALLAHVERDYAN, Rubèn Serral Gracià, Theo M. Nieuwenhuizen

Published Apr 5, 2005• 65 citations
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Abstract

We show that work can be extracted from a two-level system (spin) coupled to a bosonic thermal bath. This is possible due to different initial temperatures of the spin and the bath, both positive (no spin population inversion), and is realized by means of a suitable sequence of sharp pulses applied to the spin. The extracted work can be of the order of the response energy of the bath, therefore much larger than the energy of the spin. Moreover, the efficiency of extraction can be very close to its maximum, given by the Carnot bound, at the same time the overall amount of the extracted work is maximal. Therefore, we get a finite power at efficiency close to the Carnot bound. The effect comes from the back-reaction of the spin on the bath, and it survives for a strongly disordered (inhomogeneously broadened) ensemble of spins. It is connected with generation of coherences during the work-extraction process, and we deduced it in an exactly solvable model. All the necessary general thermodynamical relations are deduced from the first principles of quantum mechanics and connections are made with processes of lasing without inversion and with quantum heat engines.

DOI
10.1103/physreve.71.046106

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We show that work can be extracted from a two-level system (spin) coupled to a bosonic thermal bath. This is possible due to different initial temperatures of the spin and the bath, both positive (no spin population inversion), and is realized by means of a suitable sequence of sharp pulses applied to the spin. The extracted work can be of the order of the response energy of the bath, therefore much larger than the energy of the spin. Moreover, the efficiency of extraction can be very close to its maximum, given by the Carnot bound, at the same time the overall amount of the extracted work is maximal. Therefore, we get a finite power at efficiency close to the Carnot bound. The effect comes from the back-reaction of the spin on the bath, and it survives for a strongly disordered (inhomogeneously broadened) ensemble of spins. It is connected with generation of coherences during the work-extraction process, and we deduced it in an exactly solvable model. All the necessary general thermodynamical relations are deduced from the first principles of quantum mechanics and connections are made with processes of lasing without inversion and with quantum heat engines.
The spin–boson model, often used in NMR and ESR physics, quantum optics and spintronics, is considered in a solvable limit to model a spin one-half particle interacting with a bosonic thermal bath. By applying external pulses to a non-equilibrium initial state of the spin, work can be extracted from the thermalized bath. It occurs on the timescale 𝒯 2 inherent to quantum coherence. The work (partly) arises from heat given off by the surrounding bath, while the spin entropy remains constant during a pulse. This presents a new mechanism and time and temperature regimes for limiting the validity of the Clausius inequality and Thomson's formulation of the second law (cycles cost work). Apart from this, starting from a fully disordered state, coherence can be induced by employing the bath. A gain from a positive-temperature (inversion-free) two-level system is shown to be possible.
The spin‐boson model, often used in NMR and ESR physics, quantum optics and spin‐tronics, is considered in a solvable limit to model a spin one‐half particle interacting with a bosonic thermal bath. By applying external pulses to a non‐equilibrium initial state of the spin, work can be extracted from the thermalized bath. It occurs on the timescale T2 inherent to transversal (‘quantum’) fluctuations. The work (partly) arises from heat given off by the surrounding bath, while the spin entropy remains constant during a pulse. This presents a violation of the Clausius inequality and the Thomson formulation of the second law (cycles cost work) for the two‐level system.
We consider a class of quantum heat engines consisting of two subsystems interacting with a work-source and coupled to two separate baths at different temperatures Th>Tc. The purpose of the engine is to extract work due to the temperature difference. Its dynamics is not restricted to the near equilibrium regime. The engine structure is determined by maximizing the extracted work under various constraints. When this maximization is carried out at finite power, the engine dynamics is described by well-defined temperatures and satisfies the local version of the second law. In addition, its efficiency is bounded from below by the Curzon-Ahlborn value 1-radical Tc/Th and from above by the Carnot value 1-(Tc/Th). The latter is reached-at finite power--for a macroscopic engine, while the former is achieved in the equilibrium limit Th-->Tc . The efficiency that maximizes the power is strictly larger than the Curzon-Ahloborn value. When the work is maximized at a zero power, even a small (few-level) engine extracts work right at the Carnot efficiency.
In this paper, we consider a model of two-level quantum heat engine to investigate the explicit analytic expression for the thermodynamics quantities in different condition under the finite-time operation. In this engine, the working substance is composed of a spin-half particles immersed in a magnetic field. The finite-time thermodynamic processes consisting of two quantum adiabatic and two quantum isothermal processes. This processes working between two heat reservoirs with an inverse temperatures β1β_{1}β1​ and β2β_{2}β2​ (<β1<β_{1}<β1​). In this processes, we obtain the work, heat, power and efficiency at maximum power output of the model. Our result of the efficiency at maximum power agree with the universal value in the first order of Carnot efficiency.
Many work extraction or information erasure processes in the literature involve the raising and lowering of energy levels via external fields. But even if the actual system is treated quantum mechanically, the field is assumed to be classical and of infinite strength, hence not developing any correlations with the system or experiencing back-actions. We extend these considerations to a fully quantum mechanical treatment by studying a spin-1/2 particle coupled to a finite-sized directional quantum reference frame, a spin-l system, which models an external field. With this concrete model together with a bosonic thermal bath, we analyze the back-action a finite-size field suffers during a quantum-mechanical work extraction process and the effect this has on the extractable work and highlight a range of assumptions commonly made when considering such processes. The well-known semiclassical treatment of work extraction from a pure qubit predicts a maximum extractable work W=kTlog2 for a quasistatic process, which holds as a strict upper bound in the fully quantum mechanical case and is attained only in the classical limit. We also address the problem of emergent local time dependence in a joint system with a globally fixed Hamiltonian.
Quantum systems strongly coupled to many-body systems equilibrate to the reduced state of a global thermal state, deviating from the local thermal state of the system as it occurs in the weak-coupling limit. Taking this insight as a starting point, we study the thermodynamics of systems strongly coupled to thermal baths. First, we provide strong-coupling corrections to the second law applicable to general systems in three of its different readings: As a statement of maximal extractable work, on heat dissipation, and bound to the Carnot efficiency. These corrections become relevant for small quantum systems and always vanish in first order in the interaction strength. We then move to the question of power of heat engines, obtaining a bound on the power enhancement due to strong coupling. Our results are exemplified on the paradigmatic situation of non-Markovian quantum Brownian motion.