Periodic thermodynamics of open quantum systems

Kay Brandner and Udo Seifert

Phys. Rev. E 93, 062134 – Published 22 June 2016

Abstract

The thermodynamics of quantum systems coupled to periodically modulated heat baths and work reservoirs is developed. By identifying affinities and fluxes, the first and the second law are formulated consistently. In the linear response regime, entropy production becomes a quadratic form in the affinities. Specializing to Lindblad dynamics, we identify the corresponding kinetic coefficients in terms of correlation functions of the unperturbed dynamics. Reciprocity relations follow from symmetries with respect to time reversal. The kinetic coefficients can be split into a classical and a quantum contribution subject to an additional constraint, which follows from a natural detailed balance condition. This constraint implies universal bounds on efficiency and power of quantum heat engines. In particular, we show that Carnot efficiency cannot be reached whenever quantum coherence effects are present, i.e., when the Hamiltonian used for work extraction does not commute with the bare system Hamiltonian. For illustration, we specialize our universal results to a driven two-level system in contact with a heat bath of sinusoidally modulated temperature.

Received 2 April 2016

DOI:https://doi.org/10.1103/PhysRevE.93.062134

Physics Subject Headings (PhySH)

Entropy production Nonequilibrium & irreversible thermodynamics Open quantum systems Quantum master equation Quantum thermodynamics

Statistical Physics & Thermodynamics

Authors & Affiliations

Kay Brandner¹ and Udo Seifert²

¹Department of Applied Physics, Aalto University, 00076 Aalto, Finland
²II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany

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Issue

Vol. 93, Iss. 6 — June 2016

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Images

Figure 1
Illustration of a periodically driven open quantum system. The energy of the system, symbolically shown as an atom confined in a chamber, is modulated by three external controllers, each of which is represented by a reciprocating piston. Simultaneously, heat is exchanged with one cold and one hot reservoir.
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Figure 2
Two snapshots of the operation cycle of a two-level quantum heat engine. A single particle is confined in a double well potential and coupled to a thermal reservoir, whose temperature oscillates between $T^{h}$ (left panel) and $T^{c} < T^{h}$ (right panel). In a coarse-grained picture, this setup can be described as a two-level system, where the particle is localized either in the left or in the right well. Work is extracted from the system by varying a certain external control parameter, which affects both the energetic difference between the two minima of the potential and the height of the barrier separating them. This control operation, which corresponds to the nonclassical driving protocol (77), inevitably allows the particle to tunnel between the two wells. Consequently, it will typically be found in a coherent superposition of the unperturbed energy-eigenstates during the thermodynamic cycle.
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Figure 3
Maximum efficiency of a two-level quantum heat engine. The solid lines show the explicit result (86) in units of the Carnot efficiency $η_{C}$ as function of the coherence parameter $θ$ for different values of the damping parameter $α$ . The dashed lines indicate the corresponding bound (68) evaluated with the protocols (82) and the optimal phase shift (87). The remaining parameters have been chosen as $κ = 1 / 2$ and $ν = 10$ . For clarity, the legend in the lower left corner follows the order of the plotted curves from top to bottom.
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Figure 4
Dependence of the maximum power of a two-level quantum engine on the coherence parameter $θ$ for $κ = 1 / 2, ν = 10$ , and three different values of $α$ . The solid lines correspond to the optimized output (90) in units of $P_{0} \equiv T^{c} F_{q}^{2} {(ξ^{cl})}^{2} α / (12 k_{B} T)$ [89]. The dashed lines show the maximum power as a fraction of its upper bound (92). In the limit $θ \to π / 2$ , both quantities $P_{\max}$ and the bound ${\hat{P}}_{\max}$ vanish, while their ratio approaches a finite value. The legend in the lower left corner has been sorted according to the order of the plotted curves from top to bottom. This correspondence applies to dashed and solid lines, respectively.
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Figure 5
Flow chart visualizing the interdependence between properties of the Lindblad generator (left column), relations between thermodynamic quantities (central column), and bounds on the performance figures of quantum heat engines (right column). Solid arrows denote unrestricted implications, while dashed arrows require the additional condition attached to them. In the last column, we used the abbreviations $\bar{η} \equiv η / η_{C}$ and ${\bar{P}}_{\max} \equiv P_{\max} / P_{0}$ , where $P_{0} = T^{c} F_{q}^{2} L_{q q} / 4$ for the dashed arrow in the bottom line and otherwise $P_{0} = T^{c} F_{q}^{2} L_{q q}^{ins} / 4$ . An engine is considered to be time-reversal (TR) symmetric if the corresponding driving protocols fulfill the condition (64).
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