Stroboscopic two-stroke quantum heat engines

Otavio A. D. Molitor and Gabriel T. Landi

Phys. Rev. A 102, 042217 – Published 16 October 2020

Abstract

The formulation of models describing quantum versions of heat engines plays an important role in the quest toward establishing the laws of thermodynamics in the quantum regime. Of particular importance is the description of stroke-based engines, which can operate at finite time. In this paper we put forth a framework for describing stroboscopic, two-stroke engines, in generic quantum chains. The framework is a generalization of the so-called SWAP engine and is based on a collisional model, which alternates between pure heat and pure work strokes. The transient evolution towards a limit cycle is also fully accounted for. Moreover, we show that once the limit cycle has been reached, the energy of the internal sites of the chain no longer changes, with the heat currents being associated exclusively to the boundary sites. Using a combination of analytical and numerical methods, we show that this type of engine offers multiple ways of optimizing the output power, without affecting the efficiency. Finally, we also show that there exists an entire class of models, characterized by a specific type of interchain interaction, which always operate at Otto efficiency, irrespective of the operating conditions of the reservoirs.

Received 18 August 2020
Accepted 21 September 2020

DOI:https://doi.org/10.1103/PhysRevA.102.042217

Physics Subject Headings (PhySH)

Open quantum systems Quantum thermodynamics

Quantum Information, Science & TechnologyStatistical Physics & Thermodynamics

Authors & Affiliations

Otavio A. D. Molitor ^* and Gabriel T. Landi ^†

Instituto de Física da Universidade de São Paulo, 05314-970 São Paulo, Brazil

^*otavio.molitor@usp.br
^†gtlandi@if.usp.br

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Vol. 102, Iss. 4 — October 2020

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Images

Figure 1
The quantum heat engine (QHE) model is composed of two processes: (a) heat stroke ( $q$ ) and (b) work stroke ( $w$ ). During the heat stroke the internal interactions of the quantum chain are turned off while the boundary subsystems 1 and $N$ interact with a cold bath at temperature $T_{C}$ and a hot bath at temperature $T_{H}$ , respectively. The work stroke, in its turn, consists of a unitary evolution of the chain as a whole, by means of the interaction $V_{S}$ , and with the baths disconnected from the chain. The strokes are operated sequentially in a cyclic way, as shown by the arrows around (a) and (b). The result is a stroboscopic evolution of the state of the chain, which results in a discrete-time evolution of all thermodynamic quantities, as illustrated in (c).
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Figure 2
Stroboscopic evolution of the working fluid from its initial state $ρ_{S}^{0}$ until it reaches a limit cycle of two nonequilibrium steady states $ρ_{S}^{*}$ and ${\tilde{ρ}}_{S}^{*}$ .
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Figure 3
Values of (a) $Z_{1}^{n}$ (yellow circle), ${\tilde{Z}}_{1}^{n}$ (purple square), (b) $Z_{2}^{n}$ (yellow circle), ${\tilde{Z}}_{2}^{n}$ (purple square) (c) $S^{n}$ , (yellow circle), ${\tilde{S}}^{n}$ (purple square), and (d) $A^{n}$ (yellow circle), ${\tilde{A}}^{n}$ (purple square), as functions of the number of cycles $n$ . The dashed gray lines are the stationary values of the $c$ -number variables. These plots were obtained for $λ = 0.2, p = 0.99, T_{C} = 0.4, T_{H} = 0.8, ω_{1} = 0.75, ω_{2} = 1.0$ , and $g = 0.3$ . The interaction times were fixed at $τ_{q} = τ_{w} = 1$ . The initial state is assumed to be both qubits in the ground state.
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Figure 4
Plots of (a) $Q_{C}^{n}$ (blue circles), $Q_{H}^{n}$ (red diamonds), and (b) $W^{n}$ (green circles) with respect to the number of cycles $n$ . The parameters are the same as in Fig. 3. The numbers in the plot indicate the steady-state values for this specific configuration.
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Figure 5
Optimization of the output power. The figure corresponds to a plot of Eq. (42) (scaled by $10^{3}$ ) as a function of the interaction times $τ_{q}$ and $τ_{w}$ , of the heat and work strokes. The parameters are similar to those of Fig. 3; namely, $T_{C} = 0.4, T_{H} = 0.8, ω_{1} = 0.75, ω_{2} = 1 ., g = g_{C H} = 0.3$ .
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Figure 6
Operation of a $N$ -site two-stroke engine. The curves correspond to simulations for the XX [(a)–(c)] and XXZ chain [(d)–(f)] for different chain sizes $N$ . (a), (d) presents the relaxation of the heats towards the limit cycle: $Q_{C}^{n}$ blue circles ( $N = 3$ ), blue diamonds ( $N = 4$ ), and blue triangles ( $N = 5$ ); $Q_{H}^{n}$ : red circles ( $N = 3$ ), red diamonds ( $N = 4$ ), and red triangles ( $N = 5$ ) and (b), (e) shows the relaxation of the work $W^{n}$ : dark green circles ( $N = 3$ ), green diamonds ( $N = 4$ ), and light green triangles ( $N = 5$ ). Finally, (c), (f) presents he limit-cycle output power $P^{*}$ as a function of $λ$ (which is proportional to $τ_{q}$ ): red circles ( $N = 3$ ), blue diamonds ( $N = 4$ ), and green triangles ( $N = 5$ ). The on-site potentials $ω_{i}$ were chosen to interpolate linearly between $ω_{1} = 1.5$ and $ω_{N} = 2.0$ . Other parameters were $J_{x} = J_{y} = 0.8, T_{C} = 0.2, T_{H} = 0.8$ , and $τ_{w} = 0.25$ . For (a)–(c) we fixed $J_{z} = 0$ and for (d)–(f) $J_{z} = 0.7$ .
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