# Quantum thermal machines: a simple scheme with realistic bath modelling

This is a Perspective on "Periodically refreshed quantum thermal machines" by Archak Purkayastha, Giacomo Guarnieri, Steve Campbell, Javier Prior, and John Goold, published in Quantum 6, 801 (2022).

By Felipe Barra (Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile).

Thermal machines interact with two or more heat reservoirs to perform work on a third system or use work to cool down the hot reservoir. In the first case, the machine works as an engine, and in the second, as a refrigerator. Their efficiency or their coefficient of performance, respectively, are always smaller than those of an ideal reversible machine. However, as is well known, the process must be quasistatic to operate reversibly; thus such an engine delivers vanishing power.

In [1], A. Purkayastha et al. study an irreversible process in which an agent couples a system to a hot bath and a cold bath for a time $\tau$. The agent performs or extracts work in this process by switching on and off the system-bath interaction. This way, a time-periodic steady state is reached, and the machine performs non-equilibrium cycles. Similar situations have been studied before [2,3,4], but without realistic modeling of the heat baths. A macroscopic bath is usually modeled as a system with quadratic hamiltonian, and its most important property is its spectral function (determined by the coupling to the system) [5]. A mapping from a bath to a tight-binding chain [6] allows the authors of [1] to numerically investigate the impact of baths with different spectral functions on the performance of a machine. The coupling time determines the chain size affected by the interaction with the system. The rest of it rethermalizes the chain after it is disconnected from the system. Note that, for the system to reach a steady state, the bath must be in the thermal state at the beginning of each interaction. Suppose $\tau$ is smaller than the relaxation time of the baths, then at least two copies of each bath are needed. When the machine interacts with a copy of each, the other copies have time to reach equilibrium and are ready to drive the device for the next cycle. Therefore diminishing the interaction time increases the complexity of the entire setup.

Under this scheme of periodically refreshed thermal baths, the thermodynamic properties can be computed unambiguously, and the main questions addressed are, when does the system work as a heat engine or refrigerator? What are the machine’s efficiency and dissipation rate, and how do they depend on the critical parameters of the whole system, like the bath spectral functions and the coupling time $\tau$? The answers are captivating, and we list here the main observations.

Varying the parameters of the system and the bath, and the temperatures, one can find the machine working as an engine or refrigerator. Given the working regime, the main thermodynamic quantities are plotted as a function of $\tau$.

In the heat-engine regime, one observes a $\tau$ value at which the power is maximized simultaneously with the dissipation rate. At this value, the efficiency is also at its maximum value. Moreover, as the spectral function becomes narrower, the power and efficiency peak increases as the dissipation peak decrease.

In the refrigerator regime, the situation is similar. The heat extracted from the hot bath, the consumed power, the coefficient of performance, and the dissipation rate all peak basically at the same $\tau$. As the spectral function becomes narrower, the extracted heat and the efficiency increase while the consumed power and the dissipation decrease.

The simultaneous peak of efficiency and power (or extracted heat) shows that in this irreversible machine, there is no trade-off between efficiency and power, and the coincident peak in the dissipation rate indicates that the device is more efficient in the most dissipative regime. However, the efficiency is significantly lower than the Carnot efficiency of the reversible engine.

In the optimal regime of narrow spectral function, and with $\tau$ where the thermodynamic quantities peak, only one site of the a chain is affected, and the engine works as those considered in [2,3,4]. The limit of large $\tau$ reproduces that of a time-continuous autonomous machine [7].

The study of [1] contributes significantly to our understanding of quantum engines or refrigerators in deep non-equilibrium regimes and with realistic modeling of the baths and interpolates between two situations well explored in the literature.

### ► References

[1] A. Purkayastha, G. Guarnieri, S. Campbell, J. Prior, and J. Goold, Periodically refreshed quantum thermal machines, Quantum 6, 801 (2022).
https:/​/​doi.org/​10.22331/​q-2022-09-08-801

[2] F. Barra, The thermodynamic cost of driving quantum systems by their boundaries, Sci. Rep. 5, 14873 (2015).
https:/​/​doi.org/​10.1038/​srep14873

[3] G. De Chiara, G. Landi, A. Hewgill, B. Reid, A. Ferraro, A. J. Roncaglia, and M. Antezza, Reconciliation of quantum local master equations with thermodynamics, New J. Phys. 20, 113024 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aaecee

[4] G. Guarnieri, D. Morrone, B. Çakmak, F. Plastina, and S. Campbell, Non-equilibrium steady-states of memoryless quantum collision models, Phys. Lett. A 384, 126576 (2020).
https:/​/​doi.org/​10.1016/​j.physleta.2020.126576

[5] U. Weiss, Quantum Dissipative Systems, 4th ed. (World Scientific, Singapore, 2012).

[6] A. Nazir and G. Schaller, The Reaction Coordinate Mapping in Quantum Thermodynamics, in Thermodynamics in the Quantum Regime, edited by Felix Binder, Luis A. Correa, Christian Gogolin, Janet Anders, and Gerardo Adesso (Springer) Chap. chapter 23, pp. 551–577.
https:/​/​doi.org/​10.1007/​978-3-319-99046-0_23

[7] A. Purkayastha, G. Guarnieri, S. Campbell, J. Prior, and J. Goold, Periodically refreshed baths to simulate open quantum many-body dynamics, Phys. Rev. B 104, 045417 (2021).
https:/​/​doi.org/​10.1103/​PhysRevB.104.045417

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