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arXiv:2109.12648 · PRX Quantum

Geometric Optimization of Nonequilibrium Adiabatic Thermal Machines and Implementation in a Qubit System

Pablo Terrén Alonso, Paolo Abiuso, Martí Perarnau-Llobet, Liliana Arrachea

Published Feb 16, 2022• 36 citations
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Abstract

We adopt a geometric approach to describe the performance of adiabatic quantum machines, operating under slow time-dependent driving and in contact with two reservoirs with a temperature bias during all the cycle.We show that the problem of optimizing the power generation of a heat engine and the efficiency of both the heat engine and refrigerator operational modes is reduced to an isoperimetric problem with nontrivial underlying metrics and curvature.This corresponds to the maximization of the ratio between the area enclosed by a closed curve and its corresponding length.We illustrate this procedure in a qubit coupled to two reservoirs operating as a thermal machine by means of an adiabatic protocol.

DOI
10.1103/prxquantum.3.010326

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We adopt a geometric approach to describe the performance of adiabatic quantum machines, operating under slow time-dependent driving and in contact with two reservoirs with a temperature bias during all the cycle.We show that the problem of optimizing the power generation of a heat engine and the efficiency of both the heat engine and refrigerator operational modes is reduced to an isoperimetric problem with nontrivial underlying metrics and curvature.This corresponds to the maximization of the ratio between the area enclosed by a closed curve and its corresponding length.We illustrate this procedure in a qubit coupled to two reservoirs operating as a thermal machine by means of an adiabatic protocol.
A quantum thermal machine can be controlled by time-periodic perturbations that define a cycle, as in conventional macroscopic thermal machines. This paper presents a general and unified approach to the study of quantum thermal machines, including both heat engines and refrigerators, which operate under periodic adiabatic driving and are in contact with thermal reservoirs kept at different temperatures. Importantly, many observables characterizing the operation and the performance of these machines are geometric in nature.
We present a detailed analysis of slowly driven quantum thermal machines based on interacting qubits within the framework of the Lindblad master equation. By implementing a systematic expansion in the driving rate, we derive explicit expressions for the rate of work of the driving forces, the heat currents exchanged with the reservoirs, and the entropy production up to second order, ensuring full thermodynamic consistency in the linear-response regime. The formalism naturally separates geometric and dissipative contributions, identified by a Berry curvature and a metric in parameter space, respectively. Analytical results show that the geometric heat pumped per cycle is bounded by kBTNqln⁡2k_B T N_q \ln 2kB​TNq​ln2 for NqN_qNq​ non-interacting qubits, in direct analogy with the Landauer limit for entropy change. This bound can be surpassed when qubit interactions and asymmetric couplings to the baths are introduced. Numerical results for the interacting two-qubit system reveal a non-trivial role of the interaction between qubits and the coupling between the qubits and the baths in the behavior of the dissipated power. The approach provides a general platform for studying dissipation, pumping, and performance optimization in driven quantum devices operating as heat engines.
The concept of thermal machines has evolved from the canonical steam engine to the recently proposed nanoscopic quantum systems as working fluids. The latter obey quantum open system dynamics and frequently operate in nonequilibrium conditions. However, the role of this dynamics in the overall performance of quantum heat engines remains an open problem. Here we analyze and optimize the efficiency and power output of two-stage quantum heat engines fueled by nonequilibrium steady states. In a charging first stage, the quantum working fluid consisting of a qutrit or two coupled qubits is connected to two reservoirs at different temperatures, which establish a heat current that stores ergotropy in the system; the second stage comprises a coherent driving force that extracts work from the machine in finite a amount of time; finally, the external drive is switched off and the machine enters a new cycle.
We study the optimization of the performance of arbitrary periodically driven thermal machines. Within the assumption of fast modulation of the driving parameters, we derive the optimal cycle that universally maximizes the extracted power of heat engines, the cooling power of refrigerators, and in general any linear combination of the heat currents. We denote this optimal solution as ``generalized Otto cycle'' since it shares the basic structure with the standard Otto cycle, but it is characterized by a greater number of fast strokes. We bound this number in terms of the dimension of the Hilbert space of the system used as working fluid. The generality of these results allows for a widespread range of applications, such as reducing the computational complexity for numerical approaches, or obtaining the explicit form of the optimal protocols when the system-baths interactions are characterized by a single thermalization scale. In this case, we compare the thermodynamic performance of a collection of optimally driven noninteracting and interacting qubits. Remarkably, for refrigerators the noninteracting qubits perform almost as well as the interacting ones, while in the heat engine case there is a many-body advantage both in the maximum power, and in the efficiency at maximum power. Additionally, we illustrate our general results studying the paradigmatic model of a qutrit-based heat engine. Our results strictly hold in the semiclassical case in which no coherence is generated by the driving, and finally we discuss the noncommuting case.
Geometric heat pumping is the transfer of energy between reservoirs at equal temperature where the heat exchanged in an adiabatic cycle depends only on the driving contour in parameter space. We implement this mechanism on a superconducting quantum processor using a collision model, where a driven qubit interacts repeatedly with two ancilla qubits that are prepared in thermal states and emulate the reservoirs. Energy measurements before and after each interaction provide a calorimetric record of the heat exchanged with each reservoir. By reversing the driving contour, we extract a heat contribution associated with pumping. Its dependence on the period and on the contour follows exact simulations of the collision model, without adjustable parameters, and approaches in the slow-driving regime the geometric limit of the corresponding Lindblad dynamics.
The purpose of this paper is to present a study of finite-time thermodynamics applied to evaluate the ecological performance of a quantum heat engine which operates between two thermal reservoirs using the working substance of spin-1/2 systems. The quantum heat engine cycle is composed of two isothermal processes, an adiabatic process and an isomagnetic field process. A sequence of time evolution was determined from the quantum angular momentum rate based on the semigroup approach and the quantum master equation. The individual time duration is added to account for the total cycle time. The objective ecological function representing a compromise between power output and irreversibility is maximized with respect to cycle temperature ratio. Effects of thermal reservoir temperature ratio and magnetic field ratio on the ecological function have been discussed. A comparison of quantum heat engine performance under maximum ecological function and maximum power conditions is also presented.