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arXiv:2208.04917 · Physical Review A

Algebraic approach to a nonadiabatic coupled Otto cycle

Alan Duriez, D. Martínez-Tibaduiza, Antonio Zelaquett Khoury

Published Dec 19, 2022• 4 citations
arXivHTMLPDF

Abstract

Algebraic methods for solving time-dependent Hamiltonians are used to investigate the performance of quantum thermal machines. We investigate the thermodynamic properties of an engine formed by two coupled qubits, performing an Otto cycle. The thermal interaction occurs with two baths at different temperatures, while work is associated with the interaction with an arbitrary time-dependent magnetic field that varies in intensity and direction. For the coupling, we consider the 1D isotropic Heisenberg model, which allows us to describe the system by means of the irreducible representation of the $\mathrm{su}(2)$ Lie algebra within the triplet subspace. We also inspect different settings of the temperatures and frequencies of the cycle, and investigate the corresponding operation regimes. We describe how these regimes can change with the speed of the protocol. Finally, we compare the weak and strong coupling regimes, and conclude that for the weak case the efficiency is optimal and can surpass the value of the classical Otto cycle. For low temperatures of the heat baths, we show that losses due to quantum friction are reduced in the strong coupling limit.

DOI
10.1103/physreva.106.062212

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Algebraic methods for solving time-dependent Hamiltonians are used to investigate the performance of quantum thermal machines. We investigate the thermodynamic properties of an engine formed by two coupled qubits, performing an Otto cycle. The thermal interaction occurs with two baths at different temperatures, while work is associated with the interaction with an arbitrary time-dependent magnetic field that varies in intensity and direction. For the coupling, we consider the 1D isotropic Heisenberg model, which allows us to describe the system by means of the irreducible representation of the su(2)\mathrm{su}(2)su(2) Lie algebra within the triplet subspace. We also inspect different settings of the temperatures and frequencies of the cycle, and investigate the corresponding operation regimes. We describe how these regimes can change with the speed of the protocol. Finally, we compare the weak and strong coupling regimes, and conclude that for the weak case the efficiency is optimal and can surpass the value of the classical Otto cycle. For low temperatures of the heat baths, we show that losses due to quantum friction are reduced in the strong coupling limit.
We discuss a model of a closed quantum evolution of two qubits where the joint Hamiltonian is so chosen such that one of the qubits acts as a bath and thermalizes the other qubit which is acting as the system. The corresponding exact master equation for the system is derived. Interestingly, for a specific choice of parameters the master equation takes the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form, with constant coefficients representing pumping and damping of a single qubit system. Based on this model we construct an Otto cycle connected to a single qubit bath and study its thermodynamic properties. Our analysis goes beyond the conventional weak coupling scenario and illustrates the effects of finite baths, including non-Markovianity. We find closed form expressions for efficiency (coefficient of performance), power (cooling power) for the heat engine regime (refrigerator regime), and for different modifications of the joint Hamiltonian.
Quantum thermal machines make use of non-classical thermodynamic resources, one of which include interactions between elements of the quantum working medium. In this paper, we examine the performance of a quasi-static quantum Otto engine based on two spins of arbitrary magnitudes subject to an external magnetic field and coupled via an isotropic Heisenberg exchange interaction. It has been shown earlier that the said interaction provides an enhancement of cycle efficiency, with an upper bound that is tighter than the Carnot efficiency. However, the necessary conditions governing engine performance and the relevant upper bound for efficiency are unknown for the general case of arbitrary spin magnitudes. By analyzing extreme case scenarios, we formulate heuristics to infer the necessary conditions for an engine with uncoupled as well as coupled spin model. These conditions lead us to a connection between performance of quantum heat engines and the notion of majorization. Furthermore, the study of complete Otto cycles inherent in the average cycle also yields interesting insights into the average performance.
We present a detailed study of quantum thermal machines employing quantum systems as working substances. In particular, we study two different types of two-stroke cycles where two collections of identical quantum systems with evenly spaced energy levels are initially prepared at thermal equilibrium by putting them in contact with a cold and a hot thermal bath, respectively. The two cycles differ in the absence or the presence of a mediator system, while, in both cases, non-resonant exchange Hamiltonians are exploited as particle interactions. We show that the efficiencies of these machines depend only on the energy gaps of the systems composing the collections and are equal to the efficiency of equivalent Otto cycles. Focusing on the cases of qubits or harmonic oscillators for both models, we maximize the engine power and analyze, in the model without the mediator, the role of the waiting time between subsequent interactions. It turns out that the case with the mediator can bring performance advantages when the interaction times are comparable with the waiting time of the correspondent cycle without the mediator. We find that in both cycles, the power peaks of qubit systems can surpass the Curzon-Ahlborn efficiency. Finally, we compare our cycle without the mediator with previous schemes of the quantum Otto cycle showing that high coupling is not required to achieve the same maximum power.
We study finite-time quantum Otto cycles whose working medium is a harmonic oscillator undergoing a quantum Brownian motion described by the Caldeira-Leggett model when the oscillator is in contact with heat baths in isochoric processes. The time evolution of the Otto cycle is studied by analytically solving the exact Heisenberg-Langevin equations for the system variables and the interaction energy between the system and the bath. This enables us to investigate non-Markovian strong-coupling effects on the quantum Otto cycle. We obtain cyclic steady states and study the thermodynamic properties of the Otto cycle for various values of the parameters describing the heat baths and the coupling between the system and the bath. We compare our results with those obtained in the Markovian limit, where the time evolution is described by the Lindblad equation. We find that the change in the interaction energy during the isochoric process contributes to both work and heat and plays a crucial role in determining the thermodynamic behavior of the cycle. This is in contrast to the behavior of the state observables of the working medium, which show relatively small change compared to the Markovian counterparts. We find that non-Markovian effects only become significant when we consider thermodynamic fluxes such as work and heat with the system-bath interaction fully taken into account. We also compare our results with the power-efficiency trade-off relation recently proposed for the Markovian quantum Otto engine.
Abstract This paper proposes a theoretical model for a heat engine operating on the Otto cycle, employing a two-qubit Heisenberg chain as its working substance. This system is subjected to an external, time-dependent homogeneous magnetic field and a symmetric cross-spin interaction. The interaction is configured to occur along two distinct orientations, contingent upon the direction of Zeeman energy and symmetric interaction: either exclusively along the X -direction or exclusively along the Z -direction. Furthermore, the thermal reservoirs integral to the Otto engine are considered to engage with the working substance via two discrete modes: a localized interaction or a collective interaction. The two qubits exhibit weak coupling to a non-equilibrium thermal environment during the isochoric heating and cooling strokes, thereby validating the application of a Markovian model as a suitable approximation for describing the time evolution of the working substance. This study investigates the influence of various system parameters on the thermodynamic quantities and the resultant power output. A principal finding is that refrigeration functionality is the predominant operational characteristic when the system engages in collective interaction with the reservoirs. Conversely, localized interactions facilitate dual operational modes, with the sign of the Zeeman energy determining the function as either a refrigerator or a heat engine. Significantly, positive power output is achieved in the engine regime, and we find that the choice of interaction axis acts as a powerful amplification mechanism for the thermodynamic output, while the system-environment coupling scheme proves to be the most decisive factor in determining the machine’s fundamental purpose.
We study a quantum Otto cycle that uses a 2-qubit working substance whose Hamiltonian does not commute with itself at different times during unitary strokes. We investigate how the cycle responds to the loss of quantum adiabaticity when these strokes are operated with a finite duration. We find that qualitative features such as the possibility of counter-rotating cycles operating as heat engines, or a cycle efficiency that can increase with a decrease in the temperature difference between the baths, are resilient even to highly nonadiabatic strokes. However, cycle efficiency rapidly decreases, although it can still remain above the standard Otto value for small degrees of quantum nonadiabaticity.