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arXiv:2403.09154

Energy-gap modulation and majorization in three-level quantum Otto engine

Sachin Sonkar, Ramandeep S. Johal

Published Mar 14, 2024• 0 citations
arXivHTMLPDF

Abstract

A three-level quantum system having two energy gaps presents a nontrivial working medium for a quantum heat engine. Our focus lies in understanding the constraints on the ability to modulate these gaps relative to the changes in probability distributions at the two given heat reservoirs. It is seen that an Otto engine in the quasistatic limit is feasible if at least one energy gap shrinks during the first quantum adiabatic stage. We analyze operating conditions under different variations of the gaps, revealing that a definite majorization relation between the hot and cold distributions serves as a sufficient criterion for the engine when both gaps are shrinking. Further, Otto efficiency is enhanced in case majorization holds. On the other hand, majorization becomes a necessary condition when only one of the gaps is shrinking. In the special case where one gap remains fixed, majorization is both necessary and sufficient for engine operation. For an $n$-level system, we note that a well defined change in energy gaps aligns with the majorization relation, thus characterizing the operation of the engine.

DOI
10.48550/arxiv.2403.09154

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A three-level quantum system having two energy gaps presents a nontrivial working medium for a quantum heat engine. Our focus lies in understanding the constraints on the ability to modulate these gaps relative to the changes in probability distributions at the two given heat reservoirs. It is seen that an Otto engine in the quasistatic limit is feasible if at least one energy gap shrinks during the first quantum adiabatic stage. We analyze operating conditions under different variations of the gaps, revealing that a definite majorization relation between the hot and cold distributions serves as a sufficient criterion for the engine when both gaps are shrinking. Further, Otto efficiency is enhanced in case majorization holds. On the other hand, majorization becomes a necessary condition when only one of the gaps is shrinking. In the special case where one gap remains fixed, majorization is both necessary and sufficient for engine operation. For an nnn-level system, we note that a well defined change in energy gaps aligns with the majorization relation, thus characterizing the operation of the engine.
The performance of a quantum Otto engine is analyzed with regard to the constraints on the modulation of energy gaps relative to the changes in probability distributions at the two given heat reservoirs. We performed a detailed analysis with a generic three-level system (3LS), which serves as a non-trivial working medium with two energy gaps. A three-level Otto engine becomes feasible if at least one energy gap shrinks during the first quantum adiabatic stage. The operating regimes are derived for each allowed energy gap modulation, and majorization is observed to play a crucial role in determining the engine operation. This results in an enhanced Otto efficiency when the probability distributions fulfill the majorization condition. Finally, we show that our formalism applies to a swap engine based on a working medium composed of two 3LSs.
The concept of majorization is explored as a tool to characterize the performance of a quantum Otto engine in the quasistatic regime. For a working substance in the form of a single spin of arbitrary magnitude, majorization yields a necessary and sufficient condition for the operation of the Otto engine, provided the canonical distribution of the working medium at the hot reservoir is majorized by its canonical distribution at the cold reservoir. We extend our analysis for two scenarios: a three-level atom and a bipartite system consisting of spin 1/2 interacting with an arbitrary spin via an isotropic Heisenberg exchange interaction. For both these cases, we find that while a majorization condition implies positive work extraction, it only yields sufficient conditions for the engine operation. Finally, we study the local thermodynamics of spins in the case of the bipartite system and infer an upper bound on the quantum Otto efficiency using the majorization relation.
Abstract The efficiency of a quantum heat engine is maximum when the unitary strokes of the quantum Otto cycle are adiabatic. On the other hand, this may not be always possible due to small energy gaps in the system, especially at the critical point (CP) where the gap between the ground state and the first excited state vanishes and the system gets excited. With the aim to regain this lost adiabaticity, we modify one of the unitary strokes of the Otto cycle by allowing the system to first evolve with a time dependent Hamiltonian as in the case of a usual Otto cycle, followed by an additional evolution with a different time independent Hamiltonian so that the system reaches a less excited state. This will help in increasing the magnitude of the heat absorbed from the hot bath so that the work output and efficiency of the engine can be increased. We demonstrate this method using an integrable model and a non-integrable model as the working medium and discuss the generality and limitations of this method. In the case of a two spin system, the optimal value for the time till which the system needs to be freely evolved is calculated analytically in the adiabatic limit. The results show that implementing this modified unitary stroke significantly improves the work output and efficiency of the engine, especially when it crosses the CP.
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 construct a special four-level entangled quantum Otto heat engine based on the two-qubit Heisenberg XX model, in which we assume that all the energy gaps are changed in the same ratio in the two quantum adiabatic processes. Hence during the whole cycle, the relative coupling constant κ = J / B is fixed, where J and B are the coupling constant and the external magnetic field, respectively. The dependence of the basic thermodynamical quantities on the two entanglements at the end of two quantum isochoric processes with different relative coupling constants κ is studied. Our results show that in the weak coupling region, i.e. κ < 1, the heat engine can be operated in both areas where c 1 < c 2 and c 1 > c 2 , whereas when κ ⩾ 1, it only operates under the condition c 1 < c 2 . Here c 1 and c 2 are entanglements of the working substance when it comes into contact with hot and cold baths, respectively. Moreover, we find that the maximal work output for fixed κ increases with the relative coupling constant.
Abstract Recent theoretical and experimental studies in quantum heat engines show that, in the quasi-static regime, it is possible to have higher efficiency than the limit imposed by Carnot, provided that engineered reservoirs are used. The quasi-static regime, however, is a strong limitation to the operation of heat engines, since an infinitely long time is required to complete a cycle. In this paper we propose a two-level model as the working substance to perform a quantum Otto heat engine surrounded by a cold thermal reservoir and a squeezed hot thermal reservoir. Taking advantage of this model we show a striking achievement, that is to attain unity efficiency even at non-null power.