Abstract
Sadi Carnot's theorem regarding the maximum efficiency of heat engines is considered to be of fundamental importance in thermodynamics. This theorem famously states that the maximum efficiency depends only on the temperature of the heat baths used by the engine, but not on the specific structure of baths. Here, we show that when the heat baths are finite in size, and when the engine operates in the quantum nanoregime, a revision to this statement is required. We show that one may still achieve the Carnot efficiency, when certain conditions on the bath structure are satisfied; however if that is not the case, then the maximum achievable efficiency can reduce to a value which is strictly less than Carnot. We derive the maximum efficiency for the case when one of the baths is composed of qubits. Furthermore, we show that the maximum efficiency is determined by either the standard second law of thermodynamics, analogously to the macroscopic case, or by the non increase of the max relative entropy, which is a quantity previously associated with the single shot regime in many quantum protocols. This relative entropic quantity emerges as a consequence of additional constraints, called generalized free energies, that govern thermodynamical transitions in the nanoregime. Our findings imply that in order to maximize efficiency, further considerations in choosing bath Hamiltonians should be made, when explicitly constructing quantum heat engines in the future. This understanding of thermodynamics has implications for nanoscale engineering aiming to construct small thermal machines.
Highlights
Nicolas Léonard Sadi Carnot is often described as the “father of thermodynamics”
The question is : given an initial cold bath Hamiltonian HCold, what is the maximum attainable efficiency considering all possible final states ρ1Cold? In both cases of perfect and near perfect work, we find that the efficiency is only maximized whenever ρ1Cold is (A) a thermal state, and (B), when it is a thermal state, can only achieve the Carnot efficiency when the inverse temperature βf is arbitrarily close to βCold
We find all cold baths can be used in heat engines; and — remarkably — that even when one of the heat baths is as small as a single qubit, as long as certain conditions of the bath Hamiltonian are met, Carnot efficiency can still be achieved in the quasi-static limit
Summary
Nicolas Léonard Sadi Carnot is often described as the “father of thermodynamics”. In his only publication in 1824 [1], Carnot gave the first successful theory in analysing the maximum efficiency of heat engines. Unlike the large scale heat engines that inspired thermodynamics, we are able to build nanoscale quantum machines consisting of a mere handful of particles, and this has prompted many efforts to understand quantum thermodynamics [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] Given such nanoscale devices, one of the main issues addressed in quantum thermodynamics is the single-shot analysis of thermodynamical state transitions or work extraction.
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