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arXiv:1010.6029

The smallest possible heat engines

Noah Linden, Sandu Popescu, Paul Skrzypczyk

Published Oct 28, 2010• 8 citations
arXivHTMLPDF

Abstract

We construct the smallest possible self contained heat engines; one composed of only two qubits, the other of only a single qutrit. The engines are self-contained as they do not require external sources of work and/or control. They are able to produce work which is used to continuously lift a weight. Despite the dimension of the engine being small, it is still able to operate at the Carnot efficiency.

DOI
10.48550/arxiv.1010.6029

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We construct the smallest possible self contained heat engines; one composed of only two qubits, the other of only a single qutrit. The engines are self-contained as they do not require external sources of work and/or control. They are able to produce work which is used to continuously lift a weight. Despite the dimension of the engine being small, it is still able to operate at the Carnot efficiency.
We investigate the fundamental dimensional limits to thermodynamic machines. In particular, we show that it is possible to construct self-contained refrigerators (i.e., not requiring external sources of work) consisting of only a small number of qubits and/or qutrits. We present three different models, consisting of two qubits, a qubit and a qutrit with nearest-neighbor interactions, and a single qutrit, respectively. We then investigate the fundamental limits to their performance; in particular, we show that it is possible to cool towards absolute zero.
Quantum heat engines employ as working agents multilevel systems instead of classical gases. We show that under some conditions quantum heat engines are equivalent to a series of reservoirs at different altitudes containing balls of various weights. A cycle consists of picking up at random a ball from one reservoir and carrying it to the next, thereby performing or absorbing some work. In particular, quantum heat engines, employing two-level atoms as working agents, are modeled by reservoirs containing balls of weight 0 or 1. The mechanical model helps us prove that the maximum efficiency of quantum heat engines is the Carnot efficiency. Heat pumps and negative temperatures are considered.
Abstract A heat engine operating in the one-shot finite-size regime, where systems composed of a small number of quantum particles interact with hot and cold baths and are restricted to one-shot measurements, delivers fluctuating work. Further, engines with lesser fluctuation produce a lesser amount of deterministic work. Hence, the heat-to-work conversion efficiency stays well below the Carnot efficiency. Here we overcome this limitation and attain Carnot efficiency in the one-shot finite-size regime, where the engines allow the working systems to simultaneously interact with two baths via the semi-local thermal operations and reversibly operate in a one-step cycle. These engines are superior to the ones considered earlier in work extraction efficiency, and, even, are capable of converting heat into work by exclusively utilizing inter-system correlations. We formulate a resource theory for quantum heat engines to prove the results.
Heat engines constitute the major building blocks of modern technologies. However, conventional heat engines with higher power yield lesser efficiency and vice versa and respect various power-efficiency trade-off relations. This is also assumed to be true for the engines operating in the quantum regime. Here we show that these relations are not fundamental. We introduce quantum heat engines that deliver maximum power with Carnot efficiency in the one-shot finite-size regime. These engines are composed of working systems with a finite number of quantum particles and are restricted to one-shot measurements. The engines operate in a one-step cycle by letting the working system simultaneously interact with hot and cold baths via semilocal thermal operations. By allowing quantum entanglement between its constituents and, thereby, a coherent transfer of heat from hot to cold baths, the engine implements the fastest possible reversible state transformation in each cycle, resulting in maximum power and Carnot efficiency. Finally, we propose a physically realizable engine using quantum optical systems.
Brunner et al. [Phys. Rev. E 85 (2012) 05111] have claimed that, "essentially only the smallest machines can approach Carnot efficiency". We have verified this claim by raising self-contained four-qubit quantum refrigerator, and we have shown that according to concepts of virtual qubit, it can reach the maximum efficiency in other words Carnot efficiency. But its efficiency, such as self-contained three-qubit quantum refrigerator is not universal. We also investigated a special case of self-contained four-qubit quantum refrigerator, in other words self-contained four-qubit quantum refrigerator with two hot baths in the same temperature. We demonstrated that its efficiency has the form as efficiency of a self-contained three-qubit quantum refrigerator. In other words, from the perspective of efficiency, this particular model is equivalent to self-contained three-qubit quantum refrigerator. We also demonstrated the efficiency of this particular model in the Carnot limit that is independent from details of system model, but only depends on the environmental temperatures. Also, we raised a system that consists of n-qubit which acts as a refrigerator. According to self-contained four-qubit quantum refrigerator, we also investigated a special case of self-contained n-qubit quantum refrigerator — a self-contained n-qubit quantum refrigerator with (n - 2) baths in the same temperature. We considered the three different special situations of the n-qubit refrigerator and demonstrated their efficiency in three different situations which has the form as efficiency of self-contained three-qubit quantum refrigerator. In this special situations, (n - 2) qubits are in thermal contact with isothermal heat baths.
We study stochastic energetic exchanges in quantum heat engines. Due to microreversibility, these obey a fluctuation relation, called the heat engine fluctuation relation, which implies the Carnot bound: no machine can have an efficiency greater than Carnot's efficiency. The stochastic thermodynamics of a quantum heat engine (including the joint statistics of heat and work and the statistics of efficiency) are illustrated by means of an optimal two-qubit heat engine, where each qubit is coupled to a thermal bath and a two-qubit gate determines energy exchanges between the two qubits. We discuss possible solid-state implementations with Cooper-pair boxes and flux qubits, quantum gate operations, and fast calorimetric on-chip measurements of single stochastic events.