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arXiv:0908.2076 · Physical Review Letters

How Small Can Thermal Machines Be? The Smallest Possible Refrigerator

Noah Linden, Sandu Popescu, Paul Skrzypczyk

Published Sep 21, 2010• 451 citations
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Abstract

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.

DOI
10.1103/physrevlett.105.130401

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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.
A fundamental challenge in quantum thermodynamics is the exploration of inherent dimensional constraints in thermodynamic machines. In the context of two-level systems, the most compact refrigerator necessitates the involvement of three entities, operating under self-contained conditions that preclude the use of external work sources. Here, we build such a smallest refrigerator using a nuclear spin system, where three distinct two-level carbon-13 nuclei in the same molecule are involved to facilitate the refrigeration process. The self-contained feature enables it to operate without relying on net external work, and the unique mechanism sets this refrigerator apart from its classical counterparts. We evaluate its performance under varying conditions and systematically scrutinize the cooling constraints across a spectrum of scenarios, which sheds light on the interplay between quantum information and thermodynamics.
The functions of the smallest self-contained thermal machine consisting of a single qutrit are studied when the weak internal coupling assumption is relaxed. It is shown that in the presence of one target to be cooled the strong coupling is not beneficial to the refrigeration. The reason is explained by examining the effect of the strong coupling on the contributions of all eigenstates transitions to the heat current of the related thermal reservoir. When acting simultaneously on two targets, the machine can be manipulated to implement distributed tasks on them, such as cooling one target and meanwhile heating another one, by adjusting the coupling strengths between the machine with the two targets. In particular, we show that the machine can realize temperature reversal for the two qubits, namely, the qubit that is coupled to the high temperature reservoir is refrigerated to a temperature below that of the qubit contacting with the low temperature reservoir.
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.
The second and third laws of thermodynamics can be used to establish a fundamental bound for the maximum possible cooling rate in approaching the absolute zero of temperature. In modeling the behavior of the molecular refrigerators geared toward attaining ultralow temperatures, only quantum mechanical, as opposed to classical physics, models can be admissible. As a simple model, we analyze a three-level quantum refrigerator, and in particular its irreversible thermodynamic performance as absolute zero is approached.