Quantum-dot Carnot engine at maximum power Massimiliano Esposito Center for Nonlinear Phenomena and Complex Systems, Universit´ e Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium. Ryoichi Kawai Department of Physics, University of Alabama at Birmingham, 1300 University Blvd. Birmingham, AL 35294-1170, USA. Katja Lindenberg Department of Chemistry and Biochemistry and BioCircuits Institute, University of California, San Diego, La Jolla, CA 92093-0340, USA. Christian Van den Broeck Hasselt University, B-3590 Diepenbeek, Belgium. (Dated: November 9, 2018) We evaluate the efficiency at maximum power of a quantum-dot Carnot heat engine. The universal value of the coefficients at the linear and quadratic order in the temperature gradient are reproduced. Curzon-Ahlborn efficiency is recovered in the limit of weak dissipation. The purpose of a heat engine is to transform an amount of heat Q$_{h}$ , extracted from a hot reservoir at temper- ature T$_{h}$ , into an amount of work W . The efficiency η = W/Q$_{h}$ for doing so is at most equal to Carnot ef- ficiency: η ≤ η$_{c}$ , with η$_{c}$ = 1 − T$_{c}$/T$_{h}$ . Here T$_{c}$ is the temperature of a second, cold reservoir T$_{c}$ ≤ T$_{h}$ , in which the remaining energy Q$_{h}$ − W is deposited. The equality is reached for reversible operation, implying that the cor- responding power output is zero. Curzon and Ahlborn were amongst the first to study the question of efficiency at maximum power [1]. By considering a simple modi- fication of the Carnot engine and after applying the so- called endo-reversible approximation (neglecting dissipa- tion in the auxiliary system), they found the following beautiful expression for the efficiency at maximum power: η ca = 1 − √ T$_{c}$/T$_{h}$ = 1 − √$_{1}$ − η$_{c}$ = η$_{c}$/ 2 + η 2 $_{c}$/ 8 + · · · . While this formula appears to describe rather well the efficiency of actual thermal plants, and is close to the efficiency at maximum power for several model systems, it is neither an exact nor a universal result, and it is neither an upper nor a lower bound. It has therefore come as a surprise that a number of universal predic- tions can be made about the expansion of the efficiency at maximum power in terms of η$_{c}$ . In the regime of lin- ear response, i.e., at first order in η$_{c}$ , it is found that the efficiency at maximum power is at most half of the Carnot efficiency [2]. In other words, the CA efficiency is an upper bound at the level of linear response. The proof was given for systems operating under steady-state condi- tions. The upper bound is reached for so-called strongly coupled systems, i.e., systems in which the heat flux and the work-producing flux are proportional to each other. More recently, it has been shown that the quadratic co- efficient, equal to 1 / 8, is also universal for strongly cou- Our heat engine model consists of a single-level quan- tum dot interacting with a metallic lead through a tun- neling junction. The quantum dot is assumed to have a In this paper we provide a full analysis of a thermal en- gine undergoing a Carnot cycle, with the auxiliary system consisting of a single-level quantum dot that is switched between a hot and a cold reservoir. We show that the ef- ficiency at maximum power is again consistent with the above discussed universality. Furthermore, CA efficiency at maximum power is obtained exactly at all orders in η$_{c}$ in the limit of low dissipation, which is distinct but similar to the case considered in the original CA paper. The above universality predictions have been con- firmed in a number of steady-state model systems in- volving classical particles [4], fermions [5], and bosons [6]. Universality has also been observed in variants of the CA model based on Carnot cycles performed in finite time, even though finding the optimal driving protocol max- imizing work extraction can be notoriously difficult [7– 13]. The connection with the steady-state analysis has been clarified by identifying the Onsager coefficient for a finite-time Carnot cycle in the linear regime [11]. Fur- thermore, agreement with the universal quadratic coeffi- cient has also been observed in a Carnot cycle based on a (classical) Brownian particle in an harmonic trap [8]. pled systems in the presence of an additional left-right symmetry in the system [3]. Furthermore, the univer- sality of the coefficients is a direct consequence of the time-reversibility of the underlying physical laws. The coefficient 1 / 2 derives from the symmetry of the Onsager matrix. The coefficient 1 / 8 can be seen as the implication of Onsager symmetry at the level of nonlinear response. II. MODEL I. INTRODUCTION arXiv:1001.2192v1 [cond-mat.stat-mech] 13 Jan 2010