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We study the performance of a cyclic heat engine which uses a small system with a finite number of ultracold atoms as its working substance and works between two heat reservoirs at constant temperatures  T    h   and  T    c  (< T    h  ). Starting from the expression of heat capacity which includes finite-size effects, the work output is optimized with respect to the temperature of the working substance at a special instant along the cycle. The maximum-work efficiency  η    m w  at small relative temperature difference can be expanded in terms of the Carnot value    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${{\boldsymbol{\eta }}}_{{\boldsymbol{C}}}={\bf{1}}-{{\boldsymbol{T}}}_{{\boldsymbol{c}}}/{{\boldsymbol{T}}}_{{\boldsymbol{h}}}$$\end{document}η    C    =  1  −T    c/T    h,    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${{\boldsymbol{\eta }}}^{{\boldsymbol{m}}{\bf{w}}}={{\boldsymbol{\eta }}}_{{\boldsymbol{C}}}/{\bf{2}}+{{\boldsymbol{\eta }}}_{{\boldsymbol{C}}}^{{\bf{2}}}({\bf{1}}/{\bf{8}}+{{\boldsymbol{a}}}_{{\bf{0}}})+{\boldsymbol{\ldots }}$$\end{document}η    m  w    =η    C/  2   +    η    C    2    (   1  /  8   +    a    0    )  +  …, where  a   0  is a function depending on the particle number  N  and becomes vanishing in the symmetric case. Moreover, we prove using the relationship between the temperatures of the working substance and heat reservoirs that the maximum-work efficiency, when accurate to the first order of  η    C  , reads    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${{\boldsymbol{\eta }}}^{{\boldsymbol{m}}{\bf{w}}}={{\boldsymbol{\eta }}}_{{\boldsymbol{CA}}}+{\bf{O}}$$\end{document}η    m  w    =    η    CA    +  O(Δ T   2 ). Within the framework of linear irreversible thermodynamics, the maximum-power efficiency is obtained as    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${{\boldsymbol{\eta }}}^{{\boldsymbol{mp}}}={{\boldsymbol{\eta }}}_{{\boldsymbol{CA}}}+{\bf{O}}$$\end{document}η    mp    =    η    CA    +  O(Δ T   2 ) through appropriate identification of thermodynamic fluxes and forces, thereby showing that this kind of cyclic heat engines satisfy the tight-coupling condition.

Universality of maximum-work efficiency of a cyclic heat engine based on a finite system of ultracold atoms