Optimal refrigerator

We study a refrigerator model which consists of two n-level systems interacting via a pulsed external field. Each system couples to its own thermal bath at temperatures T-h and T-c, respectively (theta T-c/T-h<1). The refrigerator functions in two steps: thermally isolated interaction between the systems driven by the external field and isothermal relaxation back to equilibrium. There is a complementarity between the power of heat transfer from the cold bath and the efficiency: the latter nullifies when the former is maximized and vice versa. A reasonable compromise is achieved by optimizing the product of the heat-power and efficiency over the Hamiltonian of the two systems. The efficiency is then found to be bounded from below by zeta(CA) = 1/root 1-theta-1 (an analog of the Curzon-Ahlborn efficiency), besides being bound from above by the Carnot efficiency zeta(C)=1/1-theta-1. The lower bound is reached in the equilibrium limit theta -> 1. The Carnot bound is reached (for a finite power and a finite amount of heat transferred per cycle) for ln n >> 1. If the above maximization is constrained by assuming homogeneous energy spectra for both systems, the efficiency is bounded from above by zeta(CA) and converges to it for n >> 1.