p$_{c}$ ( t ) = 1 2 [ 1 − sin ( t τ$_{c}$ | φ$_{1}$ − φ$_{0}$ | − φ$_{0}$ )] (26a) p$_{h}$ ( t ) = 1 2 1 + sin t τ$_{h}$ | φ$_{1}$ − φ$_{0}$ | + φ$_{1}$ . (26b) S ( p ) = S ( p ) − arcsin(1 − 2 p $_{)}$√$_{K.}$ (27) P = ( T$_{h}$ − T$_{c}$ ) 2 4( √ T$_{h}$/C$_{h}$ + √ T$_{c}$/C$_{c}$ ) $_{2}$D ( p$_{0}$, p$_{1}$ ) , (33) D ( p$_{0}$, p$_{1}$ ) = Δ S 2 ( φ$_{1}$ − φ$_{0}$ ) 2 . (34) Q ∗ h = T$_{h}$ Δ S − T$_{h}$ ( φ$_{1}$ − φ$_{0}$ ) 2 C$_{h}$τ$_{h}$ (28b) ( T$_{h}$ − T$_{c}$ )Δ S ( φ$_{1}$ φ$_{0}$ ) 2 > T$_{c}$ C$_{c}$τ$_{c}$ + T$_{h}$ C$_{h}$τ$_{h}$ . (29) η ∗ = η$_{c}$ (1 + √ C$_{h}$T$_{c}$/C$_{c}$T$_{h}$ ) 2(1 + √ C$_{h}$T$_{c}$/C$_{c}$T$_{h}$ ) − η$_{c}$ , = η$_{c}$ 2 + η 2 c $_{4(1 +}$√$_{r}$$_{)}$ + η 3 c $_{8(1 +}$√$_{r}$$_{)}$ + o ( η 4 $_{c}$) (35) P = Q$_{c}$ + Q$_{h}$ τ$_{c}$ + τ$_{h}$ (30) τ ∗ c = 2( φ$_{1}$ − φ$_{0}$ ) $^{2}$T$_{c}$ (1 + √ T$_{h}$C$_{c}$/T$_{c}$C$_{h}$ ) C$_{c}$ Δ S ( T$_{h}$ − T$_{c}$ ) (31a) τ ∗ h = 2( φ$_{1}$ − φ$_{0}$ ) $^{2}$T$_{h}$ (1 + T$_{c}$C$_{h}$/T$_{h}$C$_{c}$ ) C$_{h}$ Δ S ( T$_{h}$ − T$_{c}$ ) . (31b) We have calculated the efficiency η at maximum power of a Carnot cycle with a single level quantum dot as the operational device. Our calculation is in agreement with known universality properties. In particular, the ef- ficiency at maximum power is equal to half of the Carnot efficiency in the regime of linear response, η = η$_{c}$/ 2 + ... . This optimization reflects the usual competition with the denominator of the power preferring faster operation whereas the numerator suggests a slower schedule to stay closer to Carnot efficiency. For the asymptotic expansion to be valid, the optimal operation times must satisfy the condition (25) of weak with respect to the operation times. It is easy to find that the power is a maximum when with r = C$_{h}$/C$_{c}$ . When r = 1, the efficiency (35) exactly coincides with the Curzon-Ahlborn efficiency η$_{CA}$ = 1 − √ T$_{c}$/T$_{h}$ . Note also that the efficiency is bounded below by η$_{c}$/ 2 for C$_{h}$/C$_{c}$ → ∞ and bounded above by η$_{c}$/ (2 − η$_{c}$ ) for C$_{h}$/C$_{c}$ → 0. These limits can be realized without violating the condition of weak dissipation. V. DISCUSSION So far, we have maximized the power only for the fixed operation times τ$_{c}$ and τ$_{h}$ and the boundary values p$_{0}$ and p$_{1}$ of the occupation probabilities. Now we further maximize the power This inequality is consistent with the condition of the asymptotic expansion (25) and can thus be satisfied even for a large temperature difference. Using the maximum heat (28) and the optimal time (31), we finally obtain the following remarkable result for the efficiency at the maximum power: The power reaches its maximum when D ( p$_{0}$, p$_{1}$ ) takes a maximum value, D$_{max}$ = 0 . 439 at p$_{0}$ = p$_{1}$ = 0 . 0832 or p$_{0}$ = p$_{1}$ = 0 . 9168. At these conditions, the optimal oper- ation time (31) and the maximum heat (28) both vanish. However, the power remains finite. This final optimiza- tion thus leads to a singular and unrealistic situation. We note, however, that since Eq. (34) does not depend on the system parameters, the efficiency does in fact not depend on this final optimization step. Therefore, we proceed to evaluate efficiency without further reference to optimal occupation probabilities. When the operation time is too short, the irreversible heat becomes dominant and the net heat becomes nega- tive. Equation (28) indicates that positive power can be obtained only if where Δ S = S ( p$_{0}$ ) − S ( p$_{1}$ ) is the reversible entropy change. The second term on the right hand side is the irreversible heat, which has to be small under the con- dition (25) of weak dissipation. In the quasi-static limit ( τ → ∞ ), the second term vanishes and the efficiency (21) reaches the Carnot efficiency, as expected. where Inserting the value of K , we obtain the maximum heat Expanding in a Taylor series with respect to √$_{K}$$_{, Eq.}$ (18) is approximated by the two lowest order terms as protocols: dissipation. That is, for the process III the following inequality must be satisfied: 5 This can be achieved in two ways. The first one cor- responds to the usual condition for linear irreversible thermodynamics, ( T$_{h}$ − T$_{c}$ ) /T$_{h}$ ≪ 1. The alternative is | Δ S/ ( φ$_{1}$ − φ$_{0}$ ) | ≪ 1. In this limit, our result remains valid even for large temperature differences. With the optimized operation times (31), the resulting power is written as a function of p$_{0}$ and p$_{1}$ :