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single energy level ε near the Fermi level of the lead while other levels in the dot do not contribute to the processes described below. The state of the system is specified by the occupation probability p ( t ) of having an electron in the dot. The lead plays the role of a thermal bath at temperature T and chemical potential µ . Electrons are assumed to thermalize instantaneously upon tunnelling into the lead. 

The basic problem that we address is the finite-time performance of this engine as it runs through the follow- ing four standard stages of a Carnot cycle (also see figure 1): 

## I Isothermal process 

The quantum dot is in contact with a cold lead at temperature T$_{c}$ and chemical potential µ$_{c}$ . The energy level is raised from ε$_{0}$ to ε$_{1}$ according to a certain protocol during a time interval of duration τ$_{c}$ . Both work and heat are exchanged during this process. 

## II Adiabatic process 

The quantum dot is disconnected from the cold lead, and the quantum level is shifted from ε$_{1}$ to a new level ε$_{2}$ . Since the quantum dot is thermody- namically isolated, the population of the level does not change during this process. Hence, there is no heat exchange. However, the change of the energy level releases a corresponding amount of work. We assume that the operation time of this step is very short, in particular negligibly small compared to that of the isothermal processes. 

## III Isothermal process 

The dot is connected to the hot lead at temperature T$_{h}$ and chemical potential µ$_{h}$ . The energy level is lowered from ε$_{2}$ to ε$_{3}$ based on another protocol during a time interval of length τ$_{h}$ . Just as in step I, both heat and work are exchanged. 

## IV Adiabatic process 

The system is again disconnected from the lead and the level is restored from ε$_{3}$ to the initial level ε$_{0}$ , at the cost of a corresponding amount of work. Af- terwards, the dot is reconnected to the cold lead. Again, we assume that the operation time of this process is negligibly small. 

The above procedure defines one cycle of the thermal engine, requiring a total time τ$_{c}$ + τ$_{h}$ . The protocols in steps I and III must be designed in such a way that 

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the thermodynamic state of the system, in our case the occupation probability p of the quantum level, returns to the same initial value after every cycle. Since there is no change in occupation probability during the adiabatic stages II and IV, the change in occupation probability from, say p$_{0}$ to p$_{1}$ , during process I, must necessarily be compensated by a change back from p$_{1}$ to p$_{0}$ during process III. 

The time evolution of the occupation probability p ( t ) for the state of the quantum dot in contact with a lead at temperature β − 1 ( k$_{b}$ = 1) obeys the following quantum master equation: 

where the ω$_{a}$ and ω$_{b}$ are transition rates. In the wide- band approximation, these rates are given by 

where C is a rate constant. Noting that raising the energy level is equivalent to lowering the chemical potential, we introduce an effective energy level ϵ ≡ ε − µ . The master equation (1) can now be rewritten as 

The effective level varies along the Carnot cycle as ϵ$_{0}$ = ε$_{0}$ − µ$_{c}$ → ϵ$_{1}$ = ε$_{1}$ − µ$_{c}$ → ϵ$_{2}$ = ε$_{2}$ − µ$_{h}$ → ϵ$_{3}$ = ε$_{3}$ − µ$_{h}$ . Note that the change in the chemical potential is included in the jump of the effective level during processes II and IV. 

We next turn to the thermodynamic description of the model. We use the convention that heat entering the system is (like work) positive. The internal energy of the system at time t is 

where 

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