4 smaller than Γ, cf. Fig. 1(b). We perform a systematic expansion in powers of Ω. 16,17 The order in the expansion in Ω, is indicated by a superscript: (i) (instantaneous) for zeroth order and (a) (adiabatic) for first order. While the instantaneous term for the average occupation of the dot, 〈 n 〉 ( i $^{)}$, immediately adjusts to the applied gate volt- age, its first-order correction, 〈 n 〉 ( a ) accounts for a slight lagging behind. In the ac regime, the quantum dot can be modeled by its charge relaxation resistance, R$_{q}$ , in series with its elec- trochemical capacitance, C$_{µ}$ . 3 We are interested in the frequency-dependent admittance, G (Ω), which is defined by the linear relation G (Ω) = I (Ω) V (Ω). Therefore we consider the linear regime in the gate potential control- ling the level position ϵ . The expansion of the admittance of the dot for small frequencies enables to identify these quantities and to compute the RC time. To compare with this, we insert the adiabatic expansion of the occupation number into the charge continuity equation I ( t ) = − e ∂ 〈 n 〉 /∂t (Ref. 18) to arrive at To obtain the average dot occupation, we expand the master equation in powers of Ω, We refer to Ref. 16 for the evaluation of the adiabatic correction of the kernel, W ( a $^{)}$. The instantaneous occu- pation probabilities, P ( i $^{)}$, and their first adiabatic cor- rection, P ( a $^{)}$, are obtained from the master equation Eq. (10) by matching orders in the tunnel coupling. We find that the instantaneous probabilities P ( i ) start in ze- roth order in the tunnel coupling, while P ( a ) starts in minus first order in the tunnel coupling, with P ( a, − 1) be- ing proportional to the small factor Ω / Γ. A rigorous expansion in the tunnel coupling exists for the inverse of the RC -time, γ$_{RC}$ = 1 /τ$_{RC}$ . Close to the resonances, ϵ = 0 and ϵ = − U , both the change in the in- stantaneous occupation and the adiabatic correction are governed by lowest order in the tunnel coupling, which leads to the same relaxation time as found for the charge relaxation time due to fast switching, γ (1) RC = γ (1) c . In the Coulomb-blockaded regions, however, both ∂ 〈 n 〉 ( i, $^{0)}$/∂t and 〈 n 〉 ( a, − 1) are exponentially suppressed, and the ex- pression for γ (1) RC becomes meaningless. Instead, the next- order corrections in Γ needs to be taken into account. For the time variation in the instantaneous occupation num- ber ∂ 〈 n 〉 ( i, $^{1)}$/∂t , the non-exponentially suppressed contri- bution to this higher-order correction is due to quantum fluctuations, namely cotunneling processes in which the final dot state is not changed as compared to the initial one. Their rates are W$_{0}$$_{→}$$_{σ}$$_{→}$$_{0}$ = W$_{σ}$$_{→}$$_{0}$$_{→}$$_{σ}$ = − Γ 2 β φ $^{′′}$( ϵ ) and W$_{σ}$$_{→}$$_{d}$$_{→}$$_{σ}$ = W$_{d}$$_{→}$$_{σ}$$_{→}$$_{d}$ = − Γ 2 β φ $^{′′}$( ϵ + U ). The higher- order correction 〈 n 〉 ( a, $^{0)}$, on the other hand, remains ex- ponentially suppressed. As a result, the relaxation rate diverges. This type of processes presumably contributes only to the short time dynamics after a step pulse and does not enter the long-time behavior described by the exponential relaxation. We find for the difference between the RC rate γ$_{RC}$ and the charge relaxation rate γ$_{c}$ Although this difference is only due to second-order rates, it is substantial in the Coulomb-blockade regions where first-order processes are exponentially suppressed. This is shown in Fig. 3, where the inverse RC -time and the re- laxation rate after a single switching event are compared, taking into account tunneling processes up to second or- der. This difference is a consequence of the different time scales captured in the two expansions. Whereas in the response to the step we take into account the long-time limit but omit high frequencies, the ac response is sen- sitive only to the frequency at which it is tested. When taking U = 0 this result agrees well with the respective limit of Ref. 3 in the weak-coupling regime. We calculated the relaxation times of a single-level in- teracting quantum dot contacted to a lead and driven out of equilibrium by a step-pulse potential and the RC - time extracted from the ac response. In the presence of Coulomb interaction the relaxation times depend on the position of the dot level, leading to different relaxation times for charge and spin. Corrections in second order of the tunnel coupling contain real cotunneling processes V. CONCLUSION FIG. 3: Inverse RC time, γ RC , and charge relaxation rate, γ c , up to second order in the tunnel coupling Γ for U = 10Γ as a function of the dot’s level position ϵ in units of Γ, k B T = 1 . 5Γ. G (Ω) = − i Ω C$_{µ}$ + Ω $^{2}$C2 $_{µ}$R$_{q}$ , (8) 1 τ$_{RC}$ = − ∂ 〈 n 〉 ( i $^{)}$/∂t 〈 n 〉 ( a ) . (9) 0 = W ( i ) P ( i ) (10a) d P ( i ) dt = W ( i ) P ( a ) + W ( a ) P ( i ) + ∂ W ( i $_{)}$d P ( i ) dt . (10b) γ (2) RC − γ (2) c γ (1) = Γ [ 2 − 〈 n 〉 ( i, $^{0)}$] φ $^{′′}$( ϵ ) + 〈 n 〉 ( i, $^{0)}$φ $^{′′}$( ϵ + U ) ∂ 〈 n 〉 ( i, $^{0)}$/∂ϵ . (11) -10 0 10 1 1.5 2 c (1) c $^{(1)}$+ c (2) RC $^{(1)}$+ RC (2)