3 -15 -10 -5 0 5 10 0 0.5 1 1.5 2 c (1) c $^{(1)}$+ c (2) s (1) s $^{(1)}$+ s (2) FIG. 2: Charge relaxation rate, γ c , and spin relaxation rate, γ s , in first and 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. with the equilibrium occupation number 〈 n$_{c}$ 〉 $^{eq}$. This yields the expressions for the time-dependent expectation value of the charge 〈 n$_{c}$ 〉 ( t ) = (0 , 1 , 1 , 2) · P ( t ) and the spin 〈 n$_{s}$ 〉 ( t ) = (0 , 1 , − 1 , 0) · P ( t ) on the dot with the initial occupation numbers indicated by the su- perscript ”in”. The spin relaxation time can be measured by preparing the dot in a spin-polarized state by apply- ing a magnetic field and looking at the spin relaxation after the field has been switched off. If we retain only the first-order contribution in Γ, the inverse relaxation times are given by, 14 with the Fermi functions f ( ω ). For vanishing Coulomb interaction, U = 0, the relaxation times are independent of the level position ϵ and both equal to the inverse of the tunnel coupling Γ. In the presence of a finite U , when the level position is in the interval − U < ϵ < 0 the dot is predominantly singly occupied. In this pa- rameter range, γ$_{c}$ and γ$_{s}$ differ strongly, even though in first order in the tunnel coupling a change in the spin oc- cupation goes always along with a change in the charge occupation. There are two possible states for the dot to relax to single occupation, namely | ↑〉 , | ↓〉 , and con- sequently the charge relaxation time is decreased by a factor 1 / 2. On the other hand, in first order in tunneling the spin is blocked to a value different from its equilib- rium value 0 due to the lack of spin-flipping processes leading to an enhancement of τ$_{s}$ . This is shown in Fig. (2), where the inverse relaxation times for spin, γ$_{s}$ , and charge, γ$_{c}$ , are plotted as a function of the new dot’s level position after the switching, for finite U . Note that spin γ (1) s = Γ [1 − f ( ϵ ) + f ( ϵ + U )] (5a) γ (1) c = Γ [1 + f ( ϵ ) − f ( ϵ + U )] , (5b) W$_{sf}$ = − Γ β [ Γ φ $^{′′}$( ϵ ) + Γ φ $^{′′}$( ϵ + U ) − 2 U σ $^{′}$( ϵ, U ) ] W$_{d0}$ = − 2Γ e β (2 ϵ + U ) − 1 [ Γ φ $^{′}$( ϵ ) + Γ φ $^{′}$( ϵ + U ) − 2 U σ ( ϵ, U ) ] , 〈 n$_{c}$ 〉 ( t ) = 〈 n$_{c}$ 〉 eq ( 1 − e − t/τ c ) + 〈 n$_{c}$ 〉 $^{in}$e − t/τ c 〈 n$_{s}$ 〉 ( t ) = 〈 n$_{s}$ 〉 $^{in}$e − t/τ s , σ$_{Γ}$ ( ϵ, Γ , U ) = − [ Γ φ $^{′}$( ϵ ) + Γ φ $^{′}$( ϵ + U ) − 2 U σ ( ϵ, Γ , U ) ] × 1 − f ( ϵ ) − f ( ϵ + U ) 1 − f ( ϵ ) + f ( ϵ + U ) . (6) γ (2) s = σ ( ϵ, U ) ∂ ∂ϵ γ (1) s + σ$_{Γ}$ ( ϵ, U ) γ (1) s + 2 W$_{sf}$ (7a) γ (2) c = σ ( ϵ, U ) ∂ ∂ϵ γ (1) c + σ$_{Γ}$ ( ϵ, U ) γ (1) c +2 f ( ϵ + U ) W$_{0d}$ + (1 − f ( ϵ )) W$_{d0}$ 1 − f ( ϵ ) + f ( ϵ + U ) . (7b) IV. HARMONIC MODULATION and charge have independent dynamics; this is not the case in the presence of a Zeeman splitting in the dot. In fact, in the presence of a Zeeman splitting, ϵ$_{↑}$ = ϵ$_{↓}$ , and for infinite U , the first order relaxation rates are given by 1 /τ$_{1}$ = 1 + √ f ( ϵ$_{↑}$ ) f ( ϵ$_{↓}$ ) and 1 /τ$_{2}$ = 1 − √ f ( ϵ$_{↑}$ ) f ( ϵ$_{↓}$ ). When we include tunneling contributions in second order in the tunnel coupling, we find corrections of different ori- gin to the relaxation times. Cotunneling, describing real tunneling processes via energetically forbidden interme- diate states, leads to coherent processes flipping the spin of the dot, W$_{sf}$ , or to transitions between zero and dou- ble occupation, W$_{0d}$ and W$_{d0}$ . These terms, which agree with results from a standard second-order perturbation expansion, contain expressions φ ( ϵ ) = 1 2 $_{π}$ReΨ ( 1 2 + i βϵ 2 π ) and σ ( ϵ, U ) = Γ[ φ ( ϵ + U ) − φ ( ϵ )], where Ψ( x ) is the digamma function and β = 1 / ( k$_{B}$T ) the inverse temper- ature, primes denote derivatives with respect to ϵ . These cotunneling rates are given by and W$_{0d}$ = exp[ β (2 ϵ + U )] W$_{d0}$ . In addition to the cotun- neling terms virtual tunneling processes appear, which can be captured in terms of renormalization of the level position, ϵ → ϵ + σ ( ϵ, U ), and the tunnel coupling, Γ → Γ[1 + σ$_{Γ}$ ( ϵ, U )], 15 with These corrections are due to an interplay of tunneling and interaction and vanish for U = 0. We verified that they can also be identified in the second-order linear con- ductance. The corrections to the relaxation times read The influence of the corrections on the relaxation times is shown in Fig. 2. Both relaxation rates are enhanced due to the higher order tunneling processes. Importantly, the charge and spin relaxation dynamics remain governed by a single time scale, each. We now consider a harmonic modulation of the level position ϵ with frequency Ω, which we assume to be much