2 III. FAST SWITCHING II. MODEL AND FORMALISM to the line width Γ of the single level state. The effect of Coulomb interaction is manifest already in lowest or- der in the tunnel coupling, introducing a dependence on the level position of the relaxation times. We find that charge and spin have independent dynamics with a charge relaxation time that is reduced and spin relaxation time that is enhanced by interaction. Higher-order tunneling contributions lead to a difference between the RC time extracted from the ac admittance and the relaxation time extracted from the exponential decay after a fast switch. This is in contrast to the behavior of a classical RC cir- cuit and signalizes that care must be taken in extracting the characteristic time-scales of the system. evolution is given by the generalized master equation The elements W$_{χ,χ}$ ′ ( t, t $^{′}$) of the matrix W ( t, t $^{′}$) describe transitions from state χ ′ at time t ′ to state χ at time t . We consider a quantum dot attached to a single lead by a tunneling contact, cf. Fig. 1, described by the Hamiltonian H = H$_{dot}$ + H$_{tunnel}$ + H$_{lead}$ . The single- particle level spacing in the dot is assumed to be larger than any other energy scale (temperature, Coulomb in- teraction, gate voltage and the highest frequency in the time-dependent signals) such that only one energy level is accessible. The dot can be described by the single-level Anderson model with d † $_{σ}$( d$_{σ}$ ) being the creation (annihilation) operator for an electron with spin σ on the dot, and n$_{σ}$ = d † $_{σ}$d$_{σ}$ the corresponding number operator. The level position is assumed to depend on time via a gate potential. The onsite repulsion U describes the energy cost for double occupation and stems from Coulomb interaction (within the constant interaction model $^{12}$). Tunneling of elec- trons between the dot and the lead is taken into account by H$_{tunnel}$ = ∑ k,σ V c † $_{k,σ}$d$_{σ}$ + h . c . , where we assume a momentum-independent tunnel matrix element V and define the creation (annihilation) operators c † $_{k,σ}$( c$_{k,σ}$ ) for electrons with spin σ and momentum k in the lead. The lead Hamiltonian is given by H$_{lead}$ = ∑ k,σ ϵ$_{k}$c † $_{k,σ}$c$_{k,σ}$ . We assume a constant density of states ρ in the leads and define the tunnel coupling strength Γ as Γ = 2 πρ | V | $^{2}$. Since we are not interested in the dynamics of the non- interacting leads’ degrees of freedom, we trace them out obtaining an effective description of the dot in terms of its reduced density matrix. We choose as basis of the four-dimensional Hilbert space of the reduced system the eigenstates, | χ 〉 , of the isolated dot Hamiltonian H$_{dot}$ : | 0 〉 for an empty dot, | σ 〉 for a singly occupied dot with spin σ = ↑ , ↓ and | d 〉 for a doubly occupied dot. Assum- ing spin-independent tunneling, we can restrict our in- terest to the time evolution of the diagonal elements of the reduced density matrix, p$_{χ}$ . These probabilities are arranged in the vector P = ( p$_{0}$, p$_{↑}$, p$_{↓}$, p$_{d}$ ), whose time We first consider the time evolution after a fast switch- ing event at t = t$_{0}$ , as shown in Fig. 1 (a). The am- plitude of the switching is only limited insofar as we assume that only the single level of the dot is accessi- ble, i.e. the amplitude is smaller than the level spacing. The initial value, i.e., the equilibrium distribution before changing the gate voltage, is denoted by P $^{in}$. In order to identify the relaxation time, we perform the following steps. First, we expand the probability vector P ( t $^{′}$) on the right-hand side of Eq. (2) around the final time t , i.e. P ( t $^{′}$) = ∑ ∞ n =0 1 n ! ( t ′ − t ) n d $^{n}$P ( t $^{′}$) dt ′ n |$_{t}$ ′ = $_{t}$. Second, we realize that for t ′ > t$_{0}$ , the Hamiltonian is time independent, and hence the transition matrix elements depend only on the difference of the time arguments, W ( t, t $^{′}$) → W ( t − t $^{′}$). Third, we replace the lower integration bound t$_{0}$ by −∞ . This is justified for times t that are larger than the char- acteristic time scale over which the kernel W ( t ) decays. The regime of shorter times t is not the subject of the present work. As a result of these three steps, we obtain introducing the derivatives of the Laplace transforms, ∂ n W := ∂ n ∂z n [ ∫ t −∞ dt ′ e − z ( t − t $^{′}$) W ( t − t $^{′}$) ] z =0 $_{+}$. The formal solution of Eq. (3) can be written as The matrix A always possesses the eigenvalue 0, which guarantees the existence of a stationary probability dis- tribution. The other eigenvalues of A define the relax- ation rates. In the following, we concentrate on the limit of weak tunnel coupling between quantum dot and lead. This motivates an expansion of A = A (1) + A (2) + . . . in powers of the tunnel-coupling strength Γ, indicated by the superscript ( i ). Substituting the formal solu- tion for P ( t ), Eq. (4), into Eq. (3) iteratively yields A (1) = W (1) and A (2) = W $^{(2)}$+ ∂ W $^{(1)}$· W $^{(1)}$. Note that non-Markovian contributions, i.e. contributions propor- tional to ∂ n W , enter only starting from second order in Γ. The kernel W ( z ) can be computed by the real-time diagrammatic approach developed in Ref. 13. We are interested in the inverse relaxation times of charge γ$_{c}$ = 1 /τ$_{c}$ and spin γ$_{s}$ = 1 /τ$_{s}$ . They are found as the eigenvalues of A related to the left eigenvectors v c = (0 , 1 , 1 , 2) − 〈 n$_{c}$ 〉 eq (1 , 1 , 1 , 1) and v s = (0 , 1 , − 1 , 0), P ( t ) = exp ( A t ) P $^{in}$. (4) d P ( t ) dt = ∞ ∑ n =0 1 n ! ∂ n W · d n P ( t ) dt n , (3) H$_{dot}$ = ∑ σ = ↑ , ↓ ϵ ( t ) d † $_{σ}$d$_{σ}$ + Un$_{↑}$n$_{↓}$, (1) d P ( t ) dt = ∫ t t$_{0}$ dt ′ W ( t, t $^{′}$) P ( t $^{′}$) . (2)