with the matrix ˆ h = h − iλV ( G − 〈 G 〉 ). Apart from the average charge (which is zero for ˆ χ ), both log χ and log ˆ χ generate the same cumulants. The explicit calculation of the characteristic function of the full counting statistics ˆ χ involves the eigenvalues of the matrix ˆ h , where we have introduced the difference in the equilibrium population Δ P eq = P eq 2 − P eq 1 and the difference in the (dimensionless) conductance Δ g = V ( G$_{2}$ − G$_{1}$ ) / Γ. For long times Γ t ≫ 1, the matrix exponential (16) is dominated by the eigenvalue ˆ h$_{−}$ of ˆ h with the smallest real part. 7,27,28 We obtain an explicit expression for the generating function All cumulants become linear in t , due to the fact that the autocorrelation time in the system is given by Γ − 1 and every state decays to the equilibrium state after this time. The fluctuations for Γ t ≫ 1 can be seen as a sum of independent stochastic processes and the cumulant gen- erating function log ˆ χ becomes extensive in t . Interest- ingly, it is possible to obtain an explicit relation for the cumulants ( n ≥ 2) with the Narayana numbers N$_{n,m}$ = ( n m )( n m − 1 ) /n . 29 The cumulants in (18) grow factorially in magnitude with n and oscillate as a function of Δ P eq which are generic features of high-order cumulants. 30 The first couple of cumulants assume the form Note that all the odd cumulants vanish if the process is symmetric γ$_{12}$ = γ$_{21}$ , cf. Eq. (3). The cumulants in (18) and (19) agree with the result in Ref. 5. However, starting with the 4-th cumulant a discrepancy arises; e.g., for the 4-th cumulant we obtain for arbitrary times. This result has been first derived in Ref. 11 in the context of dephasing of a qubit due to the interaction with a classical two-level fluctuator, where λ denotes the interaction of the qubit with the fluctuator. Here, we are interested in the transport properties of a point contact characterized by cumulants which are given by the Taylor expansion of log χ around λ = 0; the rela- tion between these two problems is a consequence of the generic equivalence between full counting statistics and Expanding the matrix exponential in Eq. (16) in its eigenbasis, the generator for the full counting statistics reads where the Eulerian numbers E$_{n,m}$ are defined through E$_{n,m}$ = ∑ m k $_{=0}$( − 1) $^{k}$( n +1 k ) ( m +1 − k ) $^{n}$. 29 In the short time limit, the cumulants grow like 〈〈 Q $^{n}$〉〉 ∝ t $^{n}$, i.e., higher order cumulants are suppressed at short times. The first couple of cumulants are explicitly given by with the cumulants ( n ≥ 2) given by For short times, t ≪ Γ − 1 there is no evolution of the two-level system and we can set h t = 0. The full counting statistics reads The latter result is incorrect as it misses terms due to the implicit assumption in Ref. 5 that the reduced con- ductance correlators 〈〈 G ( t$_{n}$ ) · · · G ( t$_{1}$ ) 〉〉 may only depend on the largest time difference t$_{n}$ − t$_{1}$ . Even though this hypothesis is correct for correlators up to n = 3, it fails for higher-order correlators. which is different from the result 5 4 ˆ h$_{±}$ = Γ 2 [ 1 + iλ Δ g Δ P eq ± √ 1 + 2 iλ Δ g Δ P eq − λ $^{2}$(Δ g ) 2 ] log ˆ χ$_{≫}$ ( λ ) = − ˆ h$_{−}$ t. (17) 〈〈 Q $^{n}$〉〉$_{≫}$ = n ! V $^{n}$( G$_{1}$ − G$_{2}$ ) $^{n}$Γ 1 − $^{n}$t (18) ×$^{n}$$^{−}$$^{1}$ ∑ k =1 N$_{n}$$_{−}$$_{1}$$_{,k}$ ( − 1) k $^{+1}$( P eq 1 ) $^{k}$( P eq 2 ) n − $^{k}$, 〈〈 Q $^{2}$〉〉$_{≫}$ = 2 P eq 1 P eq 2 V $^{2}$( G$_{1}$ − G$_{2}$ ) $^{2}$t Γ , (19) 〈〈 Q $^{3}$〉〉$_{≫}$ = 6 P eq 1 P eq 2 ( P eq 2 − P eq 1 ) V $^{3}$( G$_{1}$ − G$_{2}$ ) $^{3}$t Γ 2 . 〈〈 Q $^{4}$〉〉$_{≫}$ = 24 P eq 1 P eq 2 [( P eq 2 − P eq 1 ) 2 − P eq 1 P eq 2 ] V $^{4}$( G$_{1}$ − G$_{2}$ ) $^{4}$t Γ 3 , ˆ χ = ˆ h$_{+}$e − ˆ h$_{−}$t − ˆ h$_{−}$e − ˆ h$_{+}$t ˆ h$_{+}$ − ˆ h$_{−}$ (23) 〈〈 Q $^{2}$〉〉$_{≫}$ = P eq 1 P eq 2 V $^{2}$( G$_{1}$ − G$_{2}$ ) $^{2}$t$^{2}$, (22) 〈〈 Q $^{3}$〉〉$_{≫}$ = P eq 1 P eq 2 ( P eq 2 − P eq 1 ) V $^{3}$( G$_{1}$ − G$_{2}$ ) $^{3}$t$^{3}$. 〈〈 Q $^{n}$〉〉$_{≪}$ = V $^{n}$( G$_{1}$ − G$_{2}$ ) $^{n}$tn (21) ×$^{n}$$^{−}$$^{1}$ ∑ k =1 E$_{n}$$_{−}$$_{1}$$_{,k}$$_{−}$$_{1}$ ( − 1) k $^{+1}$( P eq 1 ) $^{k}$( P eq 2 ) n − $^{k}$, ˆ χ$_{≪}$ ( λ ) = 〈 0 | e iλV ( G −〈 G 〉 ) $^{t}$| 0 〉 (20) = P eq 2 e iλV ( G$_{2}$ − G$_{1}$ ) P eq 1 t + P eq 1 e iλV ( G$_{1}$ − G$_{2}$ ) P eq 2 $^{t}$, C. Asymptotic long-time limit 〈〈 Q $^{4}$〉〉$_{≫}$ = 24 P eq 1 P eq 2 ( P eq 2 − P eq 1 ) $_{2}$V $^{4}$( G$_{1}$ − G$_{2}$ ) $^{4}$t Γ 3 . D. Short times E. Arbitrary times