V. QUANTUM WIRE Considering a quantum rather than a classical wire, additional noise appears due to the probabilistic nature of the charge transport (due to partitioning) even in the absence of a fluctuating environment. Given a quantum wire with N channels characterized by their transmission eigenvalues T $^{γ}$, γ = 1 , . . . , N , and biased by a voltage po- tential V , the characteristic function of the full counting statistics is given by 3 with q the charge of the electron; this result is valid in the asymptotic limit qV t/ ℏ ≫ 1, for low-temperatures ϑ ≪ qV , and with the proviso that the energy-dependence of the transmission eigenvalues is negligible in the energy interval set by the voltage. The quantum nature of the fermions leads to the noise which disappears provided that all the channels are either closed T γ = 0 or completely open T γ = 1. Note that the noise in Eq. (33) is sub-Poissonian, i.e., the Fano factor F = 〈〈 Q $^{2}$〉〉$^{q}$/q 〈 Q 〉 q is less than 1. Here, we are interested in the case where the quan- tum wire is capacitively coupled to a two-level fluctuator such that the transmission eigenvalues T$_{γ}$ change over time; note that we neglect a possible energy dependence of the transmission eigenvalues, which corresponds to the fact that we assume that the scattering center does not produce any time delay due to the scattering event. The characteristic function log χ $^{q}$( λ ) = det Q is given by the determinant of the matrix 31 with the counting operator Q$_{t}$ = ∫ $_{I}$dx | x 〉〈 x | integrated over the interval I = [0 , v F t ] and the single-particle solution of the time-dependent Schr¨ odinger equation involving a scattering center at x = 0 with time-dependent transmission [reflection] am- plitude τ$_{γ}$ ( t ) [ r$_{γ}$ ( t )]; note that we have suppressed the transverse part of the wave function belonging to the channel index γ . Equation (35) is valid in the linear- spectrum approximation, where v F is the Fermi velocity and 0 ≤ k ≤ k F . 32 As the matrix Q is block-diagonal in γ and constitutes a Toeplitz matrix with respect to the in- dex k , its determinant can be shown (using the technique discussed in Ref. 31) to have the form here, log χ q $_{n}$( λ ) = ( qV t/ 2 π ℏ ) ∑ γ log[1 + ( e iqλ − 1) T γ $_{n}$] is the characteristic function for the FCS, Eq. (32), depen- dent on the system’s state ( n = 1 , 2) and γ runs over the channel index. The expression (39) coincides with the re- sult obtained by Jordan and Sukhorukov who considered the case of rare transitions ( ℏ Γ ≪ qV ), see Ref. 9. In their work, Jordan and Sukhorukov describe the trans- port of a conserved charge in a classical bistable system where large charge fluctuations drive transitions between stable states with different transport characteristics. The generic fluctuations present in the system’s stable states then combine with the fluctuations of the bistable charge to generate the transport statistics of the bistable sys- tem. While describing a very general situation, the spe- cific analysis in Ref. 9 is limited to those cases where the charge-switching events are rare on the time scale of the underlying fluctuations in the stable states. The transla- tion to our model may not be obvious from the start and is done by choosing the process of charge partitioning for generating the underlying fluctuations (with different transmission rates T γ $_{n}$, n = 1 , 2 for the two stable states), i.e., the generators log χ q $_{n}$( λ ) correspond to the long time generators H$_{n}$t in Ref. 9. The non-linearity leading to switching between stable states generates the transition probabilities Γ$_{1}$$_{,}$$_{2}$; the latter are determined by an instan- ton trajectory and have to be small in the case of Ref. 9. into Eq. (17) and using Eq. (16), we obtain where the average 〈·〉 is over the stochastic process of the two-level fluctuator. Equation (37) can be calculated explicitly with the method outlined in Sec. III A. Indeed, Eq. (12) goes over to Eq. (37) via replacing iλI ( t ) with ( qV/ 2 π ℏ ) ∑ γ log[1+( e $^{iqλ}$− 1) T $^{γ}$( t )]. In this mapping, the conductances G$_{n}$ are changed to ( q/ 2 πi ℏ λ ) ∑ γ log[1 + ( e iqλ − 1) T γ $_{n}$] with T γ n the transparency of the channel γ when the two-level fluctuator is in the state n . Inserting for long times qV t/ ℏ ≫ 1; the expression Eq. (36) reduces to Eq. (32) when the transmission probability does not change in time. We now add the classical two-level fluc- tuator to the system: Depending on the state x = x$_{1}$$_{,}$$_{2}$ of the nearby two-level fluctuator, the quantum wire is de- scribed by one of the two sets of transmission eigenvalues T γ 1 , $_{2}$. The total generating function χ $^{qtl}$( λ ) is an average over the individual contributions 6 log χ $^{q}$( λ ) = qV t 2 π ℏ ∑ γ log[1 + ( e iqλ − 1) T $^{γ}$] , (32) 〈〈 Q $^{2}$〉〉q = q 〈 Q 〉 q ∑ γ T $^{γ}$(1 − T $^{γ}$) ∑ γ T γ (33) Q$_{kγ,k}$ ′ γ ′ = 〈 φ$_{kγ}$ ( t ) | e iλq Q$_{t}$ | φ$_{k}$ ′ γ ′ ( t ) 〉 (34) φ$_{kγ}$ ( x, t ) = { e ik ( x − v$_{F}$t ) + r$_{γ}$ ( t + x/v F ) e − ik ( x + v$_{F}$t ) x < 0 τ$_{γ}$ ( t − x/v F ) e ik ( x − v$_{F}$t ) x > 0 (35) log χ $^{q}$( λ ) = qV 2 π ℏ ∫ t 0 dt ′ ∑ γ log[1 + ( e iqλ − 1) T$_{γ}$ ( t $^{′}$)] (36) log χ qtl $_{≫}$( λ ) = ∑ n,γ P eq n log χ q $_{n}$( λ ) (39) − Γ t 2 [ 1 + µ Δ P eq − √ 1 + 2 µ Δ P eq + µ 2 ] ; iλ Δ g → µ = qV 2 π ℏ Γ ∑ γ log [ 1 + ( e iqλ − 1) T γ 2 1 + ( e iqλ − 1) T γ 1 ] (38) χ $^{qtl}$( λ ) = 〈 χ $^{q}$( λ ) 〉 = 〈 e ( qV/ 2 π ℏ ) ∫ t 0 dt ′ ∑$_{γ}$ log[1+( e $^{iqλ}$− 1) T $^{γ}$( t $^{$^{′}$)]}$〉 , (37)