8 We recognize that since we are dealing with simple Laughlin fractions of the FQHE, ν is the inverse of an odd integer and all Keldysh component exhibit power law singularities when the quantity Ω$_{2}$ ± ( ω$_{0}$ + nω$_{AC}$ ) vanishes. As a check, it is possible to recover from these components the previous result for the symmetrized noise: 30 It is also useful to know that the standard property of Keldysh Green’s functions: applies as it should for the double Fourier transform expressions. The relation between the Fourier components of the noise correlator computed in the previous section and the current deriva- tive correlator introduced in Sec. III reads: S$_{sym}$ (Ω$_{1}$ , Ω$_{2}$) = 1 2 ( S $^{+}$− (Ω$_{1}$ , Ω$_{2}$) + S − $^{+}$(Ω$_{1}$ , Ω$_{2}$)) = 2 ( e ∗ ) $^{2}$Γ2 0 4 π $^{2}$a2 $^{+}$∞ ∑ n $^{=}$−∞ $^{+}$∞ ∑ m $^{=}$−∞$^{J$_{n}$}$ ( e ∗ V$_{1}$ ω$_{AC}$ ) J$_{m}$ ( e ∗ V$_{1}$ ω$_{AC}$ ) π Γ(2 ν ) ( a ν$_{F}$ ) 2 ν × [ | Ω$_{1}$ + ω$_{0}$ + nω$_{AC}$ | 2 $^{ν}$− $^{1}$δ (Ω$_{1}$ + Ω$_{2}$ + ( n − m ) ω$_{AC}$ ) + | Ω$_{1}$ − ω$_{0}$ − nω$_{AC}$ | 2 $^{ν}$− $^{1}$δ (Ω$_{1}$ + Ω$_{2}$ − ( n − m ) ω$_{AC}$ ) ] . (44) S $^{++}$(Ω$_{1}$ , Ω$_{2}$) + S −− (Ω$_{1}$ , Ω$_{2}$) = S − $^{+}$(Ω$_{1}$ , Ω$_{2}$) + S $^{+}$− (Ω$_{1}$ , Ω$_{2}$) (45) K ββ $^{′}$(Ω$_{1}$ , Ω$_{2}$) = − Ω$_{1}$Ω$_{2}$ S ββ $^{′}$(Ω$_{1}$ , Ω$_{2}$) . (46) k ββ $^{′}$( T, ω ) = ∫ dω$_{1}$ 2 π $_{e}$− iω$_{1}$$^{T}$Kββ $^{′}$( ω$_{1}$ 2 + ω, ω$_{1}$ 2 − ω ) . (47) C. Current derivative correlators Yet, we need to relate the noise correlator S ββ $^{′}$(Ω$_{1}$ , Ω$_{2}$) to the correlator k ( T, ω ) and ultimately, to its time average K β$_{2}$$^{β$_{1}$}$( ω ) . This is achieved using the relation: So the final result for the four averaged noise derivative cor relators reads: To be complete, we can compute all the Keldysh element in the rotated basis. This is performed in Appendix B. While the advanced and retarded contribution do not bear information on the non equilibrium nature of the transport processes taking place in the mesoscopic devices and therefore in the detector, the Keldysh component: K $^{β}$−$^{β}$( ω ) = 1 τ ∫ τ 0 dT k $^{+}$− ( T, ω ) = ( e ∗ ) $^{2}$Γ2 0 4 π $^{2}$a2 $^{+}$∞ ∑ n $^{=}$−∞$^{J}$$^{2}$ n ( e ∗ V$_{1}$ ω$_{AC}$ ) 1 Γ(2 ν ) ( a ν$_{F}$ ) 2 $^{ν}$ω 2 × [ (1 − β sgn ( ω + ω$_{0}$ + nω$_{AC}$ )) | ω + ω$_{0}$ + nω$_{AC}$ | 2 $^{ν}$− 1 + (1 − β sgn ( ω − ω$_{0}$ − nω$_{AC}$ )) | ω − ω$_{0}$ − nω$_{AC}$ | 2 $^{ν}$− 1 ] (48) K $^{ββ}$( ω ) = 1 τ ∫ τ 0 dT k $^{++}$( T, ω ) = ( e ∗ ) $^{2}$Γ2 0 4 π $^{2}$a2 $^{+}$∞ ∑ n $^{=}$−∞$^{J}$$^{2}$ n ( e ∗ V$_{1}$ ω$_{AC}$ ) 1 Γ(2 ν ) ( a ν$_{F}$ ) 2 ν e − βiπν cos( πν ) ω 2 × [ | ω + ω$_{0}$ + nω$_{AC}$ | 2 $^{ν}$− 1 + | ω − ω$_{0}$ − nω$_{AC}$ | 2 $^{ν}$− 1 ] . (49) ˜ Q K = α 2 ∫ dω 2 π [ ˜ G $^{R}$( ω ) ˜ K $^{R}$( ω ) ˜ G $^{K}$( ω ) + ˜ G $^{R}$( ω ) ˜ K $^{K}$( ω ) ˜ G $^{A}$( ω ) + ˜ G $^{K}$( ω ) ˜ K $^{A}$( ω ) ˜ G $^{A}$( ω ) ] , (50) summarizes such an information in a compact way. We recall that as an alternative to the measurement of the equal time charge correlator, Eqs. (27) and (50) are also likely to be measured directly (resolved in frequency) by an SET device. 28 This completes the calculation of the current derivative correlators. In the following section we continue with the same analysis as with the DC case$^{17}$. That is we use the contour ordered elements of the charge correlator, in particular the − + component evaluated at equal time: in Sec.V we insert the expressions for Eqs. (48) - (49) and discuss the results.