7 The final result for the real time noise correlator is then: where we introduced the chiral green function of the bosonic fields: The double Fourier transform of this quantity, which will allow to relate it to the noise correlator, reads: We now specify the periodic voltage modulation, which allows to write the tunneling amplitude in terms of a series of Bessel functions J$_{n}$ : which gives the Fourier transform of non-symmetrised noise: S ββ $^{′}$( $_{t, t}$′$_{) =}$ ( e ∗ ) 2 4 π $^{2}$a$_{2}$e 2 νG ββ $^{′}$( $^{t}$−t $^{′}$) ( A ( t ) $_{A}$∗$_{(}$$_{t}$ ′$_{) +}$ $_{A}$∗$_{(}$$_{t}$$_{)}$$_{A}$$_{(}$$_{t}$ ′$_{))}$ , (36) G ββ $^{′}$( t, t ′ ) = 〈 T$_{K}$ { φ$_{r}$ ( t $^{β}$) φ$_{r}$ ( t ′ β $^{′}$) }〉 − 1 2 〈 T$_{K}$ { φ$_{r}$ ( t $^{β}$) $^{2}$}〉 − 1 2 〈 T$_{K}$ { φ$_{r}$ ( t ′ β $^{′}$) $^{2}$}〉 . (37) S ββ $^{′}$(Ω$_{1}$ , Ω$_{2}$) = ∫ ∫ $_{dtdt}$′$_{e}$ i (Ω$_{1}$ t +Ω$_{2}$ t $^{$^{′}$)}$S ( $_{t, t}$′$_{)}$ . (38) A ( t ) = Γ$_{0}$ $^{+}$∞ ∑ n $^{=}$−∞$^{e}$$^{i}$$^{(}$$^{ω$_{0}$}$$^{+}$$^{nω$_{AC}$}$$^{)}$$^{t}$$^{J$_{n}$}$ ( e ∗ V$_{1}$ ℏ ω$_{AC}$ ) , (39) S ββ $^{′}$(Ω$_{1}$ , Ω$_{2}$) = ( e ∗ ) $^{2}$Γ2 0 2 π $^{2}$a2 $^{+}$∞ ∑ n $^{=}$−∞ $^{+}$∞ ∑ m $^{=}$−∞$^{J$_{n}$}$ ( e ∗ V$_{1}$ ω$_{AC}$ ) J$_{m}$ ( e ∗ V$_{1}$ ω$_{AC}$ ) × ∫ ∫ $_{dtdt}$′$_{e}$ i (Ω$_{1}$ t +Ω$_{2}$ t $^{$^{′}$)}$e 2 νG ββ $^{′}$( t,t $^{$^{′}$)}$cos( ω$_{0}$ ( t − $_{t}$′$_{) +}$ ω$_{AC}$ ( nt − $_{mt}$′$_{))}$ . (40) S ββ $^{′}$(Ω$_{1}$ , Ω$_{2}$) = 2 ( e ∗ ) $^{2}$Γ2 0 2 π $^{2}$a2 $^{+}$∞ ∑ n $^{=}$−∞ $^{+}$∞ ∑ m $^{=}$−∞$^{J$_{n}$}$ ( e ∗ V$_{1}$ ω$_{AC}$ ) J$_{m}$ ( e ∗ V$_{1}$ ω$_{AC}$ ) × ∫ ∫ $_{dτdτ}$′$_{e}$ i $^{(Ω$_{1}$}$−Ω$_{2}$) τ/ $^{2}$e i (Ω$_{1}$+Ω$_{2}$) τ $^{′}$/ $^{2}$e2 νG ββ $^{′}$( τ $^{)}$cos (( ω$_{0}$ + n + m 2 ω$_{AC}$ ) τ + n − m 2 ω$_{AC}$$_{τ}$′ ) . S ββ $^{′}$(Ω$_{1}$ , Ω$_{2}$) = 2 ( e ∗ ) $^{2}$Γ2 0 2 π $^{2}$a2 $^{+}$∞ ∑ n $^{=}$−∞ $^{+}$∞ ∑ m $^{=}$−∞$^{J$_{n}$}$ ( e ∗ V$_{1}$ ω$_{AC}$ ) J$_{m}$ ( e ∗ V$_{1}$ ω$_{AC}$ ) [ I$_{1}$ (Ω$_{1}$ + Ω$_{2}$ , ω ) I ββ ′ 2 (Ω$_{1}$ − Ω$_{2}$ , ω$_{0}$, ω ) − I$_{3}$ (Ω$_{1}$ + Ω$_{2}$ , ω ) I ββ ′ 4 (Ω$_{1}$ − Ω$_{2}$ , ω$_{0}$, ω ) ] . (41) ′ ′ Next, it is convenient to perform a change of variable τ = t − t ′ and τ ′ = t + t ′ : Using standard trigonometric identities, one can write this expression as a product of separate integrals over τ and τ ′ . Integrals over τ contain the (zero temperature) Green’s function of the chiral fields and can be expressed in terms of Gamma function. The result has the form: The integrals I$_{1}$ , I ββ 2 , I$_{3}$ , I ββ 4 are defined and computed in the Appendix. The final result for t he 4 Keldysh matrix elements of the noise correlator is: S $^{β}$−$^{β}$(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 $^{ν}$× [ (1 − β sgn (Ω$_{1}$ + ω$_{0}$ + nω$_{AC}$ )) | Ω$_{1}$ + ω$_{0}$ + nω$_{AC}$ | 2 $^{ν}$− 1 δ (Ω$_{1}$ + Ω$_{2}$ + ( n − m ) ω$_{AC}$ ) + (1 − β sgn(Ω$_{1}$ − ω$_{0}$ − nω$_{AC}$ )) | Ω$_{1}$ − ω$_{0}$ − nω$_{AC}$ | 2 $^{ν}$− 1 δ (Ω$_{1}$ + Ω$_{2}$ − ( n − m ) ω$_{AC}$ ) ] , (42) S $^{ββ}$(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 e − βiπν cos( πν ) × × [ | 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$_{AC}$ ) ] . (43)