13 Acknowledgments Appendix A: Keldysh noise correlator calculation We thank Alex Zazunov, Per Delsing and Tim Duty for useful discussions. T.M. and T.J. acknowledge support of an ANR grant “Molspintronics” from the French ministry of research. T.M ., T.J., E.P. and G.F. acklowledge support from a Gallileo project of the “Partenariat Hubert Curien”. E.P. thanks CPT for its hos pitality. T.M. thanks University of Catania for its hospitality. From Eq. (41) we use a standard trigonometric identity in order to factorize the noise into contributions with τ and τ ′ : with: 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 ) ] , (A1) I$_{1}$ (Ω$_{1}$ + Ω$_{2}$ , ω ) = ∫ $^{+}$∞ −∞ dτ ′ e i (Ω$_{1}$+Ω$_{2}$) τ $^{′}$/ $^{2}$cos ( n − m 2 ω$_{AC}$τ ′ ) I ββ ′ 2 (Ω$_{1}$ − Ω$_{2}$ , ω$_{0}$, ω ) = $^{+}$∞ −∞ dτe i $^{(Ω$_{1}$}$−Ω$_{2}$) τ/ $^{2}$e2 νG ββ $^{′}$( τ $^{)}$cos (( ω$_{0}$ + n + m 2 ω$_{AC}$ ) τ ) I$_{3}$ (Ω$_{1}$ + Ω$_{2}$ , ω ) = ∫ $^{+}$∞ −∞ dτ ′ e i (Ω$_{1}$+Ω$_{2}$) τ $^{′}$/ $^{2}$sin ( n − m 2 ω$_{AC}$τ ′ ) I ββ ′ 4 (Ω$_{1}$ − Ω$_{2}$ , ω$_{0}$, ω ) = $^{+}$∞ −∞ dτe i $^{(Ω$_{1}$}$−Ω$_{2}$) τ/ $^{2}$e2 νG ββ $^{′}$( τ $^{)}$sin (( ω$_{0}$ + n + m 2 ω$_{AC}$ ) τ ) , (A2) G $^{ββ}$( τ ) = − ln ( 1 + βi ν$_{F}$ | τ | a ) (A3) G $^{β}$−$^{β}$( τ ) = − ln 1 − βi ν$_{F}$ τ a ) . (A4) I$_{1}$ = 1 2 ( δ (Ω$_{1}$ + Ω$_{2}$ + ( n − m ) ω$_{AC}$ ) + δ (Ω$_{1}$ + Ω$_{2}$ − ( n − m ) ω$_{AC}$ )) . (A5) I$_{3}$ = 1 2 i ( δ (Ω$_{1}$ + Ω$_{2}$ + ( n − m ) ω$_{AC}$ ) − δ (Ω$_{1}$ + Ω$_{2}$ − ( n − m ) ω$_{AC}$ )) . (A6) ∫ $^{+}$∞ −∞ sin( ω$_{0}$τ ) dτ ( a v$_{F}$ − iητ ) µ ≈ iπη sgn( ω$_{0}$ ) | ω$_{0}$ | $^{µ}$− 1 Γ ( µ ) (A7) 1 ∫ $^{+}$∞ −∞ cos( ω$_{0}$τ ) dτ ( a v$_{F}$ − iητ ) µ ≈ π | ω$_{0}$ | $^{µ}$− 1 Γ ( µ ) . (A8) with the elements of the Keldysh Green’s function for the chiral field: I$_{1}$ and I$_{3}$ are expressed in terms of delta functions: Integrals I ββ ′ 2 and I ββ ′ 4 depend explicitly on the Keldysh indices β and β ′ . Here, we need two tabulated integrals: The results for I ββ ′ 2 and I ββ ′ 4 are: I $^{β}$−β 2 = π 2Γ(2 ν ) ( a ν$_{F}$ ) 2 ν [ ( 1 − β sgn ( Ω$_{1}$ − Ω$_{2}$ 2 − ω$_{0}$ − n + m 2 ω$_{AC}$ )) ∣ ∣ ∣$^{Ω$_{1}$}$ − Ω$_{2}$ 2 − ω$_{0}$ − n + m 2 ω$_{AC}$ ∣ ∣$^{2}$$^{ν}$$^{−}$$^{1}$ + ( 1 − β sgn ( Ω$_{1}$ − Ω$_{2}$ 2 + ω$_{0}$ + n + m 2 ω$_{AC}$ )) ∣ ∣ ∣$^{Ω$_{1}$}$ − Ω$_{2}$ 2 + ω$_{0}$ + n + m 2 ω$_{AC}$ ∣ ∣$^{2}$$^{ν}$$^{−}$$^{1}$ ] . (A9)