9 V. RESULTS A. Excess noise in the quantum Hall effect We now discuss the formulas obtained in Sec. IV. In all of the results below, we have checked that when the AC drive frequency is set to 0 , we recover the DC results for the finite frequency noise at ν = 1 [1] and ν = 1 / 3 [31]. For the QPC we focus on the voltage dominated regime where the temperature is taken to zero in the current correlator, but it nevertheless enters the detector response. We start with a discussion of the results for the non symmetrized excess PA noise. We show in Fig. 3 the curves for the current derivative correlator K − $^{+}$(Ω) (see eq. 48), which is the quantity which enters the expression of the measured noise (the charge correlator). This is displayed for two different values of the filling factor ν . We choose for our main interest the Laughlin fraction ν = 1 / 3 , which is in principle the easiest attainable Laughlin fraction of the FQHE in experiments, and ν = 1 , the integer quantum Hall effect case, which here also corresponds to the noise caracteristics of a single channel normal tunneling junction. Here we have chosen the DC voltage so that the central frequency ω$_{0}$ ≡ νeV$_{0}$/ ℏ is larger than the drive frequency ω$_{AC}$ , and the amplitude of the AC voltage ( ω$_{1}$ = νeV$_{1}$/ $^{ℏ}$) is such that ω$_{1}$/ω$_{AC}$ = 1 For ν = 1 / 3 we find divergences for K − $^{+}$(Ω) located at ω$_{0}$ and at sidebands ω$_{0}$ + nω$_{AC}$ . Sidebands with n = ± 1 , ± 2 are visible. K − $^{+}$(Ω) vanishes at zero frequency. For frequencies larger than Ω = ω$_{0}$ + 2 ω$_{AC}$ , this noise derivative correlator seems to be negligible. While the formulas for K − $^{+}$(Ω) show a power law divergence, here one has to add a regularization procedure because strickly speaking, the calculations have been performed in the weak backscattering regime. This means that the differential conductance associated with the tunneling current has to be lower than the conductance quantum (otherwise, one should examine the case of the crossover to the strong back scattering regime). The validity condition of our results have been previously derived in Eq. (24) of Ref. 30. For our purpose, it just implies that the finite frequency PA noise saturates at l ocations Ω = ω$_{0}$ + nω$_{AC}$ . In the integer quantum Hall case ν = 1 , no divergences are found for K − $^{+}$(Ω) . Instead, singularities in the derivative occur for ω$_{0}$ + nω$_{AC}$ , and the current derivative correlator seems again to be negligible again beyond Ω = ω$_{0}$ + 2 ω$_{AC}$ . However it is also interesting to plot K − $^{+}$(Ω) / Ω $^{2}$; in this way we have access to an “averaged” current correlat or (noise) because the term Ω 2 in K − $^{+}$(Ω) is in fact due to derivative operators acting on the current correlator. This is depicted in Fig. 4. For ν = 1 / 3 we again find divergences (at the same locations as for K − $^{+}$(Ω) ). The only noticeable difference with the latter curves is that the averaged noise does not vanish at zero frequency. If one ignores the side bands, the central peak reminds us clearly of the finite frequency non symmetrized noise comput ed recently for a QPC in the FQHE. 31 For ν = 1 the finite frequency noise again exhibits jumps in its deriva tive with respect to frequency, but its behavior is linear between two successive singularities. Thus, for this ratio of frequencies ω$_{0}$/ω$_{1}$ > 1 , the excess noise characteristics resembles the finite frequency noise in the absence of an AC drive: the la ter is (essentially) linear for Ω < ω$_{0}$ and vanishes beyon this. Yet the PA noise does not vanish at Ω = ω$_{0}$ , it shows a singularity in its derivative at its location, together with singularities at 0.0 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 5 6 GLYPH<CapOmega> GLYPH<Slash1> Ω$_{0}$ K GLYPH<Minus>GLYPH<Plus> GLYPH<LParen1> GLYPH<CapOmega> GLYPH<RParen1> ΝGLYPH<Equal> 1 GLYPH<Slash1> 3 0.0 0.5 1.0 1.5 2.0 2.5 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 GLYPH<CapOmega> GLYPH<Slash1> Ω$_{0}$ K GLYPH<Minus>GLYPH<Plus> GLYPH<LParen1> GLYPH<CapOmega> GLYPH<RParen1> ΝGLYPH<Equal> 1 FIG. 3: Current derivative correlator for a QPC in the fractional (left) and integer (right) quantum Hall effect ( ω 0 = 3 ω AC , ω 1 = ω AC and K $^{$_{−}$+}$(Ω) is normalized by e ∗ I B ω 2 $_{0}$)