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able to compute explicitly the response of the LC circuit via the charge correlator, and to display the results for a variety of parameters. 

Coupling of the detector to an electromagnetic environment, here modelled by an ohmic bath of oscillators, smoothens the anomalies of the detected signal. The damping parameter ought to be smaller than either the DC frequency or the AC drive frequency in order that the desired effects are observed. This observation is crucial for experiments, and broadens the scope of the results since the electromagnetic environment may also model other backaction effects on the detector. 

The second important and non trivial effect is that the measured noise become negative if we increase the temperature of the detector. Remember that we are considering excess measurements; negative noise thus means that the noise for non-zero DC and AC voltages is smaller that the noise for zero voltage. 

Given the fact that the AC modulation gives rise to satellite peaks at ± ω$_{0}$ + nω$_{AC}$ , we distinguished two limits: ω$_{0}$ > ω$_{AC}$ where the central peak at the DC voltage is surrounded by its satellites, and ω$_{0}$ < ω$_{AC}$ where the satellites of the negative DC voltage frequency can lie in the positive frequency domain of the charge correlator. Both situations can be realized in practice. This brings us to the question about optimizing the detection of the location of the central peak and its satellites. Upon varying the ration between the AC drive amplitude and the AC frequency, we have shown that one can modify the respective amplitude of such peaks. This constitutes an additional knob for detection. 

The present results constitute a step in the direction of fundamental aspects of mesoscopic physics of detection in the time domain. This is an area of growing importance in mesoscopic physics when conventional detection machinery has to be aban- doned, and novel detection schemes for high frequencies adapted to the type of experiments on wishes to perform. Granted, from the experimental point of view, the LC circuit setup which we have presented here may appear a bit naive. In the long run, indeed one should attempt to describe more precisely the connection between the output signal of the mesoscopic circuit and the the transmission lines which are connected to it. This will be the topic of further investigations. 

