Optimal non-invasive measurement of Full Counting Statistics by a single qubit

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The complete characterisation of the charge transport in a mesoscopic device is provided by the Full Counting Statistics (FCS) Pₜ(m), describing the amount of charge Q = me transmitted during the time t. Although numerous systems have been theoretically characterized by their FCS, the experimental measurement of the distribution function Pₜ(m) or its moments Qⁿ are rare and often plagued by strong back-action. Here, we present a strategy for the measurement of the FCS, more specifically its characteristic function χ(λ) and moments Qⁿ, by a qubit with a set of different couplings λⱼ, j = 1,,k, k+p, k = n/2, p ≥ 0, to the mesoscopic conductor. The scheme involves multiple readings of Ramsey sequences at the different coupling strengths λⱼ and we find the optimal distribution for these couplings λⱼ as well as the optimal distribution Nⱼ of N = ∑ Nⱼ measurements among the different couplings λⱼ. We determine the precision scaling for the moments Qⁿ with the number N of invested resources and show that the standard quantum limit can be approached when many additional couplings p≫ 1 are included in the measurement scheme.