Estimate of Minimal Noise in a Quantum Conductor

We study zero temperature fluctuations of charge flow in a metallic loop induced by time dependent magnetic flux, and solve for the optimal way of varying flux in order to minimize noise. Optimal time dependence of the flux is a sum of “solitons,” each corresponding to quantized flux change. Minimal noise coincides with that for the binomial distribution with the number of attempts equal to the number of solitons and with the probabilies defined by the scattering matrix of the system. PACS numbers: 72.10.Bg, 73.50.Fq, 73.50.Td Typeset using REVTEX 1 In the theory of noise in quantum conductors it is usually regarded as a characteristic of transport complementary to conductance. During last years the literature on quantum noise was mostly concentrated on expressing it through transmission coefficients of conduction channel [1,2,3]. Having the noise written as a sum of contributions from different channels allows one to relate it with the conductance for which such representation is known for a while, and also to compare noise in the two limits, quantum and classical. Comparison with classical shot noise leads to the understanding of quantum reduction of the noise, an important effect of Fermi statistics [4,5,6]. Beyond this there are several other interesting properties of quantum noise. One is the issue of probability distribution of charge fluctuations. The question arises naturally from the comparison with the Poisson statistics of classical shot noise. In the quantum case the statistics was found to be binomial with probabilities of outcomes related with transmission coefficients of elastic scattering in the system, and with the number of attempts weakly fluctuating near eV t/h, where V is voltage applied to the system and t is the time of measurement. Another curious property of quantum noise described recently is its phase sensitivity named ‘non-stationary Aharonov-Bohm effect’. As its simplest realization one can consider metallic wire bent in a loop with magnetic flux applied to it. In this setup, when the flux is changed from one constant value to another during fixed time interval τ , the fluctuations of charge measured during much longer time t contain a term proportional to sin(π∆Φ/Φ0) ln t/τ , where ∆Φ is total change of flux and Φ0 = hc/e. Both properties of this term, the logarithmic divergence and the periodicity in ∆Φ can be put in connection with the orthogonality catastrophe theory by relating Φ(t) with the phase of forward scattering of the part of the wire threaded by the flux. In a different setup, involving same loop geometry but the flux Φ(t) varying periodically with time, the phase sensitivity leads to singularities of the noise at integer eV/h̄Ω, where V is dc voltage and Ω is the frequency of the ac flux. The strengths of the singularities are oscillating functions of the flux amplitude and are independent of the frequency Ω. The question we address in this paper is about optimal way of changing flux that minimizes induced noise. It is clear from what has been said that for achieving minimum of the noise one should change the flux by integer amount, ∆Φ = Φt=∞ − Φt=−∞ = nΦ0 , (1) in order to suppress the logarithmic term. However, since for a given ∆Φ the noise depends on the actual function Φ(t), not just on ∆Φ, we have a variational problem to solve for the noise as a functional of the time dependence of the flux. This functional was derived for a single channel ideal conductor with one localized scatterer. For T = 0 it is given by 〈〈Q〉〉 = ge 2π D(1−D) ∫