Chaotic behavior of the Compound Nucleus, open Quantum Dots and other nanostructures

Abstract

Abstract. It is well established that physical systems exhibit both ordered and chaotic behavior. The chaotic behavior of nanostructures such as open quantum dots has been confirmed experimentally and discussed exhaustively theoretically. This is manifested through random fluctuations in the electronic conductance. What useful information can be extracted from this noise in the conductance? In this contribution we shall address this question. In particular, we will show that the average maxima density in the conductance is directly related to the correlation function whose characteristic width is a measure of the energy-or applied magnetic field-correlation length. The idea behind the above was originally discovered in the context of the atomic nucleus, a mesoscopic system. Our findings are directly applicable to graphene.

Cross-section fluctuations in open microwave billiards and quantum graphs: The counting-of-maxima method revisited

“…Still, we were not able to provide an analytical derivation. We also compared our experimental and numerical results with predictions for quantum dots [16] associated with a correlation function with the shape of a Lorentzian and a squared Lorentzian, respectively, and found a good agreement for sufficiently large tunneling probabilities, i.e., transmission coefficients Γ 0.4. For small Γ → 0, however, the product of the mean density of maxima and the mean correlation width vanishes in our experiments and RMT simulations and takes a non-zero value in quantum dots.…”

“…[16] the conductance fluctuations where investigated in quantum dots with chaotic behaviour. There, an analytical expression was derived for the product of the mean density of maxima ρ QD ε and the width γ of the generalized conductance correlation function which, similarly to C 21 ( ) in Eq.…”

“…[14,16] an analytical expression was derived for the product of the mean density of maxima ρ QD X and the width X C of the correlation function for the variation of an external perpendicular magnetic field in quantum dots, which takes the shape of a square Lorentzian in the Ericson region,…”

“…This procedure, actually, was proposed at the time [7] as an add-on to the standard analysis of the cross-section fluctuations [6] and was restricted to the Ericson region of strongly overlapping resonances. Recently, the counting-of-maxima method was extended to the regions of isolated and overlapping resonances [14][15][16]. There, an analytical expression for the dependence of the product of the number of maxima and the width of the associated cross-section correlation function as function of the energy or, more generally, of some parameter, on the tunneling probability was derived for the cases, that the latter has a Lorentzian and a squared Lorentzian shape, respectively.…”

“…The aim was to verify the applicability of the method in the region of strongly overlapping resonances [7,8] and to verify the validity of the analytical results of Refs. [14][15][16] experimentally in the regions of isolated and overlapping resonances. Note, that for the largest achieved ratios of the resonance width and spacing the fluctuation properties of the S-matrix elements are already well described by the predictions for the Ericson region [38].…”