Fractal dynamics in chaotic quantum transport

Abstract

Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis-a widely used method in time-series analysis-and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that in turn qualitatively agree with the previous experimental data.

Quantum transport in Sierpinski carpets

“…Phase-coherent electronic transport through the SC, thus, depends noticeably on the carrier energy and on the geometric matching between lead positions and profiles of the extended electronic states. The quantum conductance can reach the maximum value allowed by the number of open channels in the leads [19], depending on the lead positions and their widths, and displays fractal fluctuations [21][22][23][24][25] as a function of energy, in the absence of a magnetic field. At odds with the vast majority of the literature on fractal conductance fluctuations (CFs), which considers geometrically simple structures such as billiards, here we find that, in a SC, the fractal dimension of the sample determines the fractal dimension of the CFs [26].…”

Effects of scarring on quantum chaos in disordered quantum wells

“…13 Recently, DFA has been used in the classification of fractal magnetoconductance in chaotic quantum cavities. 14 We show that PI scarring is connected to transitions in the energy level statistics. In most cases, the scars can be detected from their surroundings in the mixed region between regular and chaotic eigenvalue statistics.…”

Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves

“…Substituting Eqs. (8), (9), and (13) into Eq. (12) and using the generating function of the Hermite polynomials [50], the coefficients can be derived as a m = (m o ) Using the property of the Schrödinger coherent state and the expressions for the coefficients a m , b n , and c , the intensity of the wave-packet state I (±) =| (±) (x,y,z,t)| 2 can be derived as…”

“…Starting from Mandelbrot's seminal discovery [1], selfsimilar and fractal structures have been observed in a variety of phenomena in nature [2] and have also been found in many branches of physics [3][4][5]. One of the prominent examples in quantum systems is the fractal conductance fluctuations in gold nanowires and in mesoscopic electron billiards [6][7][8][9][10]. Another prominent example of the self-similar phenomena is the plateau formation in the transverse Hall-resistance curve of a two-dimensional (2D) electron system at low temperatures in the presence of a strong perpendicular magnetic field, known as the quantum Hall effect [11,12].…”