Fractal dynamics in chaotic quantum transport

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.

Figure 1

Snapshots of the electron density in the model stadium system (see the text) during a transport simulation with the magnetic flux $\Phi /{\Phi }_0=20$. The input and output leads extend further to the left and right.

Figure 2

Transmission factor as a function of time and magnetic flux through the stadium.

Figure 3

Transmission factor as a function of the magnetic flux at $t=1.4$. The inset shows the scaling exponent $\alpha =1.46$ obtained from the DFA analysis.

Figure 4

(a) Fractal dimension $D$ calculated from DFA with the relation $D=2-\alpha /2$ and from the variation method, respectively, during the time propagation. Note that the fractal structure is developed only at $t\gtrsim 1$. (b) Time development of the error in the fitting procedure at $t=0.7,...,1.4$ (see the text).