[Paper Review] Fractal Conductance Fluctuations in a Soft Wall Stadium and a Sinai Billiard

Andrew Sachrajda, Roland Ketzmerick+5 more|arXiv (Cornell University)|Sep 8, 1997

Theoretical and Computational Physics1 references4 citations

TL;DR

This study experimentally demonstrates fractal conductance fluctuations in two quantum billiards—soft-wall stadium and Sinai billiards—fabricated using electrostatic gates on a high-mobility semiconductor heterostructure. The findings confirm theoretical predictions of quantum signatures in classically mixed systems, with the fractal nature of fluctuations providing direct evidence for hierarchical phase space structures at the boundary between regular and chaotic dynamics.

ABSTRACT

Conductance fluctuations have been studied in a soft wall stadium and a Sinai billiard defined by electrostatic gates on a high mobility semiconductor heterojunction. These reproducible magnetoconductance fluctuations are found to be fractal confirming recent theoretical predictions of quantum signatures in classically mixed (regular and chaotic) systems. The fractal character of the fluctuations provides direct evidence for a hierarchical phase space structure at the boundary between regular and chaotic motion.

Motivation & Objective

• To investigate quantum transport properties in classically mixed systems with coexisting regular and chaotic dynamics.
• To explore whether conductance fluctuations in quantum billiards exhibit fractal characteristics as predicted by theory.
• To provide experimental evidence linking quantum signatures to the hierarchical phase space structure at the boundary between regular and chaotic motion.
• To examine the influence of system geometry—specifically soft-wall stadium and Sinai billiard configurations—on conductance fluctuations.
• To validate theoretical predictions of fractal conductance fluctuations in mesoscopic systems using realizable semiconductor nanostructures.

Proposed method

• Fabricated quantum billiards using electrostatic gates on a high-mobility AlGaAs/GaAs heterojunction to define soft-wall stadium and Sinai billiard geometries.
• Performed magnetoconductance measurements under varying magnetic fields to probe quantum transport properties.
• Analyzed conductance fluctuations across a range of magnetic fields to detect self-similar, fractal patterns.
• Used statistical analysis and visualization techniques to identify fractal scaling in the conductance data.
• Compared experimental results with theoretical predictions of fractal conductance fluctuations in systems with mixed phase space.
• Employed high-precision measurements to ensure reproducibility and to minimize noise in fluctuation patterns.

Experimental results

Research questions

• RQ1 Do conductance fluctuations in quantum billiards with mixed classical dynamics exhibit fractal scaling?
• RQ2 Can experimental observation of fractal conductance fluctuations confirm the presence of hierarchical phase space structures?
• RQ3 How do the geometric features of soft-wall stadium and Sinai billiards influence the fractal nature of conductance fluctuations?
• RQ4 To what extent do the experimental results align with theoretical predictions of quantum signatures in classically mixed systems?
• RQ5 What is the role of quantum interference and phase coherence in generating fractal conductance patterns in these nanostructures?

Key findings

• Conductance fluctuations in both the soft-wall stadium and Sinai billiard exhibit a fractal structure, confirmed by self-similar patterns across multiple scales.
• The fractal nature of the fluctuations provides direct experimental evidence for a hierarchical phase space structure at the boundary between regular and chaotic motion.
• The observed fractal conductance fluctuations are reproducible and robust under varying magnetic fields, indicating intrinsic quantum signatures.
• The experimental data for the Sinai geometry were added and validated in the revised version, strengthening the consistency of the fractal behavior across different billiard types.
• The results are in agreement with recent theoretical predictions linking quantum transport fluctuations to classical phase space complexity.
• The study establishes a direct experimental link between quantum conductance fluctuations and the underlying classical dynamics in mixed systems.