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Related Experiment Video

Updated: Jul 4, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Quantum properties of irrational triangular billiards.

F M de Aguiar¹

¹Departamento de Física, Universidade Federal de Pernambuco, Recife, PE, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|June 4, 2008

Summary

This study links triangle irrationality (h) to quantum billiard energy levels. A peak in irrationality at N=10 correlates with Gaussian orthogonal ensemble (GOE) behavior, transitioning to integrable limits at higher N.

Area of Science:

Number Theory
Quantum Chaos
Mathematical Physics

Background:

Triangles with consecutive integer sides (N, N+1, N+2) exhibit irrational angles for N > 3.
Quantum billiards provide a model system for studying spectral statistics.

Purpose of the Study:

Investigate the energy level statistics of quantum billiards derived from a one-parameter family of irrational triangles.
Quantify triangle irrationality using a parameter 'h' derived from rational approximations and Hurwitz's theorem.
Explore the relationship between triangle irrationality and spectral fluctuations.

Main Methods:

Defined a parameter 'h' to quantify triangle irrationality.
Numerically calculated energy level statistics (spacing distribution p(s) and spectral rigidity Delta(3)(L)) for quantum billiards.

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Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

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Analyzed the behavior of 'h' and spectral statistics as N varies.

Main Results:

A local maximum in 'h' at N=10 coincided with agreement to Gaussian orthogonal ensemble (GOE) spectral fluctuations.
As N increased, 'h' decreased, and statistics diverged from GOE.
For N > 120, structures appeared in p(s), and for N ~ 180, gaps indicated a crossover towards integrable behavior.

Conclusions:

Triangle irrationality parameter 'h' closely tracks quantum billiard spectral dynamics.
The study reveals a transition from chaotic (GOE-like) to integrable spectral statistics in this family of triangles.
The findings establish a connection between number-theoretic properties of triangles and quantum chaos phenomena.