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Updated: Jul 12, 2026
09:23
Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
Published on: May 30, 2014
Short-time effects on eigenstate structure in sinai billiards and related systems
1Department of Physics and Society of Fellows, Harvard University, Cambridge, Massachusetts 02138, USA.
Summary
Eigenstates in chaotic quantum systems show nonrandom behavior, deviating from random matrix theory predictions. This deviation, particularly in Sinai systems, is linked to classical dynamics and persists even at the quantum limit.
Area of Science:
- Quantum chaos
- Statistical mechanics
- Mathematical physics
Background:
- Schnirelman's theorem addresses ergodicity in quantum eigenstates of chaotic systems.
- Random matrix theory (RMT) provides a benchmark for quantum chaotic systems.
- Previous work showed nonrandom imprints in systems approaching ergodicity slowly or exhibiting eigenstate scarring.
Purpose of the Study:
- To demonstrate the nonrandom character of eigenstates in Sinai-like systems.
- To investigate deviations from RMT predictions in these systems.
- To understand the role of classical dynamics in quantum eigenstate statistics.
Main Methods:
- Investigation of Sinai-type billiard, a quantum map, and a unitary map.
- Definition and theoretical analysis of wave function and long-time transport statistics.
- Comparison of theoretical predictions with numerical data.
Main Results:
- Sinai systems exhibit deficient mixing between channels compared to RMT.
- This mixing deficit scales with |ln(Planck's constant)| as Planck's constant approaches zero.
- The deficit is attributed to the measure-zero set of orbits avoiding the Sinai obstruction.
- Coarse-graining to macroscopic scales recovers Schnirelman's theorem.
Conclusions:
- Eigenstate statistics in Sinai-like systems are not fully captured by RMT.
- Classical dynamics leave nonrandom imprints on quantum eigenstates.
- The study reconciles Schnirelman's theorem with deviations from RMT in specific quantum chaotic systems.
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