Fractal Weyl laws in discrete models of chaotic scattering

Abstract

We analyze simple models of quantum chaotic scattering, namely quantized open baker's maps. We numerically compute the density of quantum resonances in the semiclassical régime. This density satisfies a fractal Weyl law, where the exponent is governed by the (fractal) dimension of the set of trapped trajectories. This type of behaviour is also expected in the (physically more relevant) case of Hamiltonian chaotic scattering. Within a simplified model, we are able to rigorously prove this Weyl law, and compute quantities related to the "coherent transport" through the system, namely the conductance and "shot noise". The latter is close to the prediction of random matrix theory.

Invited article in the Special Issue of Journal of Physics A on "Trends in Quantum Chaotic Scattering"

References in corpus (22)

• Shot Noise in Mesoscopic Conductors
• Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering for systems with broken time reversal invariance
• Mesoscopic Transport Through Ballistic Cavities: A Random S-Matrix Theory Approach
• Universal Quantum Signatures of Chaos in Ballistic Transport
• Truncations of random unitary matrices
• Quantum computers in phase space
• Fractal Weyl laws for chaotic open systems
• Quantum-to-classical crossover of quasi-bound states in open quantum systems
• Quantum Mechanics and Semiclassics of Hyperbolic n-Disk Scattering Systems
• Quantum Versus Classical Decay Laws in Open Chaotic Systems
• The Selberg zeta function for convex co-compact Schottky groups
• Classical 1D maps, quantum graphs and ensembles of unitary matrices
• Spectral statistics for unitary transfer matrices of binary graphs
• Numerical Study of Quantum Resonances in Chaotic Scattering
• Dynamical model for the quantum-to-classical crossover of shot noise
• Relaxation process in a regime of quantum chaos
• Decoherence for classically chaotic quantum maps
• Quantum Iterated Function Systems
• Shot noise from action correlations
• Multifractal eigenstates of quantum chaos and the Thue-Morse sequence
• Microscopic Theory for the Quantum to Classical Crossover in Chaotic Transport
• Irreversible Quantum Baker Map

Cited by in corpus (41)

• Leaking Chaotic Systems
• Semiclassical structure of chaotic resonance eigenfunctions
• Spectral problems in open quantum chaos
• Quantum-to-classical correspondence in open chaotic systems
• Resonances in open quantum maps
• Nodal portraits of quantum billiards: Domains, lines, and statistics
• Weyl asymptotics: From closed to open systems
• Distribution of resonances for open quantum maps
• Localization of resonance eigenfunctions on quantum repellers
• Quantum chaotic resonances from short periodic orbits
• Fractal Weyl laws for quantum decay in generic dynamical systems
• Scarring in open quantum systems
• Resonances for open quantum maps and a fractal uncertainty principle
• Lifetime statistics in chaotic dielectric microresonators
• Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems
• Resonance distribution in open quantum chaotic systems
• From open quantum systems to open quantum maps
• On the resonance eigenstates of an open quantum baker map
• Resonance eigenfunction hypothesis for chaotic systems
• Short periodic orbit approach to resonances and the fractal Weyl law
• Distribution of resonances in the quantum open baker map
• On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
• Formation and interaction of resonance chains in the open 3-disk system
• Faster than expected escape for a class of fully chaotic maps
• Nodal domains in open microwave systems
• Quantum chaos in the spectrum of operators used in Shor's algorithm
• Multifractality of open quantum systems
• Classical transients and the support of open quantum maps
• Universal single-mode lasing in fully chaotic two-dimensional microcavity lasers under continuous-wave operation with large pumping power
• Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map
• Transient features of quantum open maps
• Weyl law for open systems with sharply divided mixed phase space
• Semiclassical theory of chaotic quantum resonances
• Resonance spectra for quantum maps of kicked scattering systems by complex scaling
• The modular multiplication operator and the quantized bakers maps
• Resonance states of the three-disk scattering system
• Spectral behavior of contractive noise
• Counting statistics of chaotic resonances at optical frequencies: theory and experiments
• Weyl law for fat fractals
• Resonances in the Two-Centers Coulomb Systems
• Distribution of resonances for hyperbolic surfaces