Quantum Graphology

Abstract

We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wave function statistics. This operator is the analogue of the classical evolution operator on the graph. It allows us to establish a connection between the corresponding periodic orbits and the statistical properties of eigenvalues and eigenfunctions. Specifically, for the energy-averaged spectral form factor we derived an exact combinatorial expression which illustrate the role of correlations between families of isometric orbits. We also show that enhanced wave function localization due to the presence of short unstable periodic orbits and strong scarring can rely on completely different mechanisms.

Experimental and Numerical Studies of One-Dimensional and Three-Dimensional Chaotic Open Systems

“…Among the systems modeled by quantum graphs are, e.g., electromagnetic optical waveguides [2,3], mesoscopic systems [4,5], quantum wires [6,7] and excitation of fractons in fractal structures [8,9]. Recent works have shown that quantum graphs provide an excellent system for studying quantum chaos [10][11][12][13][14][15][16][17][18][19] and localization phenomena [20][21][22]. Quantum graphs with absorption, more realistic but more complicated systems, have been recently studied in [16,17,[23][24][25].…”

Experimental Investigation of Reflection Coefficient and Wigner's Reaction Matrix for Microwave Graphs

“…Among the systems modeled by quantum graphs are, e.g., electromagnetic optical waveguides [3,4], mesoscopic systems [5,6], quantum wires [7,8] and excitation of fractons in fractal structures [9,10]. Recent works have shown that quantum graphs provide an excellent system for studying quantum chaos [11][12][13][14][15][16][17][18][19][20][21]. Quantum graphs with external leads (antennas) have been analyzed in detail in [16,17].…”