A variation on Smilansky's model

Italo Guarneri

Abstract Some variants of Smilansky’s model of a particle interacting with harmonic oscillators are examined in the framework of scattering theory. A dynamical proof is given of the existence of wave operators. Analysis of a classical version of the model provides a transparent picture for the spectral transition to which the quantum model owes its renown, and for the underlying dynamical behaviour. The model is thereby classified as an extreme case of chaotic scattering, with aspects related to wave packet reduction and irreversibility.

Pierre Meystre

Fabio Franchini

Sven Gnutzmann∥, Uzy Smilansky

Abstract We generalize the scattering approach to quantum graphs to quantum graphs with piecewise constant potentials and multiple excitation modes. The free single-mode case is well-known and leads to the trace formulas of Roth (1983 C. R. Acad. Sci., Paris I 296 793–5), Kottos and Smilansky (1997 Phys. Rev. Lett. 79 4794). By introducing an effective reduced scattering picture we are able to propose new exact trace formulas in the more general settings. The latter are derived and discussed in details with some numerical examples for illustration. Our generalization is motivated by both experimental applications and fundamental theoretical considerations. The free single-mode quantum graphs are an extreme idealization of reality that, due to the simplicity of the model allows to understand a large number of generic or universal phenomena. We lift some of this idealization by considering the influence of evanescent modes that only open above threshold energies. How to do this theoretically in a closed model in general is a challenging question of fundamental theoretical interest and we achieve this here for quantum graphs.

H. Girin, X. Chécoury, B. Odouard +5 more

We present a versatile photonic waveguide network platform implementing quantum (wave) graphs, proposed by Kottos \& Smilansky [Phys. Rev. Lett. \textbf{79}, 4794 (1997)] to investigate ray-wave correspondence and quantum chaos. Realized on a silicon-on-insulator chip at telecom wavelengths, it enables a unique capability: the direct, non-invasive imaging of individual optical wave functions with unprecedented resolution via third-harmonic generation in silicon, which we use to study the localization of intensity distributions. Furthermore, by investigating two graphs with contrasting classical dynamics -- one strongly chaotic, the other one ergodic, but non-mixing -- we reveal strikingly different spectral statistics, in quantitative agreement with random matrix theory and a minimal unitary quantum-map model. This establishes silicon photonics networks as a versatile and scalable platform for investigating quantum chaos and non-linear graphs, and paves the way for optical quantum computing in complex networks.

Bruce W. Shore