Delay-time distribution in the scattering of time-narrow wave packets (II) - Quantum Graphs

TL;DR

This paper investigates the distribution of time delays in scattering processes of ultra short wave packets on complex quantum graphs, providing bounds and classical limits, with applications to specific network models.

Contribution

It extends previous work to compute time-delay distributions in quantum graphs, including bounds, classical limits, and specific decay behaviors for finite networks.

Findings

Time-delay distribution decays exponentially in finite networks.

Classical limit of the distribution is discussed and characterized.

Algebraic decay of quantum time-delay is demonstrated in a model graph.

Abstract

We apply the framework developed in the preceding paper in this series (Smilansky 2017 J. Phys. A: Math. Theor. 50, 215301) to compute the time-delay distribution in the scattering of ultra short radio frequency pulses on complex networks of transmission lines which are modeled by metric (quantum) graphs. We consider wave packets which are centered at high wave number and comprise many energy levels. In the limit of pulses of very short duration we compute upper and lower bounds to the actual time-delay distribution of the radiation emerging from the network using a simplified problem where time is replaced by the discrete count of vertex-scattering events. The classical limit of the time-delay distribution is also discussed and we show that for finite networks it decays exponentially, with a decay constant which depends on the graph connectivity and the distribution of its edge lengths. We illustrate and apply our theory to a simple model graph where an algebraic decay of the quantum time-delay distribution is established.