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arXiv:1709.08845 · Journal of Physics A Mathematical and Theoretical

Delay-time distribution in the scattering of time-narrow wave packets (II)—quantum graphs

Uzy Smilansky, Holger Schanz

Published Dec 12, 2017• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aaa0df

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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.
Abstract This is the first of two subsequent publications where the probability distribution of delay-times in scattering of wave packets is discussed. The probability distribution is expressed in terms of the on-shell scattering matrix, the dispersion relation of the scattered beam and the wave packet envelope. In the monochromatic limit (poor time resolution) the mean delay-time coincides with the expression derived by Eisenbud and Wigner and generalized by Smith more than half a century ago. In the opposite limit, and within the semi-classical approximation, the resulting distribution coincides with the result obtained using classical mechanics or geometrical optics. The general expression interpolates smoothly between the two extremes. An application for the scattering of electromagnetic waves in networks of RF transmission lines will be discussed in the next paper to illustrate the method in an experimentally relevant context.
A numerical solution to the time evolution equation of the Wigner distribution function (WDF) with an accuracy necessary to simulate the passage of a wave packet past a barrier is developed, where quantum effects require high accuracy and fine discretization. A wave packet incident on a barrier, a portion of which tunnels through, demonstrates behavior that can define various characteristic transmission and reflection delay (TARD) times useful in the simulation of electron emission. A model for the TARD times is proposed that relies only on the asymptotic maxima of the position $\ensuremath{\rho}(x,t)$ and wave number $\ensuremath{\rho}(k,t)$ densities given by the WDF and applied to a ballistic trajectory model for the (faster) transmitted and (slower) reflected parts. The dependence of the TARD times on barrier width, symmetry, and abruptness is analyzed. For symmetrical barriers with characteristics similar to field emission barrier heights and widths, TARD times are on the order of a fraction of a femtosecond. The TARD times for when tunneling predominates are contrasted to tunneling times in the literature. Use of the TARD times in simulations of field emission in nanogap devices or to model ultrashort pulses generated under rapidly changing conditions for electron sources are proposed.
Two different formulations of the problem of time-delay caused by scattering from an oscillating center of force are studied. In the first one, appropriate for local interactions, a semi-classical time-dependent solution of the Schrödinger equation is obtained by writing the wave function as a solution of two coupled equations one of which is the Hamilton–Jacobi equation. These equations are then solved using the narrow wave packet approximation. The classical motion of the center of this wave packet is found for two specific cases, a square well and a Morse potential. In the second formulation a time-independent many-channel solution of the Schrödinger equation is obtained for an exactly solvable quantum-mechanical problem with a potential consisting of a separable attractive part which oscillates and a fixed rigid wall. The results show that the time-delay is extremely sensitive to the values of the amplitude and the frequency of oscillations, and also to the energy of the incident particle.
The scattering matrix S linearly relates the vector of incoming waves to outgoing wave excitations, and contains an enormous amount of information about the scattering system and its connections to the scattering channels. Time delay is one way to extract information from S, and the transmission time delay τ_{T} is a complex (even for Hermitian systems with unitary scattering matrices) measure of how long a wave excitation lingers before being transmitted. The real part of τ_{T} is a well-studied quantity, but the imaginary part of τ_{T} has not been systematically examined experimentally, and theoretical predictions for its behavior have not been tested. Here we experimentally test the predictions of Asano et al. [Nat. Commun. 7, 13488 (2016)NCAOBW2041-172310.1038/ncomms13488] for the imaginary part of transmission time delay in a nonunitary scattering system. We utilize Gaussian time-domain pulses scattering from a two-port microwave graph supporting a series of well-isolated absorptive modes to show that the carrier frequency of the pulses is changed in the scattering process by an amount in agreement with the imaginary part of the independently determined complex transmission time delay, Im[τ_{T}], from frequency-domain measurements of the subunitary S matrix. Our results also generalize and extend those of Asano et al., establishing a means to predict pulse propagation properties of non-Hermitian systems over a broad range of conditions.
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.