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Chapman & Hall/CRC eBooks

Application of Microwave Networks to Simulation of Quantum Graphs

Michał Ławniczak, Szymon Bauch, Leszek Sirko

Published Jun 9, 2016• 2 citations
Publisher

Abstract

This chapter demonstrates that using a simple experimental setup consisting of microwave networks, one may successfully simulate quantum graphs. Quantum graphs are often used as idealizations of physical networks in the limit where the widths of the wires are much smaller than their lengths, assuming that the propagating waves remain in a single transversal mode. The analogy between quantum graphs and microwave networks is based on the equivalency of the Schrodinger equation describing a quantum system and the telegraph equation describing an ideal microwave circuit. In the case of two dimensions, the Schrödinger equation for quantum billiards is equivalent to the Helmholtz equation for microwave cavities of corresponding shape. The experimentally measured fully connected hexagon networks were described in numerical calculations by fully connected quantum hexagon graphs with one lead attached to the 6-joint vertex. The microwave networks with broken time reversal symmetry were constructed using microwave circulators.

DOI
10.1201/b20232-26

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This chapter demonstrates that using a simple experimental setup consisting of microwave networks, one may successfully simulate quantum graphs. Quantum graphs are often used as idealizations of physical networks in the limit where the widths of the wires are much smaller than their lengths, assuming that the propagating waves remain in a single transversal mode. The analogy between quantum graphs and microwave networks is based on the equivalency of the Schrodinger equation describing a quantum system and the telegraph equation describing an ideal microwave circuit. In the case of two dimensions, the Schrödinger equation for quantum billiards is equivalent to the Helmholtz equation for microwave cavities of corresponding shape. The experimentally measured fully connected hexagon networks were described in numerical calculations by fully connected quantum hexagon graphs with one lead attached to the 6-joint vertex. The microwave networks with broken time reversal symmetry were constructed using microwave circulators.
We report on high-precision measurements that were performed with superconducting waveguide networks with the geometry of a tetrahedral and a honeycomb graph. They consist of junctions of valency three that connect straight rectangular waveguides of equal width but incommensurable lengths. The experiments were performed in the frequency range of a single transversal mode, where the associated Helmholtz equation is effectively one-dimensional and waveguide networks may serve as models of quantum graphs with the joints and waveguides corresponding to the vertices and bonds. The tetrahedral network comprises T junctions, while the honeycomb network exclusively consists of Y junctions, that join waveguides with relative angles 90^{∘} and 120^{∘}, respectively. We demonstrate that the vertex scattering matrix, which describes the propagation of the modes through the junctions, strongly depends on frequency and is nonsymmetric at a T junction and thus differs from that of a quantum graph with Neumann boundary conditions at the vertices. On the other hand, at a Y junction, similarity can be achieved in a certain frequency range. We investigate the spectral properties of closed waveguide networks and fluctuation properties of the scattering matrix of open ones and find good agreement with random matrix theory predictions for the honeycomb waveguide graph.
Until about 1990 only a very small number of experiments on the quantum mechanics of chaotic systems existed, apart from the early studies of nuclear spectra [Por65]. In this context the experiments on hydrogen atoms in strong microwave fields by Bayfield and Koch [Bay74], and in strong magnetic fields by Welge and his group [Hol86, Mai86] have to be mentioned in particular. The studies of irregularly shaped microwave cavities by Stöckmann and Stein [Stö90] have introduced a new type of quantum chaos research. The microwave billiards and a number of variants to be discussed in this chapter are analogue systems, as they use the equivalence of the Helmholtz equation and the time independent Schrödinger equation. Whether there is a complete correspondence with quantum mechanics or not depends on the respective boundary conditions. As most of the phenomena discussed in the following are common to all types of waves, this does not reduce the conclusiveness of the analogue experiments. Starting with a historical review, the state of the art in billiard experiments is presented with emphasis on a general survey and the technical background. The results and their quantum mechanical implications will be presented later. The hydrogen experiments, too, will be described in the proper context. The discussion of mesoscopic systems is restricted to billiard-like structures such as antidot lattices [Wei91], quantum dots [Mar92], and tunnelling devices [Fro94].
The influence of direct processes on the cross-correlation function c 12 (ν) and the elastic enhancement factor W S , β was studied experimentally using microwave networks in the presence of absorption. Microwave networks simulate quantum graphs with and without time-reversal symmetry. Direct processes exist because, for example, of non-perfect coupling between a studied system and a measuring device. We show that direct processes strongly influence the cross-correlation function c 12 (ν) in contrast to the enhancement factor W S , β , which is not sensitive to these processes. It makes the enhancement factor W S , β an important measure of the chaoticity of a quantum system.
Several key advances in the understanding of quantum chaos have resulted from the study of billiard systems, in which a particle is confined in a two‐dimensional potential well. There has been many numerical studies of wave function characteristics, nodal line and nodal point features and current distributions and correlations. One finds that chaotic states may often be represented by Gaussian random waves which consist of a superposition of monochromatic plane waves traveling in different directions.Direct studies of quantum billiards like 2D mesoscopic ballistic semiconductor quantum dots are, however, unfeasible because of their smallness and because they are normally imbedded deeply in a semiconductor structure. The 2D quantum billiard may, however, be mapped onto a flat microwave cavity, which has allowed for direct measurements of wave functions, current distributions etc. Here we discuss an alternative unconventional realization of a quantum billiard. Thus we outline how electrical networks may be used for fundamental studies of wave function properties and transport in general and, more specifically, their mapping onto open quantum dots. In addition to the scientific case the network model has obvious pedagogical merits.