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Physica Scripta

Experimental investigation of the scattering fidelity in microwave networks simulating quantum graphs

Michał Ławniczak, Agata Nicolau-Kuklińska, Szymon Bauch, Leszek Sirko

Published Apr 1, 2014• 2 citations
Publisher

Abstract

We present the results of the evaluation of the scattering fidelity in fully connected five vertices microwave networks with absorption. The scattering fidelity was calculated on the basis of the measurements of the one- and two-port scattering matrices S of the networks performed in a frequency range 8–9 GHz. We show that the scattering fidelity depends on the length of the bonds and openness of the networks.

DOI
10.1088/0031-8949/2014/t160/014025

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We present the results of the evaluation of the scattering fidelity in fully connected five vertices microwave networks with absorption. The scattering fidelity was calculated on the basis of the measurements of the one- and two-port scattering matrices S of the networks performed in a frequency range 8–9 GHz. We show that the scattering fidelity depends on the length of the bonds and openness of the networks.
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.
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.
One of the most important characteristics of a quantum graph is the average density of resonances, ρ=(L/π), where L denotes the length of the graph. This is a very robust measure. It does not depend on the number of vertices in a graph and holds also for most of the boundary conditions at the vertices. Graphs obeying this characteristic are called Weyl graphs. Using microwave networks that simulate quantum graphs we show that there exist graphs that do not adhere to this characteristic. Such graphs are called non-Weyl graphs. For standard coupling conditions we demonstrate that the transition from a Weyl graph to a non-Weyl graph occurs if we introduce a balanced vertex. A vertex of a graph is called balanced if the numbers of infinite leads and internal edges meeting at a vertex are the same. Our experimental results confirm the theoretical predictions of [E. B. Davies and A. Pushnitski, Analysis and PDE 4, 729 (2011)] and are in excellent agreement with the numerical calculations yielding the resonances of the networks.
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.
Quantum transport on structured networks is strongly influenced by interference effects, which can dramatically modify how information propagates through a system. We develop a quantum-information-theoretic framework for scattering on graphs in which a full network of connected scattering sites is treated as a quantum channel linking designated input and output ports. Using the Redheffer star product to construct global scattering matrices from local ones, we identify resonant concatenation, a nonlinear composition rule generated by internal back-reflections. In contrast to ordinary channel concatenation, resonant concatenation can suppress noise and even produce super-activation of the quantum capacity, yielding positive capacity in configurations where each constituent channel individually has zero capacity. We illustrate these effects through models exhibiting resonant-tunneling-enhanced transport. Our approach provides a general methodology for analyzing coherent information flow in quantum graphs, with relevance for quantum communication, control, and simulation in structured environments.
The scattering amplitude in simple quantum graphs is a well-known process which may be highly complex. In this work, motivated by the Shannon entropy, we propose a methodology that associates a graph with a scattering entropy, which we call the average scattering entropy. It is defined by taking into account the period of the scattering amplitude which we calculate using the Green's function procedure. We first describe the methodology on general grounds and then exemplify our findings considering several distinct groups of graphs. We go on and investigate other possibilities, one that contains groups of graphs with the same number of vertices, with the same degree, and the same number of edges, with the same length, but with distinct topologies and with different entropies. Another possibility we investigate contains graphs of the fishbone type, where the scattering entropy depends on the boundary conditions on the vertices of degree 1, with the corresponding values decreasing and saturating very rapidly as we increase the number of elementary structures in the graphs.