PhD Studentship: Modelling Relativistic Effects in Metamaterials With Quantum Graph Theory

Quantum graph theory (QGT), defined as metric graphs endowed with differential operators, has a long history in mathematics, physics, and both theoretical chemistry and biology. The theory describes the wave properties of networks constructed from edges, endowed with the Schrödinger operator, connected by vertices that serve as scattering sites, given specific boundary conditions. A key advantage to this discrete scattering language is that the eigenvalue condition can be written in terms of a secular equation involving the determinant of a unitary matrix of dimension equal to twice the number of edges on the graph.

Quantum graph models have found a great number of applications including the study of spectral statistics, quantum chaos, modelling the vibrations of coupled plates, formulating quantum random walks, quantum search algorithms, nerve impulse transmission and model various proteins, and molecular structures. Indeed the model has been applied to both finite and infinite lattice systems. with natural applications in photonic and phononic crystals. Quantum Graph Theory has recently found utility in modelling metamaterial structures.

These materials are typically composed of subwavelength building blocks (meta-atoms), often periodically repeated in space, which can have exotic effective material properties in both space and time. Recently it has been shown that in such exotic metamaterials, analogies to black and white hole event horizons can occur, where wave energy is absorbed or created, mediated by an effective moving medium that can be described as a metamaterial with time-varying properties. This PhD proposal seeks to apply QGT to other avenues of relativistic effects, realising curved spacetimes with metamaterials through a discrete graph model.

The material properties of vacuous curved spacetime and of a certain fictitious material, known as the Tamm medium, in flat spacetime are known to be noncovariantly equivalent; various realisations of these nonhomogeneous media have been discussed in the context of metamaterials, often with regions of high anisotropy. Often the nature of these fictitious media can be accommodated by adopting a piecewise homogeneous approach, which is valid for appropriate wavelength regimes where the graph model finds strength.

This project will explore the effective materials properties of various curved-spacetime scenarios that may be impractical to explore by direct methods and provide a route to experimental realisation; QGT has found application in spectral analysis of metamaterial structures and has been experimentally verified in both acoustic and electromagnetic networks.

Advert information