LAS DIMENSIONES DEL DESARROLLO

et al. [88] so as to produce Dirac point degeneracies in the bandstructure and a perturbation, the effective external field induced by the helical waveguides, lifts the degeneracies producing a topologically non-trivial result.

Although not explicitly stated, focusing on these Floquet “quasi-energies” is similar to analysing iso-frequency surfaces of photonic crystals. Rechtsman et al. [88] are looking at the form of the quasi-invariants which evolve the field along the structure. The “quasi-energy” surface is in fact a closed surface in wavevector space meaning that the surface is more accurately described as a quasi iso-frequency sur- face. The Floquet quasi-invariants are therefore similar to the iso-frequency surface if one thinks of the latter as a surface ofkz(kx,ky). In the work of Rechtsman et al. [88] a scalar wave equation was employed, implicit in which is an assumption that the polarisation state is completely decoupled from the direction of propagation. This assumption is generally not the case; for instance polarisation mixing occurs for light propagating in anisotropic materials.

In this thesis we shall present work which is complementary to these two diverse realisations of optical topological order. As in the work of Gao et al. [87] we will eschew the reliance on a particular form of lattice patterning to produce de- generacies and instead rely on the intrinsic polarisation degeneracies of anisotropic dielectrics. We will consider both homogeneous materials, as was done in Gao et al. [87], and periodic structures. In the former instance the distinction between our work and that of Gao et al. [87] is that we will consider several different forms of biaxial anisotropy rather than uniaxial anisotropy. In the latter case the distinction, as well as considering more general anisotropic materials, is that the topology of the Brillouin zone torus differs compared to the sphere of propagation directions. Such a difference in topology will require a different polarisation texture in reciprocal space and will therefore have a different topological phase diagram to that of the homogeneous material. The work we present will be complementary to that of Rechtsman et al. [88] in the sense that we will also consider light propagating out of the periodic plane of a two dimensional photonic crystal. The distinction for us is that we will exploit the effective optical spin-orbit coupling of anisotropic materials rather than considering the decoupling of the polarisation and the direction of propagation in reciprocal space.

1.7

Light-Matter Coupling in Semiconductors & Topological

Polaritons

The concepts introduced in the previous sections do not belong to any one branch of physics. They principally emerged in electronic settings but are being rapidly

appropriated by many other diverse settings. In the previous sections we primarily focused on the relevance of topological concepts to optical systems as that shall be one of the two primary settings of the work presented in this thesis. The other setting we shall be considering is that of exciton-polariton systems.

An exciton is an elementary excitation of a semiconductor. This excitation comprises a valence band hole and a conduction band electron which are bound by their Coulomb attraction [17]. There are in fact a number of discrete exciton energy levels just below the semiconductor band-gap as well as the continuum of unbound yet interacting electron and holes above the electronic band-gap [92]. For light impinging upon a semiconductor, with a frequency just below that of the band-gap frequency, the excitons make a significant contribution to the optical response of the material [93]. The effect of the excitons is to heavily modify the linear photonic dispersion resulting in mixed light-matter modes known as polaritons [17]. These polaritons have proved an interesting platform to study myriad phenom- ena. In particular microcavity polaritons resulting from the coupling of heavy-hole quantum-well excitons to photonic cavity modes have allowed polariton condensa- tion [94] and consequently polariton lasing [95], among many other effects. Within the microcavity polariton setting there has also been several proposals [34–36] and a realisation [96] of various topological insulators in two dimensions characterised by non-zero Chern numbers. These schemes have considered the implementation of triangular two dimensional lattice potentials for either the excitonic [34, 36] or the photonic component [35] of the polaritons so as to produce Dirac point in the polaritonic band-structure. The degeneracies at the Dirac points were then lifted due to the Zeeman splitting of the bright excitons arising from the introduction of a magnetic field. The result of this is to achieve the desired set of topologically non-trivial bands.

The study of topological effects in polaritonic systems is however not as mature as in electronic and optical settings. The topological features of three dimensional dispersion relations of polaritons have yet to be explored. Similarly the possibility of polaritons as a platform to study non-Hermitian topological effects has yet to be fully examined. One of the intentions of this thesis is to focus on these inchoate ar- eas. We shall do this by considering the dispersion relations of the magneto-exciton- polaritons of bulk semiconductors. In comparison to the spectrum of quantum well excitons, the exciton spectrum of bulk semiconductors is more complex. This is the case as the excitons formed between the light-hole valence band and the conduction band electrons need to be included for the bulk semiconductor. The resulting dis- persions therefore promise to be rich in structure and consequently an interesting platform to study for three dimensional topological features and, when dissipation is included, non-Hermitian effects also.