Topological Light–Matter Design : Simulations in Polariton Profiles and Ultracold Atom Systems

Rips, Konstantin and Ruostekoski, Janne (2021) Topological Light–Matter Design : Simulations in Polariton Profiles and Ultracold Atom Systems. Masters thesis, Lancaster University.

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Abstract

This work analyses novel schemes for realizing topological light-matter in ultracold atom and polaritonic systems. The observed phenomena can be formulated in the language of space fibrations, homotopy, homology-cohomology duality, and the geometry of gauge fields. The main aspects of this research work consist of: 1. Polaritonic lattice geometries in dimension D=2 can simulate topological insulators based on symmetry class A Hamiltonians by breaking time-reversal symmetry. The competing transverse electric--transverse magnetic (TE-TM) (δt) and Zeeman splitting term (B) yield gapped phases - these are described by Grassmann manifolds G2,6(C) and  G3,6(C) with respect to the spectral-flattened Hamiltonian on the proposed lattice geometry. Valence Bloch bundles over the Brillouin zone are illustrated, and corresponding Chern numbers computed. We discover an index formula which relates the sum of valence band Chern numbers to the index of the projector onto the valence band states. This finding sheds new light on the bulk-boundary correspondence in polaritonic lattices, and allows to extrapolate some properties of single sheet 2D Chern insulators. The formulation is supported by careful numerical analysis of the gapped phases in the (δt, B)--space. Numerical evidence allows us to predict C=2-Chern insulators, which must be accompanied by topologically protected polaritonic edge mode states. 2. Laser--assisted coupling of hyperfine levels in cold Rubidium atoms provides a platform to simulate spherical spaces which host monopoles and carry a fibre bundle structure. Within the pseudo--spin 1 realization one finds non--Landau phase transitions between ground states of Chern numbers C=2,1,0 - we compute the corresponding spin texture configurations. An explanation is offered based on a topology change of the total space of the bundle, since no typical symmetry breaking occurs. Via careful combination of numerical and topological analysis we find transitions between spaces RP3, S3 and S2×U(1). Furthermore, I respond to the recent experiment of Sugawa et al. on a pseudo--spin 3/2 system: An alternative analytical derivation of the ground state second Chern number is provided by exploiting the full symmetry of the artificial non--abelian gauge field geometry. In fact, detailed bundle structure analysis reveals a symplectic geometry with a connection to an instanton type bundle (S7,S4, S3).