Line Graph Lattice Implementation in Circuit Quantum Electrodynamics using Three-dimensional Cavities

Abstract

The electronic band structure of a solid describes the allowed energies of the electrons which arise from the periodicity of a solid lattice. Electronic band gaps are interesting because they occur in discrete, periodic materials — a free electron gas would not ex hibit band gaps, but electrons in a periodic potential may have gapped energy bands – so the band gap can be exploited to build semiconductor devices. However, chemistry limits the lattice geometries available in nature and therefore constrains the range of measurable electronic band structures. A similar band structure that can be studied is the photonic energy band that arises from trapping photons in photonic lattices made in circuit quantum electrodynamics. We can observe novel band structures in these artificial lattices and study changes introduced by nonlinearities such as photon-photon interactions mediated by superconducting qubits. A good starting point for such quantum simulation experiments is a lattice without photon-photon interactions. This thesis introduces the field of quantum simulation and the study of the photonic Kagome lattice, which is made of three-dimensional microwave cavity resonators. These cavity resonators are a suitable platform for band structure quantum simulation because of their relatively low loss at room tempera ture, ease of fabrication, and high coupling strength to other nonlinear elements. Two types of microwave cavities are studied: rectangular cavities and quarter-wave coax ial resonators. The resonant modes and coupling parameters of cavity resonators are characterized, and the lattice band structure is explained using line graph theorems. Measurements from the Kagome lattice are reported and the methods of analyzing such measurements are explained. The experiments performed have implications for quantum simulation of solid state physics.