Lattices in Circuit Quantum Electrodynamics: A Platform for Nonequilibrium Quantum Simulation

Abstract

In recent years, superconducting circuits have emerged as a promising platform for quantum computation and quantum simulation. One of the main driving forces behind this progress has been the ability to fabricate relatively low-disorder, low-loss circuits with a high-degree of control over many of the circuit parameters, both in fabrication and in-situ. This coupled with advances in cryogenics and microwave control electronics have significantly improved the rate of progress. The field, which is broadly called circuit quantum electrodynamics (cQED) has become one of the cleanest and most flexible platforms for studying strong interactions between light

and matter.

This thesis builds upon previous work on small-scale interacting cQED lattices and larger-scale, non-interacting superconducting circuit lattices to investigate nonequilibrium quantum simulation using large-scale, interacting cQED lattices. In the first experiment, construct a 72-site 1D chain of CPW cavities with each cavity coupled to a single transmon qubit. Each transmon imparts a strong nonlinearity on the photons in the cavity lattice. This, coupled with the drive and dissipation in the lattice gives rise to a novel dissipative phase transition, where the system exhibits bistability with time-scales many orders of magnitude slower than any intrinsic decay rates in the

system.

The next project in this thesis involves the study of non-Euclidean lattices which can be made from CPW lattices. This work relies on the fact that the frequency of the resonators in the lattice is dependent mostly on the total length of the cavities, not the length between the ends of the cavity. This means that we can form lattices from CPW cavities where the edge distance is not the same but they are the same frequency. The first result in this direction is the fabrication and measurement of a finite piece of a hyperbolic graph, formed from a ‘regular’ tiling of heptagons. This lattice had an effective curvature which is quite large and in principle exhibited a gapped flat band.

Following up on the hyperbolic lattice work, the last result described in this thesis involves the study of exotic new lattices which have band structures that exhibit exotic features such as gapped flat bands and Dirac cones. This work involves looking at the spectra of graphs and their corresponding line-graphs, which are the graphs which are pertinent to the tight-binding Hamiltonian in CPW lattice devices. Here, we derive the mathematical relationship between the spectrum of a graph and its line graph as well as what is known as the split, or subdivision, graph. These operations allow us to exactly maximize the gap between the flat band and the rest of the spectrum for certain line graph lattices.