Hyperbolic and Flat-Band Lattices in Circuit QED

The field of circuit QED has emerged as a rich platform for both quantum computation and quantum simulation. The unique deformability of coplanar waveguide (CPW) microwave resonators enables realization of artificial photonic materials with novel lattice structures. We will present two classes of such examples.  First, we will show that these techniques can be used to produce periodic lattices in a hyperbolic space of constant negative curvature. Second, we present examples of Euclidean lattices which exhibit gapped flat bands. Because CPW resonators are one-dimensional objects, the lattices formed are line graphs. We will show that line graphs lead naturally to flat bands and that criteria for when they are gapped can be derived from graph-theoretic techniques. The resulting gapped flat-band lattices are typically forbidden in standard atomic crystallography, but readily realizable in superconducting circuits.  With the addition of high-kinetic-inductance materials or transmon qubits these systems will constitute a table-top simulator of interacting and quantum mechanical particles in strong curvature.