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A new electromagnetic cavity structure, a lattice of 3D cavities consistingof an array of posts and gaps is presented. The individual cavity elements arebased on the cylindrical re-entrant (or Klystron) cavity. We show that thesecavities can also can be thought of as 3D split-ring resonators, which isconfirmed by applying symmetry transformations, each of which is anelectromagnetic resonator with spatially separated magnetic and electric field.The characteristics of the cavity is used to mimic phonon behaviour of a onedimensional chain of atoms. It is demonstrated how magnetic field coupling canlead to phonon-like dispersion curves with acoustical and optical branches. Thesystem is able to reproduce a number of effects typical to one-dimensionallattices exhibiting acoustic vibration, such as band gaps, phonon trapping, andeffects of impurities. In addition, quasicrystal emulations predict the resultsexpected from this class of ordered structures. The system is easily scalableto simulate 2D and 3D lattices and shows a new way to engineer arrays ofcoupled microwave resonators with a variety of possible applications to hybridquantum systems proposed.

The 3D Split-Ring Cavity Lattice: A New Metastructure for Engineering Arrays of Coupled Microwave Harmonic Oscillators