  Superconducting metamaterials are a promising resource for quantum
information science. In the context of circuit QED, they provide a means to
engineer on-chip, novel dispersion relations and a band structure that could
ultimately be utilized for generating complex entangled states of quantum
circuitry, for quantum reservoir engineering, and as an element for quantum
simulation architectures. Here we report on the development and measurement at
millikelvin temperatures of a particular type of circuit metamaterial resonator
composed of planar superconducting lumped-element reactances in the form of a
discrete left-handed transmission line (LHTL). We discuss the details of the
design, fabrication, and circuit properties of this system. As well, we provide
an extensive characterization of the dense mode spectrum in these metamaterial
resonators, which we conducted using both microwave transmission measurements
and laser scanning microscopy (LSM). Results are observed to be in good
quantitative agreement with numerical simulations and also an analytical model
based upon current-voltage relationships for a discrete transmission line. In
particular, we demonstrate that the metamaterial mode frequencies, spatial
profiles of current and charge densities, and damping due to external loading
can be readily modeled and understood, making this system a promising tool for
future use in quantum circuit applications and for studies of complex quantum
systems.
