  Motivated by recent surprising experimental results for the noise output of
superconducting microfabricated resonators used in quantum computing
applications and astronomy, we develop a fully quantum theoretical model to
describe quantum dynamics of these circuits. Building on theoretical techniques
from quantum optics, we calculate the noise in the output voltage due to
two-level system (TLS) defects. The theory predicts squeezing for the noise in
the amplitude quadrature with respect to the input noise, which qualitatively
reproduces the noise ellipse observed in experiment. We show that noise
enhancement along the phase direction persists for pump frequencies away from
resonance. Our results also suggest that intrinsic TLS fluctuations must be
incorporated in the model in order to describe the experimentally observed
dependence of the phase noise on input power.
