  We report an experimentally observed anomalous doubly split spectrum and its
split-width fluctuation in an ultrastrongly coupled superconducting qubit and
resonator. From an analysis of Rabimodel and circuit model Hamiltonians, we
found that the doubly split spectrum and split-width fluctuation are caused by
discrete charge hops due to quasiparticle tunnelings and a continuous
background charge fluctuation in islands of a flux qubit. During 70 hours in
the spectrum measurement, split width fluctuates but the middle frequency of
the split is constant. This result indicates that quasiparticles in our device
seem mainly tunnel one particular junction. The background offsetcharge
obtained from split width has the 1/f noise characteristic.
