  We analyze the behavior of a Cooper Pair Box (CPB) that is coupled to charge
fluctuators that reside in the dielectric barrier layer in the box's
ultra-small tunnel junction. We derive the Hamiltonian of the combined system
and find the coupling between the CPB and the fluctuators as well as a coupling
between the fluctuators that is due to the CPB. We then find the energy levels
and transition spectrum numerically for the case of a CPB coupled to a single
charge fluctuator, where we treat the fluctuator as a two-level system that
tunnels between two sites. The resulting spectra show the usual transition
spectra of the CPB plus distinctive transitions due to excitation of the
fluctuator; the fluctuator transitions are 2-e periodic and resemble saw-tooth
patterns when plotted as a function of the gate voltage applied to the box. The
combined CPB fluctuator spectra show small second-order avoided crossings with
a size that depends on the gate voltage. Finally, we discuss how the
microscopic parameters of the model, such as the charge times the hopping
distance, the tunneling rate between the hopping sites, and the energy
difference between the hopping sites, can be extracted from CPB spectra, and
why this yields more information than can be found from similar spectra from
phase qubits.
