Spectroscopy of Charge Fluctuators Coupled to a Cooper Pair Box

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.