  We have observed a few distinct anomalous avoided level crossings and voltage
dependent transitions in the excited state spectrum of an Al/AlOx/Al
Cooper-pair box (CPB). The device was measured at 40 mK in the 15 - 50 GHz
frequency range. We find that a given level crosses the CPB spectrum at two
different gate voltages; the frequency and splitting size of the two crossings
differ and the splitting size depends on the Josephson energy of the CPB. We
show that this behavior is not only consistent with the CPB being coupled to
discrete charged "two-level" quantum systems which move atomic distances in the
CPB junctions but that the spectra provide new information about the
fluctuators, which is not available from phase qubit spectra of anomalous
avoided levels. In particular by fitting a model Hamiltonian to our data, we
extract microscopic parameters for each fluctuator, including well asymmetry,
tunneling amplitude, and the minimum hopping distance for each fluctuator. The
tunneling rates range from less than 3.5 to 13 GHz, which represent values
between 5% and 150% of the well asymmetry, and the dipole moments give a
minimum hopping distance of 0.3 to 0.8 Anstrom. We have also found that these
discrete two-level systems have a pronounced effect on the relaxation time (T1)
of the quantum states of the CPB and hence can be a source of dissipation for
superconducting quantum bits.
