Coupling and dephasing in Josephson charge-phase qubit with radio frequency readout

The Cooper pair box qubit of SQUID configuration enables the readout of the qubit states by probing the effective Josephson inductance of the SQUID. This is realized by coupling the qubit to a tank circuit which induces a small alternating supercurrent in the SQUID loop. The effect of a small (but finite) geometrical inductance of the loop on the eigenstates of the system is figured out. The effect of qubit dephasing due to quadratic coupling to the tank circuit is evaluated. It is shown that the rate of dephasing in the vicinity of the magic points is relatively low unless the Josephson junctions forming the qubit are rather dissimilar. In the vicinity of the avoided level-crossing point such dephasing is always significant.