The quantization of a charge qubit. The role of inductance and gate   capacitance
  The Hamiltonian of a charge qubit, which consists of two Josephson junctions is found within well known quantum mechanical procedure. The inductance of the qubit is included from the very beginning. It allows a selfconsistent derivation of the current operator in a two state basis. It is shown that the current operator has nonzero nondiagonal matrix elements both in the charge and the eigenstate basis. It is also shown that the interaction of the qubit with its own LC resonator has a noticeable influence on the qubit energies. The influence of the junctions asymmetry and the gate capacitance on the matrix elements of the current operator and on the qubit energies are calculated. The results obtained in the paper are important for the circuits where two or more charge qubits are coupled with the aid of inductive coil. 
Ya. S. Greenberg and W. Krech