Optimal effective current operator for flux qubit accounting for inductive effects

An optimal effective current operator for flux qubit has been investigated by taking account of the inductive effects of the circuit loop. The whole system is treated as two interacting subsystems: one is the inductance-free flux qubit consisting of three Josephson junctions and the other a high frequency LC-oscillator. As the composite system hardly affords one excessively high energy LC photon, an effective theory for the inductive flux qubit providing its physical variable operators has been achieved, which can take account of the inductive effects but does not include the additional degree of freedom for the LC-oscillator. Considering the trade-off between simplicity and accuracy, it has been revealed that the optimal effective current operator resulting in an error only on the order of $L^{3/2}$ provides an approximation of high accuracy, which is also verified numerically.