Deriving Effective Coupling Strength with Born-Oppenheimer Approximation

The development of high-fidelity quantum gates is paramount to the scalability of quantum processors. Tunable couplers have played a key role in the realization of these low error rate quantum gates in superconducting circuits. However, the derivation of the effective coupling strength between the qubits is usually enabled by a Schrieffer-Wolff transformation, which is complex and requires prior-knowledge of the proper generator for the transformation. We propose a simpler method of obtaining this effective coupling using an approach similar to the Born-Oppenheimer approximation.