Tunable coupler for superconducting Xmon qubits: Perturbative nonlinear model

We study a recently demonstrated design for a high-performance tunable coupler suitable for superconducting Xmon and planar transmon qubits. The coupler circuit uses a single flux-biased Josephson junction and acts as a tunable current divider. We calculate the effective qubit-qubit interaction Hamiltonian by treating the nonlinearity of the qubit and coupler junctions perturbatively. We find that the qubit nonlinearity has two principal effects: The first is to suppress the magnitude of the transverse XX coupling from that obtained in the harmonic approximation by about 15%. The second is to induce a small diagonal ZZ coupling. The effects of the coupler junction nonlinearity are negligible in the parameter regime considered.