A diagrammatic approach to compute effective Hamiltonians of driven superconducting circuits: Part I

Superconducting circuits submitted to microwave drives have proven to be a promising means to realize novel parametric interactions described by an effective, time-independent Hamiltonian. A key challenge in controlling such interactions is to develop a systematic and computationally efficient approach for obtaining the Hamiltonian terms beyond the rotating-wave-approximation. We have constructed a diagrammatic book-keeping tool to compute the effective Hamiltonian of a driven nonlinear oscillator to arbitrary order. The physical intuition associated with our Feynman-like  diagrams allows the corresponding Hamiltonian terms to be written down directly, whereas simple counting of topologically distinct diagrams yields the coefficients associated with those terms.  Underlying this diagrammatic approach is a quantum averaging method that treats, in a way we believe is novel,  on equal footing both classical and quantum nonlinear dynamics. In Part I, we introduce the diagrammatic approach through pedagogical examples and discuss its underlying physical intuition.