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In this work, we present a new method, based on Feynman-like diagrams, forcomputing the effective Hamiltonian of driven nonlinear oscillators. Thepictorial structure associated with each diagram corresponds directly to aHamiltonian term, the prefactor of which involves a simple counting oftopologically equivalent diagrams. We also leverage the algorithmic simplicityof our scheme in a readily available computer program that generates theeffective Hamiltonian to arbitrary order. At the heart of our diagrammaticmethod is a novel canonical perturbation expansion developed in phase space tocapture the quantum nonlinear dynamics. A merit of this expansion is that itreduces to classical harmonic balance in the limit of $\hbar\rightarrow0$. Ourmethod establishes the foundation of the dynamic control of quantum systemswith the precision needed for future quantum machines. We demonstrate its valueby treating five examples from the field of superconducting circuits. Theseexamples involve an experimental proposal for the Hamiltonian stabilization ofa three-legged Schr\"odinger cat, modeling of energy renormalization phenomenain superconducting circuits experiments, a comprehensive characterization ofmultiphoton resonances in a driven transmon, a proposal for an novelinductively shunted transmon circuit, and a characterization of classicalultra-subharmonic bifurcation in driven oscillators. Lastly, we benchmark theperformance of our method by comparing it with experimental data and exactFloquet numerical diagonalization.

A diagrammatic method to compute the effective Hamiltonian of driven nonlinear oscillators