Flux-biased loops including one or more Josephson junctions are ubiquitous elements in
quantum information experiments based on superconducting circuits. These quantum
circuits can be tuned to implement a variety of Hamiltonians, with applications ranging from
decoherence-protected qubits to quantum limited converters and amplifiers. The extrema of
the Hamiltonian of these circuits are of special interest because they govern their low-energy
dynamics. However, the theory of superconducting quantum circuits so far lacks a systematic …