  Superconducting microwave circuits incorporating nonlinear devices, such as
Josephson junctions, are one of the leading platforms for emerging quantum
technologies. Increasing circuit complexity further requires efficient methods
for the calculation and optimization of the spectrum, nonlinear interactions,
and dissipation in multi-mode distributed quantum circuits. Here, we present a
method based on the energy-participation ratio (EPR) of a dissipative or
nonlinear element in an electromagnetic mode. The EPR, a number between zero
and one, quantifies how much of the energy of a mode is stored in each element.
It obeys universal constraints--valid regardless of the circuit topology and
nature of the nonlinear elements. The EPR of the elements are calculated from a
unique, efficient electromagnetic eigenmode simulation of the linearized
circuit, including lossy elements. Their set is the key input to the
determination of the quantum Hamiltonian of the system. The method provides an
intuitive and simple-to-use tool to quantize multi-junction circuits. It is
especially well-suited for finding the Hamiltonian and dissipative parameters
of weakly anharmonic systems, such as transmon qubits coupled to resonators, or
Josephson transmission lines. We experimentally tested this method on a variety
of Josephson circuits, and demonstrated agreement within several percents for
nonlinear couplings and modal Hamiltonian parameters, spanning five-orders of
magnitude in energy, across a dozen samples.
