Superconducting microwave circuits incorporating nonlinear devices, such as Josephson junctions, are a leading platform 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 mode energy is stored in each element. The EPRs obey universal constraints and are calculated from one electromagnetic-eigenmode simulation. They lead directly to the system quantum Hamiltonian and dissipative parameters. The method provides an intuitive and simple-to-use tool to quantize multi-junction circuits. 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.

结合非线性器件（如约瑟夫森结）的超导微波电路是新兴量子技术的领先平台。增加电路复杂性进一步需要有效的方法来计算和优化多模分布式量子电路中的频谱、非线性相互作用和耗散。在这里，我们提出了一种基于电磁模式下耗散或非线性元件的能量参与比 (EPR) 的方法。EPR 是一个介于 0 和 1 之间的数字，它量化了每个元素中存储的模式能量的多少。EPR 服从通用约束，并通过一个电磁本征模仿真计算得出。它们直接导致系统量子哈密顿量和耗散参数。该方法提供了一种直观且易于使用的工具来量化多结电路。我们在各种约瑟夫森电路上对这种方法进行了实验测试，并证明了非线性耦合和模态哈密顿参数在几个百分点内的一致性，跨越了五个数量级的能量，跨越了十几个样本。