the specific case of a Josephson junction atom coupled to a multimode resonator in the Rabi model. More recently [@Adrian-Long] studied the convergence of the bare couplings between the superconducting qubits and multimode resonators in various general coupling configurations.

Combination of lumped element circuit quantization methods [@Devoret-Les-Houches; @BKD; @Burkard] with classical circuit synthesis techniques [@Foster; @Brune; @Newcomb] resulted in “blackbox quantization” methods [@BBQ-Yale; @Brune-Quantization; @Solgun] which allowed extraction of the parameters in the quantum Hamiltonian models of the superconducting circuits from the electromagnetic finite-element simulations. The simulations correspond to the linear passive part of the circuits which is usually a distributed microwave structure as seen looking into the ports defined across the Josephson junctions. Although such an approach allows an accurate treatment of very general structures consisting possibly of multiple microwave modes simulation of large multi-qubit devices might quickly become computationally demanding.

Following a similar approach we show here that for superconducting processors consisting of low anharmocity qubits like transmons the dispersive interaction parameters such as exchange coupling and Purcell decay rates of the qubits and their coupling to the voltage drives are related in a simple way to the microwave impedance response functions as seen at the “qubit ports” 