Exact Multimode Quantization of Superconducting Circuits via Boundary Admittance and Continued Fractions

Accurate extraction of linearized quantum circuit models from electromagnetic simulations is essential for the design of superconducting circuits. We present a quantization framework based on the driving-point admittance Y_in(s) seen by a Josephson junction embedded in an arbitrary passive linear environment. By taking the Schur complement of the nodal admittance matrix, we show that the linearized coupled system obeys an eigenvalue-dependent boundary condition, s Y_in(s) + 1/L_J = 0, whose roots determine the dressed linear mode frequencies. This boundary condition admits an exact continued fraction representation: any positive-real admittance can be realized as a canonical Cauer ladder, yielding a tridiagonal (Jacobi) structure that enables certified convergence bounds via interlacing theorems.For the full nonlinear Hamiltonian, we treat Josephson junctions in the charge basis, where each cosine potential is exactly tridiagonal, and couple them to cavity modes in the Fock basis; in the general multi-junction case this yields a block-tridiagonal structure solvable by matrix continued fractions, enabling systematic diagonalization across all coupling regimes from dispersive through ultrastrong and deep strong coupling. The resulting quantization procedure is: (i) compute or measure Y_in(s), (ii) solve the boundary condition to obtain dressed eigenfrequencies, (iii) synthesize an equivalent passive network, and (iv) quantize while retaining the full cosine nonlinearity of the Josephson junction. We prove that junction participation decays as mathcal{O}\(ω_n^-1\) at high frequencies for any circuit with finite shunt capacitance, ensuring ultraviolet convergence of perturbative corrections without imposed cutoffs.