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 $Y_{\mathrm{in}}(s)$ 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, $s Y_{\mathrm{in}}(s) + 1/L_J = 0$, 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 $Y_{\mathrm{in}}(s)$, (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.

💡 Research Summary

The paper presents a rigorous framework for quantizing superconducting circuits that contain one or more Josephson junctions embedded in arbitrary passive linear environments. The central object is the driving‑point admittance Y_in(s) seen at the junction port, which fully characterizes the linear response of the surrounding network. By applying a Schur‑complement reduction to the nodal admittance matrix, the authors derive an eigenvalue‑dependent boundary condition s·Y_in(s)+1/L_J=0. The roots of this equation give the dressed linear mode frequencies of the combined system, establishing a direct link to Sturm‑Liouville spectral theory.

Because any positive‑real admittance admits a canonical Cauer ladder realization, Y_in(s) can be expressed exactly as a continued‑fraction expansion. This expansion corresponds to a chain of LC sections whose quantization yields a tridiagonal (Jacobi) matrix for the linear sector. The tridiagonal structure enables certified convergence bounds via interlacing theorems, and the continued‑fraction form provides analytic expressions for the resolvent, poles, and residues.

For the nonlinear part, each Josephson junction is treated in the charge basis, where the cosine potential couples only nearest‑neighbor charge states, resulting in an exactly tridiagonal Hamiltonian. In multi‑junction circuits the overall Hamiltonian becomes block‑tridiagonal, which can be solved using matrix continued fractions. This approach works uniformly across coupling regimes—from the dispersive limit (g/|Δ|≲0.1) through strong coupling, ultrastrong coupling (g/ω_r≳0.1), and deep‑strong coupling (g/ω_r≳1)—without invoking the rotating‑wave approximation.

A key theoretical result is the proof that junction participation decays as O(ω_n⁻¹) at high frequencies provided the junction has a finite shunt capacitance. Consequently, perturbative corrections such as Kerr coefficients, Lamb shifts, and dispersive shifts converge without the need for artificial cut‑offs, guaranteeing ultraviolet (UV) regularity of the theory.

The practical quantization procedure consists of four steps: (i) obtain Y_in(ω) from electromagnetic simulation or measurement, (ii) solve the boundary condition to find dressed eigenfrequencies, (iii) synthesize an equivalent passive network via Cauer continued fractions, and (iv) quantize the full Hamiltonian while retaining the exact cosine nonlinearity. This pipeline maps measured or simulated admittance data directly onto Hamiltonian parameters, eliminating phenomenological fitting.

The authors also show how standard circuit‑QED quantities—qubit‑mode coupling g, anharmonicity α, dispersive shift χ—emerge as controlled limits of the exact Hamiltonian, clarifying the approximations involved and providing explicit validity criteria. Dissipative rates follow from the admittance combined with Fermi’s golden rule, reproducing Purcell effects and their multimode generalizations.

Overall, the work bridges classical microwave network synthesis, spectral theory of Jacobi matrices, and circuit quantum electrodynamics, delivering an exact, convergent, and computationally tractable method for multimode quantization of superconducting circuits across all coupling regimes.