Exact Multimode Quantization of Superconducting Circuits via Boundary Admittance

Abstract

We show that the Schur complement of the nodal admittance matrix, which reduces a multiport electromagnetic environment to the driving-point admittance $Y_{\mathrm{in}}(s)$ at the Josephson junction, naturally leads to an eigenvalue-dependent boundary condition determining the dressed mode spectrum. This identification provides a four-step quantization procedure: (i) compute or measure $Y_{\mathrm{in}}(s)$, (ii) solve the boundary condition $sY_{\mathrm{in}}(s) + 1/L_J = 0$ for dressed frequencies, (iii) synthesize an equivalent passive network, (iv) quantize with the full cosine nonlinearity retained. Within passive lumped-element circuit theory, we prove that junction participation decays as, we prove that junction participation decays as $O(ω_n^{-1})$ at high frequencies when the junction port has finite shunt capacitance, ensuring ultraviolet convergence of perturbative sums without imposed cutoffs. The standard circuit QED parameters, coupling strength $g$, anharmonicity $α$, and dispersive shift $χ$, emerge as controlled limits with explicit validity conditions.

Figures (10)

• Figure 1: Overview of the boundary condition framework. The measured or simulated admittance $Y_{\mathrm{in}}(\omega)$ is reduced to the junction port via Schur complement, synthesized into an equivalent circuit, and quantized to yield the exact Hamiltonian. Physical observables are extracted by numerical diagonalization or perturbative expansion. The Hamiltonian retains the full cosine nonlinearity; standard circuit QED emerges as the dispersive limit. The dashed box indicates that the framework applies uniformly from dispersive through deep strong coupling, different regimes are distinguished by which terms in the cosine expansion are numerically significant, not by modifications to the quantization procedure.
• Figure 2: Schur complement reduction. (a) A multiport network with junction port $J$ and internal nodes $\bar{J}$ (gray dots). (b) The Schur complement eliminates internal nodes, yielding the scalar driving-point admittance $Y_{\mathrm{in}}(s)$ at the junction. The positive-real property is preserved under this reduction.
• Figure 3: Quarter-wave transmission line resonator coupled to a transmon qubit. The resonator is shorted at $x=0$ and couples through capacitance $C_g$ to the transmon at $x=\ell$. The transmon consists of a Josephson junction with energy $E_J$ shunted by capacitance $C_J$.
• Figure 4: Graphical solution of the boundary condition equation $sY_{\mathrm{in}}(s) + 1/L_J = 0$. The blue curve shows the dimensionless product $\omega L_J \cdot \mathrm{Im}[Y_{\mathrm{in}}(j\omega)]$ as a function of frequency normalized to the bare resonator frequency $\omega_r$. Dressed mode frequencies occur where this curve intersects unity (red dashed line). For a transmon capacitively coupled to a resonator, two solutions emerge: the qubit mode $\omega_q$ at lower frequency and the dressed resonator mode $\omega_r'$ pushed slightly below the bare resonator frequency. The vertical asymptote at $\omega_r$ reflects the pole in the resonator admittance. Circuit parameters: $E_J/h = 12$ GHz, $C_J = 85$ fF, resonator at 7 GHz with $C_r = 200$ fF, and coupling capacitance $C_g = 8$ fF.
• Figure 5: Foster canonical form for a lossless positive-real admittance. The circuit consists of parallel branches connected to a common input node: a direct capacitor $C_\infty$ to ground capturing the high-frequency asymptotic behavior, an inductor $L_0$ to ground present when there is a pole at dc, and series $LC$ tanks tuned to the resonant frequencies $\omega_k = 1/\sqrt{L_k C_k}$. Each tank represents an environmental mode that can exchange energy with the junction.

Theorems & Definitions (9)

• Proposition 3.1: Positive-Real Preservation
• proof
• Theorem 3.2: Boundary Condition Equivalence
• Remark 3.3: Equivalence to Transmission Line Boundary Condition
• Proposition 5.1: Properties of the Generalized Eigenvalue Problem
• proof
• Theorem 6.4: Ultraviolet Suppression
• proof
• Remark 8.1