Calibration-Ready Josephson–SQUID Diagnostics for Weak Josephson-Energy Perturbations: Two-Junction Identifiability, Experimental Design, and Ramsey Closure

Abstract

This work develops a device-parameterized, calibration-ready synthetic framework for detecting and discriminating weak Josephson-energy perturbations in an asymmetric split-junction superconducting circuit. The study is motivated by the broader Residual Clock–Energy program, but its circuit-level methodology is self-contained: it reports no experimental detection, proposes no modification of the Josephson relation, and does not assume a microscopic origin for the perturbation. The single-parameter Josephson-energy susceptibility leads to the expansion $$\kappa_J(u)\zeta(u) + O(\zeta^2).$$ This replaces an unconstrained frequency-response coefficient by a circuit-derived susceptibility. The primary common-mode analysis combines an asymmetric split-transmon model, paired-flux parity structure, correlated readout covariance, finite calibration uncertainty, bounded systematic nuisance, generalized matched filtering, and an explicit false-positive reserve. Statistical calibration uncertainty is marginalized into an effective covariance, while bounded control uncertainty is propagated into a worst-case amplitude-bias certificate. For the declared synthetic device and nuisance model, the primary calibration certificate gives $$\mathrm{VIF}_{\mathrm{eff}} = 1.0868, \qquad B^* = 8.743 \times 10^{-7},$$ with a nominal systematic-bias reserve of approximately $6.07\%$ relative to the prescribed one-sided $1\%$ false-positive target. The corresponding robust $95\%$ recovery floor is $$a_{\min,95}^{\mathrm{robust}} \simeq 7.29 \times 10^{-6},$$ equivalent to a synthetic half-flux frequency-response floor of approximately $15.5 \text{ kHz}$. These quantities are sensitivity requirements of the declared synthetic calibration model, not predictions of new physics and not measured device limits. The asymmetric two-junction extension decomposes perturbations into common ($\zeta_+$) and differential ($\zeta_-$) modes, yielding $$\kappa_J(u) \left[ \zeta_+ + q_d(u)\zeta_- \right] + O(\vert{}\boldsymbol{\zeta}\vert{}^2), \qquad q_d(u) = \frac{d}{f_d(u)^2}.$$ Thus common and differential junction perturbations generate distinct flux-dependent templates, but both are even under flux reflection and are therefore not separated by parity alone. For the baseline asymmetry $d = 0.45$, the calibrated common and differential templates have correlation $$\rho_2 \simeq 0.9384,$$ leaving only about $11.9\%$ of the differential-template information orthogonal to the common mode. Simultaneously fitting both amplitudes therefore increases the marginal standard errors by a factor of approximately $2.89$. The result is not an absolute non-identifiability theorem: the two modes are distinguishable in principle, but their separation is expensive in frequency-only precision for the declared device and scan geometry. The paper therefore treats identifiability itself as an experimental-design problem. A scan over the junction-asymmetry parameter shows that reducing $d$ can substantially improve common/differential separation. Subject to the illustrative synthetic requirement $E_J(\Phi_0/2)/E_C \ge 20$, the template correlation decreases from approximately $0.938$ at $d = 0.45$ to $0.854$ at $d = 0.30$ and $0.804$ at $d = 0.25$, while the joint-fit precision penalty falls correspondingly. These lower-$d$ operating points are prospective design studies rather than replacements for the primary $d = 0.45$ calibration certificate, and they involve the expected trade-off with reduced half-flux $E_J/E_C$. The linear loop-current response introduces a complementary susceptibility structure $$\frac{(1-d^2)\sin(\pi u)\cos(\pi u)}{\cos^2(\pi u) + d^2\sin^2(\pi u)}.$$ This susceptibility is odd under $u \mapsto 1-u$, in contrast with the even frequency templates. Consequently, an externally phase-referenced, loop-current-sensitive, or otherwise phase-sensitive auxiliary observable could provide a differential-only information channel and structurally break the frequency-only common/differential degeneracy. The present work identifies this design principle but does not claim that such an auxiliary channel has been experimentally implemented. The dynamic nuisance branch uses trapping–recombination–generation dynamics for the normalized quasiparticle density $x_{\mathrm{qp}}$, rather than an arbitrary quasiparticle-like Ornstein–Uhlenbeck process. The same latent quasiparticle density drives both the qubit-frequency nuisance and a $T_1$-derived witness. In the declared synthetic population, multi-witness conditioning removes approximately $99\%$ of the modeled quasiparticle-induced frequency-noise variance. Ramsey closure is deliberately restricted. Gaussian power-spectral-density-to-Ramsey reconstruction is used only over the predeclared short-time window $$t \le 50\ \mu\mathrm{s},$$ while longer-time quasiparticle coherence is left to direct time-domain or non-Gaussian analysis. A separate quasi-static calculation provides a guardrail for non-Gaussian quadratic flux noise near the sweet spot. The inference architecture is intentionally fail-closed. Static spectroscopy cannot determine the microscopic origin of two mechanisms that generate the same effective Josephson-energy perturbation, and a nuisance component exactly parallel to the candidate template remains non-identifiable without an independent control or witness. Candidate admission therefore requires a fixed circuit transfer model, parity diagnostics, calibrated thermal, quasiparticle and pickup controls, bounded systematic-bias accounting, reserve criteria, independent witness conditioning, and confidence-calibrated recovery. This paper is the third work in a public sequence. Paper I, Residual Clock Dynamics: Operational-Time Rescaling and Effective Relaxation in a Minimal $U(1)$ Model (Zenodo DOI: 10.5281/zenodo.22735204), develops the operational-time framework. Paper II, Field-Theoretic Residual Clock Response and the Criterion for a Common Operational Time (Zenodo DOI: 10.5281/zenodo.22739058), derives the conditional parent-model chain $$X \longrightarrow \rho_X \longrightarrow \omega_C(X).$$ The present paper instead studies the independent device-level chain $$(\zeta_1,\zeta_2) \longrightarrow (E_{J1},E_{J2}) \longrightarrow \omega_{01}.$$ No microscopic derivation of $X \rightarrow \zeta_i$ is claimed. Establishing or excluding such a theory-to-device matching remains a separate open problem and is not required for the circuit-level methodology developed here. A separate reproducibility archive accompanies this version. It contains the numerical scripts, canonical output tables, experimental-design scans, generated figures, environment specification, execution instructions, regression checks, and SHA256 integrity manifest for the new two-junction and experimental-design calculations. The previously released baseline static and dynamic synthetic calculations remain archived under the earlier public reproducibility record. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords superconducting qubits; transmon qubit; Josephson junction; asymmetric SQUID; matched filtering; sensitivity calibration; nuisance dynamics; quasiparticle noise; parameter identifiability; experimental design