Amplitude-Phase Coupling and Average-Hamiltonian Cancellation in Figure-8-Driven Transmon Qubits: Theory and Phase-Resolved Experiments on Two Devices

We report phase-resolved measurements of amplitude-phase coupling (APC) in strongly driven superconducting transmon qubits, a symmetry-based mechanism analysis, and follow-up experiments on a second device. On a Rigetti Ankaa-3 transmon (1000 shots per point, five drive amplitudes, matched root-mean-square drive), the fitted optimal control phase φ*(A) drifts by more than 3 rad across the explored amplitude range for both a rectangular reference pulse and a self-intersecting Figure-8 trajectory, and the phase contrast of the two waveform families exhibits a reproducible low-/high-amplitude crossover. Theoretically, we prove that in the resonant rotating frame of the three-level Duffing model a constant-phase rectangular drive cannot produce any phase-dependent population response: CRect=0 exactly. Any measured rectangular phase contrast is therefore a probe of physics outside this model (beyond-rotating-wave carrier-phase dependence, control-chain distortion, readout imperfections). The Figure-8 waveform breaks the symmetry intrinsically through its 2:1 Fourier content; a third-order resonance ω+ω-2ω=0 mediated by the anharmonic level |2 predicts CF8∝ A⁶N² at weak drive, confirmed by direct numerical propagation (fitted log-log slope 5.97±0.02; loop-number ratio 2434 vs. predicted 2500). On the successor device Cepheus-1-108Q, the predicted phase contrast lies below the binomial shot-noise floor at the accessible amplitudes, but the same dataset reveals a stronger signature of the underlying first-order average-Hamiltonian cancellation: the phase-averaged survival probability of the Figure-8 train remains pinned at P(|0)φ≈0.985 across all tested conditions, while the isenergetic rectangular reference sweeps the full range 0.24-0.93—an approximately 70-fold suppression of the Rabi-drive response. Together these results identify APC as a measurable control-landscape feature with distinct intrinsic and extrinsic components, and demonstrate a calibration-free, amplitude-insensitive idling property of Figure-8 pulse trains.