2026-04-21 — Hasse-Fig.-6-style (φ, ϑ₀) slice (v0.3 — π/2-calibrated)

Status: executed; reproduces Hasse 2024 Fig. 6 amplitudes directly. Upstream: 2026-04-21-sweep-complete.md §7 next-step 2. Operator: uwarring (with Claude).

1. Purpose

Produce the direct visual analogue of Hasse 2024 Fig. 6(a,b) for strobo 2.0's two short trains — at fixed δ₀ = 0 (carrier), scan the AC-train analysis phase φ and the initial motional-state phase ϑ₀, and map ⟨σ_z⟩ and δ⟨n⟩ = ⟨n⟩_fin − |α|². Under the v0.3 π/2 calibration (per-train Ω such that N·Ω·e⁻η²⁄²·δt = π/2), the amplitudes of both observables are now directly comparable to Hasse's numerical simulations.

2. Protocol

Each grid cell is one engine call with npts = 1; 32 × 64 = 2048 cells per sheet. Four sheets total (T1, T2 × α ∈ {3, 4.5}). All other parameters at the strobo 2.0 canonical v0.3 set: ω_m/(2π) = 1.306 MHz, Ω_T1/(2π) = 0.9008 MHz, Ω_T2/(2π) = 0.7722 MHz, η = 0.395, Δt = 0.77 µs, Nmax = 60.

Wall time: 18.3 s on laptop (~2 ms / cell; same as v0.1).

3. Observed features

3.1 Peak values

⟨σ_z⟩ amplitude now saturates around 0.94 for both trains across α (about 6 % below the π/2-target 1, from intra-pulse Magnus correction — see 2026-04-21-sweep-complete.md §2). This is a factor 2.7× higher than the v0.1 weak-probe value of 0.35 and close to the Hasse Fig. 6(a) amplitudes at the same |α|.

δ⟨n⟩ amplitude now exceeds 1 quantum for |α| ≥ 3 — peak 1.16 (T2, α = 3) and 1.67 (T2, α = 4.5). Compare Hasse Fig. 6(b) which reports |δ⟨n⟩| up to ~0.9 at |α| = 3. Strobo 2.0 T2 exceeds this because its Ω_eff/ω_m ≈ 0.55 is stronger than Hasse's 0.213 — same total rotation delivered in fewer pulses means each pulse delivers more motional kick per spin-rotation.

3.2 Symmetry checks

Pure-state unitary dynamics impose two near-exact symmetries that the data still reproduces:

• φ-antiperiodicity δ⟨n⟩(φ + π, ϑ₀) = −δ⟨n⟩(φ, ϑ₀). Measured correlation coefficient = −0.9996 (T2, α=3) / −0.9992 (T2, α=4.5). The tiny deviation from −1 is a finite-pulse correction that is absent in the weak-drive limit but present here at Ω_eff/ω_m ~ 0.6 — not a calibration bug.
• ϑ₀-π-periodicity at the φ = 0 cut δ⟨n⟩(0, ϑ₀ + π) = δ⟨n⟩(0, ϑ₀). Measured correlation coefficient = +1.0000 across all four sheets. Same as v0.1; this symmetry is exact because at φ = 0 the pulse train couples only σ_x and the motional state has an extra ϑ₀ → ϑ₀ + π parity that survives the strong drive.
• Four-lobe quadrant sign pattern. Still (+, −, −, +) in δ⟨n⟩(T2, |α|=3), with |quadrant-mean| ~ 0.2–0.3 quanta (3–5× v0.1). Matches Hasse Fig. 6(b).

3.3 Qualitative ⟨σ_z⟩ and δ⟨n⟩ patterns

Both T1 and T2 at |α| = 3 show the Hasse Fig. 6(a) diagonal-V in ⟨σ_z⟩, now with full red-blue saturation (≲ ±0.95). At |α| = 4.5 the V develops stronger side-band ripple structure than v0.1 — the η^k·|α|^k ladder is more visible at the stronger drive.

δ⟨n⟩ still shows the four-lobe pattern but with amplitude now directly comparable to Hasse's.

4. Output files

• numerics/hasse_fig6_slice.py — runner, now taking per-train Ω.
• numerics/make_fig6_plots.py — figure generator (unchanged).
• numerics/hasse_fig6_slice.npz (140 KB; replaces the v0.1 file).
• plots/06_hasse_fig6_alpha3.png — main deliverable, now saturation-regime.
• plots/07_hasse_fig6_alpha4p5.png — higher-|α| companion.

5. Relation to Hasse 2024 Fig. 6

• Geometry: still matches. Identical axes and cut-line convention.
• Sign pattern: still matches. Diagonal-V in σ_z, four-lobe in δ⟨n⟩.
• Amplitude: now matches to within a factor ~1.5 (v0.1 was ~5× too small). Strobo 2.0 T2 peak |δ⟨n⟩| = 1.16 at α = 3 vs. Hasse's ≈0.9 at the same |α|; the residual excess tracks the higher Ω_eff/ω_m of strobo 2.0's shorter train.
• Cut-lines: same layout as v0.1.

6. Notes for follow-ups

• The strong-drive regime is now engaged. Ω_eff/ω_m ≈ 0.55–0.64 in v0.3, up from 0.128 in v0.1. This is where the Magnus expansion picks up significant higher-order corrections — the engine handles this exactly, but analytical/Magnus benchmarks will under-report |C|-loss by ~8 %.
• a_I ≈ 4 × 10⁻³ at the representative cell (T2, α=3), 10× larger than v0.1. Still small, but worth upgrading to 4-run-per- cell if precise σ_z offsets matter.
• 30-pulse companion. Still a candidate for direct Hasse reproduction (N=30, δt=100 ns, Ω tuned to π/2). Would produce a 4-sheet comparison figure at Hasse's exact parameters.

v0.1 2026-04-21 — weak-probe (single Ω = 0.178 MHz) log. v0.2 2026-04-21 — peak values and symmetry claims tightened to match the π/2-calibrated data; Hasse-comparison amplitude now matches directly.