Logbook — 2026-04-14 — 2D scan at |α|=3, ±15 MHz × [0, 2π): sideband envelope is the velocity signature

Context. User request: spin contrast as a function of motional phase (0 → 2π) and analysis-pulse detuning (±15 MHz) at fixed |α| = 3. First wide-range detuning scan under the synced-phase convention (2026-04-14-flag1-synced-phase.md).

Verdict. ±15 MHz covers 23 comb teeth (k ∈ {−11, …, +11}). The per-tooth |C| envelope is φ_α-asymmetric: at φ_α = π/2 the peak sideband amplitude shifts toward k > 0, at φ_α = 3π/2 toward k < 0, symmetric at φ_α = 0 and π. This is the clean velocity-channel signature predicted by dossier §1.4 — Doppler asymmetry of sidebands set by ⟨P̂⟩. What we reported earlier as "5% |C| φ_α-spread at carrier" was the tiny carrier slice of this envelope; the full signature only appears once multiple sidebands are resolved.

1. Scan parameters

• α = 3 (coherent amplitude), η = 0.397
• v0.9.1 Hasse-matched: N = 30 pulses, δt = 0.13·T_m, intra-pulse motion on, pulse centering on.
• Synced-phase convention: inter-pulse U_gap with motion + spin δ·σ_z.
• Rectangular pulse (envelope effect: ≤ 0.5% on lineshape per the AOM entry).
• Grid: 1001 detuning pts over ±15 MHz (step 30 kHz — one per tooth HWHM), 64 φ_α pts over [0, 2π).
• Wall time: 73.5 s. nmax = 60 (zero Fock leakage at this α).

Driver: ../numerics/run_2d_alpha3.py. Output: ../numerics/scan_2d_alpha3.h5. Plot driver: ../numerics/plot_2d_alpha3.py.

2. Overview — the comb, repeated 23 times

../plots/scan_2d_alpha3_overview.png shows the full ±15 MHz window:

• |C| heatmap: vertical bands at δ = k·ω_m, k = 0, ±1, … ±11. Tooth height decays with |k|; teeth |k| > 5 are visible but small. Between teeth |C| is effectively zero (masked).
• arg C heatmap: rich colour structure at each tooth (distinct HSV patterns per k and per φ_α slice), confirming that each tooth carries its own (|α|, φ_α)-dependent phase signature — not just a repeat of the carrier.
• σ_z heatmap: σ_z ≈ −1 dominates (off-resonance, spin unrotated); teeth appear as narrow red/orange lines where the pulse train produces an effective rotation.

3. Per-tooth envelope — the velocity signature

../plots/scan_2d_alpha3_tooth_envelope.png:

• Left panel: |C| at exact tooth centres δ = k·ω_m, as a 2D map over (k, φ_α).
• Right panel: traces through the map at selected φ_α.

Reading the right panel:

Mechanism. At φ_α = π/2 the coherent state has ⟨X̂⟩ = 0, ⟨P̂⟩ = +2|α|. Between pulses the spin picks up phase exp(−i·δ·T_gap/2); combined with the N-pulse coherent accumulation and the η-coupling that encodes position, the ⟨P̂⟩ momentum imprints itself on the which sideband carries the coherence. Positive ⟨P̂⟩ → amplitude biased toward positive k (blue-shifted sidebands); negative ⟨P̂⟩ → toward red.

This is precisely the Doppler velocity-channel signature dossier §1.4 described qualitatively. The earlier "falsification" in v0.8 missed it because (i) v0.8 used frozen-motion and suppressed the η·ω_m·δt correction entirely, and (ii) earlier scans only went to ±6 MHz, covering ≤ 4 sidebands and missing most of the envelope structure.

4. What the carrier slice was

At δ = 0 (k = 0), the 5% |C| φ_α-spread reported in the S2-revisited entry is the small remnant of the envelope — the carrier amplitude is nearly φ_α-independent because the coherent state's position moments cancel in the time-averaged carrier matrix element. The envelope's richer |α|- and φ_α-dependence lives at the sidebands.

Reading values of |C| at tooth centres for φ_α = 0° and φ_α = 90°:

At φ_α = 0°, the |C| is symmetric in k. At φ_α = 90°, there's a pronounced blue-shift bias: |C|(k=+5) = 0.220 vs |C|(k=−5) ≈ 0.02. Ratio > 10 — clean directional signature. (Numbers read off the 30-kHz grid; sub-kHz refinement at tooth centres unchanged from coarse sample.)

5. Implications for WP-E

This is the strongest direct observation of the velocity channel in the WP's data. It reframes the v0.9.1 story:

• The position channel (arg C at the carrier) is the amplitude and phase of the central tooth — i.e., one point in a much richer pattern.
• The velocity channel is the asymmetric envelope of sideband amplitudes — not a width of the carrier, but a k-asymmetry visible only in wide-range scans.
• An experimenter wanting to read out |α| AND φ_α needs to measure at multiple sidebands, not just scan detuning around δ = 0.

For v0.4:

• Add this 2D scan as a deliverable figure — it's the cleanest single visual of the full forward map.
• Redefine "velocity channel" to mean the k-asymmetry of the sideband envelope, not a broadening of a single peak.
• The injectivity map from (|α|, φ_α) to the observable now uses multiple tooth amplitudes + central arg C, which is a richer measurement than previously envisioned.

6. Open questions / next moves

• |α|-scaling of the envelope asymmetry. At |α| = 1 and 5 the envelope width should scale; the asymmetry-vs-|α| relation would validate the analytical picture. Rerun 2D at {1, 5}? Cheap (~75 s each).
• Tooth envelope closed form. The left panel is reminiscent of a Bessel / Gaussian pattern weighted by a φ_α-phase factor. Likely analytic form: |C|(k, φ_α) ≈ |J_k(2η|α| sin φ_α_effective)| or similar; worth checking against the data at one |α|.
• AOM envelope on the 2D scan. Previous AOM logbook showed lineshape robustness at ±500 kHz. Over ±15 MHz the edges may have more visible effect on high-|k| sideband amplitudes. Not critical; revisit if v0.4 needs it.
• v0.4 figure choice. scan_2d_alpha3_tooth_envelope.png is the single strongest "velocity channel" figure WP-E now has. Consider as headline alongside the carrier-summary figure.

7. Files added

• ../numerics/run_2d_alpha3.py
• ../numerics/plot_2d_alpha3.py
• ../numerics/scan_2d_alpha3.h5
• ../plots/scan_2d_alpha3_overview.png
• ../plots/scan_2d_alpha3_sideband_zooms.png
• ../plots/scan_2d_alpha3_tooth_envelope.png
• This entry.

Engine unchanged.

Next: optional |α| ∈ {1, 5} 2D scans to complete the envelope-scaling story; or proceed to v0.4 with this as the headline.