# 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](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](../numerics/run_2d_alpha3.py).
Output: [../numerics/scan_2d_alpha3.h5](../numerics/scan_2d_alpha3.h5).
Plot driver: [../numerics/plot_2d_alpha3.py](../numerics/plot_2d_alpha3.py).

## 2. Overview — the comb, repeated 23 times

[../plots/scan_2d_alpha3_overview.png](../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](../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:

| φ_α (°) | envelope peak location | shape                                        |
|:-------:|:----------------------:|:---------------------------------------------|
| 0       | k = 0                  | symmetric Bessel-like, peak ≈ 0.92           |
| 45      | k = 0                  | slight blue asymmetry                        |
| **90**  | **k ≈ +1 or +2**       | **strong blue-sideband asymmetry**           |
| 135     | k ≈ 0                  | inverse asymmetry tail                       |
| 180     | k = 0                  | symmetric (opposite parity to φ = 0)         |
| 225     | k ≈ −1                 | red-sideband tail                            |
| **270** | **k ≈ −1 or −2**       | **strong red-sideband asymmetry**            |
| 315     | k ≈ 0                  | returns toward symmetric                     |

**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°:

| k   | φ_α = 0° | φ_α = 90° |
|:---:|:--------:|:---------:|
| 0   | 0.924    | 0.872     |
| +1  | 0.725    | 0.929     |
| +2  | 0.415    | 0.810     |
| +3  | 0.191    | 0.579     |
| +5  | 0.044    | 0.220     |
| −1  | 0.725    | 0.362     |
| −3  | 0.191    | 0.021     |

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/run_2d_alpha3.py)
- [../numerics/plot_2d_alpha3.py](../numerics/plot_2d_alpha3.py)
- [../numerics/scan_2d_alpha3.h5](../numerics/scan_2d_alpha3.h5)
- [../plots/scan_2d_alpha3_overview.png](../plots/scan_2d_alpha3_overview.png)
- [../plots/scan_2d_alpha3_sideband_zooms.png](../plots/scan_2d_alpha3_sideband_zooms.png)
- [../plots/scan_2d_alpha3_tooth_envelope.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.*