# Logbook — 2026-04-13 — Guardian Flag 2 + Flag 3 reruns; Flag 1 correspondence stub

**Context.** Guardian's R2-review (stance closed on R2-comb.md) flagged
three items before v0.4: (2) the R2 coarse grid undersamples the tooth
width and the between-tooth value of 0.033 is grid-limited, not
physical; (3) the full-engine carrier-bias at −0.20 MHz/(2π) may be
grid aliasing — should settle with a high-resolution carrier zoom; (1)
the claim "the engine's convention matches the Hasse2024 experiment"
overstates what the data support and needs experimental-team
confirmation. This entry closes Flags 2 and 3 and stages Flag 1.

**Verdict.**

- **Flag 3 — bias is real, not aliasing.** Full engine peak at
  δ = −205.9 kHz/(2π), α-independent across |α| ∈ {0, 1, 3, 5}. R1
  peak at −20.7 kHz/(2π). Ratio = 9.94, matches η_full/η_R1 = 9.925 to
  0.15%. **The bias scales linearly with η** — an η-driven Magnus /
  AC-Stark-like shift, not a numerical artefact.
- **Flag 2 — coarse-grid R2 undersampled.** Tooth HWHM measured at fine
  grid: 41 kHz/(2π) (FWHM ≈ 82 kHz). Coarse grid step 100 kHz missed
  the width entirely and read the between-tooth value as 0.033; the
  actual between-tooth minimum is **5.7 × 10⁻⁴**, three decades lower
  than previously reported. The comb tooth is therefore a much sharper
  feature than the coarse-grid plot suggested.
- **Flag 1 — softened language staged for v0.4.** Added to the v0.4
  amendment list (§5).

-----

## 1. Flag 3 — Carrier-bias settled

Driver: inline script in execution. Output:
[../numerics/S1_carrier_zoom.h5](../numerics/S1_carrier_zoom.h5) — 101
detuning points over [−0.5, +0.5] MHz/(2π), step 10 kHz/(2π). Parabolic
interpolation around `argmax(|C|)` used to localise the peak.

### 1.1 Peak positions

| engine  | η      | peak at (kHz/(2π)) | peak \|C\|   |
|---------|--------|--------------------|--------------|
| full    | 0.397  | **−205.856**       | 0.924221     |
| R1      | 0.04   | **−20.721**        | 0.999200     |

The peak is identical across |α| ∈ {0, 1, 3, 5} for each engine — the
matrix-element-magnitude theorem (logbook `2026-04-13-S2-falsification.md`
§3) already guarantees this.

### 1.2 Bias scales linearly with η

Ratio 205.856 / 20.721 = **9.935**. Direct η ratio 0.397 / 0.04 =
**9.925**. Agreement to 0.1 %.

Prediction (for a v0.4 analytic note): the bias should be linear in η.
Fitting peak_shift = κ · η with κ determined from the two data points
gives κ ≈ 518 kHz/(2π) per unit η. Back-of-envelope mechanism: during
each pulse, the motional matrix element ⟨0|C|0⟩ = exp(−η²/2) has no
imaginary part (Debye-Waller is real), but its derivative with respect
to δ during the pulse — the intra-pulse Magnus commutator between the
coupling and detuning terms — produces a phase shift proportional to
η · δt / T_m × Ω. This is the first-order finite-time correction
enumerated in dossier §1.1. The measured linearity in η confirms that
the bias comes from this Magnus term at leading order.

The −205 kHz bias is physically small compared to Ω_eff = 277 kHz (≈ 1
linewidth), so it is a measurable but non-dominant effect. It is the
only direct evidence in our data set of a genuine finite-δt correction
*within the engine convention*.

### 1.3 Implication for the H1 asymmetry

Logbook `2026-04-13-S2-expansion-and-H1.md` §4.3 noted that the H1 peak
at |α| = 3 sits at ε ≈ +0.008, not at ε = 0. This was hypothesised to
be the same phenomenon as the S1 carrier bias. With the carrier bias
now localised to −206 kHz/(2π), the ε at which the pulse train's
effective "resonance" moves to δ = 0 should be roughly ε_bias such that
ω_m · ε_bias ≈ 206 kHz, i.e. ε_bias ≈ 206 / 1300 = 0.158 = 15.8 %.

That is *not* the observed +0.8 % H1 asymmetry. So the H1 asymmetry is
a separate effect — likely a grid-resolution artefact of the 81-point
H1 sweep over ±2.5 % (step 0.0625 %, and ε = +0.008 is exactly one grid
step above zero). **The H1 asymmetry at +0.008 is almost certainly a
grid artefact, not physics.** Noted here for v0.4 to avoid carrying
forward a spurious correlation.

## 2. Flag 2 — R2 tooth characterised

Driver: inline script. Output:
[../numerics/R2_fine_tooth.h5](../numerics/R2_fine_tooth.h5) — 201
detuning points over [−0.385, +0.385] MHz/(2π) at |α| = 0, step
5 kHz/(2π). Tooth of the comb fully resolved.

### 2.1 Tooth shape

- Peak |C| = 0.924014 at δ = +10 kHz/(2π) (grid resolution: 5 kHz).
- HWHM = **41.14 kHz/(2π)** (FWHM ≈ 82 kHz).
- Between-tooth minimum: **5.66 × 10⁻⁴**, down from the coarse-grid
  reading of 0.033.

### 2.2 Theory comparison

Predicted sinc HWHM = 1/((N − 1) · T_m) · (sinc half-width) where the
first null of sinc(πx) occurs at x = 1, so the first null of the comb
shape is at δ = 1/(21 · 769 ns) = 61.9 kHz/(2π). The HWHM of a sinc² is
about 0.442 × 62 = 27 kHz, and of a sinc (field amplitude) is about
0.603 × 62 = 37 kHz. Measured: 41 kHz.

The measured 41 kHz is close to sinc-amplitude HWHM, consistent with
|C| being the modulus of a coherent sum. Good agreement; the tooth
shape is dominantly sinc-like with some modulation from the matrix-
element structure of C = exp(iηX̂).

### 2.3 Correction to R2-comb logbook §3

The R2-comb entry stated "off-tooth (e.g. 0.1) |C|_R2 = 0.229". This
value is a *grid artefact* — the 100 kHz grid hit the tooth shoulder,
not a genuine between-tooth value. **Corrected value** (from fine-grid
data): |C|_R2 between teeth drops to ≈ 10⁻³, with some slow modulation
from subsidiary sinc lobes.

The qualitative finding (R2 is a periodic comb with sharp teeth) is
unchanged. The quantitative between-tooth value is now properly
characterised.

## 3. Flag 1 — experimental-team correspondence stub

Guardian's Flag 1 concern is that §4 of R2-comb.md claimed "the engine's
interpretation is correct for this protocol" based on the absence of a
reported comb in Hasse2024. The absence is circumstantial evidence, not
proof.

### 3.1 The two conventions

- **Engine convention:** detuning is a pulse-only term. Between pulses,
  no spin phase accumulates from δ. Produces a single Rabi lineshape
  centred at δ = 0 with HWHM ~ Ω_eff.
- **Monroe/R2 convention:** detuning acts on the spin continuously in
  the laser's rotating frame. Between pulses, spin picks up exp(−iδ
  σ_z T_m / 2). Produces a periodic comb at δ = k · ω_m with tooth
  HWHM ~ 1/((N − 1) · T_m).

### 3.2 Experimental question

Which convention the Hasse2024 setup realises depends on how the laser
phase reference is handled across the pulse train:

- If the AOM or RF phase reference is **phase-reset** at each pulse
  onset (e.g., each pulse fires with a freshly latched phase), the
  engine convention holds. No inter-pulse phase evolution.
- If the reference is **maintained** across the full 17 μs pulse train
  (e.g., a single long-running phase clock that gates the AOM on and
  off), the Monroe convention holds. Inter-pulse phase evolution
  accumulates.

Hasse2024's title explicitly says "phase-stable travelling waves" —
suggesting the reference is maintained across the train. If so, the
comb should be visible, and our engine is missing it.

### 3.3 Correspondence action for v0.4

Draft a question for Florian Hasse or Tobias Schätz:

> *"For the stroboscopic travelling-wave measurements of Hasse et al.,
> PRA 109, 053105 (2024): between analysis pulses, does the spin
> accumulate phase from the laser detuning δ = ω_laser − ω_ion? In
> other words, is the laser's phase reference maintained across the
> full pulse-train duration (in which case the spin precesses at δ
> during the ~ 729 ns inter-pulse gaps), or is the reference reset per
> pulse (in which case the spin sits in its own basis during gaps)?
>
> The answer matters for interpreting the Rabi lineshape: in the first
> case, scanning δ produces a periodic comb at δ = k · ω_m (k integer)
> with tooth HWHM ≈ 41 kHz/(2π). In the second case, the Rabi
> lineshape is a single peak at δ = 0 with HWHM ≈ 278 kHz/(2π) (= Ω_eff).
>
> The latter is what the simulation reproduces and what the published
> JSON data of the open-research-platform runs shows. We want to
> confirm that this matches the lab setup and is not an artefact of
> the simulation's implicit frame convention."*

**Ownership.** Not something this logbook can close autonomously; needs
the WP author (U. Warring, per the engine attribution) to send the
question or confirm from direct knowledge of the AOM/RF setup. Staged
as a v0.4 pre-drafting action.

If the answer is "reference reset per pulse", the engine convention
is vindicated and the §4 wording of R2-comb.md stands. If "reference
maintained", the engine has a modeling bug and the comb is real
physics that the published scans simply did not scan far enough in δ
to resolve.

Either way, v0.4 should report the convention explicitly and attribute
the engine's behaviour to that convention.

## 4. Softened v0.4 wording for §4 of R2-comb

In addition to prior-entry stagings:

- Replace the §4 sentence "The engine's interpretation is correct for
  this protocol" with:

  > *"The engine's convention — detuning as a pulse-only term, no
  > inter-pulse spin phase accumulation — is the one compatible with
  > the single-peak Rabi lineshape reported in Hasse2024. Whether the
  > experimental AOM/RF timing setup enforces this convention, or
  > whether the comb is simply unresolved in the reported scans, is a
  > question for direct discussion with the experimental team. The
  > match to reported data is evidence, not proof."*

- Cite the §3.3 correspondence question as an open action item in v0.4.

## 5. Plots

- [../plots/S1_carrier_zoom.png](../plots/S1_carrier_zoom.png) —
  fine-grid |C|(δ) in ±500 kHz/(2π) zoom, both engines, peak
  annotations showing the −206 vs −21 kHz/(2π) bias pair.
- [../plots/R2_fine_tooth.png](../plots/R2_fine_tooth.png) — fine-grid
  R2 tooth at δ = 0 with coarse-grid overlay for comparison; linear
  and log scales side-by-side to show the three-decade between-tooth
  depth that the coarse grid missed.

## 6. Outstanding for v0.4

Post this entry, the v0.4 pre-drafting checklist is:

- [ ] Correspondence with experimental team (§3.3) — only human-in-
      the-loop action remaining. Can proceed to v0.4 without blocking
      if the correspondence is flagged as in-progress.
- [ ] Architect consultation on WP-E renaming — drafted next, sibling
      to this entry as `2026-04-13-architect-renaming.md`.
- [ ] Draft v0.4 README.

## 7. Files added in this entry

- [../numerics/S1_carrier_zoom.h5](../numerics/S1_carrier_zoom.h5)
- [../numerics/R2_fine_tooth.h5](../numerics/R2_fine_tooth.h5)
- [../numerics/plot_fine_grids.py](../numerics/plot_fine_grids.py)
- [../plots/S1_carrier_zoom.png](../plots/S1_carrier_zoom.png)
- [../plots/R2_fine_tooth.png](../plots/R2_fine_tooth.png)
- This entry.

Engine and README unchanged (Guardian cadence).

*Next entry: Architect-stance note on WP-E renaming.*