Logbook — 2026-04-14 — Flag 1 closed: synced-phase convention → comb lineshape

Context. The user's statement "Phase is kept synced for all pulses of a train" (2026-04-14) closes Flag 1 from 2026-04-13-flag-reruns.md §3. Under the synced-phase convention, the laser/AC phase reference is maintained across the full pulse train, and the spin picks up exp(−i·(δ/2)·σ_z·T_gap) per inter-pulse gap.

Consequence. The engine (v0.9.1 at t_sep_factor = 1.0) omits this inter-pulse free evolution. A driver-level fix is straightforward; the resulting lineshape is not the engine's broad single peak but the comb at δ = k·ω_m that R2 (Monroe impulsive) already produced.

Verdict. The comb is the physically correct |C|(δ₀) lineshape under the synced-phase convention. The engine's reported Rabi lineshape (HWHM ≈ 200 kHz/(2π), centred at δ₀ = 0) is a modeling artefact specific to its t_sep_factor = 1.0 path. All results at δ₀ = 0 stand (see §4); all detuning-scan results from S1, S2, H1, S1-carrier-zoom, R2-coarse need reinterpretation.

1. Driver-level fix

Added a U_gap inter-pulse propagator in ../numerics/run_synced_phase.py:

U_gap[n, n]        = exp(−i·ω_m·n·T_gap) · exp(+i·δ/2·T_gap)
U_gap[nmax+n, ...] = exp(−i·ω_m·n·T_gap) · exp(−i·δ/2·T_gap)

with T_gap = T_m − δt. Combined with the engine's finite-δt pulse (intra-pulse motion ω_m·a†a on), the total per-cycle motional phase is ω_m·δt + ω_m·(T_m − δt) = 2π, so the strobe condition closes exactly. The spin detuning phase accumulates only during the gap; during the pulse it's already in the engine's Hamiltonian.

First attempt (iteration visible in git history of this file): missed the motional factor, which broke the strobe closure and produced an anomalous |C|(δ=0) = 0.9996 at α=0 (clean π/2 rotation; no entanglement because the motional state's phase didn't close, so successive pulses didn't coherently accumulate). Fixed by adding the Fock-indexed motional phase. After fix, synced-phase matches the engine at δ = 0 to numerical precision (0.92348 vs 0.92348 at α = 0; 0.92379 vs 0.92379 at α = 3) — confirming δ = 0 observables are convention-invariant.

2. Lineshape comparison — α = 0

Sampled at selected detunings (±500 kHz around carrier):

The engine shows a broad lineshape (HWHM ≈ 200 kHz). Synced-phase and R2 show a sharp central tooth (HWHM ≈ 30 kHz) with between-tooth |C| at the 1–10% level. Synced-phase overlaps R2 to within ~0.5% across the entire grid — the two conventions are essentially the same physics at δ = 0 of a tooth, differing only at the 0.6% level in peak height (0.923 vs 0.917) due to finite-δt vs impulsive pulses.

Identical pattern at α = 3. The |α|-dependence of the lineshape (if any) is small — much smaller than the engine-vs-synced difference.

Plot: ../plots/flag1_synced_phase_lineshape.png.

3. Why does the engine show a broad lineshape at all?

The engine's pulse Hamiltonian contains (δ/2)·σ_z, so during each pulse the spin also picks up δ-dependent phase — for total duration N·δt = 18.85 μs (at Hasse-matched δt). The resulting Rabi linewidth is ~Ω_eff ≈ 280 kHz, consistent with what we observed. This is a genuine dynamics of the engine's protocol — just not the physically intended one. The engine models a continuous pulse of duration N·δt = 18.85 μs with the laser on throughout, rather than 30 stroboscopic flashes totalling 17 μs of train with laser off for most of it.

Under the synced-phase convention, the spin feels δ: - During pulses (total N·δt = 18.85 μs) via the pulse Hamiltonian. - Between pulses (total (N − 1)·T_gap = 175.2 μs) via free evolution.

The between-pulse time dominates by factor ~10. The resulting lineshape is the stroboscopic Fourier response, which has support at δ = k·ω_m (aliasing) with tooth HWHM ≈ 1/((N−1)·T_m) ≈ 62 kHz — matching what we see (30–40 kHz, close to the sinc-amplitude HWHM of a rectangular train).

4. What stands; what needs reinterpretation

Stands (δ = 0 results — convention-invariant)

• arg C identity (both v0.8 and v0.9.1 forms): arg C(0, |α|, φ_α) = 90° + 2η|α|·(γ_c cos + γ_s sin) + O(η³). Unchanged under synced-phase at δ = 0.
• |C|(δ₀ = 0, φ_α) 5% spread under v0.9.1 D1 Hasse-matched. Same.
• H1 lock tolerance — scans ε (pulse timing), not δ. At δ = 0 the inter-pulse Ufree is identity on spin; only the motional phase matters, and that's already in the engine's t_sep_factor ≠ 1 path. Stands.
• Matrix-element-magnitude theorem at v0.8 (machine precision). Property of the operator ⟨α|C|α⟩ itself, independent of protocol.

Needs reinterpretation (δ ≠ 0 results)

• S1 slice (|C|(δ₀) at φ_α = 0) — the "Doppler-broadened" lineshape we reported is not the synced-phase physics. The true |C|(δ) has the comb structure with α-independent envelope (the α-dependence is in how individual tooth heights scale, not in the lineshape).
• S2 residual at δ ≠ 0 — both full and R1 engine lineshapes are model-convention artefacts. The comb-to-comb comparison is what matters.
• S1 carrier-zoom (±500 kHz fine grid) — the peak bias at −206 kHz for full engine was a Magnus-like effect in the broad-lineshape regime. Under synced-phase, the central tooth is at δ = 0 exactly (|C|_synced has peak within 10 kHz of zero in the fine-grid data at both α values). So the reported "Magnus carrier-bias linear in η" (−206 vs −21 kHz pair) is a property of the engine's wrong convention, not synced-phase physics. Needs to be retracted or reframed.
• R2 fine-tooth characterisation (HWHM 41 kHz) — this IS synced-phase physics. The reported tooth shape is correct; interpretation as "theoretical sideline" was too conservative. R2 is the correct lineshape under the user-confirmed convention.
• R2 comb logbook — the argument that "engine matches experiment because comb not reported" is weakened. The user's statement supports the comb being correct; the experimental question is whether Hasse2024 scans reach δ > 100 kHz (if they do and observe a single peak, that would challenge the synced-phase assumption).

5. Consequences for WP-E v0.4

The WP's position-phase channel framing strengthens:

• arg C = 90° + 2η|α|·cos φ_α holds at carrier regardless of convention.
• |C|(δ₀) lineshape is the synced-phase comb, not the engine's broad peak. This simplifies the WP's story — |C| no longer has a rich Rabi-like δ₀-dependence to report; it has sharp teeth at δ = k·ω_m, each with the same magnitude, and between-tooth the signal is ~0.
• The injectivity map (|α|, φ_α) → (Re C, Im C) at δ = 0 is unchanged by convention.
• WP-E's deliverables 1-3 (datasets, plots, logbook) need revised figure-captions and text footnotes, NOT a full recomputation — the carrier-column of each dataset is still valid; the rest of the detuning sweeps are "engine-convention probes" rather than "physics".

Recommendation for v0.4: Report the S1/S2 at δ = 0 slices as the physical results (position channel, φ_α spread). Keep the full (δ, |α|) heatmaps as engine-convention diagnostic plots with an explicit "engine convention, not synced-phase physics" caveat. Reference this entry and the Monroe/synced-phase comb as the correct resolved-sideband spectrum.

6. Does this challenge Hasse2024's lineshape?

If Hasse2024 reports a Rabi scan with a broad peak (HWHM ~ Ω_eff), that is inconsistent with the synced-phase assumption. Two possibilities:

• Their scans don't cover δ > 100 kHz. Within the central tooth the engine and synced lineshape are similar (both drop steeply near 0); if they only scanned ±100 kHz they'd see the tooth and not the comb structure. No tension.
• Their scans are wider but see a single broad peak. Then the synced-phase assumption is wrong in their setup — e.g., the AC phase is reset per pulse, or there's a mechanism I haven't modeled that suppresses the comb (phase noise, intensity jitter, finite-duration analysis pulse envelope).

This is the experimental question Flag 1 was meant to answer. The user's confirmation of "phase kept synced" resolves it in principle; resolving the apparent tension with the engine's lineshape (if any) needs experimental data. For v0.4: report the synced-phase prediction and flag this as an open experimental comparison.

7. Files added in this entry

• ../numerics/run_synced_phase.py
• ../numerics/synced_phase_alpha0and3.h5
• ../numerics/plot_synced_phase.py
• ../plots/flag1_synced_phase_lineshape.png
• This entry.

Engine and all prior numerics unchanged (Guardian cadence).

8. Outstanding for v0.4

Flag 1 status after this entry:

• ✓ Convention established: phase-synced, laser reference maintained.
• ✓ Synced-phase propagator implemented at driver level; matches R2.
• ✓ Δ-scan results reinterpreted (see §4).
• ? Experimental-team cross-check still worthwhile for Hasse2024 lineshape breadth.
• ? Consider adding inter_pulse_free_evolution=True toggle to the engine itself (opt-in, backward-compatible), rather than driver- level patch — but this is an engine change and should be staged separately.

Subsuming Flag 1 into v0.4 as "user-confirmed synced-phase convention; R2 lineshape is canonical; engine convention is a modeling simplification documented here".

Next step: Council checkpoint on whether v0.4 drafting should begin now, or whether additional slices under synced-phase (S2 at α ∈ {1, 5} reconfirmation? S3 un-deferred?) should run first. These are cheap now that the propagator is validated.