# 2026-04-21 — Strobo 2.0 Kickoff

**Status:** executed 2026-04-21. Parameter-plan record of the main sweep;
outcomes are in the companion log
[2026-04-21-sweep-complete.md](2026-04-21-sweep-complete.md).
**Author/operator:** uwarring (with Claude)

**v0.2 edits (2026-04-21, post-preflight).** §1 observable list, §4.2,
§4.3, §4.4, §5 grid count, §8 execution plan and §9 open questions
updated after preflight invalidated the original single-run |C|
identity (see §4.2 for details) and to reflect the as-implemented
data format (one combined `.npz` + manifest) and observable count
(five reported observables, not three). The original v0.1 text is
preserved in git history.

-----

## 1. Purpose

Map the joint dependence of five observables on two control axes for two
short AC pulse-train configurations:

- **Observables** (all defined precisely in §4)
  - *Coherence contrast* `|C| = √(a_x² + a_y²)` — amplitude of the σ_z
    fringe when scanning the train's analysis phase φ, matching the
    Hasse Fig. 2(b) "0.76(3)" contrast metric. Extracted from two
    runs per grid cell (initial spin |+x⟩ and |+y⟩).
  - *Analysis phase* `arg C = atan2(a_y, a_x)` — the value of φ that
    maximises σ_z.
  - *Spin polarisation* ⟨σ_z⟩ at the canonical analysis phase φ = 0
    (i.e., `sz_A` from Run A).
  - *Back-action* δ⟨n⟩_A at φ = 0 (Hasse 2024 App. D / Fig. 6b).
  - *Back-action* δ⟨n⟩_B at φ = π/2 (companion analysis phase).
- **Axes**
  - Train detuning δ₀ relative to the atomic carrier,
    δ₀/(2π) ∈ [−10, +10] MHz.
  - Initial motional-state phase ϑ₀ ∈ [0, 2π).
- **Parameterised by**
  - Coherent displacement amplitude |α| ∈ {1, 3, 4.5}.
  - Train configuration: T1 = (N = 3, δt = 100 ns) and
    T2 = (N = 7, δt = 50 ns).

Total: 5 observables × 3 α-values × 2 trains = 30 heatmap panels,
published as 5 figures of 2 × 3 panels each.

## 2. Physical parameters (v0.3, 2026-04-21 — π/2-calibrated)

**Device-scale parameters (train-independent):**

| Symbol        | Value                | Meaning                                |
|---------------|----------------------|----------------------------------------|
| ω_m/(2π)      | 1.306 MHz            | Motional (LF) frequency                |
| T_m           | 0.7657 µs            | Motional period = 2π/ω_m               |
| η             | 0.395                | Effective Lamb–Dicke parameter         |
| e⁻η²⁄²        | 0.9250               | Debye–Waller factor                    |
| Δt            | 0.770 µs             | Inter-pulse spacing                    |
| Δt / T_m      | 1.0056 (+0.56 %)     | **Slightly super-stroboscopic**        |

**Train configurations**, with Ω chosen per train so that
`N·Ω·e⁻η²⁄²·δt = π/2` (full π/2 analysis rotation on ground motion,
matching the Hasse 2024 App. D convention):

| Train | N | δt (ns) | Ω/(2π) (MHz) | θ_pulse = Ω·δt·e⁻η²⁄² (rad) | N·θ_pulse (rad) | Total duration       |
|-------|---|---------|--------------|-----------------------------|-----------------|----------------------|
| T1    | 3 | 100     | **0.9008**   | 0.5236  (30.0°)             | 1.5708 = π/2    | 2·Δt + 3·δt = 1.84 µs|
| T2    | 7 | 50      | **0.7722**   | 0.2244  (12.86°)            | 1.5708 = π/2    | 6·Δt + 7·δt = 4.97 µs|

Per-pulse rotation is **moderate** for T1 (0.52 rad ≈ π/6) and **weak**
for T2 (0.22 rad ≈ π/14). The trains share the same accumulated
rotation but distribute it very differently in time. Ω_eff/ω_m ≈ 0.64
(T1) / 0.55 (T2) — both well into the strong-drive regime where a
plain Magnus expansion misses important structure; the engine handles
this exactly via Fock-basis matrix exponentiation.

**Near-stroboscopic-resonant caveat.** Δt is 0.56 % longer than T_m. Over a
7-pulse train this accumulates to Δφ_strobo = 2π·7·0.0056 ≈ 14.1° of
motional-phase slip from first to last pulse — not negligible for T2.
For T1 (2 inter-pulse intervals) the slip is ≈ 4.0°. Flagged for
interpretation.

**History note.** The initial v0.1 run used a single pre-calibration
value Ω/(2π) = 0.178 MHz, delivering only N·θ_pulse ≈ π/9 per train
(weak probe; max |C| ≈ 0.35). That dataset is superseded by the
π/2-calibrated sweep of this v0.3 configuration. The
[Rabi-rate reconciliation memo](2026-04-21-rabi-reconciliation.md) is
updated accordingly.

## 3. Sequence layout

```
  t = 0:   prepare |+x⟩ ⊗ |α e^{i ϑ₀}⟩
  then:    AC pulse train (N pulses, duration δt, spacing Δt, detuning δ₀)
  readout: ⟨σ_x⟩, ⟨σ_y⟩, ⟨σ_z⟩, ⟨n⟩
```

- **Initial spin:** |+x⟩ = (|↓⟩ + |↑⟩)/√2 for Run A; |+y⟩ for Run B.
  Experimental equivalent: a perfectly-synchronised MW π/2 that we
  absorb into state prep.
- **Initial motion:** coherent state |α e^{i ϑ₀}⟩ produced by D(α) with
  α = |α| e^{i ϑ₀} (convention anchored to the engine; see §6).
- **Analysis phase φ:** the engine's ac_phase_deg is fixed at 0 for
  all simulations. The experimental φ-scan is recovered by the two-run
  extraction in §4.2 (init |+x⟩ → a_x; init |+y⟩ → a_y), not by a
  single-run Bloch-magnitude readout.
- **Each train is calibrated to a full π/2 analysis rotation** on the
  ground motional state, matching the Hasse 2024 App. D convention.
  Because T1 (3 × 100 ns) and T2 (7 × 50 ns) differ in N·δt, the
  required Rabi rate is train-specific:

  ```
  Ω_T{k}/(2π)  =  1 / (4 · N_k · δt_k · e^{−η²/2})
  ```

  giving Ω_T1/(2π) = 0.9008 MHz and Ω_T2/(2π) = 0.7722 MHz (see §2).
  This supersedes the single-Ω sweep of the initial strobo 2.0 run,
  which used the pre-calibration value Ω/(2π) = 0.178 MHz and
  delivered only N·θ_pulse ≈ π/9. The σ_z(φ) = a_I + a_x cos φ +
  a_y sin φ fringe structure in §4.2 is kinematic and holds at any
  N·θ; at the π/2 calibration the fringe amplitude saturates at ≈ 1
  on ground motion and the numerical value of |C| becomes directly
  comparable to the Hasse 2024 "0.76(3)" contrast at |α| ≈ 3.
- **No separate closing π/2.** The train is the only spin-coupling
  element between state prep and readout — the train *is* the
  analysis π/2.

## 4. Observable definitions

### 4.1 Back-action (Hasse 2024 definition)

```
δ⟨n⟩(δ₀, ϑ₀; |α|, N, δt) = ⟨n⟩_fin − ⟨n⟩_ini
```

with ⟨n⟩_ini = |α|² exactly (coherent initial state) and ⟨n⟩_fin computed
from the reduced motional density matrix after tracing out spin. Matches
Hasse 2024 App. D and Fig. 6b. The full definition (canonical analysis
phase, reporting choice, and units) is in §4.4.

### 4.2 Coherence and the φ-scan (Protocol-A, Hasse-style)

In Hasse 2024 the AC pulse train is calibrated to a full π/2 analysis
rotation (30 × 100 ns), and scanning the train's global phase φ produces
a near-unity Ramsey-like fringe. Strobo 2.0 applies the same calibration
rule to its shorter trains (§3): each of T1 and T2 is tuned in Ω so
that N·Ω·e⁻η²⁄²·δt = π/2. By the lab-frame equivalence
`U(φ) = R_z(φ) U(0) R_z(φ)†`, scanning the train's global phase φ is
equivalent to scanning the initial-spin azimuth by −φ, so

```
⟨σ_z⟩(φ) = a_I(δ₀, ϑ₀) + a_x(δ₀, ϑ₀) cos φ + a_y(δ₀, ϑ₀) sin φ.
```

The **coherence contrast** (directly comparable to the "0.76(3)" in
Hasse Fig. 2(b), since both are measured under a π/2-calibrated train)
is the fringe amplitude:

```
|C|(δ₀, ϑ₀)   ≡ √(a_x² + a_y²),
arg C(δ₀, ϑ₀) ≡ atan2(a_y, a_x)      (the φ value that maximises σ_z).
```

On the ground motional state at δ₀ = 0 this saturates at sin(π/2) = 1;
finite |α| and finite-δ₀ departures from saturation then produce the
structure mapped by this WP.

**Preflight 2026-04-21 disproved the single-run identification**
`|C| = |⟨σ_x⟩ + i⟨σ_y⟩|` that appeared in the kickoff v0.1 draft. That
identity holds only for a Ramsey scheme with a *separate* closing π/2
at phase φ; the AC-train-as-π/2 protocol of this WP requires two runs
per grid cell to isolate a_x and a_y. Preflight measured, at
(|α| = 3, δ₀ = 0, ϑ₀ = 0, T2):

- ⟨σ_x⟩+i⟨σ_y⟩ at φ_laser = 0, init |+x⟩  →  magnitude 0.956
- Fringe amplitude √(a_x² + a_y²) from 8-point φ-scan   →  0.352

These differ by a factor of ~2.7 — the first is the spin Bloch
x-y projection after the train, the second is the Hasse coherence.
**We adopt the second.**

**Two-run evaluation per (δ₀, ϑ₀) cell.**

- Run A: spin in |+x⟩ (theta_deg = 90, phi_spin_deg = 0), engine
  ac_phase_deg = 0. Record `sz_A, nbar_A`.
- Run B: spin in |+y⟩ (theta_deg = 90, phi_spin_deg = 90), engine
  ac_phase_deg = 0. Record `sz_B, nbar_B`.

Then (assuming `a_I ≈ 0`, to be checked):

- `a_x ≈ sz_A`,  `a_y ≈ sz_B`.
- `|C| = √(sz_A² + sz_B²)`.
- `arg C = atan2(sz_B, sz_A)`.

**a_I offset check.** At the anchor and at one α = 4.5 point, also run
init |−x⟩ (phi_spin_deg = 180) and form `a_I = (sz_A + sz(−x))/2`,
`a_x_corr = (sz_A − sz(−x))/2`. Proceed with two-run evaluation only if
|a_I| ≤ 1 × 10⁻² throughout. If the offset is larger anywhere, promote
to a four-run-per-cell evaluation (spins at {+x, +y, −x, −y}).

### 4.3 σ_z report

σ_z is reported at fixed φ = 0, i.e. `sz_A` from Run A (initial spin
|+x⟩, engine ac_phase = 0). This is the direct read-out at the canonical
analysis-phase choice; the `arg C` heatmap of §4.2 records the phase at
which σ_z is instead maximised.

### 4.4 Back-action report

δ⟨n⟩ is reported at the canonical analysis phase φ = 0 and at its
orthogonal companion φ = π/2:

```
δ⟨n⟩_A (δ₀, ϑ₀) = nbar_A − |α|²   (at φ = 0,   from Run A)
δ⟨n⟩_B (δ₀, ϑ₀) = nbar_B − |α|²   (at φ = π/2, from Run B)
```

Hasse Fig. 6b shows δ⟨n⟩ over the two-axis (φ, ϑ₀) plane; our sheets
are two orthogonal cuts through the analogous surface.

**Units.** δ⟨n⟩ is reported in raw quanta in the dataset and plot
labels; the fractional form δ⟨n⟩/|α|² (Hasse-style, ±1 range) can be
recovered by element-wise division at read time, since |α| is
recorded in the manifest per sheet. This supersedes the kickoff-v0.1
plan (§9 Q4) of auto-normalising to fractional — keeping the raw
numbers in the archive avoids numerical blow-up as |α| → 0 and keeps
the reported amplitude directly comparable to the engine's nbar
output.

## 5. Scan grid (confirmed)

| Axis                | Range                       | Points |
|---------------------|-----------------------------|--------|
| δ₀/(2π)             | [−10, +10] MHz, uniform     | 81     |
| ϑ₀                  | [0, 2π), uniform            | 64     |
| |α|                 | {1, 3, 4.5}                 | 3      |
| Train               | {T1 = 3×100 ns, T2 = 7×50 ns} | 2   |

**Grid cells:** 81 · 64 · 3 · 2 = **31 104 cells**.
**Engine calls:** **2 · 31 104 / 81 = 768 calls** to `run_single`, each
sweeping 81 detunings internally (2 initial-spin runs × 64 ϑ₀ × 3 α ×
2 trains). Fock cutoff: N_max = 60 (ample for |α| = 4.5 with
⟨n⟩ = 20.25; see §8 step 2).

Two distinct reference points are used:

- **Cross-check anchor:** (δ₀ = 0, ϑ₀ = 0, |α| = 0). Reduces to a pure
  carrier-driven train on the ground motional state; independent check
  against an analytic LD-expansion prediction (§8 step 1).
- **Convergence anchor:** (δ₀ = 0, ϑ₀ = 0, |α| = 4.5), i.e. the
  highest-⟨n⟩ cell in the grid. N_max sensitivity is evaluated here as
  the worst case (§8 step 2).

## 6. Conventions (engine anchor)

Following [scripts/stroboscopic_sweep.py](../../scripts/stroboscopic_sweep.py)
(the intended execution engine):

- Motional state: `α = |α| · exp(i · ϑ₀)` in `D(α) = exp(α a† − α* a)`.
- Position quadrature: `X = a + a†`. → ϑ₀ = 0 prepares the ion on +X̂
  (⟨X̂⟩ > 0, ⟨P̂⟩ = 0); ϑ₀ = π/2 prepares on +P̂.
- Per-pulse coupling: `(ℏΩ/2) · [C(η, a, a†) σ₋ + c.c. σ₊]` with
  `C(η, a, a†) = exp(i η (a† + a))`, evaluated in the frame rotating at
  the carrier — detuning δ₀ enters as a free σ_z/2 · δ₀ term during both
  pulses and inter-pulse intervals.
- Inter-pulse free evolution: unitary `U_free(Δt)` with `H_free = ω_m a†a
  + (δ₀/2) σ_z` for duration Δt.

## 7. Expected qualitative features (sanity-check predictions)

Predictions to verify before accepting the sweep data.

- **δ₀-axis resonances.** The ±10 MHz window spans ±7.66 × ω_m, so we
  expect structure at δ₀ = 0 (carrier) and δ₀ = ±k · ω_m for
  k = 1, 2, … , 7 (sidebands). Carrier width ≈ Ω_eff/(2π) = 0.178 ·
  exp(−η²/2)/(2π) ≈ 0.165 MHz; individual sideband widths scale as
  η^k · Ω.
- **ϑ₀ symmetry.** At |α| = 0 the maps are ϑ₀-independent (no displaced
  state to have a phase). At |α| > 0, periodicity in ϑ₀ is 2π on the
  σ_z / C observables and **π** on δ⟨n⟩ (Hasse Fig. 6b shows the
  characteristic "four-lobe" quadrant pattern at |α| = 3).
- **Back-action sign.** δ⟨n⟩ can be either sign: stroboscopic coupling can
  cool or heat the motion depending on (δ₀, ϑ₀). Max-|δ⟨n⟩| scales roughly
  with η · |α| on resonance.
- **Coherence envelope.** |C| is bounded by 1; its maximum on the grid
  (reached off-resonance at δ₀ ≫ Ω_eff) is ≈ 1. Its minimum at
  carrier + optimal ϑ₀ is set by N · θ_pulse (≈ 0.34–0.39 rad) and |α|.
- **T1 vs T2 cross-check.** Both trains have near-identical N·θ_pulse
  (0.335 vs 0.391 rad) but different pulse durations. T2's longer total
  duration (4.97 vs 1.84 µs) means the ϑ₀ argument drifts more during
  the train at non-zero detunings; residuals between T1 and T2 at fixed
  N·θ_pulse isolate this finite-train-duration effect.

## 8. Execution plan

As executed 2026-04-21 (see [2026-04-21-sweep-complete.md](2026-04-21-sweep-complete.md)):

1. **Preflight.** At the cross-check anchor (δ₀ = 0, ϑ₀ = 0, |α| = 0),
   verify that the two-run extraction yields real C near the
   weak-pulse-limit prediction sin(N·θ_pulse). Implemented in
   [numerics/preflight.py](../numerics/preflight.py).
2. **Convergence check.** At the convergence anchor
   (|α| = 4.5, δ₀ = 0, ϑ₀ = 0), run N_max ∈ {50, 60, 70, 80} and
   record δ⟨n⟩ convergence. Target: absolute change ≤ 10⁻³ between
   N_max = 60 and N_max = 80.
3. **φ-scan spot-check.** At (|α| = 3, δ₀ = 0, ϑ₀ = 0, T2), run
   8 values of φ_laser and fit to a_I + a_x cos φ + a_y sin φ.
   Confirm max residual ≤ 10⁻¹⁰ and that the two-run extraction
   (spin {+x, +y}) matches the fit coefficients to ≤ 10⁻³.
4. **Main sweep.** 31 104 grid cells over the §5 grid, 768 engine
   calls total. Save as one combined
   [numerics/strobo2p0_data.npz](../numerics/strobo2p0_data.npz)
   (keyed by `{train}_alpha{|α|}_{observable}`) with one
   [numerics/strobo2p0_manifest.json](../numerics/strobo2p0_manifest.json)
   recording parameters, observable contract, and runtime.
5. **Plots.** Five figures in [plots/](../plots/), each a 2 × 3 panel
   array (rows: T1, T2; cols: |α| = 1, 3, 4.5), for a total of
   30 heatmap panels covering |C|, arg C, ⟨σ_z⟩, δ⟨n⟩_A, δ⟨n⟩_B.
6. **Logbook.** This kickoff, plus follow-up entries (open-ended;
   one per substantive decision or sub-deliverable). As of
   2026-04-21 the WP has four logbook entries total — see the
   [README](../README.md) tree:
   - this file (parameters, observables, grid);
   - [2026-04-21-sweep-complete.md](2026-04-21-sweep-complete.md)
     (main sweep + qualitative findings);
   - [2026-04-21-rabi-reconciliation.md](2026-04-21-rabi-reconciliation.md)
     (three-candidate Ω read-off protocol);
   - [2026-04-21-hasse-fig6-slice.md](2026-04-21-hasse-fig6-slice.md)
     ((φ, ϑ₀) analogue of Hasse Fig. 6 + symmetry checks).

## 9. Open questions / decisions to confirm

1. **Rabi rate reconciliation.** *Resolved 2026-04-21 (v0.3)*: each
   train is given its own Ω such that `N·Ω·e⁻η²⁄²·δt = π/2`
   (Ω_T1/(2π) = 0.9008 MHz, Ω_T2/(2π) = 0.7722 MHz). Matches the
   Hasse 2024 App. D convention. See
   [2026-04-21-rabi-reconciliation.md](2026-04-21-rabi-reconciliation.md)
   for the five candidate values considered and
   [plots/00_rabi_calibration.png](../plots/00_rabi_calibration.png)
   for the |C|_vacuum curve.
2. **Inclusion of a pre-train MW π/2.** Treated as perfectly absorbed into
   state prep (|+x⟩). Include it explicitly if experimentally the MW
   π/2 duration is comparable to Δt and its free-evolution contribution
   to the motional phase matters. Default: absorbed.
3. **Δt = 0.77 µs vs T_m = 0.7657 µs.** Deliberate choice or round value?
   If deliberate, this WP reports the "near-stroboscopic-slip" regime;
   if round, a second run at Δt = T_m exactly would isolate the slip
   effect.
4. **δ⟨n⟩ normalisation for publication.** Resolved in §4.4: archive
   stores raw quanta; fractional form is a read-time derivation. The
   plots in this WP also display raw quanta; a Hasse-style fractional
   comparison figure can be generated in a follow-up pass if needed.
5. **Noise model.** Pure-state unitary for this WP. If experimental
   comparison becomes a goal, add decoherence per WP-E §4a D2 protocol.

## 10. Provenance

- Engine: extension of
  [scripts/stroboscopic_sweep.py](../../scripts/stroboscopic_sweep.py)
  (Float64, exact Fock-basis matrix exponentiation via scipy.linalg.expm).
- Reference paper: Hasse, Palani, Thomm, Warring, Schaetz,
  Phys. Rev. A **109**, 053105 (2024) — App. C (Hamiltonian) and
  App. D (back-action definition + Fig. 6 qualitative comparison).
- Sibling WPs:
  [wp-phase-contrast-maps/](../../wp-phase-contrast-maps/) — N = 22 weak
  pulses, δt = 40 ns, phase-contrast forward map.

-----

*v0.1 2026-04-21 — initial draft.*
*v0.2 2026-04-21 — observable contract corrected after preflight.*
*v0.3 2026-04-21 — per-train π/2 calibration: Ω is train-specific,
§2 rewritten accordingly, §3 sequence layout reframed as
"π/2 analysis pulse per train," §4.2 caveat removed (|C| is now
directly comparable to Hasse 2024 Fig. 2(b)), §9 Q1 resolved.*