Dossier: Hasse et al.

We mapped the "coastline" of the stroboscopic measurement — the boundary between two regimes where the same apparatus reads out very different kinds of information. This is work package WP-C (Coastline): see wp-strong-weak-coastline/, commit 528267b, run time 15.5 s.

The two regimes are strong binding (ion tightly confined relative to the pulse timescale — the measurement resolves discrete motional-frequency teeth) and weak binding (pulses too broad to resolve individual teeth — the measurement sees a Doppler-like continuum). See the Headline findings below for the plain-English story. The technical summary:

• The boundary turns out to be a 1D curve, not a 2D surface. Once we tune the drive so that N pulses deliver exactly a π/2 net rotation per cell, the number of pulses N drops out — only pulse duration δt relative to the motional period T_m matters.
• A simple conjecture — that three independent spectral widths (train length, pulse duration, Doppler) combine as a root-sum-square (χ-collapse) — is falsified. What the data show instead is physics worth naming.
• The tooth-visibility figure of merit, V (see panel below), is non-monotone in the coherent-state amplitude |α|: at long pulses we see min V ≈ 0.10 at |α| ≈ 3, local max V ≈ 0.40 at |α| ≈ 4.75. A smooth single-cycle oscillation robust under pulse-number changes. Closed-form Lemma B below establishes the oscillation is a finite-δt correction to the impulsive floor; the Debye–Waller-class vs. Jaynes–Cummings-like attribution remains open (memo §5.3).
• The impulsive-kick limit (infinitely short pulses) has a closed-form floor V_imp = ½(1 + exp(−2η²)) = 0.86482 at η = 0.397, independent of N and |α| (for |α| ≳ 2) — matches the numerical overlay to 10⁻⁴ at attainable grid points.
• The engine is valid wherever the effective drive stays well below the motional frequency (Ω_eff/ω_m ≤ 0.3). A second, stricter Lamb–Dicke condition (η√⟨n⟩ ≲ 1) bounds only the interpretive picture, not the numerics.
• Strong negative result — the (V low, P low) quadrant of the rubric is empty. Extending the detuning ladder to seven points between δ = 0 and δ = 2·ω_m and scanning the full 6×6 grid at four |α| values: in every cell where V is "low" (V < 0.3), off-tooth coherence P_mid,min ≥ 0.966. No cell reaches the Doppler-merging corner. Why: the strobe gap locked to T_m combined with the π/2 net-rotation pin makes the train a stroboscopic heterodyne; in the interaction picture at H₀ = ω_ma†a + (δ/2)σ_z, gaps contribute nothing and N-fold Doppler-amplification collapses to a single-pulse residual. This suppresses the Doppler-accumulation budget from 𝒪(N) to 𝒪(1) — the residual is visible in small-N/large-δt high-V cells (P_mid,min down to 0.93) but rubric-irrelevant. This quantitatively validates a robustness the Hasse 2024 scheme relies on implicitly — and had not been tested until now.

Analytic promotion (2026-04-21, memo v0.4.7): two closed-form lemmas in wp-strong-weak-coastline/notes/analytic-reference.md, verified numerically:

• Lemma A — stroboscopic motional heterodyne (interaction-picture form). In the interaction picture at H₀, (i) the gap Hamiltonian vanishes exactly for any gap duration and (ii) the coupling operator is stroboscopically stationary at pulse-start times t_j = j·T_m. Consequence: N-pulse Doppler accumulation → single-pulse residual (𝒪(N) → 𝒪(1) suppression), not exact vanishing. The "empty in principle" framing of the v0.4.6 draft overreached — a small finite-δt per-pulse Doppler residual survives the heterodyne, but is bounded below 0.07 and does not reach the rubric's low-P corner.
• Lemma B — impulsive V floor. In the impulsive limit at net rotation π/2 with t_sep = T_m, tooth visibility V_imp = ½(1 + exp(−2η²)) independent of |α| and N (for |α| ≥ π/(4η) ≈ 1.98). Derived from a 2×2 spin-block identity plus displacement-operator algebra. What the lemma does establish: the impulsive limit has no α-structure, so any observed V(|α|) oscillation at finite δt must be a finite-δt correction. What it does not establish: whether that correction is "Debye–Waller-class" or JC-like — both enter at order ω_m·δt. Memo §5.3 attribution remains "finite-δt correction, mechanism open".

Full council memo: council-memo-2026-04-21 (v0.4.7 — walked back from v0.4.6 after reviewer caught overclaims in the Lemma-A SP formulation and the Lemma-B corollary). Probes: α-recovery, Doppler, analytic.

A short-train variant of the Hasse protocol, mapped end-to-end at the same π/2 analysis-rotation calibration used in Hasse 2024 App. D. Two pulse-train configurations — N = 3 × 100 ns and N = 7 × 50 ns — at Δt = 0.77 µs ≈ T_m (near-stroboscopic, +0.56 % slip), swept over train detuning δ₀/(2π) ∈ [−10, +10] MHz and initial motional phase ϑ₀ ∈ [0, 2π) at |α| ∈ {1, 3, 4.5}. 31 104 grid cells, 768 engine calls, ~100 s wall time. See Strobo 2.0 progress mirror.

Calibration. Each train is given its own Rabi rate so that N·Ω·e^−η²/2·δt = π/2 on the ground motional state: Ω_T1/(2π) = 0.9008 MHz, Ω_T2/(2π) = 0.7722 MHz. This matches the analysis-rotation convention of Hasse 2024 Fig. 2(b) (0.76(3) coherence) and Fig. 6 directly.

Five observables reported per cell (logbook kickoff §4):

• Coherence contrast |C| = √(a_x² + a_y²) — amplitude of the σ_z fringe under scanning analysis phase φ, extracted from two runs per cell (initial spin |+x⟩ and |+y⟩). Now directly comparable to Hasse Fig. 2(b)'s "0.76(3)".
• Analysis phase arg C = atan2(a_y, a_x) — the φ that maximises σ_z.
• Spin polarisation ⟨σ_z⟩ at φ = 0, saturating near ±0.94.
• Stroboscopic back-action δ⟨n⟩ = ⟨n⟩_fin − |α|² at φ ∈ {0, π/2} — peak ±1.67 quanta at |α| = 4.5 (Hasse 2024 Fig. 6(b) definition, amplitudes now comparable within ~1.5× to Hasse's numerics).

Three qualitative findings:

• Carrier saturates plus rich sideband comb. ϑ₀-averaged |C| on T2 at |α| = 3 drops to < 0.05 between sidebands; peaks are resolved up to ~6·ω_m. This is the strong-drive regime (Ω_eff/ω_m ≈ 0.55–0.64), a new feature vs the v0.1 weak-probe run.
• |C| peak is still |α|-independent (within ~2 %) across |α| ∈ {1, 3, 4.5}. Peak |C| ≈ 0.94, about 8 % below sin(π/2) = 1 due to intra-pulse Magnus correction (ω_m·δt ≈ 0.4–0.8 rad per pulse) — an intrinsic physical feature of short-train π/2 calibration.
• Hasse Fig. 6 geometry reproduced at matching amplitude. At δ₀ = 0 the (φ, ϑ₀) slice gives the diagonal-V in ⟨σ_z⟩ (range ±0.94) and four-lobe (+,−,−,+) pattern in δ⟨n⟩ (peak ±1.1 at α = 3, ±1.67 at α = 4.5). Symmetries δ⟨n⟩(φ+π) = −δ⟨n⟩(φ) and δ⟨n⟩(0, ϑ₀+π) = δ⟨n⟩(0, ϑ₀) preserved to correlations ±0.9996 and +1.0000. See 2026-04-21-hasse-fig6-slice.md.

Rabi-rate open question is now resolved. The earlier three-candidate reconciliation (0.178 / 0.300 / 0.446 MHz) has been replaced by a per-train π/2 calibration — see 2026-04-21-rabi-reconciliation.md v0.2. Lab-feasibility caveat: the required Ω ≈ 0.77–0.90 MHz is 2.6–3× the Hasse Table II AC-beam value; if the lab apparatus cannot reach this, a weak-probe v0.1-style calibration would be needed for direct comparison.

At the original Hasse parameters (N = 30 pulses, δt/T_m = 0.13, η = 0.397, |α| = 3), two of the three dimensionless ratios are order unity: pulse bandwidth 1/(ω_mδt) ≈ 1.22 and Doppler halfwidth η|α| ≈ 1.19, while train bandwidth 1/N ≈ 0.03 is small. The protocol is neither deeply impulsive nor deeply resolved. That is precisely why a quantitative coastline is worth mapping rather than assuming.