Tutorial

Strong vs. Weak Binding: Two Limiting Perspectives

The stroboscopic protocol fires N short laser pulses at the ion, one per motional period. What the resulting spin state tells us depends on how the pulse duration compares with the motion: if the ion moves very little during a pulse, the measurement reads out discrete motional-frequency components; if the ion moves appreciably, it reads out a Doppler-like velocity distribution. We call these the strong-binding and weak-binding limits. The WP-C (Coastline) numerical study maps the boundary between them along the natural axes N (number of pulses) and δt/T_m (pulse duration in units of the motional period T_m = 2π/ω_m).

From these we build three dimensionless ratios whose relative sizes select the applicable limit:

• Train bandwidth 1/N: width of each comb tooth. Long trains (large N) have sharp teeth.
• Pulse bandwidth 1/(ω_mδt): spectral width of a single pulse envelope. Short pulses are broadband.
• Doppler halfwidth η|α|: how much the ion's coherent motion Doppler-broadens each tooth. Grows with motional amplitude.

Coastline criterion — which limit applies?

Headline findings from WP-C (Coastline)

1. The coastline is a 1D curve, not a 2D map. Once we tune the drive so that the total spin rotation across all N pulses stays fixed at π/2 (N·Ω_eff·δt = π/2 — called the "recalibrated-Ω" or "option (a)" calibration), the number of pulses N drops out of V to better than 0.5%. Only the pulse-duration ratio δt/T_m controls where on the coastline we sit. N-dependence returns only if we instead hold the Rabi drive at its Hasse value and let the rotation vary (control slice), where it appears as a Rabi-envelope oscillation — a power artefact, not a coastline feature.
2. A plausible width-composition rule ("χ-collapse") is falsified. We tested whether the three widths combine in a root-sum-square, χ² = (1/N)² + (1/(ω_mδt))² + (η|α|)², with V ∼ exp(−χ²). The data refute it: at χ ≈ 1.2, V ranges from 0.10 at |α| = 3 to 0.85 at |α| = 1. The three widths do not compose as independent Gaussian factors. What remains after the failed collapse is genuine structure, not noise — which is where the next two findings enter.
3. V oscillates in the motional amplitude |α|. At the long-pulse corner δt/T_m = 0.80, tooth visibility is non-monotone: V ≈ 0.10 at |α| ≈ 3, climbing to a local maximum V ≈ 0.40 at |α| ≈ 4.75 before falling again past |α| ≈ 6 (α-recovery probe: logbook). The shape is robust to within 10⁻³ across N ∈ {24, 48, 96}, confirming this is physics of the finite-pulse propagator, not a Rabi-envelope artefact of a particular N. Lemma B below establishes that the oscillation is a finite-δt correction to the impulsive floor (the impulsive limit carries no α-structure). Whether the correction is best named "Debye–Waller-class" or has a JC-like component is open — both enter at order ω_mδt, and the α-recovery v2 Test B showed JC-style interference is a real feature of the system under option-(b). Mechanism attribution deferred to the closed-form finite-δt calculation (WP-C v0.2).
4. A new rubric row — "encoder-sensitivity revival" — replaces some pulse-broadening cells. At the V minimum near |α| ≈ 3 we also measured P = 1.000 to three decimals. So those cells are not Doppler-broadened; the pulse train simply produces a spin coherence |C|(ϑ₀) that happens to be almost independent of ϑ₀ for that particular (α, δt). The measurement reads out information about everything except the motional phase — a blind spot of the encoder map, not a loss of coherence.
5. The impulsive limit is a clean, universal floor — now in closed form. In the vanishing-pulse-duration limit, V = V_imp uniformly across N and |α|. Lemma B (analytic reference) evaluates this exactly: V_imp = ½(1 + exp(−2η²)), derived from Σ² = 𝟙 for the 2×2 spin block plus the standard displacement-operator overlap identities. At η = 0.397 this is 0.86482, valid for |α| ≥ π/(4η) ≈ 1.98. The numerical impulsive-overlay on the v0.1.1 heatmaps matches the lemma to four decimals. Everywhere else on the grid, the coastline measures the δt-driven departure from this Debye–Waller floor — and Lemma B says that departure is necessarily a finite-δt correction, since the impulsive reference itself carries no α- or N-structure.
6. Engine-validity is controlled by the drive Lamb–Dicke ratio only. The simulator stores the spin–motion coupling as the exact matrix exponential exp[iη(a + a^†)] with no small-η expansion, so it remains numerically valid even when the motional LD parameter η√(⟨n⟩+1) exceeds 1 (at |α| = 3 it is 1.26; at |α| = 5, 2.03). What is required is that the rotating-wave approximation on the drive hold — encoded as Ω_eff/ω_m ≤ 0.3. Motional-LD breaches therefore show up as a physical-picture caveat (the discrete-sideband interpretation may not apply), not an engine error, and are rendered as a distinguishable hatching on WP-C heatmaps.
7. The (V low, P low) quadrant of the rubric is empty — a strong negative result. The v0.1 sweep measured the off-tooth coherence P at one detuning only (δ = ½ω_m). The follow-up Doppler probe extended it to seven detunings spanning δ ∈ {0, ¼, ½, ¾, 1, 1½, 2}ω_m — covering on-sideband, mid-sideband, and second-sideband points where Doppler broadening would show up. Across the full 6×6 × 4|α| × 7-detuning grid (≈ 65 k evolutions), every rubric-relevant cell (V < 0.3) has P_mid,min ≥ 0.966. No cell reaches the Doppler-merging corner. A smaller, rubric-irrelevant residual is visible: small-N/large-δt cells with high V can carry P_mid,min down to 0.93 — a 𝒪(1) per-pulse finite-bandwidth signature that Lemma A predicts survives the heterodyne cancellation.
   
   Why this happens — Lemma A (analytic reference, interaction-picture formulation): in the interaction picture at H₀ = ω_ma†a + (δ/2)σ_z, (i) the gap Hamiltonian vanishes exactly for any gap duration — no motional phase can accumulate between pulses — and (ii) because the pulse period t_sep = T_m equals one motional period, the coupling operator is stroboscopically stationary at pulse-start times (ω_mt_j = 2πj). The consequence for Doppler broadening is a reduction of the accumulation budget from 𝒪(N) to 𝒪(1): a single-pulse finite-bandwidth residual survives but is not amplified by N-fold repetition. This is the mechanism behind the stroboscopic protocol's robustness to Doppler broadening of individual pulses — a robustness the Hasse 2024 scheme implicitly relies on, now a quantitatively tested and analytically characterised property.
   
   What changed in v0.4.7. The v0.4.6 draft of this note framed Lemma A in the Schrödinger picture at the idealisation T_gap = T_m and concluded "Doppler merging unreachable in principle". A reviewer caught that the executed code runs at T_gap = T_m − δt, so the SP gap propagator is not identically unity. The IP formulation above is exact for the executed train and gives the correct (weaker but still rubric-sharp) statement: 𝒪(N) → 𝒪(1) suppression, not strict vanishing.
   
   The immediate consequence for the rubric: the fourth row (Doppler merging) is not a regime the protocol occupies under the recalibrated-Ω calibration on this grid; it would only re-emerge if the 𝒪(1) per-pulse residual were allowed to grow — e.g. by breaking the strobe-gap lock or abandoning the π/2 net-rotation pin.

The Doppler Mechanism: Primary Momentum Readout

The analysis pulse is a two-photon Raman transition with k_eff = 2π/λ_eff. An ion moving with velocity v sees a Doppler-shifted transition:

For a coherent state |α⟩ at ω_m, the velocity at the turning point is v_peak = ω_m x₀ √2 |α|, giving a peak Doppler shift:

The detuning spectrum P_↑(δ) directly encodes the velocity distribution. Scanning the analysis-pulse frequency sweeps a narrow velocity filter across the distribution.

Analytic Estimates

Using η_LF = 0.40, ω_LF/(2π) = 1.30 MHz, Ω/(2π) = 0.30 MHz:

The Rabi instrument function

The analysis pulse acts as a narrowband frequency discriminator. For a nominal π/2 pulse of duration τ = π/(2Ω_R), the spin-flip probability at effective detuning δ is:

Here Ω_R = Ω · exp(−η²/2) ≈ 0.277 MHz is the Debye–Waller-suppressed carrier Rabi rate (not the bare Ω = 0.300 MHz). This sets the width of the velocity filter: Doppler shifts δ_D ≫ Ω_R are strongly suppressed.

Small-η expansion

Finite-Pulse Effects: What "Blur" Actually Means

The Doppler estimates above assume instantaneous pulses. In reality, each flash has duration δt ≈ 40 ns, during which the ion moves. The heuristic phase blur Δφ ~ k_eff v δt = δ_D δt is a useful scale, but in a full model finite-δt effects split into two distinct mechanisms:

Corrected experimental parameters

The stroboscopic pulse train accumulates a π/2 rotation over ~22 pulses (1 per motional cycle), not in a single kick. Key numbers:

Intra-pulse phase blur (corrected)

During each 40 ns flash, the coherent peak motion produces a Doppler phase evolution Δφ = δ_D · δt = 2π · [δ_D/(2π)] · δt. Note the factor of 2π when δ_D/(2π) is quoted in MHz:

The baseline finite-time parameter is ω_mδt ≈ 0.33 rad, present even at α = 0. The zero-point Doppler contribution adds an additional ~0.13 rad RMS. At α = 5, intra-pulse blur (1.3 rad) exceeds the motional phase per pulse (0.33 rad) — position readout is genuinely degraded, strengthening the case for treating the analysis pulse as a Doppler spectroscope rather than a phase-sensitive position probe.

Reading the Numerical Results

Interactive plots of the simulation data are on the Numerics Simulate Code page. Below is guidance for interpreting the detuning spectra at α = 0, 1, 3, 5.

What the coherence envelope tells us

The coherence (Bloch vector length in the equatorial plane) directly reveals the Doppler velocity distribution. At each detuning δ₀, only the velocity class satisfying k_eff·v ≈ −δ₀ contributes coherently; other velocity classes dephase. The coherence envelope is a proxy for the velocity distribution convolved with the Rabi instrument function. Its progressive broadening from α = 0 → 5 confirms the Doppler mechanism quantitatively.

σ_z contrast: two data sources

The σ_z contrast encodes the frequency-integrated Doppler velocity distribution. The non-monotonic trend in the HDF5 data (0.61 → 0.71 → 0.84 → 0.75) arises from the interplay of two effects. For the ground state (α = 0), even zero-point Doppler width (σ_D/Ω = 1.73) partially dephases the σ_z fringe. As α increases, the stroboscopic sampling catches the ion at different velocities across its oscillation cycle; the time-averaged σ_z signal reflects the overlap between the velocity distribution and the Rabi instrument function. The initial contrast rise from α = 0 → 3 and the subsequent drop at α = 5 encode information about the velocity distribution's shape.

The uniform contrast (≈ 0.56) in the 22-pulse stroboscopic runs reflects the different parameter regime of that simulation. Explaining the quantitative difference between the two methods is itself a diagnostic target for WP-B.

Sideband structure as Doppler fingerprint

The σ_x, σ_y traces show sideband oscillations with spacing ~ω_m. As α increases, the pattern becomes asymmetric and increasingly complex. In the sideband picture, this reflects higher-order transitions. In the Doppler picture, it reflects frequency-dependent coupling of different velocity classes. The two descriptions are equivalent; the Doppler interpretation is more transparent for reading off momentum information.

Entropy as spin–motion entanglement map

The von Neumann entropy peaks where spin–motion entanglement is strongest. The entropy broadening with α maps the growing range of Doppler-accessible transitions.

Connecting Numerics to Analytics

Council Action Plan

Parallel WP track — parametric coastline (WP-Coastline / WP-C)

Standing Question

"Is the phase+Doppler readout best formalised as sampling χ(ξ) (tomography), or as estimating (x, v) with calibration (metrology) — and under which noise model does either remain stable?"

Practical Starting Points

1. Reproduce the coherence envelope analytically

The simulation coherence profiles (see Numerics Simulate Code) are benchmarks. Compute the stroboscopic average of the Rabi instrument function convolved with the time-dependent velocity distribution for each α. Matching the envelope shape and width is the first quantitative test of the Doppler model (WP-A.1).

2. Reconcile the two σ_z contrast datasets

The HDF5 adaptive-learner data yields a non-monotonic sequence 0.61 → 0.71 → 0.84 → 0.75; the 22-pulse stroboscopic JSON runs yield a uniform ≈ 0.56. Both are concrete numerical targets for WP-B — an analytic model should explain both from velocity-distribution overlap, without free parameters. The cross-method difference itself is a diagnostic: understanding why the two methods produce different contrast values will close R5.

3. Coherent vs. thermal discrimination

Run the same simulation for a thermal state at n̄ = 9 (matching α = 3). The coherence envelope should be symmetric and broader. This is the one-shot diagnostic.

4. Sideband–Doppler consistency

Compute the detuning spectrum from both sideband (s = 0–3) and Doppler (velocity convolution) pictures for α = 1. Discrepancies at the tails identify where the full exponential coupling matters.