Doppler mechanism, analytic estimates & work packages
The analysis pulse is a two-photon Raman transition with keff = 2π/λeff. An ion moving with velocity v sees a Doppler-shifted transition:
For a coherent state |α⟩ at ωm, the velocity at the turning point is vpeak = ωm 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.
Using ηLF = 0.40, ωLF/(2π) = 1.30 MHz, Ω/(2π) = 0.30 MHz:
| Quantity | α = 0 | α = 1 | α = 3 | α = 5 |
|---|---|---|---|---|
| ⟨n⟩ = |α|² | 0 | 1 | 9 | 25 |
| δDpeak/(2π) | 0 | 1.04 MHz | 3.12 MHz | 5.16 MHz |
| δDpeak / Ω | — | 3.5 | 10.4 | 17.2 |
| RMS Doppler width σDzp/(2π) = ηωm/(2π) | 0.52 MHz | 0.52 MHz | 0.52 MHz | 0.52 MHz |
| σDzp / Ω | 1.73 | 1.73 | 1.73 | 1.73 |
Even zero-point motion produces Doppler shifts comparable to Ω (ratio 1.73). At α = 1 the peak shift is 3.5× Ω; at α = 5 it reaches 17.2×. The Doppler mechanism is not a correction — it is the momentum measurement.
The analysis pulse acts as a narrowband frequency discriminator. For a nominal π/2 pulse of duration τ = π/(2ΩR), the spin-flip probability at effective detuning δ is:
Here ΩR = 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 δD ≫ ΩR are strongly suppressed.
| δ/(2π) | δ/Ω | Origin | P↑ |
|---|---|---|---|
| 0 | 0 | Resonance | 0.500 |
| 0.30 MHz | 1.0 | δ = Ω | 0.401 |
| 0.52 MHz | 1.7 | Zero-point RMS | 0.250 |
| 1.04 MHz | 3.5 | Peak vel., α = 1 | 0.007 |
| 3.12 MHz | 10.4 | Peak vel., α = 3 | 0.008 |
| Order | Term | η = 0.4 | vs. linear |
|---|---|---|---|
| 1 (linear) | η | 0.40 | ref. |
| 2 (quadratic) | η²/2 | 0.080 | 20% |
| 3 (cubic) | η³/6 | 0.011 | 2.7% |
Linear only: ~20% error. Through η²: ~3%. Through η³: sub-percent.
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 ~ keff v δt = δD δt is a useful scale, but in a full model finite-δt effects split into two distinct mechanisms:
Geometric phase averaging — set by how much keff·x(t) changes over the pulse. Scales with both ωmδt (fraction of motional cycle) and keffv δt (depending on motional state). Controls the spatial averaging of the travelling-wave pattern.
Doppler-detuning distortion — controlled by δD/ΩR, i.e. the Doppler shift relative to the Rabi linewidth. Changes the effective rotation angle and axis for each velocity class.
The stroboscopic pulse train accumulates a π/2 rotation over ~22 pulses (1 per motional cycle), not in a single kick. Key numbers:
| Parameter | Value | Note |
|---|---|---|
| Mode frequency ωm/(2π) | 1.30 MHz | |
| Motional period Tm | 769 ns | |
| Effective η | 0.397 | AC Raman, LF mode |
| Eff. carrier Rabi ΩR/(2π) | 0.277 MHz | = Ω · exp(−η²/2) |
| Pulse duration δt | 40 ns | |
| δt/Tm | 0.052 | |
| Rotation per pulse θ | 0.070 rad (4.0°) | = ΩR · δt |
| Pulses for π/2 | ~22 | 1 per motional cycle |
| Total π/2 time | 17.0 μs | = 22 × Tm |
| Duty cycle | 5.2% | = δt/Tm |
During each 40 ns flash, the coherent peak motion produces a Doppler phase evolution Δφ = δD · δt = 2π · [δD/(2π)] · δt. Note the factor of 2π when δD/(2π) is quoted in MHz:
| α | δDpeak/(2π) | Δφpeak (rad) | Significance |
|---|---|---|---|
| 1 | 1.04 MHz | 0.26 | Small–moderate |
| 3 | 3.12 MHz | 0.78 | Moderate |
| 5 | 5.16 MHz | 1.30 | Order-unity (~1 rad, tens of degrees) |
The baseline finite-time parameter is ωmδt ≈ 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.
This protocol is a Floquet-synchronised quadrature probe, not a softened Monroe ultrafast kick. Information comes from coherent accumulation over ~22 phase-locked flashes, not single-pulse impulsiveness. The stroboscopic synchronisation trades the Monroe requirement δt ≪ Tm for a phase-stability requirement over N ≈ 22 pulses: Δωm/ωm ≪ 1/(Nπ) ≈ 0.7%. See the Sail essay (§3) for full analysis.
Interactive plots of the simulation data are on the Numerics page. Below is guidance for interpreting the detuning spectra at α = 0, 1, 3, 5.
The coherence (Bloch vector length in the equatorial plane) directly reveals the Doppler velocity distribution. At each detuning δ0, only the velocity class satisfying keff·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.
The σz contrast follows 0.61 → 0.71 → 0.84 → 0.75. This non-monotonic trend arises from the interplay of two effects. For the ground state (α = 0), even zero-point Doppler width (σD/Ω = 1.73) partially dephases the σz fringe. As α increases, the stroboscopic sampling catches the ion at different velocities across its oscillation cycle; the time-averaged σz 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 σx, σy traces show sideband oscillations with spacing ~ωm. 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.
The von Neumann entropy peaks where spin–motion entanglement is strongest. The entropy broadening with α maps the growing range of Doppler-accessible transitions.
| Quantity | Analytic | Simulation | |
|---|---|---|---|
| Debye–Waller | exp(−η²/2) = 0.924 | Ωeff/Ω = 0.277/0.300 = 0.923 | ✓ |
| σDzp/Ω | 1.73 | σz contrast 0.61 at α = 0 | ✓ |
| Coherence width | ∝ √(2⟨n⟩ + 1) | Monotonic broadening α = 0 → 5 | ✓ |
| Sideband complexity | Higher s for larger α | Increasing asymmetry | ✓ |
The σz contrast modulation (0.61 → 0.71 → 0.84 → 0.75) is the frequency-integrated version of the Doppler velocity distribution. The full detuning scan preserves the spectral structure. For state reconstruction, the detuning spectrum is the primary observable; contrast alone discards information.
WP-A.3 prerequisite
Backaction Scaling
Purity, quadrature disturbance, coherent-state overlap. Residual backaction vs. η with η², η³ terms.
Backaction budget table (η, α, Ωδt, NS). Resolves L5.
WP-A.1
Coherent-State Signal Model (Doppler + Phase)
Phase: position → fringe shift. Doppler: detuning spectrum for α = 0, 1, 3, 5 including velocity convolution. Compare against simulation coherence envelopes.
Two-channel analytic model. Resolves L2.
WP-A.2
Sideband Truncation Bounds
Weights to s = 3 with Laguerre structure. Truncation failure map. Cross-reference sideband and Doppler descriptions.
Truncation validity map.
WP-B
Numerical Anchoring
Confirm δD/Ω ratios against simulation. δt sweep. Reproduce σz contrast trend (0.61→0.71→0.84→0.75) from analytic Doppler model.
Verification memo. Resolves R5.
WP-C gated on WP-A
Tomographic Stress-Test
C.1: M: ρ → (phase, Doppler spectrum) for Fock, squeezed, cat. C.2: Injectivity + noise (hard gate). C.3: Fidelity.
Injectivity certificate. Fidelity table. Resolves L4, L7.
WP-D gated on A+C
BAE Forward View
Test L6. Characterise or archive. No rescue.
BAE feasibility or archival.
"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?"
The simulation coherence profiles (see Numerics) 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).
The non-monotonic sequence 0.61 → 0.71 → 0.84 → 0.75 is a concrete numerical target for WP-B. An analytic model that reproduces this from velocity-distribution overlap, without free parameters, would close R5.
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.
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.