Framework

Interaction Hamiltonian & measurement channels

Interaction Hamiltonian

H_TI = ℏ(ω_z/2)σ_z + ℏω_m a†a + ℏ(Ω/2)[C σ₋ + C† σ₊]

where C(η, a, a†) = exp(iη(a + a†))

Exponential coupling in the position quadrature — not truncated. Source of both power (Doppler-resolved mapping, higher sidebands) and complications (backaction, nonlinear decoding).

Where η ≈ 0.4 Enters

AC Raman: λ_eff ≈ 140 nm, η_LF ≈ 0.40, η_MF ≈ 0.23, η_HF ≈ 0.18. Not in the deep Lamb–Dicke regime. Debye–Waller factor exp(−η²/2) ≈ 0.924. Sideband amplitudes from |0⟩: carrier 0.923, 1st 0.369, 2nd 0.074, 3rd 0.010.

Measurement: Two Channels

Position → phase shift

Travelling wave imprints spin-dependent phase ∝ k_eff·x(t).

Velocity → Doppler detuning

Moving ion sees δ_D = k_eff·v. Peak Doppler shift 2η·ω_m·|α| exceeds Ω for all α ≥ 1. Detuning spectrum is the primary momentum channel.

Semiclassical Baseline

⟨α|exp(iη(a+a†))|α⟩ = exp(−η²/2) · exp(i 2η Re α)

Backaction & BAE

exp(iη(a+a†)) at η ≈ 0.4 couples both quadratures. Whether a perturbative linear regime exists is open (L6).

Tomographic Potential

Phase shift (position) + Doppler spectrum (momentum) together access full phase-space. Characteristic function sampling ⟨exp(ik x)⟩ provides complementary formalism.

Lock-Key Assignments

Element	Type	Status
Stroboscopic coupling maps phase-space → spin.	Lock	Established.
Position → phase; p-proxy → Doppler.	Key	Under test (L2).
Decoder faithful under noise.	Key	Under test (L4).
Near-BAE.	Key	Undetermined (L6).
Nonclassical tomography.	Key	Gated (L7).