# Quantum inertial-sensor physics (cold-atom interferometer)
Most of Kshana drives its quantum sensors from **published Allan/noise coefficients**
— datasheet lookups (see [`QUANTUM-MODELS.md`](QUANTUM-MODELS.md) and
[`PROVENANCE.md`](PROVENANCE.md)). This document covers the one place where the engine
instead computes sensor performance **from first principles**: the cold-atom
interferometer (CAI) accelerometer model in
[`src/inertial/quantum_imu.rs`](../src/inertial/quantum_imu.rs).
It is deliberately a *minimal, honest* physics layer — the quantum-projection-noise
floor and the interferometer scale factor — not a full instrument simulator. The
sections below state exactly what is and is not modelled.
## What is modelled
A three-pulse (π/2–π–π/2) Mach–Zehnder atom interferometer (Kasevich & Chu 1991;
Peters, Chung & Chu 2001), the standard cold-atom accelerometer geometry:
| Effective wavevector | `k_eff = 4π/λ` | Two-photon Raman; Rb-87 D2 (780.241 nm) → `k_eff ≈ 1.611×10⁷ rad/m`. |
| Interferometer phase | `Φ = k_eff · a · T²` | Uniform specific force `a` along `k_eff`, pulse separation `T`. The `T²` scaling is the dominant sensitivity lever (microgravity buys long `T`). |
| Quantum projection noise | `σ_Φ = 1/(C·√N)` | Per shot, fringe contrast `C`, atom number `N` — the shot-noise limit of a two-port population readout. |
| Per-shot acceleration sensitivity | `σ_a = σ_Φ / (k_eff·T²)` | Phase noise referred to acceleration. |
| Shot-noise-limited ASD | `n_a = σ_a·√T_c` | Sampling every cycle time `T_c`; units (m/s²)/√Hz. |
| Velocity-random-walk PSD | `q_va = n_a²` | **The coefficient the classical `AccelModel` consumes** — now *derived*, not supplied. |
| Contrast decay | `C(t) = C₀·exp(−t/τ_c)` | Decoherence over the interrogation. |
| Vibration transfer function | `\|H(ω)\| = (4/ω²)·sin²(ωT/2)` | Acceleration→phase response of the ideal three-pulse geometry (Cheinet et al. 2008); DC limit `T²`. |
| Vibration-limited phase | `σ_Φ² = k_eff²·S_a·T³/3` | Flat acceleration PSD `S_a` along the Raman axis; `∫₀^∞\|H\|²dω = (2π/3)T³`. |
| Vibration-limited accel | `σ_a = √(S_a/(3T))` | Per shot; note `k_eff` cancels — set only by the platform PSD and interrogation time. |
| Axis projection | `a_∥ = k̂_eff · a` | First-order coupling is rank-1: only the along-beam component enters the phase. |
| Coriolis / rotation phase | `Φ_cor = 2·k_eff·v_⊥·Ω·T²` | Rotation systematic for a moving vehicle (Lan et al. 2012); equivalent bias `2·v_⊥·Ω` = the classical Coriolis term. |
| AC-Stark (light-shift) phase | `Φ_LS = (δ_LS,1 − δ_LS,3)/Ω_eff` | One-photon light shift; a *constant* shift cancels by π/2–π–π/2 symmetry (Peters 2001; Gauguet 2008). |
The closing of the loop is the point: `CaiAccelerometer::q_va()` produces exactly the
white-acceleration PSD that the rest of the inertial stack already integrates into a
velocity/position error — so a quantum sensor's noise can be traced to its atom number,
interrogation time, and contrast rather than to a datasheet line.
A worked figure (Rb-87, `T = 10 ms`, `N = 10⁶`, `C = 0.5`, `T_c = 0.5 s`): `Φ(1 g) ≈
1.58×10⁴ rad`, `σ_Φ = 2×10⁻³ rad`, `σ_a ≈ 1.24×10⁻⁶ m/s²` (≈ 0.13 µg) per shot, and a
shot-noise floor `n_a ≈ 0.09 µg/√Hz`. With a modest platform vibration PSD `S_a =
10⁻¹⁰ (m/s²)²/Hz` the **vibration-limited** per-shot floor is `σ_a ≈ 5.8×10⁻⁵ m/s²`
(≈ 5.9 µg) — ~45× the shot-noise floor, showing why real devices are vibration-, not
projection-, limited.
## What is NOT modelled (and why the floor is optimistic)
The shot-noise floor above is a **fundamental lower bound**. Real CAI accelerometers sit
well above it (≈ 1–50 µg/√Hz). The dominant term — **vibration coupling** — and the two
leading deterministic systematics — **Coriolis/rotation** and the **AC-Stark light shift** —
are now modelled (the transfer-function, Coriolis and light-shift rows above). The
remaining gap is systematics this layer still does not include:
- **Wavefront aberration** and higher-order beam-pointing systematics — **not modelled**.
- **Mach–Zehnder fringe ambiguity** (the phase is periodic; large accelerations alias) —
**not modelled**; the model returns the wrapped phase, not an unwrapped estimate.
Mapping to the literature: Groves, *Principles of GNSS, Inertial, and Multisensor
Integrated Navigation Systems* §12.5 (quantum technology); Cheinet et al., *IEEE Trans.
Instrum. Meas.* 57 (2008) for the interferometer sensitivity/transfer function; Freier et
al., *J. Phys.: Conf. Ser.* 723 (2016) for the mobile-gravimeter error budget; CARIOQA-PMP
for the space-accelerometer parameter regime.
## Status
This is the **P2 quantum-physics-layer** item from [`ROADMAP.md`](../ROADMAP.md): the
Mach–Zehnder phase, projection noise, scale factor, derived `q_va`, contrast decay, **and
the vibration-coupling transfer function / white-PSD variance** are implemented and
unit-tested against hand computation (including a numeric band-integral cross-check of the
transfer function against its analytic `T³` result).
The model is also **wired into the runnable inertial scenario**: an accelerometer in an
inertial-deadreckoning scenario resolves to `ImuKind::QuantumCai` when it carries a `[cai]`
block (wavelength, pulse separation `T`, atom number `N`, contrast `C`, cycle time `T_c`,
and an optional platform `vibration_psd`), and its velocity-random-walk PSD `q_va` is then
**derived** from the interferometer physics — the shot-noise floor plus, when a vibration
PSD is given, the vibration-limited contribution in quadrature — rather than supplied as a
datasheet coefficient. Scenarios without a `[cai]` block are classical and byte-unchanged.
The Coriolis (`coriolis_phase` / `coriolis_accel_bias`) and AC-Stark light-shift
(`ac_stark_phase`) systematics are implemented and unit-tested (the Coriolis equivalent
bias is checked against the classical `2·Ω×v`; the AC-Stark phase against its symmetric
cancellation). A cycle-time drift sweep (`cai_drift_sweep`) reports the quantum-CAI
dead-reckoning position drift versus cycle time — the computational core of a
quantum-vs-classical comparison. The model is **validated against a published device**:
for representative GAIN parameters the modelled quantum-projection-noise floor lies below
— and within ~2 orders of — the 96 nm/s²/√Hz short-term noise of the Freier et al. 2016
mobile gravimeter (arXiv:1512.05660), consistent with that real device being
vibration/technical-limited rather than at the standard quantum limit.
The remaining follow-ons are wavefront/beam-pointing systematics, fringe-ambiguity
resolution, a numerically exact reproduction of the CARIOQA-PMP Monte-Carlo and
Boeing/AOSense GPS-denied flight-test budgets (which need the published platform PSDs and
per-shot SNR), and a JS playground quantum-vs-classical comparison preset on top of
`cai_drift_sweep`.