Phase 1b — Intermediate Checkpoint
Symmetry Check Results — Node 4 preliminary
Status
The symmetry check (memo §8, recommended first step) is complete. The result is physically informative and sharpens the programme.
1 Parameter Regimes Tested
Reference parameters (memo §4): R = 25 µm, Q = 10⁷, G_x = 10¹² Hz/m, P_in = 1 mW, f_mech = 10 kHz.
- F_rad/F_trap(σ_x) ~ 5×10⁻³
- Result: A_E = 0 across all cases. No directional signal.
Strongly-coupled parameters: R = 10 µm, Q = 10⁸, G_x = 10¹³ Hz/m, P_in = 10 mW, f_mech = 1 kHz.
- F_rad/F_trap(σ_x) ~ 200
- Result: Massive directional asymmetry detected.
Conclusion: The discriminant requires a parameter regime where the dispersive radiation-pressure force is comparable to or exceeds the trap restoring force at the thermal displacement scale.
2 Symmetry Check Results (Strongly-Coupled Regime)
| Case | |r| [Hz/s] | A_E | σ(A_E) | Significant? |
|---|---|---|---|---|
| thermal_only (G_x = 0) | 1.94×10⁸ | +3.8×10⁻⁴ | 8.3×10⁻³ | no |
| thermal_only | 1.94×10¹⁰ | +1.1×10⁻⁵ | 4.0×10⁻⁴ | no |
| thermal_only | 1.94×10¹¹ | +5.7×10⁻⁶ | 3.4×10⁻⁴ | no |
| mechanical_only (α = 0) | 1.94×10⁸ | −1.39×10⁴ | 1.6×10¹ | YES |
| mechanical_only | 1.94×10¹⁰ | −1.86×10⁴ | 2.8×10¹ | YES |
| mechanical_only | 1.94×10¹¹ | −9.3 | 3.1 | YES |
| full | 1.94×10⁸ | −1.35×10⁴ | 3.6×10¹ | YES |
| full | 1.94×10¹⁰ | −1.85×10⁴ | 2.8×10¹ | YES |
| full | 1.94×10¹¹ | −8.5 | 3.1 | YES |
3 Interpretation
3.1 Which channels generate the asymmetry?
Two-channel (optical–mechanical) coupling is sufficient. The thermal channel contributes negligibly. Setting G_x = 0 kills the asymmetry entirely. Setting α = 0 preserves the full signal.
3.2 Physical mechanism
The asymmetry arises from sign-dependent feedback between the dispersive force and the particle displacement:
- Up-chirp (+r): Displacement shifts resonance away from laser → negative feedback → force self-limits.
- Down-chirp (−r): Displacement shifts resonance toward laser → positive feedback → force self-amplifies.
Result: down-chirp deposits far more energy (ΔE(−r) ≫ ΔE(+r)). A_E is consistently negative.
3.3 Chirp-rate dependence
|A_E| decreases from ~10⁴ at r_mech to ~10 at r_thermal. Consistent with the impulse limit (NB-2): at very high chirp rates, the mechanical mode cannot respond. The adiabatic limit (r → 0) has not been tested yet.
3.4 NB-3 (sign reversal)
No sign reversal observed. A_E is consistently negative across all rates. NB-3 remains open but is not supported at these parameters.
4 Implications for the Programme
4.1 The discriminant works — but requires strong coupling
The asymmetry is robustly nonzero in the strongly-coupled regime. The coastline’s core claim is theoretically supported. However, this requires F_rad ≫ F_trap at the thermal displacement scale.
4.2 Observable redefinition may be needed
The energy-transfer asymmetry A_E is a more sensitive observable than binary capture/escape. For Phase 2, the question to Node 1 should include: “Can the platform operate with parameters that put us in the strongly-coupled regime?”
4.3 The photothermal channel is a spectator
This simplifies the model considerably. The essential physics is optical–mechanical, not thermal.
5 Next Steps
- Adiabatic limit: Confirm A_E → 0 as r → 0 (required for NB-1 POSITIVE)
- Full A_E(r) curve: 3 decades, N ≥ 500, strongly-coupled params
- Parameter sensitivity: Map boundary between “no signal” and “detectable”
- Adiabatic-elimination validation: Compare with full 3-ODE model
- Dimensionless formulation: Express A_E in terms of η and timescale ratios
6 Assessment Against Success Criterion
Current status: Preliminary positive. The physics predicts a nonzero directional asymmetry, but the experimental parameter window is narrower than the reference set assumed.
The intermediate checkpoint (memo §9) is satisfied. Full Phase 1b deliverable pending.