Computational Results

Phase 1b computational engine: symmetry check, analytical derivation, and figure generation.

Analytical Derivation

The asymmetry is governed by a single dimensionless coupling parameter:

η = G_x² · P_in / (r² · m · κ²) (up to numerical prefactors)

Regime	η	Behaviour
Strong back-action	η ≫ 1	Large asymmetry, saturated
Maximum sensitivity	η ~ 1	Optimal chirp-rate window
Perturbative	η ≪ 1	A_E → 0 (impulse limit)
Adiabatic	r → 0	A_E → 0 (system tracks equilibrium)

Force Landscape

Radiation-pressure force, optical spring constant, and effective damping as functions of detuning. The Lorentzian force profile creates sign-dependent feedback during chirp traversal.

η Scaling

Dimensionless coupling parameter η as a function of chirp rate, showing the transition from strong-coupling to perturbative regimes.

Dimensionless coupling parameter η vs chirp rate |r|. The shaded band indicates the tested chirp-rate range. η = 1 marks the crossover between strong-coupling and perturbative regimes.

Symmetry Check

Three subcases tested in the strongly-coupled regime (R = 10 µm, Q = 10⁸, G_x = 10¹³ Hz/m, P_in = 10 mW):

Case		r	[Hz/s]	A_E
thermal_only	1.94×10⁸	+3.8×10⁻⁴	8.3×10⁻³	—
thermal_only	1.94×10¹⁰	+1.1×10⁻⁵	4.0×10⁻⁴	—
thermal_only	1.94×10¹¹	+5.7×10⁻⁶	3.4×10⁻⁴	—
mechanical_only	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

Key result

Two-channel (optical–mechanical) coupling is sufficient. The thermal channel is a spectator. Setting G_x = 0 kills the asymmetry; setting α = 0 preserves the full signal.

Physical Mechanism

Up-chirp (+r): Laser approaches resonance from below. Particle displacement shifts the resonance away from the laser — negative feedback. Force self-limits.

Down-chirp (−r): Displacement shifts the resonance toward the laser — positive feedback. Force self-amplifies.

Result: A_E < 0 consistently (down-chirp heats more than up-chirp). No sign reversal at tested parameters.

Scaling Summary

The mechanism is the correlation between accumulated mechanical displacement and the Lorentzian force gradient, with opposite sign for up-chirp vs down-chirp.

Zeroth order: Integrated impulse symmetric in sign(r). No asymmetry without back-action.
First order: Back-action breaks symmetry. Scales with η.
Scaling: η ∝ G_x² · P_in / (r² · m · κ²)
Thermal channel: Not required at leading order.

Code

All source code is in the repository:

Script	Function	Runtime
`src/phase1b_analytical.py`	Analytical derivation + figure generation	~5 s
`src/phase1b_strong_coupling.py`	Symmetry check (strongly-coupled regime)	~30 s
`src/phase1b_engine_v2.py`	Energy-transfer model (reference params)	~15 s

# Quick verification
pip install numpy scipy matplotlib
python src/phase1b_strong_coupling.py   # expect A_E ≠ 0 for mechanical_only and full
python src/phase1b_analytical.py        # generates figures/