Phase 1b Memo — Theoretical Model for Capture Asymmetry
Interface document for Node 4 (Open-System Theory)
0 Purpose
This memo defines the minimum coupled-mode model that Node 4 must construct before Phase 2 (platform discussion with Node 1) can proceed. The model’s function is to predict the capture asymmetry A(r) as a function of chirp rate r, identify whether a non-empty chirp-rate window exists for physically realistic parameters, and provide a go/no-go signal for experimental resource commitment.
The model acts as a pre-filter. It does not advocate for the programme; it tests whether the programme survives minimal theoretical scrutiny.
1 System Definition
Three coupled degrees of freedom:
Intracavity field amplitude a(t). The optical field inside the WGM mode:
da/dt = [ i·2π·Δ(t) − κ/2 ] · a(t) + √κ_ex · s_in
Photothermal resonance shift δν_th(t). Delayed-feedback channel:
dδν_th/dt = −(δν_th − α · |a|²) / τ_th
Centre-of-mass position x(t). Mechanical response:
m · d²x/dt² = −mω_mech² · x − Γ_mech · m · dx/dt + F_rad(a, x)
The resonance frequency seen by the probe:
ν_WGM(t) = ν_WGM,0 + δν_th(t) + G_x · x(t)
2 Drive Protocol
Linear chirp: ν(t) = ν₀ + r · t
The model must be solved for both signs of r at each |r|.
3 Required Outputs
3.1 Primary deliverable: A(r) curve
A plot of A as a function of |r|, scanned over at least 3 decades of chirp rate.
3.2 Window identification
- r_min: chirp rate below which A → 0 (adiabatic limit)
- r_max: chirp rate above which A → 0 (impulse limit)
- |A|_max: the peak asymmetry magnitude
3.3 Channel attribution
For each rate regime, identify which physical channel dominates: radiation-pressure backaction (instantaneous), photothermal feedback (delayed, τ_th), or mechanical inertia (slow, τ_mech).
3.4 Sign-reversal assessment
State whether the model predicts a sign change in A(r) at any critical rate r*.
3.5 Parameter sensitivity
Identify which parameters A is most sensitive to.
4 Reference Parameter Set
| Parameter | Symbol | Reference value | Range to scan |
|---|---|---|---|
| Microsphere radius | R | 25 µm | 15–40 µm |
| WGM quality factor | Q | 10⁷ | 10⁶–10⁸ |
| Cavity linewidth | κ/2π | ~20 MHz | scales with Q |
| Thermal relaxation time | τ_th | 10 µs | 1–100 µs |
| Photothermal coefficient | α | 1 MHz per mW | order-of-magnitude |
| Mechanical frequency | ω_mech/2π | 10 kHz | 1–100 kHz |
| Mechanical damping | Γ_mech/2π | 1 Hz | 0.1–100 Hz |
| Probe power (input) | P_in | 1 mW | 0.1–10 mW |
| Chirp rate | |r| | — | 10⁵–10¹² Hz/s |
5 Success Criterion
Phase 1b is complete when the model delivers one of:
(i) Predicted non-empty window. |A| > 0.05 for some range of chirp rates within the platform’s accessible range.
(ii) Robust null. A = 0 across all accessible chirp rates and across the full parameter range, for physical reasons (symmetry or timescale separation).
Either outcome is a valid completion. Outcome (i) sharpens Phase 2. Outcome (ii) saves experimental resources.
6 Model Scope and Boundaries
- Classical treatment. The discriminant A is a classical observable.
- Single-mode approximation. One WGM mode coupled to one mechanical mode.
- Single-exponential thermal model. Sufficient to determine whether asymmetry exists at all.
- Back-action included. Mechanical displacement shifts WGM resonance via G_x — the channel through which all three clocks interact.
7 Interface to Phase 2
| Phase 1b output | Phase 2 question it enables |
|---|---|
| Predicted r_min, r_max | Can the platform operate in this range? |
| |A|_max and parameter dependence | What shot-to-shot reproducibility is needed? |
| Channel attribution | Which timescale must the platform match? |
| Sign-reversal prediction | Does the platform span the crossover rate? |
| Sensitive parameters | Which must be controlled to ≤ factor-of-2? |
8 Suggested Numerical Approach
Symmetry check (recommended first step). Test whether fewer than all three channels can produce A ≠ 0. Set G_x = 0 (thermal + optical only), then α = 0 (optical + mechanical only), then retain all three.
Direct integration. Solve the three coupled ODEs numerically with an ensemble of thermal initial conditions.
Adiabatic elimination. If κ ≫ |r|/κ, eliminate the cavity field to reduce to two coupled equations.
Dimensionless formulation. Express in terms of ratios r·τ_th, r/κ², ω_mech·τ_th.
9 Timeline and Coordination
- Intermediate checkpoint (2–3 weeks): Single-parameter-set result. Is A nonzero at any chirp rate?
- Full deliverable (4–8 weeks): Complete A(r) curve, parameter sensitivity, channel attribution, sign-reversal assessment.