coastline v0.5 · Stable v0.5 · April 2026

Frequency-Trajectory Control of Resonant Systems

Stewarded by U. Warring

0 Load-Bearing Statement

A laser system capable of programmable frequency trajectories ν(t) on timescales comparable to a resonant system’s dynamical response constitutes a control resource distinct from static detuning, provided the direction of frequency traversal produces measurable asymmetry in the system’s response.

This coastline defines and bounds that claim.

1 External Constraints (citation-only)

This coastline does not replicate the following; it references them as boundary conditions.

1.1 Cavity optomechanics

The radiation-pressure interaction between an optical cavity mode and a mechanical degree of freedom is governed by the linearised optomechanical Hamiltonian with detuning Δ = ω_laser − ω_cavity as the primary control parameter. Standard treatments assume Δ is constant or varies slowly relative to the cavity linewidth κ. [Aspelmeyer, Kippenberg, Marquardt, Rev. Mod. Phys. 86, 1391 (2014)]

1.2 Whispering-gallery-mode resonators

Dielectric microspheres and microtoroids support high-Q WGM modes with thermal bistability arising from photothermal and thermo-optic effects. The bistability boundary in detuning–power space is well characterised for static or quasi-static frequency scans. [Carmon, Yang, Vahala, Opt. Express 12, 4742 (2004); Fomin et al., JOSA B 22, 459 (2005)]

1.3 Optical levitation

Dielectric particles can be optically trapped in vacuum with centre-of-mass oscillation frequencies in the kHz–MHz range. Feedback cooling to the motional ground state has been demonstrated for nanoparticles. [Delić et al., Science 367, 892 (2020); Gonzalez-Ballestero et al., Science 374, 6564 (2021)]

1.4 Electro-optic chirped photonics

Electro-optic modulation in lithium niobate and related materials enables programmable frequency sweeps with Δν ~ 10–80 GHz on ns–100 ns timescales, with electrical-to-optical waveform transfer. [Representative capability; specific platform parameters to be confirmed in Phase 2]

1.5 Landau–Zener and non-adiabatic traversal

Sweep-rate-dependent transition probabilities in two-level systems are governed by the Landau–Zener formula. Dissipative extensions exist but are not standard. The analogy to chirped resonance traversal is structural, not quantitative. [Zener, Proc. R. Soc. A 137, 696 (1932); Wubs et al., New J. Phys. 7, 218 (2005)]

1.6 Near-critical sensing (boundary framework)

Systems near boundaries (resonance, instability, critical coupling) exhibit enhanced sensitivity with reduced invertibility. Traversal of such boundaries differs qualitatively from static probing. [Warring, “Amplifiers at the Boundary,” T(h)reehouse +EC, v0.4.0]

1.7 Trapped-ion non-adiabatic control

Fast gates and waveform-engineered pulses in trapped-ion systems demonstrate that non-adiabatic protocols can outperform quasi-static approaches when control waveforms are matched to system timescales. [Leibfried et al., Rev. Mod. Phys. 75, 281 (2003); Schäfer et al., Nature 555, 75 (2018)]

1.8 Open-system dynamics under non-equilibrium driving

Dissipative quantum and classical systems driven through resonances at finite rate exhibit history-dependent responses governed by the competition between drive rate and relaxation timescales. [Zurek, Nature 317, 505 (1985); Nalbach and Thorwart, Phys. Rev. Lett. 103, 220401 (2009); Colla and Breuer, Phys. Rev. A 105, 052443 (2022)]

2 Scope and Definitions

2.1 Frequency trajectory

A time-dependent laser frequency ν(t) applied to a resonant system, where the functional form of ν(t) is the control variable. The simplest case is a linear chirp ν(t) = ν₀ + r·t with chirp rate r = dν/dt.

2.2 Traversal vs probing

  • Probing (static): Laser frequency fixed at detuning Δ; system responds at that operating point.
  • Traversal (dynamic): Laser frequency moves through the resonance; the system’s response depends on the rate and direction of crossing.

This coastline concerns the regime where traversal produces outcomes that static probing at any single detuning cannot replicate.

2.3 Primary system

Optically levitated dielectric microsphere supporting WGM modes, with coupled degrees of freedom: (a) intracavity field, (b) thermal state of the resonator, (c) centre-of-mass motion.

2.4 Control resource

A physical degree of freedom qualifies as a control resource (in the sense of this coastline) if its variation produces measurable outcomes that are inaccessible to any protocol that does not explicitly encode the rate and sign of frequency traversal.

3 Novel Boundaries (falsifiable, versioned)

### 3.1 Boundary NB-1: Chirp-direction asymmetry (v0.2, frozen) **Claim:** For a levitated WGM resonator, the capture probability depends on the sign of the chirp rate *r*. Specifically, the capture asymmetry > *A* ≡ ( *P*\_capture(+*r*) − *P*\_capture(−*r*) ) / ( *P*\_capture(+*r*) + *P*\_capture(−*r*) ) satisfies \|*A*\| > 0 for at least one value of \|*r*\|, **and** *A* → 0 as *r* → 0 (quasi-static limit). **Falsification:** *A* = 0 across all accessible chirp rates → NULL. *A* ≠ 0 but persists at *r* → 0 → attributable to generic thermal hysteresis, not trajectory control. **Status:** Untested. Discriminant sheet v0.2 specifies the minimal experimental protocol.
### 3.2 Boundary NB-2: Chirp-rate window (v0.1, open) **Claim:** The asymmetry *A*(*r*) is nonzero only within a finite window of chirp rates, bounded by: - *r* → 0: adiabatic limit, *A* → 0 - *r* → ∞: impulse limit, *A* → 0 **Falsification:** If *A* is monotonically increasing with *r* up to the maximum accessible rate, NB-2 is not supported. **Status:** Untested. Requires chirp-rate scan over ≥ 2 decades.
### 3.3 Boundary NB-3: Anomalous sign reversal (v0.1, open) **Claim:** For systems with competing response channels, the sign of *A* may reverse at a critical chirp rate *r*\*, indicating a crossover in the dominant coupling mechanism. **Falsification:** If *A*(*r*) is single-signed across the entire accessible window, NB-3 is not supported. **Status:** Speculative. Elevated to boundary status because a sign reversal, if observed, would constitute the strongest evidence for a multi-channel trajectory-control regime.

4 Architectural Notes

4.1 Relation to “Amplifiers at the Boundary”

The near-critical sensing essay identifies a general pattern: systems near boundaries exhibit sentinel behaviour (high gain, low invertibility). This coastline instantiates that pattern for a specific boundary — the WGM resonance in a coupled optomechanical system — and adds a specific claim: that traversal rate and direction are control variables that interact with the boundary in ways that static operation does not access.

4.2 Regime classification

Regime System Role of ν(t) Status
Free particle atoms, positronium Doppler tracking (cooling) Established
Bound emitter NV, ions spectral addressing / RAP Established
Cavity system WGM (fixed) detuning control Established
Hybrid system levitated WGM moving boundary + internal resonance This coastline

4.3 Collaboration geometry

Node 1 — Trajectory generation (electro-optic photonics). A platform capable of programmable ν(t) with ns-scale waveform control, GHz bandwidth, and electrical-to-optical transfer.

Node 2 — System and measurement (precision particle control). Trapped or levitated particle expertise, precision quantum control infrastructure, phase-sensitive detection.

Node 3 — Precision validation (metrology-grade systems). Provides the precision boundary condition: whether trajectory-dependent protocols survive metrological scrutiny.

Node 4 — Theoretical prediction (open-system dynamics). Dissipative dynamics and numerical modelling for the coupled-mode system under chirped drive.

Phased engagement:

  • Phase 1a (Internal alignment): Discriminant sheet, system identification.
  • Phase 1b (Theoretical model): Node 4 constructs coupled-mode model for A(r).
  • Phase 2 (Platform discussion): Present discriminant and model predictions to Node 1.
  • Phase 3 (External validation): Approach Node 3 with falsifiable test and predictions.

4.4 What this coastline does not claim

  • It does not claim that chirped light is superior to static detuning for all optomechanical tasks.
  • It does not claim quantum advantage or quantum control; the discriminant is classical.
  • It does not claim that the levitated WGM system is the only or best platform; it is the sharpest test case.
  • A positive NB-1 does not imply a universal control primitive. It implies a new control primitive for systems with coupled internal resonance and delayed dynamical response.

5 Open Items

  1. Joint figures of merit across nodes not yet defined.
  2. Theoretical model for A(r) — Phase 1b deliverable (in progress).
  3. Wavelength compatibility: Node 1 platform wavelength range must overlap with accessible WGM resonances.
  4. Vacuum and optical access: Experimental feasibility assessment pending (Node 2).
  5. Prospective chart-transition test: This coastline as candidate for Chart-Transition Protocol.
  6. Node 3 engagement timing: Phase 3 should not be entered before Phase 2 provides concrete numbers.
  7. Precision benchmark question: Does trajectory-dependent control outperform composite pulses and optimal control?
  8. Independent replication: Confirm whether alternative microresonator platforms exhibit the same dynamics.

6 Version History

Version Date Changes
v0.1 April 2026 Initial coastline. Three novel boundaries.
v0.2 April 2026 Three-node collaboration geometry. Phased engagement.
v0.3 April 2026 Capture definition added to NB-1. NB-3 reproducibility clause.
v0.4 April 2026 Node 4 added. Four-node geometry. Phase 1b inserted.
v0.5 April 2026 Full depersonalisation. Theory success criterion. Preparation-noise control.