Atom-coupling objective

Decision

The first atom-aware production objective is the projected electric Green LDOS at the trapped atom, normalized to the same discrete source in homogeneous cladding. For the current configuration the assumed transition is centered at the origin, z polarized, and resonant at normalized frequency f_a = 1 (780 nm).

This is the correct directly computable objective for spontaneous-emission enhancement and weak-coupling cavity cooperativity. It is not, by itself, a measurement of the vacuum Rabi coupling g, quasinormal-mode volume, guided beta factor, trap quality, or tweezer scattering.

Continuous definition

For atom position \(\mathbf r_a\), unit transition dipole \(\hat{\mathbf d}\), and angular frequency \(\omega_a\), the projected electric LDOS is proportional to

\[ \rho_d(\mathbf r_a,\omega_a) \propto \operatorname{Im}\left[ \hat{\mathbf d}\cdot \mathbf G(\mathbf r_a,\mathbf r_a;\omega_a) \hat{\mathbf d} \right]. \]

The normalized optimization quantity is

\[ F_a = \frac{\rho_d^{\mathrm{device}}}{\rho_d^{\mathrm{hom}}}. \]

In the weak-coupling, single-mode limit, resonant LDOS scales as \(g^2/\kappa\). Therefore maximizing \(F_a\) improves the Purcell rate and cooperativity for fixed atomic linewidth. Maximizing bare \(g\) instead would require a pole-residue or rigorously normalized QNM objective; that is a later stage.

Discrete estimator and normalization

The reciprocal FDFD solve is

\[ A(f_a,p)e=d_a, \qquad h_a=d_a^T e, \qquad F_a=\frac{\operatorname{Im}h_a} {\operatorname{Im}h_{a,\mathrm{hom}}}. \]

d_a is the reflection-even centered z dipole. Its support is entirely in the immutable air strip. The homogeneous denominator uses the same grid, source discretization, PML, and longitudinal surface-Green machinery, with n_core == n_clad. This cancels the arbitrary discrete dipole amplitude and makes the reported enhancement dimensionless on a given grid. It does not remove the need for grid and PML convergence.

Gradient

The optimized scalar is \(J=\log F_a\). Since the homogeneous denominator is independent of topology,

\[ \frac{\partial J}{\partial p} =\frac{1}{\operatorname{Im}h_a} \operatorname{Im}\left[\frac{\partial h_a}{\partial p}\right]. \]

Reciprocity and complex symmetry give

\[ \frac{\partial h_a}{\partial\rho} =(2\pi f_a)^2\Delta\epsilon\, W^T(T_e\odot e^2), \]

followed by the exact transpose projection/filter backpropagation. No extra adjoint solve is needed for this colocated dipole objective.

The production proposal is a multiobjective trust direction, not a change to the reported LDOS definition. A fresh 29-frequency fixed-subspace Loewner model supplies \(\nabla\log Q\). After separately normalizing that gradient and \(\nabla\log F_a\), a deterministic positive-weight search chooses a bounded direction whose first-order slopes for both quantities are positive, including after clipping variables at zero and one. This local construction does not accept the step; it only makes a trial more likely to satisfy the two independent improvement gates.

The accelerated production path supersedes that fixed-weight direction after resonance nucleation. It evaluates normalized LDOS at the moving real part of the retarded-real-axis Loewner pole, includes both device and homogeneous frequency derivatives, and uses a constrained moving-asymptote subproblem. The actual fixed-atom-frequency LDOS remains an independent strict acceptance gate. See SHIFTED_COOPERATIVITY_ARCHITECTURE.md for its continuous definition, discrete estimator, gradient, normalization, detuning constraint, and validation contract.

Independent acceptance test

The differentiable LDOS objective never accepts a topology by itself. For every proposed step, atom_ldos_optimizer.py:

rebuilds a wide full-cluster sweep and a dense, sheet-consistent local seven-probe real-frequency sweep;
runs the combined shared-pole validator using the current tracked pole and predeclared plus/minus-two-halfwidth target interval;
requires the atom LDOS enhancement to improve;
requires independently fitted shared-pole Q to improve strictly;
limits checkpoint-to-checkpoint pole motion in units of the old half-width;
verifies maximum density in the protected air strip is exactly zero; and
backtracks or rejects the update when any condition fails.

For the shifted constrained-MMA production path, acceptance additionally requires directly recomputed normalized LDOS at the authoritative shared pole's real part to improve and atom detuning to remain within two pole half-widths (or monotonically restore feasibility). Three provisional gradient/MMA refreshes remain inside one outer trust box and never count as accepted steps. The next topology can begin only after pole-centered promotion and a fingerprint-matched seven-case PML gate pass.

A positive-step command additionally requires a metrics file from a passed sheet-local-v2 combined shared-pole transverse-PML campaign. The local acquisition must contain at least 65 points so the independent order-29/32 ensemble is possible. Spectral, PML, and grid evidence levels remain separate.

An already accepted wide/local dataset may seed the current side of the first restartable step only when its stored raw vector is bitwise equal to the warm start. The atom-frequency gradient is still solved freshly, and every proposed topology is always rebuilt across both spectral windows. Reuse saves duplicate baseline work but never validates a trial topology.

Demonstrations required before a long run

Full regression suite before and after the objective change.
Same-grid homogeneous structure returns F_a = 1.
Directional finite difference of log(F_a) on the smallest useful 3D grid.
Source support lies wholly inside the exact air strip.
Combined spectral validation of the warm start with an explicit, non-stale reference pole.
Combined transverse-PML validation of that same pole identity.
One bounded atom-LDOS step that improves both F_a and shared-pole Q and survives a completely rebuilt combined validator.
A second grid/domain point before interpreting the enhancement physically.

Production command template

Do not fill these pole/window values from an older grid. Use values from the accepted reacquisition on the current grid.

The qualified fine-grid command is fixed in scripts/run_fine_cooperativity_step.sh; see UNATTENDED_RUNBOOK.md. For a new grid, rebuild its wide/local/confirmatory and PML evidence before updating that runner. Do not fill pole/window values from an older discretization.

run_atom_live.sh publishes a running state before the solve. It exits nonzero on a rejected step, and scripts/finalize_atom_run.py publishes a complete state only after independently reading the saved metrics, verifying strict objective and Q improvement, exact air density, the sheet-local spectral gate, and rebuilding the accepted geometry/field views. The public live page does not bypass any scientific stop condition.

Next physical extensions

Before a device claim, add an atom-position/orientation ensemble for trap uncertainty, a pole-residue or QNM normalization for bare g, guided-port projections for beta factor and useful outcoupling, realistic material loss, and fabrication/trap/tweezer robustness.