# XYZ-symmetric y-atom SiN air-strip volume-constrained-Q campaign

Date: 2026-08-31  
Campaign: `sin500_xyz_y_atom_strip_16x2_temporal_qv_constrained_v2`

## Goal

The corrected design goal is high cavity lifetime without allowing the mode
to leave the y-oriented atom. Stage B maximizes Q subject to one frozen
atom-local mode-volume ceiling:

\[
 V_a\le(1+0.002)V_h,
 \qquad
 C=\frac{3}{4\pi^2}\frac{Q_h}{V_a/\lambda_p^3}.
\]

Here the atom is in air, so the normalization is `lambda_pole^3`, not the SiN
wavelength cubed. `Q_h` is the harmonic consensus of independent atom-field
and total-energy decay estimates. `V_a` is evaluated from Ey at the atom for
that same fitted pole. `V_h` is measured once at the exact temporal handoff;
it never ratchets upward. The 0.2% margin is an estimator-noise allowance, not
a geometric budget. The failed collection-probability branch is preserved as
a negative ablation: it raised Q from 268 to 10,027 while physical atom volume
rose from 4.289 to 100.02 um3, leaving Purcell only 3.76.

## Immutable physical geometry

- Vacuum discovery wavelength: 780 nm.
- Material: nondispersive SiN with index 2 in air.
- Full design region: 16 um along x and 2 um along y. Stage A is exactly
  0.5 um thick; Stage B may optimize one global design-region thickness from
  0.2 to 0.7 um.
- Atom: at the origin, polarized along y.
- Atom access: the entire through-device strip `-0.5 um < x < +0.5 um` is
  immutable air for every y. On the 50 nm grid this is exactly ten retained-x
  columns and becomes a one-micron strip after unfolding.
- Ports: a 600 nm-wide x-directed SiN feedthrough on both sides. The width is
  intentionally different from the 500 nm thickness so its fundamental
  TE-like mode is not a polarization-degenerate square-guide doublet.
- Symmetry: x magnetic, y electric, and z magnetic mirror planes,
  `symmetry=(1,-1,1)`. The retained source amplitude is `1/sqrt(2)` and x
  unfolding supplies the coherent counterpropagating partner, giving unit
  total incident power. Reduced energy is multiplied by eight.
- Parameterization: 3,200 stored direct 50 nm pixels in the retained 8 um by
  1 um quadrant before hard air, outer guard, and port pins are removed. No
  spatial filter, projection, binarization, periodic template, or density
  penalty is active. At the Stage-B handoff, one bounded scalar for the
  globally uniform slab thickness is appended to these in-plane coordinates.
- Initialization: density zero on every trainable pixel and throughout the
  atom-access strip. Only the pinned outer feedthrough is fixed to density one.
  The temporal gradient must therefore construct an air-mode cavity rather
  than perturbing a midpoint-permittivity pad.

The non-PML interior is 17.1 um by 2.5 um by 1.7 um. Ten 50 nm PML cells are
added per full-domain face. The 1.7 um z interior leaves 0.60 um of air between
each 500 nm device face and the z PML, retaining the previously audited
slab-face padding rather than moving the absorber closer to the thicker slab.
The reduced Maxwell grid is 181 by 35 by 27.

## Stage A: absolute temporal architecture discovery

Let `s_k` be the analytically delayed complex Gaussian bright-channel field at
the atom. The exact discrete target pole is

\[
 a_{k+1}=e^{-(\gamma+i\omega_0)\Delta t}a_k
          +(1-e^{-\gamma\Delta t})s_k,
 \qquad \gamma=\frac{\omega_0}{2Q_t}.
\]

The fixed target uses `Q_t=10,000`, beta one, `V_t=0.1 um^3`, and perfect
y-polarized overlap. Its trace is the prompt bright-mode response plus the
causal cavity response. It is never shifted, projected, or rescaled to match
the simulated amplitude. The complete vector objective is

\[
 J_t=-\frac{\sum_k\|\mathbf E(\mathbf r_a,t_k)-\hat y E_{t,k}\|^2}
              {\sum_k|E_{t,k}|^2}.
\]

Raw Adam uses learning rate 0.01. One 2 ps forward/reverse pair applies one
pixel update. Stage A runs for 50 updates rather than 150. Handoff also
requires a valid pole, field/energy Q agreement within 25%, and conservative
Q at least 100; iteration 50 is the earliest permitted handoff, not permission
to use an untrusted fit.

## Same-pole Q, V, and Purcell measurement

After the pulse clears, Ey at the atom is fitted, recentered, and fitted again.
Total electromagnetic energy is fitted independently at the same pole
frequency. The trusted lifetime is guarded by both estimates, and their
harmonic consensus is

\[
 Q_h=\frac{2Q_EQ_U}{Q_E+Q_U}.
\]

The same three source-off windows determine

\[
 V_a=\frac{8\langle U_{\rm reduced}\rangle}
           {\epsilon_{\rm air}\langle|E_y(\mathbf r_a)|^2\rangle},
 \qquad
 C=\frac{3}{4\pi^2}\frac{Q_h}{V_a/\lambda_p^3}.
\]

The mode-volume window spread must remain below 25%. Thus Q, V, wavelength,
and C always describe the same fitted y-polarized pole at the atom.

## Stage B: pure Q under the frozen atom-volume constraint

The differentiated objective is only the log harmonic lifetime consensus,

\[
 J_Q=\log\left(\frac{2Q_EQ_U}{Q_E+Q_U}\right).
\]

At handoff the normalized thickness coordinate `u_t` is enabled, with

\[
 t(u_t)=0.2\ {\rm um}+0.5\ {\rm um}\,u_t,\qquad 0\le u_t\le1.
\]

On the retained positive-z half-domain, cell `k` receives the exact planar
fill fraction

\[
 f_k(t)=\operatorname{clip}\!\left(\frac{t/2-z_k^-}{\Delta z},0,1\right),
 \qquad
 \epsilon_{ijk}=1+(\epsilon_{\rm SiN}-1)\rho_{ij}f_k.
\]

Thus 500 nm is exactly five full retained-z cells, while a moving face has one
grey subpixel layer rather than jumping by 100 nm after symmetry unfolding.
The external feedthrough and its injected mode remain 500 nm thick; the raw
in-plane pixels may form a short transition to the optimized design-region
thickness. This avoids changing the source basis while the cavity is being
optimized.

At each fresh geometry, one joint derivative gives `grad(J_Q)` and another
gives `grad(log V_a)` with respect to all density pixels and `u_t`. The
ten-pair L-BFGS-B Q direction is projected into the local
non-increasing-volume half-space,

\[
 \nabla\log V_a\cdot d\le0.
\]

Box projection is alternated with this half-space projection when constructing
the joint density/thickness trial. The adaptive trust radius acts on the
normalized thickness coordinate exactly as it acts on a raw density
coordinate. This first-order protection is not trusted alone. Every candidate
receives a full forward replay and is rejected unless

\[
 V_a^{\rm candidate}\le(1+0.002)V_h.
\]

Accepted candidates must also increase Q strictly, pass both pole estimators,
keep the three-window volume spread below 25%, and agree with the trust model.
Rejected radii reuse the cached Q and volume gradients and cost only a forward
replay. Frequency follows the measured pole rather than being constrained.
No Q-dependent density cap, objective weight, or cumulative volume
allowance.

The production update-42 migration was audited before activation. At the
unchanged 500 nm geometry, the expanded-support material map reproduced field
Q, energy Q, and physical atom volume bit-for-bit. For a +0.001 normalized
coordinate perturbation (+0.5 nm physical thickness), the forward difference
gave `d log(Q)/du_t = 1.60980` versus the adjoint value `1.61117`; it gave
`d log(V_a)/du_t = -0.766754` versus the independent volume-adjoint value
`-0.766791`. Both complete joint gradients were finite.

The worker and dashboard are persistent services with an independent
five-minute stale-checkpoint watchdog. The optimizer runs indefinitely until
explicitly paused. The exact handoff density is stored in the v2 artifact so
future objective ablations never require another deterministic replay.

## Restart snapshots are part of the optimizer contract

The rolling artifact is saved atomically after every optimizer turn. In
addition, every 50 completed parameter updates (`total_updates = 50, 100,
150, ...`) creates an immutable, full-state snapshot under
`benchmarks/artifacts/invdes_atom_hole_sin500_xyz_y_atom_strip_16x2_temporal_qv_constrained_v2_snapshots/`.
Each snapshot contains the parameters, Adam moments or L-BFGS history, trust
state, source/pole tracking state, objective telemetry, and history needed for
an exact restart. A later save at the same update must never overwrite the
numbered snapshot. Future agents must branch small objective or controller
experiments from the nearest numbered snapshot (or the explicit handoff
parameters) instead of replaying or restarting the cavity from scratch.