Manuscript Hawk02-B@2 · Hawk02-B.tex · section · 95ea744f

Exact invariant-sector matrix representation

Exact invariant-sector matrix representation · lines 1702–1730

The full physical radiation space is $(C^{d + 1})^{\otimes N}$ . The ideal prepared trajectory occupies exactly

H_{R, \leq 1} = span {∣ vac ⟩, ∣ a; j ⟩},

of dimension $1 + N d$ .

The total one-excitation sector has ordered basis

∣ m, a ⟩ = ∣ a ⟩_{M} ∣ vac ⟩_{R}, ∣ r, j, a ⟩ = ∣ 0 ⟩_{M} ∣ a; j ⟩_{R} .

Its dimension is $d (N + 1)$ , and

S_{j} = a \sum (∣ m, a ⟩ ⟨ r, j, a ∣ + ∣ r, j, a ⟩ ⟨ m, a ∣)

on that sector. There $H_{0} = ω I$ . This is an exact invariant-sector representation, not a probabilistic cutoff.

Arbitrary perturbations leaving this sector require the common full-space comparison in Theorem thm:B6. The reported detuning was sector preserving.

Displayed formulas in this passage

unlabeled · line 1707
$H_{R, \leq 1} = span {∣ vac ⟩, ∣ a; j ⟩},$
unlabeled · line 1714
$∣ m, a ⟩ = ∣ a ⟩_{M} ∣ vac ⟩_{R}, ∣ r, j, a ⟩ = ∣ 0 ⟩_{M} ∣ a; j ⟩_{R} .$
unlabeled · line 1719
$S_{j} = a \sum (∣ m, a ⟩ ⟨ r, j, a ∣ + ∣ r, j, a ⟩ ⟨ m, a ∣)$