Manuscript Em03-B@2 · Em03-B-v2.tex · section · c1a0977d

The saturating carrier

The B1–B3 mathematical interface › The saturating carrier · lines 472–490

Em03-B@2#eq:carrier

Fix integers $N \geq 1$ and, later, $K \geq N$ . The reported saturation level $N$ and the control-domain cutoff $K$ are different parameters.

Let

K_{N} = span {∣ 0 ⟩_{C}, \dots, ∣ N ⟩_{C}}, P_{N} = n = 0 \sum^{N} ∣ n ⟩ ⟨ n ∣, Q_{N} = I - P_{N} .

Let $V_{N}$ identify the number bases on $P_{N} H_{h}$ and $K_{N}$ . Define

C_{N} (X) = V_{N} P_{N} X P_{N} V_{N}^{†} + tr (Q_{N} X) ∣ N ⟩ ⟨ N ∣_{C} .

Displayed formulas in this passage

Em03-B@2#eq:carrier · line 485
$C_{N} (X) = V_{N} P_{N} X P_{N} V_{N}^{†} + tr (Q_{N} X) ∣ N ⟩ ⟨ N ∣_{C} .$
unlabeled · line 479
$K_{N} = span {∣ 0 ⟩_{C}, \dots, ∣ N ⟩_{C}}, P_{N} = n = 0 \sum^{N} ∣ n ⟩ ⟨ n ∣, Q_{N} = I - P_{N} .$