Manuscript Em03-E@1 · Em03-E-v1.tex · section · c4b07e6a

Occupation emission

E1: explicit internal and emission Hamiltonians › Occupation emission · lines 531–562

Em03-E@1#eq:numberemission

For each $i$ , choose a probability vector

w_{i, n} \geq 0, n = 0 \sum^{J_{i}} w_{i, n} = 1.

Its semiclassical calibration is specified in Section sec:matching. Define

∣ E_{i} ⟩ = ∣ e ⟩_{A_{i}} ∣ 0 ⟩_{O_{i}}, ∣ F_{i} ⟩ = n = 0 \sum^{J_{i}} w_{i, n} ∣ d_{n} ⟩_{A_{i}} ∣ n ⟩_{O_{i}} .

These are orthonormal. Set

G_{num, i} = ∣ F_{i} ⟩ ⟨ E_{i} ∣ + ∣ E_{i} ⟩ ⟨ F_{i} ∣, \int g_{num, i} (t) d t = π /2.

The exact emission is

∣ e ⟩_{A_{i}} ∣ 0 ⟩_{O_{i}} ⟼ - i n = 0 \sum^{J_{i}} w_{i, n} ∣ d_{n} ⟩_{A_{i}} ∣ n ⟩_{O_{i}} .

The generator is a norm-one finite matrix. Its terms, including the zero-occupation daughter term, are part of the specification.

Displayed formulas in this passage

Em03-E@1#eq:numberemission · line 553
$∣ e ⟩_{A_{i}} ∣ 0 ⟩_{O_{i}} ⟼ - i n = 0 \sum^{J_{i}} w_{i, n} ∣ d_{n} ⟩_{A_{i}} ∣ n ⟩_{O_{i}} .$
unlabeled · line 534
$w_{i, n} \geq 0, n = 0 \sum^{J_{i}} w_{i, n} = 1.$
unlabeled · line 539
$∣ E_{i} ⟩ = ∣ e ⟩_{A_{i}} ∣ 0 ⟩_{O_{i}}, ∣ F_{i} ⟩ = n = 0 \sum^{J_{i}} w_{i, n} ∣ d_{n} ⟩_{A_{i}} ∣ n ⟩_{O_{i}} .$
unlabeled · line 546
$G_{num, i} = ∣ F_{i} ⟩ ⟨ E_{i} ∣ + ∣ E_{i} ⟩ ⟨ F_{i} ∣, \int g_{num, i} (t) d t = π /2.$