Manuscript Hawk02-C@2 · Hawk02-C.tex · section · bb332d66

Exact emission, Volterra dynamics, and recovery

Exact emission, Volterra dynamics, and recovery · lines 722–764

Hawk02-C@2#eq:sourceeq, Hawk02-C@2#eq:fieldeq, Hawk02-C@2#eq:fieldsolution, Hawk02-C@2#eq:volterra

Define

e^{- i t h_{g}} (1, 0) = (A (t), f_{t}) .

The prepared vector belongs to $D (h_{g})$ , which is invariant under its unitary group. Thus the component equations hold strongly:

i A ˙ (t) i f ˙_{t} = ω_{0} A (t) + ⟨ g, f_{t} ⟩, = M_{ν} f_{t} + g A (t), A (0) = 1, f_{0} = 0.

Variation of constants gives, as an $L^{2}$ identity,

f_{t} (ν) = - i g (ν) \int_{0}^{t} e^{- i ν (t - s)} A (s) d s .

Consequently,

A ˙ (t) = - i ω_{0} A (t) - \int_{0}^{t} K (t - s) A (s) d s, K (u) = \int_{0}^{\infty} ∣ g (ν) ∣^{2} e^{- i ν u} d ν .

The kernel is bounded and continuous. These are exact equations, not a Markov approximation.

For $f \in k$ , write

∣ f, a ⟩ = a^{†} (f \otimes e_{a}) Ω.

Set

q (t) = ∣ A (t) ∣^{2}, p (t) = 1 - q (t), L_{t} ∣ a ⟩_{C} = ∣ f_{t}, a ⟩ .

Displayed formulas in this passage

Hawk02-C@2#eq:sourceeq · line 731
$i A ˙ (t) i f ˙_{t} = ω_{0} A (t) + ⟨ g, f_{t} ⟩, = M_{ν} f_{t} + g A (t), A (0) = 1, f_{0} = 0.$
Hawk02-C@2#eq:fieldeq · line 731
$i A ˙ (t) i f ˙_{t} = ω_{0} A (t) + ⟨ g, f_{t} ⟩, = M_{ν} f_{t} + g A (t), A (0) = 1, f_{0} = 0.$
Hawk02-C@2#eq:fieldsolution · line 739
$f_{t} (ν) = - i g (ν) \int_{0}^{t} e^{- i ν (t - s)} A (s) d s .$
Hawk02-C@2#eq:volterra · line 745
$A ˙ (t) = - i ω_{0} A (t) - \int_{0}^{t} K (t - s) A (s) d s, K (u) = \int_{0}^{\infty} ∣ g (ν) ∣^{2} e^{- i ν u} d ν .$
unlabeled · line 725
$e^{- i t h_{g}} (1, 0) = (A (t), f_{t}) .$
unlabeled · line 756
$∣ f, a ⟩ = a^{†} (f \otimes e_{a}) Ω.$
unlabeled · line 760
$q (t) = ∣ A (t) ∣^{2}, p (t) = 1 - q (t), L_{t} ∣ a ⟩_{C} = ∣ f_{t}, a ⟩ .$