Manuscript Em04-A@2 · Em04-A-v2.tex · proposition · f5b28abc

Proposition: Two-site closed-form benchmark

Transient entanglement and controlled comparisons · lines 880–910 · proof included or linked

Em04-A@2#prop:twosite

For the localized preparation, $N = 2$ , and equal detunings, let $κ = 2 (1 - r) / χ$ and define

F_{κ} (τ) = ⎩ ⎨ ⎧ sin (ω τ) / ω, τ, sinh (ν τ) / ν, κ < 4, ω = 4 - κ^{2} /4, κ = 4, κ > 4, ν = κ^{2} /4 - 4 .

Then

C_{01} (τ) = 2 e^{- 2 r τ / χ} e^{- κ τ /2} ∣ F_{κ} (τ) ∣.

For $r = 1$ , this is $e^{- 2 τ / χ} ∣ sin (2 τ) ∣$ .

Proof. Write $S_{01} = i y$ , $z = S_{00} - S_{11}$ . The equations are

z^{'} = - 4 y, y^{'} = z - κ y, z (0) = 1, y (0) = 0.

Hence $y^{''} + κ y^{'} + 4 y = 0$ , $y^{'} (0) = 1$ , whose solution is $y = e^{- κ τ /2} F_{κ} (τ)$ . Use $C = 2 q ∣ y ∣$ .

Displayed formulas in this passage

unlabeled · line 884
$F_{κ} (τ) = ⎩ ⎨ ⎧ sin (ω τ) / ω, τ, sinh (ν τ) / ν, κ < 4, ω = 4 - κ^{2} /4, κ = 4, κ > 4, ν = κ^{2} /4 - 4 .$
unlabeled · line 895
$C_{01} (τ) = 2 e^{- 2 r τ / χ} e^{- κ τ /2} ∣ F_{κ} (τ) ∣.$
unlabeled · line 905
$z^{'} = - 4 y, y^{'} = z - κ y, z (0) = 1, y (0) = 0.$