Diploma thesis

This Taylor series expansion is performed at a point with perfect phasematching
(∆k0 = 0). Inserting Equations 3.44 to 3.45 into Formula 3.43 evaluates to:
� ∞ � ∞
|ψs,i〉 = A
0 0 � ∞ � ∞
= A
0
0
dωs dωie − (νs+ν i) 2
2σ 2 e
γL2
− 4
(τsνs+τiνi) 2
â † s(ωs)â †
i (ωi) |0〉
1
−
dωs dωie 2σ2 (ν2 s +ν2 γL2
i +2νsνi)− 4 (τ 2 s ν2 s +τ 2 i ν2 i +2τsτiνsνi) †
â s(ωs)â †
i (ωi) |0〉 .
From this equation, we obtain the factorizability condition [21]:
2
σ 2 + γL2 τsτi = 0,
(3.48)
2
σ 2 + γL2 (k ′ p − k ′ s)(k ′ p − k ′ i) = 0. (3.49)
The utilized crystal and laser have to meet this specific requirements to generate
uncorrelated photon pairs.
Further on with this formula, it is possible to determine the slope of ∆k = 0 for
every particular point of the contour in the {ωs, ωi}-plane (see Figure 3.25).
τsνs + τiνi = 0 (3.50)
� �
�
τs
k ′
p − k ′ �
s
⇒Θ = − arctan
τi
= − arctan
k ′ p − k ′ i
Figure 3.25: Definition of the phasematching angle
(3.51)
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