Diploma thesis

the rotating wave approximation [18]:
(ω−ωp)
−
αp(ω) = Ape 2σ (3.14)
By substituting Equations 3.9, 3.10, 3.13 and 3.14 into 3.8, we obtain for the Hamiltonian:
ˆH (�r, t) = χ (2)
�
dV
� ∞ � ∞ � ∞
(ω−ωp)
−
dω dωs dωiApe 2σ As(ωs)Ai(ωi)
V
0
0
0
e i(� kp(ωp)− � ks(ωs)− � ki(ωi))�r e −i(ωp−ωs−ωi)t â † s (ωs) â †
i (ωi) + h.c. (3.15)
.The parameters for the envelope of signal and idler frequencies have been merged
into As,i.
We now assume that our initial state |ψ(t = 0)〉 is the vacuum state |0〉. The
corresponding two-photon-state |ψs,i〉 evaluates to:
|ψs,i〉 = |0〉 +
� 1
i� χ(2)
psi
� ∞ � ∞ � ∞
0
0
0
�� t
(ω−ωp)
− 2σp dω dωs dωiApe As(ωs)Ai(ωi) dt
0
′ e −i∆ωt′
��
·
V
dV e i∆� �
k(ωp,ωs,ωi)�r
â † s (ωs) â †
i (ωi)
�
+ h.c.
(3.16)
The functions As,i and Ap are slowly varying with frequency, we hence treat them
as constant and combine them with the susceptibility into the constant A ′ . Furthermore,
the leading vacuum term is of no particular interest to us and will therefore
be omitted in future calculations. The hermitian conjugate part covers the reverse
process. Since only few photons are downconverted the reversal process is neglected.
This leads to the following state:
|ψs,i〉 = A ′ V
� ∞ � ∞ � ∞
0 0
� �
1
·
V
0
�� t
dω dωs dωiα(ω)
dt
0
′ e −i∆ωt′
V
dV e i∆� �
k(ωp,ωs,ωi)�r
â † s (ωs) â †
i (ωi) |0〉 . (3.17)
In this formula ∆k represents the momentum mismatch already introduced in the
SHG treatment (Equation 3.3).
∆k = kp(ωp) − ks(ωs) − ki(ωi) (3.18)
We extent the integration time to ±∞, since we are interested in the steady state.
With this simplification, the time integration is easily solveable and yields:
� ∞
dt ′ e −i∆ωt′
= 2πδ(∆ω) (3.19)
−∞
The result is the well known energy conservation condition, with ∆ω = ωp − ωs − ωi,
that replaces the frequency mismatch. We employ the energy conservation condition
to get rid of the ω integration.
�
9
�
|0〉