Spatial Characterization Of Two-Photon States - GAP-Optique

2. Correlations and entanglement
q s x
q s y
q i x
y
qi s
i
qs x
a
h
i
j
k
l
q s
y
h
b
m
n
p
r
qi y
i
m
c
s
t
u
Figure 2.5: The signal-idler cross terms in matrix A are responsible for the correlations
between the generated photons. By suppressing these cross terms, and by making
permutations over columns and rows, the matrix becomes a two block matrix, where
each block contains the information of one photon.
ing the following terms negligible
i =w 2 p + γ 2 L 2 tan ρ0 2
j =γ 2 L 2 sin ϕi tan ρ0
l =γ 2 L 2 tan ρ0(Np − Ni cos ϕi)
m = − γ 2 L 2 sin ϕs tan ρ0
q i
x
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi
t =γ 2 L 2 tan ρ0(Np − Ns cos ϕs)
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs
z =γ 2 L 2 N 2 p − γ 2 L 2 Np(Ni cos ϕi + Ns cos ϕs) + γ 2 L 2 NsNi cos ϕs cos ϕi
j
n
s
d
v
w
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi. (2.15)
As an example, a configuration with a negligible Poynting walk-off fulfills this
condition always when
cos ϕs = Np
2Ns
cos ϕi = Np
2Ni
w 2 p
γ 2 L 2 = tan ϕs tan ϕi
T 2 0 =N 2 p γ 2 L 2
wp ≪ws, wi
s
k
p
t
v
f
z
i
l
r
u
w
tan ϕs 2 tan ϕi 2 − 1
4
z
g
(2.16)
When the correlations are totally suppressed by satisfying these or other conditions,
the two-photon state is separable, and each photon is in a pure state.
Therefore, the two-photon mode function can be written as a product of two
mode functions, one for each photon,
24
Φ(q s, ωs, q i, ωi) = Φs(q s, Ωs)Φi(q i, Ωi). (2.17)