Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
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2. Correlations and entanglement<br />
the spatial two-photon state is<br />
ˆρq = T rΩ[ρ]<br />
<br />
=<br />
<br />
=<br />
dΩ ′′<br />
s dΩ ′′<br />
i 〈Ω ′′<br />
s , Ω ′′<br />
i |ˆρ|Ω ′′<br />
s , Ω ′′<br />
i 〉<br />
dqsdΩsdqidΩidq ′ sdq ′ i<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)|qs, qi〉〈q ′ s, q ′ i|, (2.10)<br />
while the purity of this state, defined as T r[ˆρ 2 q], is given by<br />
T r[ˆρ 2 <br />
q] =<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i). (2.11)<br />
It is possible to solve these integrals analytically by using the exponential<br />
character of the mode function given by equation (1.35). The purity becomes<br />
T r[ˆρ 2 q] = det(2A)<br />
det(B) . (2.12)<br />
As seen in appendix A, B is a positive-definite real 12 × 12 matrix<br />
B = 1<br />
⎛<br />
⎜<br />
2 ⎜<br />
⎝<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
l<br />
0<br />
0<br />
0<br />
0<br />
k<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
r<br />
0<br />
0<br />
0<br />
0<br />
p<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
u<br />
0<br />
0<br />
0<br />
0<br />
t<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
w<br />
0<br />
0<br />
0<br />
0<br />
v<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
2z<br />
k<br />
p<br />
t<br />
v<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
2g<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
0<br />
0<br />
0<br />
k<br />
l<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
0<br />
0<br />
0<br />
0<br />
p<br />
r<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
0<br />
0<br />
0<br />
0<br />
t<br />
u<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
0<br />
0<br />
0<br />
0<br />
v<br />
w<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
k<br />
p<br />
t<br />
v<br />
0<br />
0<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
l r u w 0 0 l r u w 2z 2g<br />
defined by the equation<br />
N 4 <br />
exp − 1<br />
2 Xt <br />
BX = Φ(qs, Ωs, qi, Ωi)<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
(2.13)<br />
× Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i), (2.14)<br />
where vector X is the result of concatenation of x and x ′ .<br />
In order to see the effect of each of the spdc parameters on the spatiofrequency<br />
correlations, some typical values are used in next subsection to calculate<br />
T r[ˆρ 2 q] numerically.<br />
2.2.3 Numerical calculations<br />
As first example, consider a degenerate type-i spdc process characterized by the<br />
parameters in the second column of table 2.1. A pump beam, with wavelength<br />
20