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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A. The matrix form of the mode function<br />
writes<br />
T r[ρ 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). (A.7)<br />
where the dimension increases as new primed variables appear. Using the<br />
matrix notation the integrand becomes<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i)<br />
= N 4 <br />
exp − 1<br />
2 Xt <br />
BX , (A.8)<br />
where the vector X is the result of concatenation of x and x ′ , such that<br />
⎛<br />
⎜<br />
X = ⎜<br />
⎝<br />
↑<br />
x<br />
↓<br />
↑<br />
x ′<br />
↓<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
(A.9)<br />
and the new matrix B is given by<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 />
⎞<br />
⎟ .<br />
⎟<br />
⎠<br />
l r u w 0 0 l r u w 2z 2g<br />
(A.10)<br />
In an analogous way, the integrand on the expression for the signal photon<br />
purity<br />
<br />
T r[ρ 2 signal] =<br />
is written in a matrix notation as<br />
66<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i). (A.11)<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i)<br />
= N 4 exp<br />
<br />
− 1<br />
2 Xt <br />
CX . (A.12)