Spatial Characterization Of Two-Photon States - GAP-Optique

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