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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APPENDIX B<br />
Integrals of<br />
the matrix mode function<br />
Given a n×n symmetric and positive definite matrix A, and two n order vectors<br />
x and b<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = (2π)n/2<br />
<br />
exp −<br />
det(A) btA−1 <br />
b<br />
.<br />
2<br />
(B.1)<br />
To proof this result consider that, since A is positive definite, there exist another<br />
matrix O such that OAO t = D where D is a diagonal matrix. With the<br />
transformation y = Ox, the integral in equation B.1 becomes<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = dy exp<br />
<br />
− yt Dy<br />
2 + iyt b ′<br />
<br />
,<br />
(B.2)<br />
with b ′ = O −1 b.<br />
As the argument of the exponential, in the right side of equation B.1, is<br />
given by the matrix product<br />
− 1 <br />
y1<br />
2<br />
y2 . . . yn<br />
⎛<br />
d1<br />
⎜ 0<br />
⎜<br />
⎝ .<br />
0<br />
d2<br />
.<br />
. . .<br />
. . .<br />
. ..<br />
0<br />
0<br />
.<br />
⎞ ⎛<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎠ ⎝<br />
0 0 0 dn<br />
y1<br />
y2<br />
.<br />
yn<br />
⎞<br />
⎛<br />
⎟<br />
⎠ + i ⎜<br />
y1 y2 . . . yn ⎜<br />
⎝<br />
the integral can be written in a polynomial form as<br />
<br />
dx exp − xtAx 2 + ixt <br />
b<br />
<br />
<br />
= dy1dy2...dyn exp − y2 1d1<br />
2 + iy1b ′ 1 − y2 2d2<br />
2 + iy2b ′ 2... − y2 ndn<br />
2 + iynb ′ <br />
n<br />
(B.3)<br />
(B.4)<br />
69<br />
b ′ 1<br />
b ′ 2<br />
.<br />
b ′ n<br />
⎞<br />
⎟<br />
⎠ ,