Spatial Characterization Of Two-Photon States - GAP-Optique

1.3. Approximations and other considerations
operator in mode p, and the creation operators in modes s and i. Assuming
constant χ (2) , the two-photon state reduces to
(2) τ
iɛ0χ
|Ψ〉 = dt dV
0 V
Ê(+) p (rp, ωp) Ê(−) s (rs, ωs) Ê(−) i (ri, ωi)|1〉p|0〉s|0〉i,
(1.10)
which can be written as
|Ψ〉 ∝ dks dkiΦ(ks, ωs, ki, ωi)a † (ks)a † (ki)|0〉s|0〉i; (1.11)
where Φ(ks, ωs, ki, ωi) is known as the mode function and, assuming that the
pump has certain spatial distribution Ep(kp), is given by
τ
Φ(ks, ωs, ki, ωi) ∝ dt dV dkpEp(kp)
0
V
× exp [ikp · rp − iks · rs − iki · ri − i(ωp − ωs − ωi)t]
× â(kp)|1〉p. (1.12)
The mode function contains all the information about the generated two-photon
system in space and frequency, not only about their individual state but about
their correlations. This function is therefore highly complex, and to calculate
any feature of the two-photon state it is necessary to simplify it, as the next
section shows.
1.3 Approximations and other considerations
Analytical calculations of any features of the down converted photons require
simplification of the mode function in equation 1.12. This section lists the most
important approximations used in this thesis, as well as the restrictions that
they impose.
Separation of transversal and longitudinal components
A field with a wave vector k almost parallel to its propagation direction spreads
only a little in the transversal direction. It is then possible, to consider the
longitudinal and transversal components of the wave vector separately,
k = kzˆz + q. (1.13)
As the wave vector’s magnitude k = ωn/c is bigger than the transversal component’s
magnitude q = |q| = (k 2 x + k 2 y) 1/2 , the magnitude of the longitudinal
component kz is always a real number,
kz = k 2 1
− q
2 2 . (1.14)
Assuming that the pump, the signal, and the idler are such fields, the mode
function becomes
τ
Φ(qs, ωs, qi, ωi) ∝
0
dt dV
V
dqpEp(qp) exp [ik z pzp − ik z szs − ik z i zi]
× exp [iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i − i(ωp − ωs − ωi)t]
× â(q p)|1〉p. (1.15)
where r ⊥ n = xnˆxn + ynˆyn is the transversal position vector.
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