Spatial Characterization Of Two-Photon States - GAP-Optique

1. General description of two-photon states
When the amplitude of the electric field increases, the higher orders term in
equation 1.2 become relevant, and then a nonlinear response of the material
to the field appears. Optical nonlinear phenomena resulting from this kind
of interaction include the generation of harmonics, the Kerr effect, Raman
scattering, self-phase modulation, and cross-phase modulation [34].
Spontaneous parametric down-conversion, and other second order nonlinear
processes, result from the second order polarization, defined as the first
nonlinear term in the polarization tensor
P (2) = ɛ0χ (2) EE. (1.4)
The quantization of the electromagnetic field leads to a quantization of the
second order polarization, so that the nonlinear polarization operator ˆ P (2)
becomes
ˆP (2) = ɛ0χ (2) ( Ê(+) + Ê(−) )( Ê(+) + Ê(−) ), (1.5)
where Ê(+) and Ê(−) are the positive and negative frequency parts of the field
operator [36]. The positive frequency part of the electric field operator is a
function of the annihilation operator â(k), and is defined at position rn
xnˆxn + ynˆyn + znˆzn and time t as
=
Ê (+)
1/2
ωn
n (rn, t) = ien
2ɛ0v
dkn exp [ikn · rn − iωnt]â(kn), (1.6)
where the magnitude of the wave vector k satisfies k 2 = ωn/c. The volume
v contains the field, and en is the unitary polarization vector. The negative
frequency part of the field is the Hermitian conjugate of the positive part,
Ê (−)
n (rn, t) = Ê(+)† n (rn, t). Thus, the negative frequency part is a function of
the creation operator â † (k).
Following references [37] and [38], in first order perturbation theory, the
interaction of a volume V of a material with a nonlinear polarization ˆ P (2) and
the field Êp(rp, t), produces a system described by the state
|ΨT 〉 ∝ |1〉p|0〉s|0〉i − i
τ
0
dt ˆ HI|1〉p|0〉s|0〉i, (1.7)
where τ is the interaction time, |1〉p|0〉s|0〉i is the initial state of the field, and
the interaction Hamiltonian reads
ˆHI = − dV ˆ P (2) Êp(rp, t). (1.8)
V
The first term at the right side of equation 1.7 describes a one photon system,
while the second term describes a two-photon system. In what follows we will
consider only the second term as we are mainly interested in the generation of
pairs of photons. The two-photon system state is given by
|Ψ〉 = i
τ
0
dt ˆ HI|1〉p|0〉s|0〉i, (1.9)
or ˆρ = |Ψ〉〈Ψ| in the density matrix formalism. Seven of the eight terms
that compose the interaction Hamiltonian vanish when they are applied over
the state |1〉p|0〉s|0〉i. The remaining term is proportional to the annihilation
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