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