Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
x<br />
y<br />
z<br />
5.1. The quantum state of Stokes and anti-Stokes photon pairs<br />
zas yas<br />
xas<br />
pump<br />
anti Stokes<br />
Stokes<br />
ys<br />
control<br />
Figure 5.2: According to energy and momentum conservation, the generated photons<br />
counterpropagate. Equation 5.7 describes the relation between their propagation<br />
direction and the propagation direction of the pump and control beams, where due<br />
to the phase matching the angle of emission of the anti-Stokes photon is ϕas = π−ϕs.<br />
Because we consider a pump and a control beam with the same central frequency<br />
and k0 s k0 as, the phase matching conditions allow any angle of emission<br />
ϕas = π − ϕs, if it is not forbidden by the transition matrix elements [60].<br />
This assumption is valid only for cold atoms, since for warm atoms the process<br />
is highly directional, that is all photons are emitted along a preferred direction<br />
as proven in reference [61].<br />
There are two ways to describe the generated two-photon quantum state: it<br />
can be described by using two coupled equations in the slowly varying envelope<br />
approximation for the Stokes and anti-Stokes electric fields [65], or alternatively<br />
by using an effective Hamiltonian of interaction and first order perturbation<br />
theory [66, 67]. As the latter approach is analogous to the formalism used in<br />
chapter 1, it will be used in what follows to calculate the Stokes and anti-Stokes<br />
state.<br />
The effective Hamiltonian in the interaction picture HI describes the photonatom<br />
interaction, and is given by<br />
<br />
HI = ɛ0<br />
ϕas<br />
ϕ s<br />
xs<br />
zs<br />
zc<br />
yc<br />
xc<br />
dV χ (3) Ê − as Ê− s Ê+ c Ê+ p + h.c. (5.4)<br />
where χ (3) is the effective nonlinearity, independent of the beam intensity since<br />
the pump and the control are non-resonant [65]. Assuming a Gaussian distribution<br />
of atoms in the cloud the effective nonlinearity χ (3) can be written<br />
as<br />
χ (3) <br />
(x, y, z) ∝ exp − x2 + y2 R2 z2<br />
−<br />
L2 <br />
(5.5)<br />
where R is the size of the cloud of atoms in the transverse plane (x, y) and L<br />
is the size in the longitudinal direction.<br />
53