Spatial Characterization Of Two-Photon States - GAP-Optique

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