Single-Photon Atomic Cooling - Raizen Lab - The University of ...

momentum at the time of release. The final atomic distribution is therefore
a convolution of the initial atomic distribution with the initial momentum
distribution.
This can be made more concrete by considering atoms originating from
r0 and arriving at r after a time-of-flight t. If we assume that the velocity
distribution for these particles is ζ(v) then we expect to find the number of
atoms in a volume element d 3 r located at position r at time t to be [80]
dn = G(r,t;r0)d 3 r = ζ(v = [r − r0/t])d 3 v, (2.108)
where G(r,t;r0) is the boundary Green’s function describing the evolution
of the spatial distribution of the atoms. If the atoms have a 3-D Maxwell-
Boltzmann velocity distribution initially, then the Green’s function is given by
[80]
G(r,t;r0) =
m
2πkBTt 2
3/2 e −m|r−r0| 2 /2kBTt2 . (2.109)
If we assume that the atomic sample has a Gaussian spatial distribution ini-
tially, given by
n0(r) =
1
(2πσ 2 n) 3/2e−r2 /2σ 2 n, (2.110)
then we can solve for the spatial distribution after an expansion time t by
evaluating
The result is
n(r,t) =
n(r,t) =
G(r,t;r0)n0(r0)0d 3 r0. (2.111)
1
(2πσ 2 c) 3/2e−r2 /2σ 2 c, (2.112)
82