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

where n is the number density of atoms. Direct integration of this equation
yields the transverse intensity profile of the beam at a position z.
I(x,y) = I0(x,y)e −σ n(x,y,z)dz , (2.105)
This equation shows us that the intensity of the beam after passing through
the atomic sample is proportional to the negative exponential of the scattering
cross section multiplied by the integrated atomic column density. Therefore
by measuring I(x,y) and I0(x,y) it is possible to extract the atomic column
density. I(x,y) and I0(x,y) are measured by a CCD camera, the former is
taken with the atoms of interest present in the beam path and the latter is
taken with no atoms present. A computer then calculates the optical density
of the sample, defined as
Dopt(x,y) = −ln
I(x,y)
= σ
I0(x,y)
n(x,y,z)dz, (2.106)
to yield a quantity directly proportional to the integrated column density. This
quantity is easily manipulated numerically to give the total number of atoms
N present in the sample.
N =
Dopt(x,y)dxdy, (2.107)
This imaging technique is also used to probe the temperature of atomic
samples through the Time-of-Flight (TOF) method [80, 81]. The idea is to
release an atomic sample from all trapping potentials and then measure its
subsequent expansion. In the absence of inter-atomic interactions, the re-
leased atoms follow ballistic trajectories with velocities proportional to their
81