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==Magnetic Trapping==
The conservative potential seen by an atom with magnetic moment <math>\boldsymbol{\mu}</math> is 
<math>U(\mathbf{r})=-\boldsymbol{\mu}\cdot\mathbf{B}(\mathbf{r})</math>.
==Optical Trapping==
The conservative potential is seen by an atom of DC polarizability 
<math>\alpha\,</math> is 
<math>U(\mathbf{r})=-\frac{1}{2}\alpha|\mathbf{E}(\mathbf{r})|^2</math>.  

==Evaporative Cooling==

==Absorption Imaging==

This summary relates one person's experience with absorption imaging of trapped atoms.   Please edit for clarity as deemed necessary. 


We want to calculate the number of atoms <math>N\,</math> collected in a
<math>CO_2\,</math> dipole trap via the analysis of destructive near-resonant absorption imaging.


We start with a collimated probe beam approximately 5mm in diameter, which is passed through an entrance window to interact with the trap.  The shadow cast by the trapped atoms is at the focus of a light-gathering lens.  Per Babinet's principle, this shadow is gathered and collimated, and is then imaged by a second lens onto an array of CCD pixels.


The lens positions are calibrated in a two-step process.  First, the imaging lens is installed, and placed at the focal distance <math>f\,</math> from the CCD array by imaging a distant object: an obscene drawing, say, tacked to the far laboratory wall. 
 

The objective lens can then be attached, at which point a ruler is imaged to establish a rough length scale.  This crude process establishes a scale per pixel, which is hopefully consistent with the lens ratio of the telescope and the known size of the CCD's pixels.


Later on, gravity can be used to more exactly calibrate the length scale. ''more on this later.'' 


It's mathematically convenient if your probe beam is well below any possible saturation intensity. 


As the probe beam passes through the cloud, it obeys Beer's Law:


<center>
<math>
I(x,y) = I_0(x,y)e^{-D(x,y)} \,
</math>
</center>


where  <math>D(x,y)\,</math> is the optical depth along the probe beam's path at point <math>(x,y)\,</math> in the plane perpendicular to the probe's propagation, and is given by


<center>
<math>
D(x,y)=\sigma \int_{-\infty}^{\infty} {n(x',y',z')}|_{x,y}  dz'
</math>
</center>


where <math>\sigma\,</math> is the absorption cross-section, and where <math>n(x',y',z')\,</math> is the atomic density distribution typically given by something like


<center>
<math>
n=n_0 e^{-\frac{r^2}{2\sigma_r^2}}e^{-\frac{z^2}{2\sigma_z^2}}
</math>
</center>


If one is solely interested in atom number <math>N\,</math>, namely the integral over all space of the density distribution, then in principle we do not need to know the details of <math>n(x,y,z)\,</math>, only the total integrated optical depth.  Playing around a little, we find


<center>
<math>

D(x,y) = -\log{\frac{I}{I_0}}=\sigma \int_{-\infty}^{\infty} {n(x',y',z')}|_{x,y}dz'
</math>
</center>


where <math>z\,</math> is the line of sight.  Practically, the 2-D data array <math>D(x,y)\,</math> is simply obtained by dividing the image of the probe beam with the trap present by the image of the probe beam some
time later.  


''Discuss eliminating dark counts''


Integrating over the image plane, we then find that


<center>
<math>

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} D(x,y) dx dy =

\sigma  \int_{-\infty}^{\infty}  \int_{-\infty}^{\infty}

\int_{-\infty}^{\infty}  {n(x',y',z')}|_{x,y}dz'dx dy = \sigma N
</math>
</center>


where <math>N\,</math> is the total number of atoms.  Now, since optical depth is actually presented in the form of an array, we recast the left-hand-side of the above relation into a sum, and thus obtain


<center>
<math>
N = \frac{A}{\sigma}\sum_{ij} D_{ij}
</math>
</center>


with <math>A\,</math> being the scaled pixel area.


The cross-section can be found in Dan Steck's [[Rubidium]] D-line data, and is given by


<center>
<math>

\sigma= \frac{\sigma_0}{1+4 (\Delta/\Gamma)^2 + (I/I_s)}
</math>
</center>


where <math>\sigma_0\,</math> is the on-resonance low-intensity cross-section


<center>
<math>
\sigma_0=\frac{\hbar \omega \Gamma}{2 I_s}
</math>
</center>

One thus obtains a conversion factor between integrated optical depth and atom number.  


''There should also be correction factors involved for optical depth saturation, and intensity saturation through the cloud.''

''Diode lasers also have an off-resonant 'pedestal' which might need to be taken into account.''

==References==

*Lewandowski Thesis, Chapter 2, [http://jilawww.colorado.edu/pubs/thesis/lewandowski/ch2.pdf PDF]
*[http://www.colorado.edu/physics/2000/bec/index.html Colorado Introduction]


[[Category:Laser Systems]]
[[Category:How To]]
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