Optical Dipole Trap

General Concept

By shining high a intensity beam into a cloud of atoms we cause huge AC stark shifts. These stark shifts look like what I derived in Two Level System since I took the far detuned limit from the new eigenvalues after introducing the laser light. Then this stark shift looks like a Gaussian in the lower state which traps the atoms. Here is a diagram, note the energy sign is wrong, thanks Gemini

Formulas

Trap Depth

A Gaussian beam has intensity

$I(r,z)=\frac{2P}{\pi w_0^2}(\frac{w_0}{w(z)}{)}^2{e}^{\frac{-2r^2}{w(z{)}^2}}$

where $w_0$ is the beam waist, $P$ is the total power of the beam and

$w(z)=w_0\sqrt{1+(\frac{z}{z_R}{)}^2}$

where $z_R$ is the Rayleigh range of the beam given by

$z_R=\frac{\pi w_0^{2}n}{\lambda }$

This is the place where the beam size is $\sqrt{2}$ the beam waist $w_0$. $n$ is the index of refraction, $n=1$ for air. Now we know from Two Level System the far detuned limit gives and AC stark shift of

${\lambda }_{\pm }=\pm (\frac{\hslash \delta }{2}+\frac{\hslash {\Omega }^2}{4\delta })$

An we know the rabi frequency is given by

$\Omega =\frac{E{d}_{eg}}{\hslash }$

Now we want the change in energy of the level from the non perturbed eigenvalue so lets focus on the second term$\frac{\hslash {\Omega }^2}{4\delta }$

Now we can plug in the definition of the Rabbi frequency and we get

$\frac{E^2{d}_{eg}^2}{4\delta \hslash }$

Now lets write this in a more telling way

$U_0=\frac{1}{4}\frac{d_{eg}^2}{\delta \hslash }{E}^2=\frac{1}{4}\alpha E^2$

Which is exactly the potential for the ODT (note typically you would write a minus sign in front of this). Now this isn’t particularly useful, we talk about intensity in the lab so lets change this to intensity. We know

$I=\frac{1}{2}c{ϵ}_0{E}^2$

$\frac{2I}{c{ϵ}_0}=E^2$

so plugging this into the potential we get

$U_0=\frac{1}{4}\alpha \frac{2I}{c{ϵ}_0}=\frac{\alpha }{c{ϵ}_0}I$

Now using the equation for the intensity of a gaussian beam we have

$U_0=\frac{\alpha }{c{ϵ}_0}\frac{2P}{\pi w_0^2}(\frac{w_0}{w(z)}{)}^2{e}^{\frac{-2r^2}{w(z{)}^2}}$

But we actually only care about the deepest part of this which is when

$U_0=\frac{\alpha }{c{ϵ}_0}\frac{P}{\pi w_0^2}$

It should be noted that the derivation from the far detuned AC stark shift isn’t actually necessary for this but I think its useful to see the connection.

Trap Frequencies

Here we want to Taylor expand around the bottom of the well and set the potential for a harmonic oscillator equal to the second order part of the approximation. So the full potential is given by

$U_0=\frac{\alpha }{c{ϵ}_0}\frac{2P}{\pi w_0^2}(\frac{w_0}{w(z)}{)}^2{e}^{\frac{-2r^2}{w(z{)}^2}}$

So Taylor expanding this gives

References

arXiv:physics/9902072v1 physics.atom-ph 24 Feb 1999

Wikipedia Gaussian Beam