# Laser cooling of atoms: more area or more power?

I want to optimise a Magneto-Optical trap. Laser beams come from the the x, y and z directions (positive and negative) and slow the atoms down.

Would it be better to increase the beam spot-size (reducing the intensity) therefore slowing more atoms, or to have a smaller cooling region but with higher power and therefore more effective cooling?

Where should I start for a quantitative calculation?

• How do you load your MOT? Do you use a Zeeman slower? For which atomic specie? – dolun Dec 17 '15 at 12:17

Optimizing the MOT beams

Since the maximal force created by each beam on the atoms is limited by the saturation intensity $I_\text{sat}$ : $$\tag{1} F_\text{max}=\frac{\Gamma}{2}\hbar k\quad\text{when}\quad\frac{I}{I_\text{sat}}\geq 1$$ you will always want to have enough power in the beams of your MOT so that you hit the saturation regime. Here, $k$ is the wavelength of your laser beam, $\Gamma$ is the linewidth of your atomic transition.

The second thing you wanna do if you want to maximize the number of atoms catched in the MOT is to have the largest MOT beams as possible. But since intensity decreases as $I(r)\sim P/r^2$ for a given optical power $P$, you should take care that you have enough power in order to stay in the saturation regime.

Let say that $\sigma$ is the waist of your MOT beams. What I mean is that the appropriate situation would be that : $$I\sim\frac{P}{\sigma^2}>I_\text{sat}$$

Optimizing the magnetic gradient of the MOT

The other thing you can play with is the magnetic field gradient $b$ of your MOT. The resonance of an atom with a beam of your MOT at fixed detuning $\delta$ of the atomic transtion will occur for each couple of atomic position/velocity $(x,v)$ such that : $$\tag{2} \delta\pm kv=\mp\frac{\mu bx}{\hbar}$$ $\mu$ being the magnetic moment of your atom.

Let's considere a situation where atoms are coming from $x=-\infty$ on the MOT. Since the distance on which you can catch atoms is finite and limited by the size $\sigma$ of your beams, an optimal situation seems to be when an atom with zero velocity at the position $x=+\sigma/2$ (the "left" edge of the MOT if you prefere) is resonant with the beam pushing it towards the center of the MOT $x=0$. This implies from (2) that : $$\tag{3} |\delta|=\frac{\mu b\sigma}{2\hbar}$$ On the other side of the capture zone in $x=-\sigma/2$, that same beam is resonant with the atoms with velocity $v_i$ such that : $$\tag{4} |\delta|=kv_i-\frac{\mu b\sigma}{2\hbar}$$ This implies from (3) : $$\tag{5} kv_i=2|\delta|=\frac{\mu b\sigma}{\hbar}$$ Moreover the max incident velocity $v_i^\text{max}$ that you can catch with the MOT will be also limited by the saturation as the force applied on the atom is limited such that : $$\tag{6} F=m\frac{\mathrm{d}v}{\mathrm{d}t}=mv\frac{\mathrm{d}v}{\mathrm{d}x}=mv\frac{\mu b}{\hbar k}<F_\text{max}$$ where $m$ is the mass of the atom. This implies from (1) that : $$b<\frac{\Gamma(\hbar k)^2}{2\mu mv}$$ with a maximum of capture when : $$\tag{7} b=\frac{\Gamma(\hbar k)^2}{2\mu m v_i^\text{max}}$$ Replacing this expression of $b$ in the relation (5), you get : $$v_i^\text{max}=\sqrt{\frac{\hbar k \Gamma\sigma}{2m}}$$

• Thanks for the answer! Is your final expression an expression for the maximal trapping speed? Like the critical velocity? – SuperCiocia Dec 22 '15 at 21:23
• Yes but don't take those numbers too seriously, they are just here to give you an idea of what is happening. But all these little calculations had been done under the assumption that the total force applied on the atoms in the MOT is the sum each force exerced by each MOT beam on the atoms, which is wrong above saturation $I>I_\text{sat}$. – dolun Dec 23 '15 at 12:29
• Conclusion : the more power the better, the bigger beams the better. And then if I was you I would just turn the buttons of your 2D MOT coils power supplies in order to maximize the number of atoms loaded in the 3D MOT, which is really the only thing you have to maximize in order to make science with your atoms. – dolun Dec 23 '15 at 12:29
• Just two more questions: 1) Why do you say that the formula is wrong above Isat? What other effects come into play? 2) why do you choose $x = \sigma /2$ for the velocity to be 0? Why not just $\sigma$ Is that arbitrary? – SuperCiocia Jan 6 '16 at 16:09
• 1) Because above saturation forces created by each beam of the MOT are no longer independant from each other. For instance, an atom can absorb a photon from one beam and emit it by stimulated emission in an other beam. Plus, when the power in the MOT beams is high enough, the interferences between the beams produce a lattice (standing wave in each direction of the space) which changes the mean absorption rate of the atoms. – dolun Jan 13 '16 at 10:30

Check out this paper:

Improved magneto-optic trapping in a vapor cell. K.E. Gibble, S. Kasapi and S. Chu. Opt. Lett. 17 no. 7, 526 (1992), PSU eprint.

You'll notice that the rate at which atoms below the capture velocity enter the volume of your cooling region increases with the beam diameter squared. There is also a plot of the number of atoms trapped vs intensity. Since it saturates at high intensity, you'll win in many cases with larger beams. The beams that I used for my Rb MOT in grad school were more than 2 cm in diameter. The saturation intensity for Rb is pretty low , so it was a win in this case.

• What is a vapour cell trap? – SuperCiocia Dec 18 '15 at 15:26
• Also, doesn't the intensity of the beam go down as $d^2$? – SuperCiocia Dec 18 '15 at 15:27
• The Vapor is produced by running current through a vacuum getter, which releases the vapor throughout the chamber. Unlike an oven, you can load your MOT from this background vapor by cooling the low velocity tail of the vapor. For your case, you'll probably use the time spent traversing your 2D MOT as the figure of merit. The question in this case being how many photon scatters you can expect to get in that time. – tmwilson26 Dec 18 '15 at 15:29
• Yes, but if you are above saturation intensity, your scattering rate will not scale directly with the intensity. – tmwilson26 Dec 18 '15 at 15:30
• I suppose what I was trying to say about the 2D MOT comment was that you'll want to try to maximize the number of photon scatters, and higher intensity doesn't necessarily lead to more photon scatters because of this saturation. If you have a nearly closed transition (like Rb), you can use the two-level approximation in order determine the scattering rate with intensity. So expanding the beam may help to produce more total photon scatters. – tmwilson26 Dec 18 '15 at 15:35