What is the Festina Lente Regime? I am studying BEC formation without evaporative cooling, so realized only thrpugh optical means.
One of the problems to face is "photon reabsorption": an atom absorbs a photon of the laser used for cooling and spontaneously re-emits it, then if the atomic sample is very dense, the photon can be reabsorbed by other atoms, heating the sample.
The Festina Lente Regime should suppress this heating, and consists in setting a trap frequency $\omega_0$ than the spontaneous emission rate $\Gamma_s$.
$$
\omega_0\gg \Gamma_s
$$
The process is described here, and also here in a more qualitative way.
I really can not understand how this festina lente regime works in a "simple way" from these articles, why it succeedes in supressing heating.
I ask for a simple explanation of this regime, or a reference in which it is explained clearly. Calculations are welcome, but also just a qualitative picture is appreciated.
 A: The "festina  lente" (FL) regime is when the spontaneous emission/absorption rate $R_{\mathrm{sat}}$ is smaller than the trap frequency $\omega$.
A qualitative and intuitive way of understanding this is as follows. If the absorption rate is smaller than the trap frequency, then the time $\tau_s$ taken to absorb one photon from the trap light is much longer than the time to perform an oscillation in the trap $\tau = 2\pi/\omega$, so $\tau \gg \tau_s$. That is, you need to "make" a lot of trap oscillations before you absorb a photon and hence see the ensuing heating effects. This just meant you have to wait quite a while to start seeing the effects of absorption, and so you can usually neglect them.
To be more mathematical, the spontaneous emission/absorption rate $R_{\mathrm{sat}}$ is given by:
$$ R_{\mathrm{sat}} = \frac{\Gamma}{2}\frac{I/I_{\mathrm{sat}}}{1+I/I_{\mathrm{sat}}+(2\Delta/\Gamma)^2} ,$$ where $\Gamma$ is the natural linewidth of the atomic state, $I$ the incident intensity, $I_{\mathrm{sat}}$ the saturation intensity for the transition, and $\Delta$ the detuning between laser frequency and atomic transition.
You can already see that is goes as $1/\Delta^2$, i.e. far-detuned light will cause very little absorption. That's why, for traps, you choose very far detuned light.
For a typical cold atoms / BEC experiment, let's use the $D2$ line of $^{87}$Rb, and a trap light of $1064$ nm. Typically, a BEC emerges for trap frequencies on the order of $\omega \sim 10-100$ Hz.
Using a $1$ W and $100$ µm waist beam, typical for optical dipole traps, the scattering rate $R_{\mathrm{sat}}$ is $\sim 1.6\times 10^{-5}$ Hz.
So indeed, $\omega \gg R_{\mathrm{sat}}$.
