 # 1 Theory

In ultracold atoms we must reach a temperature of the order of 30 nk. By laser cooling techniques we are limited by the rate of heating of the atom by the light, around 30 μk .
Thus was developed the method of Evaporative cooling. The principle of this method is to remove atoms fastest (high energy) and by this dramatically lower the temperature of each ensemble.
The force on atom in a magnetic trap is
F =  − μbmjgj(dB0(z))/(dz)
We do this for atoms in a state of |9 ⁄ 2, 9 ⁄ 2. If we can give it enough energy so it could switch to a different spin and if we were to switch to a diffrent ml it would feel a upside down potential and therefore would not be in a potential well, as a free particle.
Due to the exerted magnetic field the energy levels in atoms are split Figure 1 Hyperfine energy shift for the D1manifold (2P1 ⁄ 2) of 40K as a function of magnetic field
Zeeman shift makes the energy required to stimulate and create an atomic transition to be:
Ec = (2)/(9)mfμb(ν − 1286MHz)/(Λmfmf) − B0
Λmfmf =  − (9μb(mf + mf))/(2h)

# 2 My program

So if we give the atom enough energy of the appropriate frequency, the atom will then transition to |7 ⁄ 2, 7 ⁄ 2 . Then the atom will be free and will not be placed in our atoms cloud (and actually go out of our ensemble).Then we can calculate the change in the number of atoms and internal energy by:
dN = (1)/(2λ(ωr)3)Ec(ϵ2)/(e(ϵ)/(kT) + 1)dϵ
dU = (1)/(2λ(ωr)3)Ec(ϵ3)/(e(ϵ)/(kT) + 1)dϵ
where ωr are oscillator frequencies of the laser and λ = (ωz)/(ωr) .
So if we divide the process into segments of time and each segment is considered how many atoms and energy got lost we can find the number of atoms and the temperature at the end of the process.
The equations describing the dynamics are:
Ni  =  1 − (dt)/(lifetime)Ni − 1 − dNi
ui = ui − 1 + 38 + 8(Ni − 1)/(106)10 − 9kbNi − 1dt − dui
were the second term in U is heating rate of the atoms.
In the program this is done with # 3 Results

in the end of the program you get the NfinalandTfinal. You can see the dynamics on the plot at each time. I run the program several time and I get those results:
 a [(MHz)/(sec)] ν0 [MHz] Tfinal [μK] Nfinal 5 1160 45 2*107 6 1160 46.4 2.32*107 7 1160 55.5 2.54*107 5 1180 45.38 2.25*107 5 1190 44.3 2.5*107 5 1200 45.3 2.67*107 3 1220 19.2 2.38*107

The main problem is that the life-time in the trap is 40sec therefore, the time cannot run over the 25sec of evaporation.

Some of my plot are here: Figure 2 plot of evaporation cooling in 40K with ν(t) = 1160 + 5t [MHz] Figure 3 plot of evaporation cooling in 40K with ν(t) = 1160 + 7t [MHz] Figure 4 plot of evaporation cooling in 40K with ν(t) = 1190 + 5t [MHz] Figure 5 plot of evaporation cooling in 40K with ν(t) = 1220 + 3t [MHz]