Project on Force Evaporation cooling in ^{40}K
15/3/2016
Theory
My program
Result
Download
Reference
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 = − μ_{b}m_{j}g_{j}(dB_{0}(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
m_{l} 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
Zeeman shift makes the energy required to stimulate and create an atomic transition to be:
E_{c} = (2)/(9)m_{f}μ_{b}⎡⎣(ν − 1286MHz)/(Λ_{mfm’f}) − B_{0}⎤⎦
Λ_{mfm’f} = − (9μ_{b}(m_{f} + m’_{f}))/(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(ℏω_{r})^{3})^{∞}⌠⌡_{Ec}(ϵ^{2})/(e^{(ϵ)/(kT)} + 1)dϵ
dU = (1)/(2λ(ℏω_{r})^{3})^{∞}⌠⌡_{Ec}(ϵ^{3})/(e^{(ϵ)/(kT)} + 1)dϵ
where ω_{r} are oscillator frequencies of the laser and λ = (ω_{z})/(ω_{r}) .
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:
N_{i}
=
⎛⎝1 − (dt)/(lifetime)⎞⎠N_{i − 1} − dN_{i}
u_{i} = u_{i − 1} + 3⎛⎝8 + 8(N_{i − 1})/(10^{6})⎞⎠10^{ − 9}k_{b}N_{i − 1}dt − du_{i}
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 N_{final} and T_{final}. 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)]

ν_{0 }[MHz]

T_{final} [μK]

N_{final}

5

1160

45

2*10^{7}

6

1160

46.4

2.32*10^{7}

7

1160

55.5

2.54*10^{7}

5

1180

45.38

2.25*10^{7}

5

1190

44.3

2.5*10^{7}

5

1200

45.3

2.67*10^{7}

3

1220

19.2

2.38*10^{7}

The main problem is that the lifetime in the trap is 40sec therefore, the time cannot run over the 25sec of evaporation.
Some of my plot are here:
4 Download the program
The program runs in MATHEMATICA 10, you can download it
here.
At the Technion there is MATHEMATICA 10 on all the computers. If you have a problem to get it you can ask help from the computer center
here .
To run the program you need to choose Evaluationevaluate Notebook.
For more details go to MATHEMATICA home page
here .
if you want to change something you can do it but then run all the notebook again.
5 Reference