This simulation presents, in a very simplified way, the operation of a magneto-optical trap to trap and cool atoms to a few millikelvins of temperature. The model is based on an ensemble of non-interacting gases that are subject to the forces of six light beams, three of which are counter-propagating pairs. The simulation is performed in three dimensions, but only the projections of positions and velocities on the $xz$ plane are shown.

Buttons:

• Restarts the simulation.
• Configures other trap parameters: laser detuning, magnetic field intensity, number of atoms, most likely speed (which regulates the initial temperature of the sample) and the mass of the atoms.
• Lasers: Turns the light beams on and off.
• Magnetic field: Turns the magnetic field generated by the pair of Helmholtz coils on and off.

The force in the $j$ direction exerted on the atoms due to the pair of beams in that direction is given by

$$ F_j = F_j^+ + F_j^-, $$

where

$$ F_j^\pm = \pm \hslash k \dfrac{\gamma_{22}}{2} \times $$ $$ \times \dfrac{ I/I_0 }{ 1 + I/I_0 + 4(\delta \mp kv_j \mp \beta_j r_j)^2/\gamma_{22}^2 }. $$

$\gamma_{22}$ is the rate of spontaneous decay of atoms from state |2⟩ to state |1⟩, $k$ is the wave number, $I$ is the intensity of the laser beam, $I_0$ is the saturation intensity of the transition |1⟩ - |2⟩, $\delta$ is the laser detuning with respect to the transition resonance, $v$ is the speed of the atom and $\beta_j$ is defined by

$$ \beta_j = \dfrac{g\mu_B}{\hslash} \dfrac{\partial B_j}{\partial j}. $$

$g$ is the gyromagnetic factor, $\mu_B$ is the Bohr magneton and $B_j$ the $j$ component of the magnetic field generated by the pair of Helmholtz coils.

The dynamics of atoms is based on Newton's second law and a random variable $\epsilon = 0.5$ m/s was added in order to simulate the random fluctuations in atomic velocities due to spontaneous emissions:

$$ v(t) \rightarrow v(t) + \epsilon. $$

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