Theory - Magneto-Optical Trap (Atom trapping)

11 - Magneto-Optical Trap (Atom trapping)

Atom-light interaction theory

We detail a semi-classical approximation for the following case of an atom interacting with a light field.

Consider the effect of a laser beam with frequency $\omega$, detuning $\Delta$, and wavevector $k$, interacting with a two-level atom, where $\omega$ is tunable. The detuning of the laser $\Delta$ is defined as $\Delta=\omega-\omega_0$, where $\hbar\omega_0$ is the energy difference between the two states, $\omega$ is the laser frequency, and the wavevector $k$ is defined as $k=2\pi/\lambda$, where $\lambda$ is the wavelength of the laser photons. The detuning $\Delta$ in magneto-optical traps is negative and on the order of $\Gamma$ such that the laser is nearly resonant with the atomic transition.

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The two-level atom interacts with the laser in a two-step cyclic process.

1. The atom absorbs a photon and gains momentum $p=\hbar k$ in the direction of laser propagation

2. The atom spontaneously emits a photon after some time, on the order of the lifetime of the excited level $\tau$.

The spontaneous emissions are random and isotropic, so after a large number of cycles, there is no net contribution to the atomic momentum. Thus the main contribution to the momentum is the from the photon absorption process, which continues until the net atomic velocity gained pushes the atom off-resonant with the laser light. The net force per photon absorption transferred to the atom from the laser is equal to the spontaneous emission times the excited sstate fraction times the momentum gained by the atom.

F=\hbar k \Gamma \rho_{ee} = \hbar k \frac{\Gamma}{2} \frac{I/I_\text{sat}}{1+I/I\text{sat}+ \left \[ \frac{2(\Delta - kv)}{\Gamma} \right \]^2}

where $\rho_{ee}$ is the fraction of atoms in the excited state, $I$ is the intensity of the laser beam, and $I_\text{sat}$ is the saturation intensity. This is also called the scattering or radiation presure force.

Optical molasses

So far, we have looked at the effect of a single laser beam. What happens if we apply more beams? Consider the situation of a counter-propagating beam where the frequencies are sufficiently detuned that the light fields do not interfere, we can add the radiative forces together to give

$$\vec{F}_\mathrm{sc} = \frac{\hbar \vec{k} \Gamma}{2} \left( \frac{s_0}{1 + s_0 + [2(\Delta-kv)/\Gamma]^2} - \frac{s_0}{1 + s_0 + [2(\Delta+kv)/\Gamma]^2} \right)$$

For small velocities, we can approximate the equation as,

$$\vec{F}_{sc} \approx \frac{8\hbar k^2 \Delta s_0}{\Gamma(1 + s_0 + 4\Delta^2 / \Gamma^2)^2} \vec{v} \equiv - \beta \vec{v}.$$

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As displayed in the figure, for small velocities and negative detuning (red-detuned light), the forces looks like a friction force with damping coefficient $\beta$. For this reason, this configuration is called optical molasses. An atom moving in a given direction will see the counter-propagating light shifted towards resonance and will preferentially absorb photons, slowing down from photon momentum kicks, cooling down.

We note here that a limitation is that the atoms experience a velocity-dependent frictional force but are not trapped. In order to confine the atoms, we usually combine a velocity-dependent force with magnetic fields that provide a position-dependent force for spatial confinement.

A Magneto-optical trap

Let us now consider the 1D molasses setup shown above, this time adding a magnetic field $B(z)=-bz$. The axis $z$ is the one along which light is propagating, and $b$ is the magnetic field gradient. This field can be produced, for example using an anti-Helmholtz coil pair. Let us look at the effect of the field on the atomic levels. For simplicity, we will consider an atom whose ground state has total angular momentum $J_g=0$ and excited state $J_e = 1$. The ground state is not affected by the presence of the magnetic field. On the other hand, the excited state splits into three Zeeman sublevels with a spatially dependent shift,

$$\Delta E = \mu_\mathrm{B} g_{J_e} m_{J_e} b z$$

with the Lande'{e} g-factor $g_{J_e}$ and the angular momentum projection of the excited state along the quantization axis $m_{J_e}$. Circularly polarized laser beams will produce a spatially dependent force that will push the atoms towards $z=0$. As shown, red-detuned light $\sigma^+$ coming from the left addresses $m_{J_e}=+1$ atoms while $\sigma^-$ will act on $m_{J_e}=-1$. The result is a force that pushes atoms towards the center.

Such a spatial dependent force transforms the equation into

F_{\text{sc}} = \frac{\hbar k \Gamma}{2} \left( \frac{s}{1 + s + \[ 2(\Delta - kv - \mu_{\text{B}} g_{J_e} m_{J_e} b z/h)/\Gamma \]^2} - \frac{s}{1 + s + 2\[\Delta + kv + \mu_{\text{B}} g_{J_e} m_{J_e} b z/h)/\Gamma\]^2} \right)

For small velocity and displacement, the force can be approximated as

$$F_\text{sc} = -\beta v - \kappa z, \qquad \kappa = \frac{g_{J_e} \mu_\mathrm{B} b}{\hbar k_\text{L}} \beta.$$

This equation describes a one-dimensional (1D) damped harmonic oscillator where the magnetic field provides a conservative potential. It can be easily extended to a two-dimensional magneto optical trap, or a three-dimensional magneto-optical trap.