Theory of laser cooling

Consider a two level atom with ground state $\vert g\rangle$ and excited state $\vert e \rangle$. There are three processes taking place when this atom is put in a resonant electro-magnetic field (see fig. 1.1). The absorption of one photon excites the atom from the $\vert g\rangle$ to the $\vert e \rangle$ state, and a momentum of $\hbar k$ is transferred to the atom in the direction of the photon, with $k = \vert \vec k\vert = 2 \pi / \lambda$, and $\lambda$ the wavelength of the photon.

Figure 1.1: Absorption and emission of a two level atom

If the atom is in the $\vert e \rangle$ state, it can emit a photon by stimulated emission, which would cause a momentum transfer of $\hbar k$ in the direction opposite to the photon. The third process is called spontaneous emission. The atom falls back to its ground state, emitting a photon in a random direction. An absorption/stimulated emission cycle has no net momentum transfer to the atom, since the momentum transfers are equal and opposite to each other. When an atom goes through $N$ absorption/spontaneous emission-cycles, it will experience a net momentum transfer of $N \hbar k$ due to the absorption, and no net momentum transfer due to the spontaneous emission, since these photons are emitted in a random direction. With this technique it is possible to alter the velocity of neutral atoms by using laser light with a frequency of $\omega_0 = (E_e-E_g)/\hbar$, with $E_e$ the energy level of the excited state, and $E_g$ the energy level of the ground state.

The maximum force exerted on an atom by a resonant electro-magnetic field with frequency $\omega = ck$ is given by:

$$\vec F_{max} = \frac {\hbar \vec k \Gamma} {2}$$ (1.1)

in which $\Gamma$ = $1/\tau$ is the spontaneous decay rate. If the light intensity is low, the absorption profile of the transition is given by the Lorentz profile $\cal{L}$:

$${\cal L}(\omega - \omega_{0}, \Gamma) = \frac{1}{1+4(\frac{\omega - \omega_0}{\Gamma})^2}$$ (1.2)

with $\omega$ the frequency in the rest frame of the atom, and $\omega_0$ the transition frequency. The chance that a photon with frequency $\omega$ will be absorbed by the atom, is proportional to the value of the Lorentz profile. The width of this transition is defined as the full width half maximum value:

$$\Delta \omega_{FWHM} = \Gamma$$ (1.3)

For larger values of the intensity $I$ of the laser light, the transition width is broadened according to:

$$\Delta \omega_{FWHM} = \sqrt{1+s_0}\Gamma$$ (1.4)

Here $s_0 = I/I_0$ is the saturation parameter, with $I_0$ the saturation intensity, and $I$ the intensity of the laser light. For large intensities of the laser light, this can be approximated by $\sqrt{s_0}$.

The force on the atom is dependent on the intensity and the frequency of the laser light used [3]:

$$\vec F = \frac {s_0} {1 + s_0 +2 (\delta - \vec k \cdot \vec v / \Gamma)^2} \frac {\hbar \vec k \Gamma} {2}$$ (1.5)

with $\delta = \omega_{laser} - \omega_0$ is the detuning of the laser. The term $-\vec k \cdot \vec v$ is the Doppler shift.

When the atom is placed in a magnetic field, the energy levels of the atom will shift by an amount of $\Delta E_{m_{j}}$ [4]:

$$\Delta E_{m_j} = g_j \mu_B B m_j$$ (1.6)

due to the Zeeman effect. Here $B$ is the applied magnetic field, $\mu_B$ the Bohr magneton, and $g_j$ the Landé factor, depending on the quantum numbers $j$, $s$ and $l$:

$$g_j = 1 + \frac{j(j+1)+s(s+1) - l(l+1)}{2j(j+1)}$$ (1.7)

This change in the energy levels causes a shift in the frequency $\omega_0$, so that the total detuning of the laser becomes:

$$\delta_{total} = \omega_{laser} - \omega_0 -\vec k \cdot \vec v -(g_e m_e -g_g m_g) \frac {\mu_B B} {\hbar}$$ (1.8)

Figure 1.2: Two level scheme of triplet He

The two state atom used in our experiments is He*, or metastable helium, and the two states are He($2^3$S$_1$) (ground state) and He($2^3$P$_2$) (excited state)(see fig. 1.2). This transition is called the D2 transition. The transition from He($2^3$S$_1$) to He($2^3$P$_1$) is called the D1 transition, and has an energy difference between the two states very close to the energy difference of the D2 transition.

The reason that He($2^3$S$_1$) can be used as a ground state is that it has a lifetime of 8000 s, which is much longer than the flight time of the atoms in our setup (5-10 ms). By choosing the polarization of the laser light, the atoms can be optically pumped to the $2^3$S$_1$(m$_g=+1$) ($\sigma^+$ light) or to the $2^3$S$_1$(m$_g=-1$) ($\sigma^-$ light), because the spin selection rules demand that $\Delta m=+1$ and $\Delta m=-1$ for $\sigma^+$ resp. $\sigma^-$ light.

The wavelength of the laser light used for the D2 transition is 1083.034 nm, which is in the infrared regime. The saturation intensity for this transition is $I_0 = 1.67$ W/m$^2$, and $\Gamma = 1.63$ MHz. With the diode laser used for our experiments, we could detune a few gigahertz, so that we could also see the D1 transition, which lies 2.299 GHz above the D2 transition.