@ everexpanding

Laser cooling techniques allow us to cool atoms or molecules to extremely low temperatures, approaching absolute zero. By creating ultracold atoms, we can gain insights into fundamental physics, but also develop practical applications such as precise atomic clocks. Unique quantum mechanical effects such as Bose-Einstein condensation can be observed near absolute zero. So far, laser cooling has mostly been used on atoms, but more recently, more complex systems such as di-atomic molecules have been cooled in this way.

Laser slowing of atomic beams

When a photon is absorbed by an atom, the momentum of the photon is transferred to the atom. This momentum transfer exerts a force on the atoms, which we call scattering force or radiation force. So if we irradiate a beam of atoms with a laser beam in the opposite direction, we can slow down the atoms in the beam.

Each ground state atom in the beam can absorb one photon and is slowed down by the recoil velocity $v_\mathrm{rec} = \hbar k/m$, where $k$ is the wavenumber of the photons and $m$ is the mass of the atom. Before the atom can be slowed down again, it must return to its ground state by emitting a photon. Since the emission is in a random direction, the average momentum change vanishes and can be neglected for the moment.

To describe the slowing process more mathematically, we consider a simple model where we have two-state atoms with a ground state $\ket{g}$ and an excited state $\ket{e}$ separated by the energy $\hbar \omega_0$ (resonance frequency $\omega_0$). The excited state has a natural linewidth $\Gamma$ corresponding to its rate of spontaneous emission of a photon. The total photon scattering rate for a laser with intensity $I$ and frequency $\omega_L$ is then given by

$\Gamma_\mathrm{scatt} = \frac{\Gamma}{2} \frac{I/I_\mathrm{sat}}{1 + I/I_\mathrm{sat} + 4 \delta^2/\Gamma^2},$.

with the saturation intensity $I_\mathrm{sat}$ and the frequency detuning $\delta = \omega_L - \omega_0$ of the laser from atomic resonance. The scattering rate has a Lorentzian shape in $\delta$ and in the high intensity limit $I\gg I_\mathrm{sat}$ we find the maximum scattering rate $\hat{\Gamma}_\mathrm{scatt} = \Gamma/2$.

The scattering force is simply the photon momentum times the scattering rate. So it can now be written as

$F_\mathrm{scatt} = \hbar k \Gamma_\mathrm{scatt}$

and hence the maximum force exerted on the atoms is $\hat{F}_\mathrm{scatt} = \hbar k \Gamma/2$.

This scattering force corresponds to a maximum deceleration for an atom with mass $m$ of

$\hat{a} = \frac{\hat{F}\mathrm{scatt}}{m} = v\mathrm{rec}\frac{\Gamma}{2}.$

Typically, the declaration is half the maximum, $a=\hat{a}/2$. So in order to stop an atom with initial velocity $v_0$ we need to irradiate the atom over a distance, called the stopping distance, of

$L_0 = \frac{v_0^2}{\hat{a}}.$

For typical elements used for laser cooling, such as sodium or rubidium, we have stopping distances of about a metre, which is a convenient length. However, we have assumed a constant deceleration, which is only true over a small range of velocities $\Delta v \sim \Gamma/k$. Due to the Doppler effect, the shift in the effective laser frequency changes as the atom slows down, taking the atoms out of resonance. This reduces the number of scattering events and hence the scattering force. The atoms become 'blind' to the laser. There are two techniques that solve this problem. One is called chirp cooling and the other is Zeeman cooling (or Zeeman slower).

Another problem that makes the atoms 'blind' to the laser is that of optical pumping. This is due to the fact that atoms are not perfect two-level atoms, but can have two ground state hyperfine levels. So we set the cooling laser at a frequency that excites electrons from the second ground state. When the excited electron is de-excited, it may end up in the first ground state instead of the second. Then it can no longer be excited by the laser and essentially falls out of the cooling cycle. So the scattering rate decreases over time and eventually disappears. The obvious solution is to use a second so-called repumper laser, which excites the electron from the wrong ground state so that it can decay back to the right state and take part in the cooling cycle again.

Chirp cooling

In chirped cooling, the laser frequency changes to compensate for the changing Doppler shift as the atoms slow down. A chirped laser pulse sweeps the frequency rapidly, so that a range of different frequencies is used for the different states the atoms may be in. The required sweep time can be calculated from the number of scattering events $\mathcal{N} = v_0/v_\mathrm{rec}$ needed to stop the atoms at the initial velocity $v_0$. Then, at half the maximum deceleration, the sweep time would be $4\mathcal{N}/\Gamma$. For sodium atoms with 34000 scattering events this would mean that the frequency would have to sweep in 2 ms over a range of more than a GHz.

Zeeman slower

In Zeeman cooling, instead of changing the laser frequency to stay in resonance, a magnetic field is used to perturb the energy levels of the atoms to stay in resonance with a laser of fixed frequency. The underlying effect that causes the magnetic field perturbation is called the Zeeman effect.

The velocity at a distance $z$ from the starting point is given by

$v(z) = v_0 \sqrt{1-\frac{z}{L_0}},$

where $L_0$ is the stopping distance. Then to counteract the changing Doppler shift we need to satisfy the condition

$\omega_0 + \frac{\mu_B B(z)}{\hbar} = \omega_L + kv(z).$

The left-hand side corresponds to the increase of the atomic resonance frequency $\omega_0$ due to the Zeeman shift from the magnetic field $B(z)$. The right-hand side gives the Doppler shift of the laser frequency $\omega_L$. Thus, for $0 \leq z \leq L_0$ the required magnetic field has the profile

$B(z) = B_0 \sqrt{1-\frac{z}{L_0}} + \tilde B,$

with $B_0 = \hbar k v_0/\mu_B$ and the bias field $\tilde B = \hbar(\omega_L-\omega_0)/\mu_B$. Such a field can be created with a tapered solenoid. If we have $\mu_B \tilde B \approx \hbar (\omega_L - \omega_0)$, then the atoms will come to a complete stop at the end of the solenoid, but in general we would like to have some small rest-velocity for the atoms, so that they can travel out of the solenoid to a region where we can perform experiments. This setup is called a Zeeman slower.

This technique has some important and advantageous properties over the chirp cooling technique. Firstly, the Zeeman slower reduces the velocity of a large fraction of the atoms in the beam to a low final velocity $v_\mathrm{f}$. It slows every atom with initial velocity in the range between $v_0$ and $v_\mathrm{f}$ since they all interact with the radiation at some point. Secondly, with this technique the optical pumping problem can be easily avoided, since the atoms are always in a strong axial magnetic field. Thus there is a well-defined axis of quantisation, which allows to make use of the selection rules for radiative transitions by using circularly polarised lasers which avoids the undesired optical pumping.

Laser cooling and trapping of atoms

Instead of a beam, we now consider a vapour or gas of atoms. In the case of the beam, we had a predefined single direction in which the atoms were moving, and a single laser was enough to slow them down. In a vapour, however, the atoms are moving in all directions and we need lasers in all three axes to slow them down, a total of six lasers.

Doppler cooling and optical molasses

First, we consider only one direction and irradiate the atom gas from both sides. The two counter propagating laser beams are tuned slightly below the atomic resonance frequency (red-detuned). For a stationary atom the scattering forces from the two beams balance each other out. But for a moving atom the Doppler effect introduces an imbalance in the forces. If the atom moves in the same direction as the laser the effective frequency decreases by $kv$ (red shift). Thus the laser becomes more off resonance and the scattering rate decreases which leads to a weaker scattering force. On the other hand, if the atom moves opposite to the laser the effective frequency increases by $kv$ (blue shift). Therefore the laser becomes more on resonance and the scattering force gets stronger. Hence, overall, the atom feels a stronger force from laser it travels to, which slows down the atom. Because of the important role of the Doppler effect, this process is called Doppler cooling.

By putting two counter propagating red-detuned laser beams in all three orthogonal, so in total six lasers, we get a so-called optical molasses. In each of the three dimensions we have the Doppler cooling process.

Mathematically, for every direction $i=1,2,3$ and assuming low velocities $kv_i \ll \Gamma$, the force acting on an atom is

$F_\mathrm{molasses} = F_\mathrm{scatt}(\omega_L - \omega_0 - kv_i) - F_\mathrm{scatt}(\omega_L - \omega_0 + kv_i) \equiv -\alpha v_i,$

with the laser's detuning $\delta = \omega_L - \omega_0$.[^1] So the light exerts a damping (frictional) force on the atom like on a particle in a viscous fluid. This analogy lead to the name molasses. The damping coefficient is

$\alpha = 2k \frac{\partial F_\mathrm{scatt}}{\partial \delta} \approx -4\hbar k^2 \frac{I}{I_\mathrm{sat}} \frac{2\delta/\Gamma}{\left(1 + (2\delta/\Gamma)^2\right)^2},$

for intensities $I$ well below the saturation $I_\mathrm{sat}$ where each beam's force acts independently. A damping requires $\alpha > 0$ which is the case for red-detuned lasers $\delta = \omega_L - \omega_0 < 0$.

Doppler cooling limit

Due to random fluctuations of the absorption and emission of photons by the atoms, the cooling is accompanied by a heating process. The random addition of momenta produces a random walk of the atoms which increases the mean square velocity. Performing conservation of energy and momentum for the absorption and spontaneous emission of an photon, one finds that on average the change of kinetic energy per scattering process due to heating is twice the recoil energy $E_\mathrm{rec} = m v_\mathrm{rec}^2/2 = \hbar k/2$. Considering direction $i=1,2,3$ with the two counter propagating lasers, the change of kinetic energy due to heating is given by

$\left(\frac{\mathrm{d}W}{\mathrm{d}t}\right)\mathrm{heating} = 2E\mathrm{rec}\;2\Gamma_\mathrm{scatt},$

and due to the cooling by

$\left(\frac{\mathrm{d}W}{\mathrm{d}t}\right)_\mathrm{cooling} = -\alpha \braket{v_i^2}.$

The lower temperature limit is reached when the overall change in kinetic energy vanishes, i.e. $0=\mathrm{d}W/\mathrm{d}t=(\mathrm{d}W/\mathrm{d}t)\mathrm{cooling} + (\mathrm{d}W/\mathrm{d}t)\mathrm{heating}$. Thus, we find the mean square velocity

$\braket{v_i^2} = 2 E_\mathrm{rec} \frac{2 \Gamma_\mathrm{scatt}}{\alpha}.$

for every direction $i=1,2,3$. According to the equipartition theorem this is related to the temperature $T$ by $m \braket{v_i^2} = k_B T$, where $k_B$ is Boltzmann's constant, which results in

$T = \frac{\hbar \Gamma}{4 k_B} \frac{1+ (2\delta/\Gamma)^2}{-2\delta/\Gamma}.$

That function has a minimum at $\delta=-\Gamma/2$ of

$T_\mathrm{D} = \frac{\hbar \Gamma}{2 k_B}$

which is called the Doppler cooling limit. It gives the lowest temperature we expects to reach with the optical molasses. We can correspond a velocity

$v_\mathrm{D} \approx \sqrt{\frac{k_B T_\mathrm{D}}{m}} = \sqrt{\frac{\hbar k}{m} \frac{\Gamma}{k}} \equiv \sqrt{v_\mathrm{rec} v_\mathrm{cap}}$

with this Doppler cooling limit. The velocity $v_\mathrm{cap} \sim \Gamma/k$ estimates the capture velocity of the optical molasses. This is the regime where the scattering force is significant and the optical molasses technique effectively cools the atoms. For sodium, for example, the Doppler cooling limit is $240~\text{µK}$ and the capture velocity is $6\mathrm{~m~s^{-1}}$.

This theory described the optical molasses technique well until experiments found much lower temperatures than the Doppler cooling limit under certain conditions. This simple two-level atomic model cannot explain this sub-Doppler cooling. However, an explanation for it can be found in the so-called Sisyphus cooling.

Magneto-optical trap

With the optical molasses technique we cannot confine the atoms in space, therefore it is not a trap, since there is no restoring force which would keep the atoms in the molasses. There is just a viscous inhibition of their escape. We can however introduce a positional dependence in the force by adding a magnetic field gradient and choosing the right set of polarisation for the lasers. In each direction we have the two counter propagating red-detuned laser beams with opposing circular polarisation ($\sigma^+$ and $\sigma^-$). Due the spatially dependent Zeeman shift the lasers are in resonance with the atom at different places, which results in a restoring force towards the point where the magnetic field vanishes, $B=0$. This is the trap part and the red-detuning provides the cooling part as discussed above. This scheme in all three dimension is a robust trap called magneto-optical trap (MOT), which combines cooling and trapping. Typically, the magnetic field is generated by two coils with opposite currents in a Helmholtz arrangement, creating a quadrupole field.

We consider an atom with the simple transition from $J=0$ to $J=1$. In the vicinity of the centre of the MOT, where $B=0$, is a non-zero uniform field gradient which perturbs the atomic energy levels, due to the Zeeman effect. The three sublevels with $M_J=0,\pm 1$ of $J=1$ vary linearly with the atom's position in each direction. For an atom which, for example, is displaced from the centre in positive $z$-direction the $\Delta M_J = -1$ transition becomes closer to resonance with the red-detuned laser frequency. The selection rules result in an absorption of a $\sigma^-$-polarised photon which gives a scattering force pushing the atom towards the centre. Similarly for an atom moving in the opposite direction ($z<0$), the $\Delta M_J = +1$ becomes closer to resonance with the other laser beam and the atom absorbs a $\sigma^+$-polarised photon. The same process happens along all three axes and the trapped atoms are confined at the point where $B=0$.

Mathematically, the force along an axis $i=1,2,3$, which includes the Zeeman effect in the force found for the optical molasses, is described by

$F_\mathrm{MOT} = F_\mathrm{scatt}^{\sigma^+}(\omega_L - kv_i - (\omega_0 + \beta_i x_i)) - F_\mathrm{scatt}^{\sigma^-}(\omega_L + kv_i - (\omega_0 - \beta_i x_i)) \equiv -\alpha v_i - \frac{\alpha \beta_i}{k} x_i,$

with the Zeeman shift $\beta_i x_i = ({g_J \mu_B}/{\hbar}) ({\mathrm{d}B}/{\mathrm{d}x_i}) x_i$.

The terms $\omega_0 \pm \beta_i x_i$ are respectively the resonance absorption frequencies for $\Delta M_J = \pm 1$ transitions at position $x_i$. We see, therefore, that the imbalance of the scattering force due to the Zeeman effect leads to a restoring force with the spring constant $\alpha \beta /k$. Hence, ordinarily the atoms undergo a simple, over-damped harmonic motion when loaded into the MOT.

The atoms in the MOT have a higher temperature than in the optical molasses. Among other things, this is because when the Zeeman shift exceeds the Doppler shift, the sub-Doppler cooling mechanism breaks down. That is why a typical MOT collects atoms from a slowed atomic beam and when sufficient atoms accumulated in the trap, the MOT is turned off to cool the atoms by the optical molasses before further experiments are carried out.

The advantage of MOT over optical molasses is the much higher capture rate. For sodium, the capture velocity of the MOT is about $70\mathrm{~m~s^{-1}}$, which is much higher than that of molasses with about $6\mathrm{~m~s^{-1}}$. This relatively high capture rate allows the MOT to be loaded directly from the vapour at room temperature in the case of heavy alkali atoms such as rubidium, without the additional slowdown.

Sub-Doppler cooling

As mentioned above, the Doppler cooling limit proves in fact not to be the lower temperature limit for the optical molasses. The simple picture of two-state atoms, in which the scattering forces from each of the six lasers add up independently, cannot explain this sub-Doppler cooling. The ground state of an atom usually has a Zeeman structure inside, which provides additional complexity and thus enables new processes. The most significant mechanism is the so-called Sisyphus cooling, in which the atoms lose energy as they move through a standing wave.

Sisyphus cooling

The principle of Sisyphus cooling is based on the dynamic Stark effect (or AC Stark effect), in which the atomic energy levels shift in the light field.

We consider two counter propagating laser beams with orthogonal linear polarisation which generate a standing wave in polarisation but not in intensity. The resulting polarisation depends on the relative phase between the lasers and on the position in space. Along the standing wave, the polarisation changes from linear to $\sigma^-$ to linear (turned by 90$^\circ$) to $\sigma^+$ over the course of $\lambda/2$ and repeats itself flipped. Where the lasers have a phase difference of $\pm\pi/2$ we get a circular polarisation ($\sigma^\pm$) and where we have no difference we get a linear polarisation.

We now consider an atom with two levels of angular momentum $J=1/2$ and $J'=3/2$ respectively. As the atom moves through the standing wave, the polarisation gradient causes a periodic modulation of the dynamical Stark shifts of the states in the lower level $J=1/2$. For the field-free case, all energy levels $M_J$ are degenerate, however, in the presence of circularly polarised light, the dynamical Stark effect lifts this degeneracy. The direction and amount of the lifting depend on the polarisation in place. This polarisation dependence is used to introduce a spatially dependent deceleration force.

In the presence of $\sigma^-$-polarisation, the $M_J=1/2$-level is lowered and the $M_J=-1/2$-level is raised and vice versa for $\sigma^+$-polarisation. So along the standing wave the $M_J$ levels rise and fall periodically, where if $M_J = 1/2$ is a valley, $M_J=-1/2$ is a hill. If the atom were to wander through this potential landscape remaining in the same state, its total energy would not change. However, we can now dissipate energy from the atom by changing the atomic state at the right moments. In this case, the atom absorbs a photon when it is at the top of a hill and then spontaneously falls back to the bottom of a valley. For this to work, the decay into the valley must have a higher probability than the reverse process. In this mechanism, therefore, the kinetic energy that the atom converts into potential energy when it climbs the hill is lost when it spontaneously decays into the valley. The atom therefore moves more slowly.

At each such transition, the atom loses an energy $U_0$ approximately equal to the height of the hill and proportional to the laser intensity per detuning, $U_0 \propto I/|\delta|$. The equilibrium temperature for the Sisyphus cooling is reached when the energy loss due to climbing balances that of the recoil energy of the spontaneous emission, i.e. $U_0 \approx E_\mathrm{rec}$. When performing the calculations, we find the lower temperature limit for the Sisyphus cooling at

$T_\mathrm{rec} = \frac{\hbar^2 k^2}{k_Bm},$

which is called the recoil limit. For a sodium atom the recoil limit is $2.4~\text{µK}$, thus ten times lower than the Doppler cooling limit. Practically the limit is a few times this limit due to the high sensitivity to external magnetic field.

Summary

The basic principle is to use the scattering force of photons to cool atoms. Doppler cooling uses Doppler shifts to bring atoms into resonance with a slowing laser. While optical molasses is efficient at cooling atoms, it lacks the ability to trap them. In the magneto-optical trap (MOT), a magnetic field gradient creates a restoring force that allows atoms to be trapped. Sisyphus cooling involves atoms losing energy via a standing wave, resulting in a lower Doppler cooling limit. These techniques represent important advances in the field of laser cooling and trapping, facilitating progress in atomic physics research.

[^1]: Here, we used the fact that $\partial f/\partial x \approx (f(x \pm h) - f(x))/(\pm h)$ for a very small $h$. Compare with the definition of the derivative.

---

This article is the written version of a talk I gave in my university's 'Low Energy Particle Physics' seminar. So it's rather technical, but I hope it's still comprehensible enough for non-physicists.

v0 @ 789670; v1 @ 789774: corrected missing $\hbar$ in photon energy formula