March 19, 2009

In this report I will summarize some of the techniques currently used to take a cloud of atoms moving at more than 100 m/s and slow them down to a few millimeters per second, which is equivalent to temperatures below 1 μ K. At such temperatures, the atoms kinetic energy is so low that they will have no choice but to occupy the lowest energy levels available. If those particles are bosons, a macroscopic fraction of them will occupy the same quantum state: this forms a Bose-Einstein condensate (BEC), a macroscopic quantum state of matter with amazing properties.

The first step is to cool the atoms with laser light.
This process is illustrated in Fig. 1. When a laser beam hits an atom,
it exerts a force on it which is called the resonant radiation pressure.
Suppose the atom's spectrum only has 2 energy levels E_{a} and
E_{b} such as E_{b} - E_{a} = h
ν_{0}. The atom then interacts with the electromagnetic wave
by absorbing and reemitting photons. The interaction is maximum at the
resonance frequency, and decreases with detuning. If an atom is moving
between 2 beams of the same frequency, it will feel a higher frequency
for the upstream beam, due to the Doppler effect. The force exerted by
the upstreaming beam is therefore bigger, and the atom slows down. The
potential energy is effectively

So we see that if the field isn't uniform, it exerts
a force **F** on the atom:

One can show, using the optical Bloch equations, that
if ν_{0} - ν > 0, α is positive and so the atom is
pushed towards the regions of space where the intensity is higher.

Obviously, one can use 6 laser beams to cool down the atoms along each of the 3 directions, however one can only reach temperatures down to 100 μ K with this technique, which is still too hot to reach the BEC regime. Indeed, when one uses Doppler cooling there is a residual motion of the atoms one can't get rid of: whenever an atom absorbs or reemits a photon, it acquires an additionnal momentum in a random direction: even if the Doppler cooling is improved the atoms will keep moving a little.

Fig. 3: Illustration of evaporative cooling. |

The next step, shown in Fig. 3, is to use evaporative cooling. Suppose we manage to trap the atoms in a potential well. If an atom is too fast, he has the required kinetic energy to escape the trap. If the depth of the well slowly decreases, the most energetic atoms escape and the remaining particles in the trap are colder.

How can we make such a potential well? The idea,
shown in Fig. 4, is to use a configuration of Helmholtz coils in order
to have a minimum of the magnetic field at the point where we want to
trap the atoms. In this configuration the magnetic field grows linearly
with the distance from the center. The energy of a magnetic dipole
moment **μ** in a field **B** is: U = - **μ** ⋅
**B**. So the atoms whose **μ** is antiparallel to the field
are pushed towards the regions of space where B is lower.

Fig. 4: Illustration of a magnetic
trap. |

Fig. 5: Illustration of trapping atoms in
an optical lattice. |

This way one can cool the gas down to temperatures below μ K: at those temperatures a macroscopic fraction of the bosons occupy the same ground state, which means that the particles are in the same quantum state.

In optics, we know how a transmission grating works: each slit of the grating acts like a point source, the waves reemitted by each slit are phase coherent so far from the grating we get a superposition of each amplitude which interfere constructively or not, depending on the optical path difference. The same phenomenon happens for an expanding BEC trapped in an optical lattice.

The idea, illustrated in Fig. 5, is to trap the BEC in a periodic potential. One can easily create a standing wave by aligning 2 mirrors. In such a configuration there is a spatial periodicity of the intensity of light. We've seen that if the laser has the correct detuning the atoms are trapped in the regions of space where the intensity is higher. So with the same set of mirrors along each direction of space, we can make a trapping potential which is periodic along each axis:

In such a potential the BEC looks like a pseudo
crystal with atom trapped at each lattice site. We see that when the
potential depth V_{0} is big the atoms are tightly confined at
each lattice site, so they don't "see" each other. The coherence of the
condensate is then lost and the atoms can't tunnel from one site to
another. On the other hand, if V_{0} isn't too big, the
condensate keeps its coherence and the atoms can hop from one lattice
site to another. So when the potential depth increases one can see a
transition from a coherent superfluid state to a so called Mott
insulator state, where the atoms are strictly confined to each lattice
site.

We won't have time to describe here how powerful this tool is to probe the different quantum phases of matter, the idea is just that optical lattices enable us to design "home made" pseudo-crystals by controlling parameters such as the lattice spacing and the potential depth. Let's just see how we can check if the cold gas in the optical lattice is still a coherent BEC or not.

Suppose one suddenly turns off the trapping potential. Then the cloud freely expands, and the wavefunctions from different lattice sites interfere. This is exactly like the interference pattern one gets with a grating diffracting light. The different slits are phase coherent so we observe constructive interference at Bragg points. The only conceptual difference between the 2 experiments is that in optics the square of the amplitude is the light intensity whereas here we observe the square of the wave function, which is the number of particles. So if the lattice sites are phase coherent, after expansion one should observe particles only in a sharp region of space around Bragg points. If the coherence has been lost, on the other hand, for example after the transition to the Mott insulator state, we won't observe such an interference pattern. For further details see reference [3].

Fig. 6: Illustration of diffraction test of
atomic coherence. |

© 2009 F. Amet. The author grants permission to copy, distribute and display this work in unaltered form, with attributation to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] C. Cohen-Tannoudji, J. Dupont-Roc and G.
Prynberg, *Atom-Photon Interactions: Basic Processes and
Applications* (Wiley, 1998).

[2] C. Cohen-Tannoudji, "Course on Laser Cooling Techniques, Leçons du Collège de France," (unpublished).

[3] M. Greiner *et al.*, "Quantum Phase
Transition from a Superfluid to a Mott insulator in a Gas of Ultracold
Atoms", Nature **415**, 39 (2002).

[4] K. B.Davis *et al.*, "Bose-Einstein
Condensation in a Gas of Sodium Atoms", Phys. Rev. Lett. **75**, 3369
(1995).