Ultra-cold quantum gases : Bose-Einstein condensates and degenerate Fermion gases

The third lesson focused on the physics of quantum gases, Bose Einstein condensates (BECs) and degenerate fermion gases (DGFs). The properties of a BEC are reminiscent of those of a laser beam in which photons are replaced by indistinguishable atoms, with the wealth of physical effects induced by interactions between particles (whereas these are zero between photons). A GFD behaves like an electronic Fermi sea, with interactions between fermions leading to their pairing in the form of composite bosons. These bosons can be molecular dimers forming condensates, or "Cooper pairs" of great spatial extension giving GFDs properties analogous to those of electronic superconductivity.

CBEs and GFDs possess superfluid properties, manifested by the appearance of vortices or vortexes when they are rotated. The lesson reviewed some of the key properties of EBCs and GFDs observed over the last fifteen years. It began with a reminder of the general mechanisms behind the condensation of very cold boson gases. After an initial phase of laser cooling, spin-oriented atoms are generally confined in a magnetic trap. They are then subjected to a complementary "evaporative" cooling process by applying an adiabatically varied radio-frequency field to flip the spin of progressively lower-energy atoms and expel them from the trap. Under the effect of interatomic collisions, the remaining confined atoms are "thermalized" to a lower and lower temperature, down to a few hundred nanoKelvins. The de Broglie wavelength of the atoms then becomes of the order of their mutual distances, and the gas of bosons condenses in the ground state of the trap. CBE is usually detected by cutting the trap and observing the shadow of the freely expanding gas in a resonant laser beam.

The cooling of fermions, on the other hand, leads to a situation where the particles each occupy a quantum state up to the Fermi energy. The behavior of cold bosons and fermions can be compared by observing isotopes of the same element (e.g. ^Li6 and ^Li7). A gas of degenerate fermions is larger in size than that occupied by the same number of condensed bosons (Fermi pressure). Fermions are generally cooled in mixtures of bosons and fermions, with the former cooled by evaporation and the latter by collision with the former.

In a second part, the lesson described the interactions between cold atoms, recalling the definition of the diffusion length. This parameter, which can be positive (repulsive effective interactions) or negative (attractive interactions), fully characterizes binary collision processes between cold atoms. The diffusion length can be varied by applying a magnetic field to the atoms which, for certain values of this field, brings the collision entry channel into resonance with a molecular state of the two-atom system (Feshbach resonances). Away from resonance, a gas of condensed bosons is described by a mean-field model using the Gross-Pitaevskii equation, which takes the form of a Schrödinger equation including a non-linear term proportional to the scattering length. At resonance, this diffusion length diverges and the mean-field model is no longer valid.

The third part of the lesson described experiments on EBCs that were very similar to optical experiments: generation of coherent matter waves, interference between these waves, four-wave matter mixtures demonstrating effects similar to those observed in optics when three laser beams interact with each other in a transparent medium whose refractive index has a non-linear term. In the case of matter waves, the interaction between the waves is produced by the effect of collisions between atoms and is described by the non-linear term of the Gross-Pitaevskii equation.

In the fourth part of the lesson, the superfluidity phenomena of EBCs and GFDs were qualitatively described. These collective phenomena are analogous to those previously observed in much denser systems, in solid or liquid physics (superfluidity of Helium, superconductivity of metals). Studying these effects in quantum gases offers the advantage of realizing situations in which we can vary parameters (in particular the interaction between particles) that have a fixed value in Quantum Condensed Matter Physics. In this way, we can move seamlessly from a situation where inter-particle interactions are weak and mean-field theory is valid, to a strongly interacting N-body physics whose theory is still largely open. We can even reach a universal regime where the behavior of systems no longer depends on the particle species considered, and thus use the physics of ultra-cold atoms as a test bed for understanding phenomena occurring in other systems, with very different orders of magnitude (neutron stars, for example). The most spectacular manifestation of the superfluidity of EBCs and GFDs is the appearance of vortices in these rotating systems. In this case, the physics is very similar to that observed in type II superconducting metals subjected to a magnetic field.