Immanuel Bloch has already written a very good popular-science article about quantum gasses in optical lattices, so I will not re-write it here, but I will take the opportunity to discuss some of the theoretical approaches to modelling such systems. I been asked questions about the Bose-Hubbard model by interested experimentalists, so I will start there:

When there are relatively few atoms in each lattice site, and the lattice is sufficiently strong compared with the energy of the atoms, then the Hubbard (for Fermions) or Bose-Hubbard (for Bosons) model describes the system effectively. I will concentrate on the Bosonic case, since that is closest to my interests. In this model, the atoms’ behaviour is governed by a tunneling parameter between neighbouring lattice sites, and a parameter describing the interaction between atoms in the same site (see Dieter Jaksch et al. Phys. Rev. Lett. 81 3108 for details).

In the Bose-Hubbard system there is a quantum phase transition between superfluid behaviour (the atoms flow freely between lattice sites) and Mott insulator behaviour (atoms become trapped in individual sites) when the tunneling between lattice sites becomes sufficiently weak. The insulating phase is useful when we want, for instance, one atom to be localised in each lattice site to use as a quantum register.

When the number of atoms becomes large, the Hilbert space of the Bose-Hubbard model may become too large to be practical. In this case, in the superfluid (weak lattice) phase, the Gross-Pitaevskii equation describing the mean-field of all the atoms as a single wavefunction is a good model (see Morsch and Oberthaler's review article). I will take this opportunity to publicise my boss's paper on what happens when, for large atom numbers, you start with a superfluid and then ramp up the lattice potential. It turns out that Mott insulator states are hard to achieve in this case. It may be possible to achieve a Mott insulator with large atom numbers, but it looks like doing so will be a challenge.

Much more physics has been done, of course, in optical lattices than I have described. However, I hope this post has given a flavour of the subject.

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