There are now many new and intriguing experiments investigating the quantum coherent dynamics of ultracold atoms trapped in standing waves of light, or optical lattices. This is somewhat different from the cold-atom experiments we studied in quantum chaos, which were in the microKelvin regime. Ultracold experiments are at the nano-Kelvin temperatures; as the atoms become colder the de Broglie wavelength grows to become of the same order as the inter-atom separation. Either way whereas the microKelvin quantum chaos experiments were essentially testing the single-particle Schrodinger-equation dynamics, the ultracold atoms permit new effects due to the collective dynamics. An aspect of great interest is how the interactions between atoms modify the dynamics. The picture on the right illustrates an optical lattice where the atoms are localized, one in each-well, like eggs in an eggbox.

It is possible to vary the strength of the optical lattice periodically in time. This is what we call AC-driving. In the more extreme version of the kicked atom, the lattice is switched on for a very very short pulse, then turned off between kicks. The atoms move freely in between kicks. This has been shown to give the effect called "Dynamical Localization", the freezing of the momentum distribution of cold atoms. But you can also get DL with atoms in a permanently on optical lattice. Atoms trapped in a lattice or spin waves in a spin chain have an effective kinetic energy J cos P where P is the momentum and J a tunneling amplitude. It is easy to show that if they experience kicks from a parabolic potential, the resulting time-evolution operator also gives Dynamical Localization and accelerator modes, effects studied in the kicked atoms. In Ref[1] we proposed localization and accelerator modes as a means to transmit information along a spin chain without spreading. This is illustrated in the figure alongside.

People working wih AC-driven atoms usually simply vary the potential sinusoidally. Here a complete different interesting effect emerged. Confusingly, it was called "Dynamic Localization" but is quite unrelated to "Dynamical Localization". The former is the control of tunneling. Under AC-driving, the tunneling amplitude takes an effective value, so its magnitude and even sign can be controlled. In Ref[2] it was shown that in the presence of interactions, if the energy gap between singly and doubly occupied lattice sites is an exact multiple of the photon frequency, the well-know phase transition (Mott Insulator to Superfluid transition) can be controlled by a series of very sharp resonances as shown in the figure alongside