Geometrical Structure of Laplacian Eigenfunctions

One of the most intriguing phenomena in physics is the localization of waves in disordered media. This phenomenon was originally predicted by Anderson, fifty years ago, in the context of transport of electrons in crystals. Anderson localization is actually a much more general phenomenon, and it has been observed in a large variety of systems, including light waves. However, it has never been observed directly for matter waves. Ultracold atoms open a new scenario for the study of disorder-induced localization, due to high degree of control of most of the system parameters, including interaction. Here we employ for the first time a noninteracting Bose-Einstein condensate to study Anderson localization. The experiment is performed with a onedimensional quasi-periodic lattice, a system which features a crossover between extended and exponentially localized states as in the case of purely random disorder in higher dimensions. Localization is clearly demonstrated by investigating transport properties, spatial and momentum distributions. We characterize the crossover, finding that the critical disorder strength scales with the tunnelling energy of the atoms in the lattice. Since the interaction in the condensate can be controlled at will, this system might be employed to solve open questions on the interplay of disorder and interaction and to explore exotic quantum phases.

A method for confining electrons to artificial structures at the nanometer lengthscale is presented. Surface state electrons on a copper(111) surface were confined to closed structures (corrals) defined by barriers built from iron adatoms. The barriers were assembled by individually positioning iron adatoms with the tip of a 4-kelvin scanning tunneling microscope (STM). A circular corral of radius 71.3 A was constructed in this way out of 48 iron adatoms. Tunneling spectroscopy performed inside of the corral revealed a series of discrete resonances, providing evidence for size quantization. STM images show that the corral's interior local density of states is dominated by the eigenstate density expected for an electron trapped in a round two-dimensional box.