Atomic Bose-Einstein Condensates in Optical Lattices with Variable Spatial Symmetry

Abstract

Optical lattices for atomic Bose-Einstein condensates raised enormous interest, as they mirror features known from solid state physics to the field of atom optics. In perfect solid state crystals atoms are arranged in a regular array creating a periodic potential for the electrons inside. Felix Bloch was one of the first who investigated in his dissertation (1928) the quantum mechanics of individual electrons in such crystalline solids. In the independent electron approximation interatomic and interelectronic interactions are neglected. Each electron obeys the one electron Schrödinger equation with a periodic potential V (x + a) = V(x) with period a. According to Bloch’s theorem the stationary eigenstates ψ _n,q (r) are plane waves modulated by a periodic function revealing the periodicity of the atom lattice [1]. With proper periodicity and boundary conditions the eigenstates are quantized, characterized by the band index n = 0, 1,…. The plane waves propagate in the direction of the wave vector q with the associated quasimomentum ħq, which it is sometimes referred to as the crystal or lattice momentum. The energy levels E _n (q) are periodic continuous functions of the wave vector q forming the energy bands. Pictures of the energy bands showing the bandstructure are conventionally restricted the first Brillouin-zone of the reciprocal lattice −ħk ≤ q ≤ ħk. One milestone of Bloch theory and the band structure of particles is the finding of a natural physical explanation of the some 20 orders of magnitude difference in electrical conductivity between an insulator and a good conductor [2].