Effects of chaotic energy-band transport on the quantized states of ultracold sodium atoms in an optical lattice with a tilted harmonic trap

We investigate the classical and quantum properties of ultracold sodium atoms in a one-dimensional optical lattice and a three-dimensional harmonic trap. The energy versus crystal momentum dispersion relation for the lowest energy band of the optical lattice, together with the harmonic potential, define an effective Hamiltonian that we use to calculate classical atom paths. When one of the symmetry axes of the harmonic trap is aligned with the optical lattice, the atoms follow stable trajectories. But tilting this symmetry axis away from the optical lattice direction creates an unusual type of mixed stable-chaotic classical dynamics, which originates from the intrinsically quantum-mechanical nature of energy bands. In this regime, the density of quantized energy levels for the system exhibits periodic fluctuations associated with both stable and unstable periodic classical orbits. One of the unstable orbits also produces well-defined scar patterns in a subset of eigenfunctions. Wave functions with distinct spatial forms are identified and related directly to particular parts of the classical phase space using a Wigner function analysis.