Transition between extended and localized states in a one-dimensional incommensurate optical lattice

Abstract

The quantum transport properties of a system are intimately related to the underlying symmetries of the Hamiltonian. In a perfectly periodic system all the eigenfunctions are extended Bloch waves @1#, while for a random potential in a quasi-one-dimensional system all the eigenfunctions are localized @2#. These properties can be experimentally studied through the dispersion of an initially localized wave packet; in the former case it grows ballistically, while in the latter the dispersion remains constant. In between these two extreme cases lie incommensurate and quasiperiodic systems. In these, the spectrum can range from having all extended states to all localized states or even to mixed behavior, as the parameters describing the system ~incommensurability, potential amplitude, etc.! are varied. For instance, in the Fibonacci lattice @3# all states are critical ~neither localized nor extended!, leading to anomalous dispersion. Another known example is the Harper model @4#, which models electrons in a two-dimensional lattice in the presence of a transverse magnetic field. The eigenfunctions for this model undergo a transition from localized to extended behavior as the amplitude of the potential of the lattice is decreased. These quantum properties could be related to the quite anomalous transport properties of quasicrystals @5#. These materials show large resistivity and a decreasing temperature dependence; this behavior is enhanced in cleaner samples, thus showing that impurities improve the transport instead of degrading it. A study of transport in a defect-free incommensurate system can then shed some light on the physics underlying quasicrystals.