Time evolution of initially localized states in two-dimensional quantum dot arrays in a magnetic field

The evolution of non-stationary localized states |Ψ(t=0)〉|Ψ(t=0)〉 is investigated in two-dimensional tight binding systems of N potential wells with and without a homogeneous field perpendicular to the plane. Most results are presented in analytical form, what is almost imperative if the patterns are as complex as for rings in a magnetic field, where the qualitatively different features arise depending on rational or irrational numbers. The systems considered comprise finite linear chains (N=2,3)(N=2,3), finite rings (N=3N=3–6), infinite chains, finite rings (N=3N=3–6) in a magnetic field, and rings with leads attached to each ring site. The position of the particle at time t is described by the projection of the wave function Pm(t)=|〈m|Ψ(t)〉|2Pm(t)=|〈m|Ψ(t)〉|2 onto the localized basis function at site m. For finite chains and rings with N=3,4,6N=3,4,6 the time evolution is periodic, whereas it is non-periodic for N=5N=5 and N greater then 6. Rings in a magnetic field show a rich spectrum of different features depending on N and the number of flux quanta through the ring, including periodic oscillation and rotation of the charge as well as non-periodic charge fluctuations.