An algebraic solution of driven single band tight binding dynamics

Abstract

The dynamics of the driven tight binding model for Wannier-Stark systems is formulated and solved using a dynamical algebra. This Lie algebraic approach is very convenient for evaluating matrix elements and expectation values. It is also shown that a dynamical invariant can be constructed. A classicalization of the tight binding model is discussed as well as some illustrating examples of Bloch oscillations and dynamical localization effects.Comment: 13 pages; revised version (changed title and sections 6,7, added references

Quantum mechanics on a circle: Husimi phase-space distributions and semiclassical coherent state propagators

“…The derivation of the semiclassical propagator on the circle follows similar lines as the one for the propagator on the real line [22]. Finally, the propagator on the circle can be represented as an infinite sum over propagators on the real line, representing different winding numbers of paths in U (1), which nicely demonstrates the influence of the global topological properties of phase space on the quantum dynamics.…”

“…Their general properties have been exhibited in [11][12][13]. For G = SU (2), the complexifier coherent states are used for the semiclassical analysis of loop quantum gravity [11], and the states for G = U (1) and G = U (1) 3 are employed in the semiclassical analysis of quantum cosmology [14] and linearized quantum gravity [7].…”

Discrete diffraction and shape-invariant beams in optical waveguide arrays

“…As the problem of discrete diffraction in waveguide arrays with curved axis or transverselly-imposed index gradients is analogous to the problem of one-dimensional or two-dimensional Bloch oscillations of electrons in periodic potentials with an applied electric field or of cold atoms in optical lattices, some results are already available in the literature. In particular, in recent works [28][29][30] an algebraic approach has been developed, capable of providing rather general results for wave packet center of mass evolution and wave packet spreading in certain lattice models. In this approach, after the introduction of a dynamical Lie algebra, an explicit form of the evolution operator is first derived, and then the expectation values of operators are calculated in the Heisenberg picture.…”

“…Though similar results have been previously published in Refs. [28][29][30] using an algebraic operator approach, they are here re-derived for the sake of completeness using the method of moments, which does not require the explicit calculation of the evolution operator and the formulation of the problem in terms of a Lie algebra. In the subsequent section a family of shapeinvariant discrete beams will be introduced, proving that their propagation in a generally-curved waveguide array is simply described by the evolution of a complex-q beam parameter, which plays an analogous role of e.g.…”

Two-dimensional Bloch oscillations: a Lie-algebraic approach

“…Bose-Einstein condensates in optical lattices has motivated new theoretical studies in this field (see [4] and the references given there). Bloch oscillations are one of the striking phenomena in such systems and an example for a counterintuitive behaviour in quantum mechanics [4][5][6][7][8][9][10][11]. While [4][5][6][7] discuss the one-dimensional case analytically in a single-band tight-binding model, in [8][9][10][11] two-dimensional Bloch oscillations are considered, however, only partially in an analytical treatment.…”