Effective time-independent description of optical lattices with periodic driving

For a periodically driven quantum system an effective time-independent Hamiltonian is derived with an eigen-energy spectrum, which in the regime of large driving frequencies approximates the quasi-energies of the corresponding Floquet Hamiltonian. The effective Hamiltonian is evaluated for the case of optical lattice models in the tight-binding regime subjected to strong periodic driving. Three scenarios are considered: a periodically shifted one-dimensional (1D) lattice, a two-dimensional (2D) square lattice with inversely phased temporal modulation of the well depths of adjacent lattice sites, and a 2D lattice subjected to an array of microscopic rotors commensurate with its plaquette structure. In case of the 1D scenario the rescaling of the tunneling energy, previously considered by Eckardt et al. in Phys. Rev. Lett. 95, 260404 (2005), is reproduced. The 2D lattice with well depth modulation turns out as a generalization of the 1D case. In the 2D case with staggered rotation, the expression previously found in the case of weak driving by Lim et al. in Phys. Rev. Lett. 100, 130402 (2008) is generalized, such that its interpretation in terms of an artificial staggered magnetic field can be extended into the regime of strong driving.Comment: 10 pages, 5 figure

Lie Theory of Differential Equations and Computer Algebra

Introduction The aim of this contribution is to show the possibilities for solving ordinary differential equations with algorithmic methods using Sophus Lie's ideas and computer means. Our material is related especially to Lie's work on transformations and differential equations---essential ideas are already contained in his first paper on transformation groups [5]---and to his article on differential invariants [6]. Very good modern surveys on such questions as are discussed here and on related problems are found in [8,9]. Lie's first intentions were to create a theory for solving differential equations with means of group theory in analogy with the Galois theory for algebraic equations. With respect to typical elements of Galois theory---fields, groups, automorphisms and relations betweeen them---this concept is realized today in the so-called Picard-Vessiot theory for linear ordinary differential equations.

Lie theory of differential equations and computer algebra, Seminar Sophus Lie 1

The aim of this contribution is to show the possibilities for solving ordinary differential equations with algorithmic methods using Sophus Lie’s ideas and computer means. Our material is related especially to Lie’s work on transformations and differential equations—essential ideas are already containe