12 research outputs found
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
Involutive divisions in Mathematica: implementation and some applications
In this paper we consider different involutive divisions and describe their implementation in Mathematica together with algorithms for the construction of involutive bases for monomial ideals. As a straightforward application, we consider computation of the Hilbert function and the Hilbert polynomial for a monomial ideal, or for a polynomial one represented by its Gröbner basis. This allows one, in particular, to determine the index of regularity of the ideal
Generalization of Risch's Algorithm to Special Functions.
Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch’s algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They can also be used to compute linear relations among integrals and to find identities for special functions given by parameter integrals. The aim of this presentation is twofold: to introduce the reader to some basic ideas of differential algebra in the context of integration and to raise awareness in the physics community of computer algebra algorithms for indefinite and definite integration