Two exact lattice propagators

Yellin, Joel

doi:10.1103/physreve.52.2208

Cited by 10 publications

(4 citation statements)

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“…which can be obtained from (1) through the following definitions: t = − τ /2µa 2 , α(t) = 2µa 3 E(τ )/ 2 and ψ n (t) = e −it/2 φ n (τ ). Some years ago, Yellin [12] found the propagator of (3) for the case α = constant by means of the algebraic properties of the hamiltonian. For our problem, the hamiltonian operator reads…”

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Time-dependent Stark ladders: exact propagator and caustic control

Sadurní¹

2013

J. Phys. A: Math. Theor.

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Abstract.In this note we present a new propagator for a particle in discrete space under the influence of a time-dependent field. With this result we are able to control the shape of caustics emerging from a point-like source, as the explicit form of the wavefronts can be put in terms of the external field.PACS numbers: 03.65. Db, 02.30.Gp, 42.25.Fx arXiv:1302.4008v1 [quant-ph] Feb 2013Time dependent Stark ladders: Exact propagator and caustic control 2 Similar propagation phenomena in time domain can be found in quantummechanical systems, electromagnetic waves and sound waves. The analogies between their wave equations in controlled and well designed situations have allowed the emulation of crystalline structures in settings with periodic symmetry, reaching recently a realization of graphene with microwave cavities [1], [2] and photonic crystals [3]. For discrete systems without periodic symmetry, the Stark ladder introduced by Wannier more than fifty years ago [4] offers itself as an interesting example. In this respect, we note that the emulation of electrons under the influence of a constant force has been achieved as well: The Wannier-Stark ladder in vibrations of aluminum rods [5] and the observation of Bloch oscillations in photonic structures [6] seem to be the simplest realizations. Among the most sophisticated, we may single out the propagation of BoseEinstein condensates in periodic optical traps [6,7,8] with the possibility of producing a Stark ladder by means of a gravitational field.In this note we study the more general case of a homogeneous force field modulated by an arbitrary time-dependent intensity, with the aim of offering another interesting possibility to the already existing configurations and emphasizing the external control of the system through such a field. We shall refer to it as a time-dependent Stark ladder, pointing out to its generalization through the time dependence of the potential and not merely to an equispaced spectrum. Our task is therefore to find the corresponding propagator in closed form. In the case of emulations outside of the quantum regime, we may simply refer to our result as the Green's function. Armed with the result, we shall proceed to characterize the behaviour of caustics emerging from a point-like initial condition. We shall also find the explicit relation between the propagation of the corresponding wavefronts and the time-dependent modulation of the discrete potential, giving the opportunity to discuss the maximal speed of propagation of a signal and how it can be controlled within the restrictions of the Lieb-Robinson bound [9]. Two recent studies in theoretical [10] and experimental [11] grounds exemplify the relevance of these ideas.We start with the problem of finding the propagator for a discrete Schrödinger equation in the presence of a time-dependent field. One possible approach for introducing such an equation is by using the central discretization of derivative operators, i.e.where a is the lattice spacing, µ is the mass of the particle and ...

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“…Some years ago, Yellin [12] found the propagator of (3) for the case α = constant by means of the algebraic properties of the Hamiltonian. For our problem, the Hamiltonian operator reads…”

mentioning

confidence: 99%

Time-dependent Stark ladders: exact propagator and caustic control

Sadurní¹

2013

J. Phys. A: Math. Theor.

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“…which turn out to be Bessel functions [22,23]. The energy levels are determined by ελ 2n and the eigenfunctions are combinations of Wannier functions of the form…”

Section: An Exactly Solvable Examplementioning

confidence: 99%

Self-similar analogues of Stark ladders: a path to fractal potentials

Sadurní

Castillo

2017

Phys. Atom. Nuclei

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We treat the eigenvalue problem posed by self-similar potentials, i.e. homogeneous functions under a particular affine transformation, by means of symmetry techniques. We find that the eigenfunctions of such problems are localized, even when the potential does not rise to infinity in every direction. It is shown that the logarithm of the energy displays levels contained in families that are analogous to Wannier-Stark ladders. The position of each ladder is proved to be determined by the specific details of the potential and not by its transformation properties. This is done by direct computation of matrix elements. The results are compared with numerical solutions of the Schrödinger equation.

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“…In this respect we should note that continuous limits in lattices can be obtained in a variety of ways, which may lead to relativistic as well as non-relativistic systems: see e.g. [16,26], where restrictions have been imposed specifically to yield the gaussian propagators typical of non-relativistic theories. In the following we show that other ways of specifying continuous limits lead naturally to effective relativistic kernels for hexagonal and linear lattices.…”

Section: The Dirac Limit Viewed In Space and Timementioning

confidence: 99%

Propagators in two-dimensional lattices

Sadurní¹

2013

Preprint

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This paper is devoted to the computation of discrete propagators in two-dimensional crystals and their application to a number of time dependent problems. The methods to compute such kernels are provided by a tight-binding representation of Dirac matrices and the generalizations of Bessel functions. Diffusive effects of point-like distributions on crystalline sheets are studied in a second quantization scheme. In the last part, a compendium of propagators is presented. The cases of square, triangular and hexagonal arrays are covered.

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Two exact lattice propagators

Cited by 10 publications

References 25 publications

Time-dependent Stark ladders: exact propagator and caustic control

Time-dependent Stark ladders: exact propagator and caustic control

Self-similar analogues of Stark ladders: a path to fractal potentials

Propagators in two-dimensional lattices

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