A Discrete Exterior Calculus Approach to Quantum Transport and Quantum Chaos on Surface

Abstract

We address the problem of computing transport observables and spectral characteristics of quantum dynamics on arbitrary surfaces. Our approach is based on discrete exterior calculus (DEC) and applies to both open and closed quantum systems. We present an efficient algorithm for the calculation of the recursive Green’s functions (for open systems) and the full set of eigenfunctions and eigenvalues (for closed systems) using numerical tools available for DEC. Our approach is applied to the calculation of the conductance of a non-flat quantum device coupled to electron reservoirs and to obtain the spectra of ballistic cavities defined on curved surfaces. In both cases we found numerical evidences of a curvature induced integrable-chaotic crossover.

Time-harmonic electromagnetics with exact controllability and discrete exterior calculus

“…The magnetic current density is − b14 , the electric current density is − b11 , and magnetic and electric charges are modeled by − ũ15 and − ũ16 . A Maxwell-like system is also obtained from rows 3, 4, 7, and 8 of Equation (2).…”

“…Wave equations are important for modeling acoustic, elastic, electromagnetic, and quantum mechanical systems in different fields of science, engineering, and technology. There has recently been growing interest in differential form -based approaches for considering wave equations (see, e.g., [1][2][3]). There are several reasons why the framework is a good alternative for the traditional vector field presentation.…”

“…That is, instead of the vector presentation E = (E 1 , E 2 , E 3 ) for the electric field, we present a 1-form 3 , where d is the exterior derivative. Further, the electric flux density is considered, instead of the vector D = (D 1 , D 2 , D 3 ), as a 2-form 2 , where ∧ is a generalization of the cross product, known as the wedge product or exterior product. We also present a 1-form H for the magnetic field and a 2-form B for the magnetic flux density.…”

GPU-accelerated time integration of Gross-Pitaevskii equation with discrete exterior calculus

“…In DEC the only source of discretization error is the discrete Hodge star operator, which defines the relations between primal and dual discrete differential forms, and thus is determined by the relation of the primal and dual spatial grids. DEC has previously been successfully utilized in elastodynamics [16], fluid dynamics [17], electromag-netism [18], and in quantum mechanics in the form of the standard linear Schrödinger equation [19]. Our work is the first one to apply DEC to the non-linear Schrödinger equation.…”

Discrete exterior calculus for photonic crystal waveguides

“…10 The method is pioneered in electromagnetic simulations by Bossavit and Kettunen, 11 while the groundwork presented by Marsden's group was applied in computer vision and image processing 12 and delivered the PyDEC software library. 13 During the last decade, the DEC has been applied to, for example, elastostatics, 14 electromagnetics, 15 quantum mechanics, 16,17 and flow problems. 18 Chen and Chew 2 applied the method for PC band structure computations in frequency domain.…”