WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations

WavePacket is an open-source program package for the numerical simulation of quantum-mechanical dynamics. It can be used to solve time-independent or time-dependent linear Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semiclassical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry.The graphical capabilities allow visualization of quantum dynamics 'on the fly', including Wigner phase space representations. Being easy to use and highly versatile, WavePacket is well suited for the teaching of quantum mechanics as well as for research projects in atomic, molecular and optical physics or in physical or theoretical chemistry.The present Part I deals with the description of closed quantum systems in terms of Schr\"odinger equations. The emphasis is on discrete variable representations for spatial discretization as well as various techniques for temporal discretization.The upcoming Part II will focus on open quantum systems and dimension reduction; it also describes the codes for optimal control of quantum dynamics.The present work introduces the MATLAB version of WavePacket 5.2.1 which is hosted at the Sourceforge platform, where extensive Wiki-documentation as well as worked-out demonstration examples can be found

Variations on PIGS: Non-standard approaches for imaginary-time path integrals

The second Rényi entropy has been used as a measure of entanglement in various model systems, including those on a lattice and in the continuum. The present work focuses on extending the existing ideas to measurement of entanglement in physically relevant systems, such as molecular clusters. We show that using the simple estimator with the regular Path Integral Ground State (PIGS) distribution is not effective, but a superior estimator exists so long as one has access to other configuration sectors. To this end, we implement the ability to explore different sectors in the Molecular Modelling Toolkit (MMTK) and use it to obtain the entanglement entropy for a test system of coupled harmonic oscillators. The Semiclassical Initial Value Representation (SC-IVR) method for real-time dynamics using the Herman-Kluk propagator is known to be an effective semiclassical method. In the present work, we combine this approximate real-time propagator with exact and approximate ground state wavefunctions in order to find ground state survival amplitudes. The necessary integrals are first performed numerically on a grid (which is feasible for only low-dimensional systems) and then stochastically using MMTK (which has applicability to high-dimensional systems). The stochastic approach is used to compare two estimators, and it is again demonstrated that better results are obtained in a specialized configuration sector

Technical Matter Wave Optics - Imaging devices for Bose condensed matter waves - an aberration analysis in space and time

Cold atomic gases are the ultimate quantum sensors. Embedded in a matter-wave interferometer, they provide a platform for high-precision sensing of accelerations and rotations probing fundamental physical questions. As in all optical instruments, these devices require careful modeling. Sources of possible aberrations need to be quantified and optimized to guarantee the best possible performance. This applies in particular to high-demanding experiments in microgravity with low repetition rates. In this thesis, we present a theoretical (3+1)d aberration analysis of expanded Bose-Einstein condensates. We demonstrate that the Bogoliubov modes of the scaled mean-field equation serve as good basis states to obtain the corresponding aberration coefficients. Introducing the Stringari polynomials, we describe density and phase variations in terms of a multipole decomposition analogous to the Zernike wavefront analysis in classical optics. We apply our aberration analysis to Bose-Einstein condensates on magnetic chip traps. We obtain the trapping potential using magnetic field simulations with finite wire elements. Using the multipole expansion, we characterize the anharmonic contributions of the Ioffe-Pritchard type Zeeman potential. Used as a matter-wave lens for delta-kick collimation, we determine the wavefront aberrations in terms of \say{Seidel-diagrams}. Supported by (3+1)d Gross-Pitaevskii simulations we study mean-field interactions during long expansion times. Matter-wave interferometry with Bose-Einstein condensates can also be performed in guiding potentials. One of the building blocks are toroidal condensates in a ring-shaped geometry. The required light field patterns are obtained by using the effect of conical refraction or with programmable digital micromirror devices. For the former, we study equilibrium properties and compare them with experimental data. We investigate the collective excitations in the two-dimensional ring-shaped condensate. Our result is compared to the numerical results of the Bogoliubov-de Gennes equations. The latter is used to find signatures in the excitation spectrum during the topological transition from simply connected harmonic to multiply connected ring traps. Changing the topology dynamically leads to radial excitations of the condensate. We propose a damping mechanism based on feedback measurements to control the motion within the toroidal ring

Aircraft noise prediction program theoretical manual: Rotorcraft System Noise Prediction System (ROTONET), part 4

This document describes the theoretical methods used in the rotorcraft noise prediction system (ROTONET), which is a part of the NASA Aircraft Noise Prediction Program (ANOPP). The ANOPP code consists of an executive, database manager, and prediction modules for jet engine, propeller, and rotor noise. The ROTONET subsystem contains modules for the prediction of rotor airloads and performance with momentum theory and prescribed wake aerodynamics, rotor tone noise with compact chordwise and full-surface solutions to the Ffowcs-Williams-Hawkings equations, semiempirical airfoil broadband noise, and turbulence ingestion broadband noise. Flight dynamics, atmosphere propagation, and noise metric calculations are covered in NASA TM-83199, Parts 1, 2, and 3

An arbitrary curvilinear coordinate particle in cell method

A new approach to the kinetic simulation of plasmas in complex geometries, based on the Particle-in-Cell (PIC) simulation method, is explored. In this method, called the Arbitrary Curvilinear Coordinate PIC (ACC-PIC) method, all essential PIC operations are carried out on a uniform, unitary square logical domain and mapped to a nonuniform, boundary fitted physical domain. We utilize an elliptic grid generation technique known as Winslow\u27s method to generate boundary-fitted physical domains. We have derived the logical grid macroparticle equations of motion based on a canonical transformation of Hamilton\u27s equations from the physical domain to the logical. These equations of motion are not seperable, and therefore unable to be integrated using the standard Leapfrog method. We have developed an extension of the semi-implicit Modified Leapfrog (ML) integration technique to preserve the symplectic nature of the logical grid particle mover. We constructed a proof to show that the ML integrator is symplectic for systems of arbitrary dimension. We have constructed a generalized, curvilinear coordinate formulation of Poisson\u27s equations to solve for the electrostatic fields on the uniform logical grid. By our formulation, we supply the plasma charge density on the logical grid as a source term. By the formulations of the logical grid particle mover and the field equations, the plasma particles are weighted to the uniform logical grid and the self-consistent mean fields obtained from the solution of the Poisson equation are interpolated to the particle position on the logical grid. This process coordinates the complexity associated with the weighting and interpolation processes on the nonuniform physical grid. In this work, we explore the feasibility of the ACC-PIC method as a first step towards building a production level, time-adaptive-grid, 3D electromagnetic ACC-PIC code. We begin by combining the individual components to construct a 1D, electrostatic ACC-PIC code on a stationary nonuniform grid. Several standard physics tests were used to validate the accuracy of our method in comparison with a standard uniform grid PIC code. We then extend the code to two spatial dimensions and repeat the physics tests on a rectangular domain with both orthogonal and nonorthogonal meshing in comparison with a standard 2D uniform grid PIC code. As a proof of principle, we then show the time evolution of an electrostatic plasma oscillation on an annular domain obtained using Winslow\u27s method