Accelerating the Fourier split operator method via graphics processing units

Current generations of graphics processing units have turned into highly parallel devices with general computing capabilities. Thus, graphics processing units may be utilized, for example, to solve time dependent partial differential equations by the Fourier split operator method. In this contribution, we demonstrate that graphics processing units are capable to calculate fast Fourier transforms much more efficiently than traditional central processing units. Thus, graphics processing units render efficient implementations of the Fourier split operator method possible. Performance gains of more than an order of magnitude as compared to implementations for traditional central processing units are reached in the solution of the time dependent Schr\"odinger equation and the time dependent Dirac equation

Quantum simulations of attosecond physics

We develop a numerical simulator to solve the time-dependent Schrödinger equation (TDSE) to simulate strong field ionization of a single electron under the influence of a one-dimensional separable potential. We use the split-step method to solve TDSE, obtain the ground state using the imaginary time method and benchmark it with the exact diagonalization of Hamiltonian. We perform the stability analysis in both real-time and imaginary-time propagation and obtain the fact that the usual model of separable orbital is not suitable in this method and demonstrate how a Gaussian model can solve this issue. Finally, we perform a comparative analysis between the analytical ground state and an adapted version of ground state adjusted according to the time-step of TDSE to justify the origin of the numerical error

A time-splitting pseudospectral method for the solution of the Gross-Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions

We present a method for numerically solving a Gross-Pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one- and two-component Bose-Einstein condensates. The method we develop maintains spectral accuracy by employing Fourier or spherical harmonics in the angular coordinates combined with generalised-Laguerre basis functions in the radial direction. Using an error analysis, we show that the method presented leads to more accurate results than one based on a sine transform in the radial direction when combined with a time-splitting method for integrating the equations forward in time. In contrast to a number of previous studies, no assumptions of radial or cylindrical symmetry is assumed allowing the method to be applied to 2D and 3D time-dependent simulations. This is accomplished by developing an efficient algorithm that accurately performs the generalised-Laguerre transforms of rotating Bose-Einstein condensates for different orders of the Laguerre polynomials. Using this spatial discretisation together with a second order Strang time-splitting method, we illustrate the scheme on a number of 2D and 3D computations of the ground state of a non-rotating and rotating condensate. Comparisons between previously derived theoretical results for these ground state solutions and our numerical computations show excellent agreement for these benchmark problems. The method is further applied to simulate a number of time-dependent problems including the Kelvin-Helmholtz instability in a two-component rotating condensate and the motion of quantised vortices in a 3D condensate

Electrically controlled focusing of de Broglie matter waves by Fresnel zone plate

The evolution from classical to quantum matter wave optics has been influenced by transformative optical devices. Fresnel zone plates (FZP), initially designed for light manipulation, have now found expanded applications in matter waves. In this study, focusing of helium atoms by an electrically biased FZP is investigated numerically. The $n$th opaque zone of the FZP is subject to electrostatic biasing using three ways: (i) $V_n=V_1$, where $V_1$ is the biasing voltage applied to the central zone, (ii) $V_n=V_1 \sqrt{n}$, and (iii) $V_n = V_1 \sin (k_E n)$, with $k_E$ being the radial modulation factor. The effect of biasing the FZP on the transmission coefficient ($T_c$), focal length ($f$), size of the focused wave packet ($\sigma_F$), transverse intensity profile, and focusing efficiency ($\eta$) is investigated. The study reveals that the electrical biasing of the FZP modulates the diffractive focusing of neutral atoms by altering the atom-surface interaction with induced polarization potential. It is observed that biasing with $V_n=V_1$ induces multi-focusing of the FZP, reducing wave packet transmission and focusing efficiency. Biasing with $V_n=V_1 \sqrt{n}$ significantly enhances the transmission coefficient by $23.7\%$, increases the focal length $f$ by $103\%$, and improves the focusing efficiency from $10\%$ to $20.17\%$, indicating enhanced focusing performance. Biasing with $V_n=V_1 \sin(k_E n)$ offers increased controllability in focusing matter waves through the parameters $k_E$ and $V_1$. In this case, a highly intense focused wave packet with a better efficiency of $20.3\%$ is observed compared to the other cases. The findings will be helpful in various emerging applications of atom optics, such as improving the performance of helium microscopes, enabling control in cold atom trapping on atom chips, and high-precision atom lithography for quantum electronic devices.Comment: 20 pages, 13 figure

The hierarchy of rogue wave solutions in nonlinear systems

Oceanic freak waves, optical spikes and extreme events in numerous contexts can arguably be modelled by modulationally unstable solutions within nonlinear systems. In particular, the fundamental nonlinear Schroedinger equation (NLSE) hosts a high-amplitude spatiotemporally localised solution on a plane-wave background, called the Peregrine breather, which is generally considered to be the base-case prototype of a rogue wave. Nonetheless, until very recently, little was known about what to expect when observing or engineering entire clusters of extreme events. Accordingly, this thesis aims to elucidate this matter by investigating complicated structures formed from collections of Peregrine breathers. Many novel NLSE solutions are discovered, all systematically classifiable by their geometry. The methodology employed here is based on the well-established concept of Darboux transformations, by which individual component solutions of an integrable system are nonlinearly superimposed to form a compound wavefunction. It is primarily implemented in a numerical manner within this study, operating on periodically modulating NLSE solutions called breathers. Rogue wave structures can only be extracted at the end of this process, when a limit of zero modulation frequency is applied to all components. Consequently, a requirement for breather asymmetry ensures that a multi-rogue wavefunction must be formed from a triangular number of individual Peregrine breathers (e.g. 1, 3, 6, 10, ...), whether fused or separated. Furthermore, the arrangements of these are restricted by a maximum phase-shift allowable along an evolution trajectory through the relevant wave field. Ultimately, all fundamental high-order rogue wave solutions can be constructed via polynomial relations between origin-translating component shifts and squared modulation frequency ratios. They are simultaneously categorisable by both these mathematical existence conditions and the corresponding visual symmetries, appearing spatiotemporally as triangular cascades, pentagrams, heptagrams, and so on. These parametric relations do not conflict with each other, meaning that any arbitrary NLSE rogue wave solution can be considered a hybridisation of this elementary set. Moreover, this hierarchy of structures is significantly general, with complicated arrangements persisting even on a cnoidal background

Estudio numérico de la generación y dinámicas de pulsos múltiples en un láser de fibra de amarre de modos pasivo

248 páginas. Doctorado en Ciencias e Ingeniería de Materiales.En el presente trabajo de investigación, se establece un estudio numérico de la generación de pulsos ultracortos en un láser de fibra de amarre de modos pasivo bajo un estricto control de polarización. En particular, se validan los principios de operación involucrados en cada una de las secciones que conforman a la cavidad a partir del modelo de propagación de la ecuación de Schrödinger no lineal (NLSE, por sus siglas en inglés nonlinear Schrödinger equation), el cual se centra en el problema de propagación de un pulso en fibra óptica bajo el formalismo de la óptica no lineal. En este caso, debido a que en la operación de la cavidad son fundamentales los efectos en polarización, se estableció un modelo de propagación descrito por NLSEs acopladas en la base de polarización circular. En particular, para garantizar que las soluciones numéricas obtenidas del modelo de la NLSE reproduzcan adecuadamente los principios de operación del láser de fibra de amarre de modos pasivo, se propusieron tres metodologías enfocadas en la evaluación del grado de convergencia y estabilidad de los métodos numéricos reportados para integrar numéricamente la NLSE, entre los cuales se evaluaron a diversos métodos de diferencias finitas y métodos de división de paso, a estos últimos también se les conoce como métodos pseudo-espectrales. Como resultado, se valida un adecuado nivel de convergencia y estabilidad en algunos de los métodos de división de paso evaluados, mientras que los métodos de diferencias finitas exhiben una pérdida de convergencia y estabilidad al no reproducir la respuesta lineal y no lineal de la fibra óptica. Adicionalmente, en un análisis comparativo de los métodos de división de paso, se exhiben las ventajas de la implementación de los métodos embebidos de Fourier de división de paso optimizado con un control del tamaño del paso iterativo, el cual ajusta el orden de exactitud demandado, disminuyendo o aumentando el tamaño del paso iterativo, en cada una de las etapas que conforman a un proceso en particular conducido por la contribución de efectos lineales y no lineales de la fibra óptica. Adicionalmente, a nivel de simulación numérica, se desarrolla una caracterización en polarización de un láser de fibra de amarre de modos pasivo con una cavidad de anillo constituida por 19 m de fibra convencional SMF-28 y 1 m de fibra dopada con erbio, esta última implica que la cavidad opere con un espectro centrado en 1550 nm. Como resultado, a partir de diferentes ajustes en polarización introducidos por un controlador de polarización en la cavidad, se establece un mapeo en polarización distinguiendo tres regímenes de pulsos: gas de solitones vectoriales, moléculas de solitones vectoriales y pulsos de ruido. En particular, cada uno de estos regímenes se estudian a partir de diferentes representaciones numéricas que son comparables con distintas técnicas de mediciones experimentales enfocadas en la caracterización de pulsos ultracortos. Así mismo, se validan las condiciones necesarias en la operación del láser para alcanzar un régimen de pulsos en particular, estableciendo una correlación entre el mecanismo de absorbedor saturable de la cavidad y la dinámica que describen los pulsos. Como consecuencia, la importancia de este trabajo de investigación reside en que, además de predecir y comprender el proceso de formación de pulsos, puede ser una guía en el desarrollo de técnicas de medición avanzadas enfocadas en la caracterización de pulsos ultracortos, así como en la correcta interpretación de los resultados experimentales.Consejo Nacional de Ciencia y Tecnología (CONACYT) por el apoyo otorgado mediante la beca de doctorado (CVU: 687612)