A basis-set based Fortran program to solve the Gross-Pitaevskii Equation for dilute Bose gases in harmonic and anharmonic traps

Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross-Pitaevskii equation (GPE). GPE is a nonlinear Schroedinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simple-harmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way.Comment: 3 figures (included), to appear in Computer Physics Communication

A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation

We present a new numerical system using classical finite elements with mesh adaptivity for computing stationary solutions of the Gross-Pitaevskii equation. The programs are written as a toolbox for FreeFem++ (www.freefem.org), a free finite-element software available for all existing operating systems. This offers the advantage to hide all technical issues related to the implementation of the finite element method, allowing to easily implement various numerical algorithms.Two robust and optimised numerical methods were implemented to minimize the Gross-Pitaevskii energy: a steepest descent method based on Sobolev gradients and a minimization algorithm based on the state-of-the-art optimization library Ipopt. For both methods, mesh adaptivity strategies are implemented to reduce the computational time and increase the local spatial accuracy when vortices are present. Different run cases are made available for 2D and 3D configurations of Bose-Einstein condensates in rotation. An optional graphical user interface is also provided, allowing to easily run predefined cases or with user-defined parameter files. We also provide several post-processing tools (like the identification of quantized vortices) that could help in extracting physical features from the simulations. The toolbox is extremely versatile and can be easily adapted to deal with different physical models

A minimisation approach for computing the ground state of Gross\u2013Pitaevskii systems

In this paper, we present a minimisation method for computing the ground stateof systems of coupled Gross\u2013Pitaevskii equations. Our approach relies on a spectral decomposition of the solution into Hermite basis functions. Inserting the spectral representation into the energy functional yields a constrained nonlinear minimisation problem for the coefficients. For its numerical solution, we employ a Newton-like method with an approximate line-search strategy. We analyse this method and prove global convergence. Appropriate starting values for the minimisation process are determined by a standard continuation strategy. Numerical examples with two and three-component two-dimensional condensates are included. These experiments demonstrate the reliability of our method and nicely illustrate the effect of phase segregation

Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity

International audienceThe aim of this chapter is first to give an introduction to the derivation of the Gross-Pitaevskii Equations (GPEs) that arise in the modeling of Bose-Einstein Condensates (BECs). In particular, we describe some physical problems related to stationary states, dynamics, multi-components BECs and the possibility of handling stochastic effects into the equation. Next, we explain how to compute the stationary (and ground) states of the GPEs through the imaginary time method (also called Conjugate Normalized Gradient Flow) and finite difference or pseudo-spectral dis-cretization techniques. Examples are provided by using GPELab which is a Mat-lab toolbox dedicated to the numerical solution of GPEs. Finally, we explain how to discretize correctly the time-dependent GPE so that the schemes are physically admissible. We again provide some examples by using GPELab. Furthermore, extensions of the discretization schemes to some classes of stochastic (in time) GPEs are described and analyzed

Development and Application of New Numerical Extensions to the Coupled Coherent States Family of Multidimensional Quantum Dynamics Methods

The coupled coherent states method has demonstrated itself as an accurate and efficient method of studying the quantum dynamics of various systems. In recent years, its applicability has been extended by incorporating a number of new numerical expansions and modifications to generate a closely related family of methods. In this thesis, two new augmentations are developed to further broaden the scope of problems that are able to be treated. The first of these is a 2-layer extension of coupled coherent states, capable of providing an increased mathematical description of a degree or degrees of freedom within a quantum mechanical system, as well as beneficial numerical and scalability properties. The newly developed method is tested on a model system-bath Hamiltonian consisting of a tunnelling mode governed by an asymmetric double well potential coupled to a harmonic bath. It is found to compare well to previous methods of studying the Hamiltonian, as well as a benchmark calculation on the system conducted in this thesis, and demonstrate the beneficial numerical and scalability properties expected. The second development is to extend coupled coherent states to treat systems of indistinguishable bosons in the second quantisation representation. The method is tested on the same Hamiltonian as the 2-layer coupled coherent states scheme, where the harmonic bath is second quantised as it is comprised of oscillators of the same frequency, so they may be thought of as indistinguishable. Exploiting this symmetry property is found to be extremely advantageous, with remarkable agreement to the benchmark calculation. The method is then tested on a model Hamiltonian consisting of 100 bosons in a shifted harmonic trap, with oscillations in the 1-body density calculated. The results are found to compare favourably with a multiconfigurational time-dependent Hartree for bosons calculation that is equivalent to the Gross-Pitaevskii equation, providing impetus for future studies on systems of Bose-Einstein condensates. The existing ab initio multiple cloning extension of coupled coherent states for nonadiabatic dynamics is also used to study the ultrafast photodissociation of 2-ethylpyrrole. The results are compared to experimental data, and a novel insight into the dissociation mechanism is obtained, with it shown to be composed of a two step process. Firstly, molecules that are able to dissociate immediately over the barrier along the N-H coordinate do so in < 50 fs, and this is followed by a second slower dissociation process from molecules that must sample the potential energy surface before finding a way around the barrier. This is not observed experimentally due to the temporal widths of the laser pulses obscuring the dynamics in the < 50 fs window

Dynamics of spinor fermions

Ultracold atomic gases have established themselves as quantum systems, which are clean and offer a high degree of control over crucial parameters. They are well isolated from their environment and thus offer the possibility to study coherent many-body dynamics. In this thesis, we address the dynamics of ultracold Fermions with large spin. Fermionic spinor gases differ from the typical situation in condensed matter physics, due to both the presence of the trap and the possibility of having fermions with large (>1/2) spin. Compared to the spin-1/2 case, large spin fermions must have one of two possible new properties. Either they obey an enhanced SU(N) symmetry, or they feature spin-changing collisions and a quadratic Zeeman shift. Here, we address the latter case. In the weakly interacting scenario, there are three different regimes. For very weak interactions, the system is in the collisionless regime and interactions can be taken into account on a mean-field level. For stronger interactions, collisions ensure local equilibrium and the system is described by hydrodynamic equations. For the intermediate regime however, there is no simple description. Moreover, the scattering cross-section for spin-changing and spin-conserving collisions can be different for large-spin fermions and we find a situation, where the system is hydrodynamic with respect to one process but not the other. In this thesis, a semi-classical Boltzmann equation with full spin coherence is developed, which allows to interpolate between the collisionless and hydrodynamic regime in the presence of the trap and for large spins. This approach goes beyond mean-field theory and treats the single-particle dynamics as an open system coupled to an environment given by all other particles. We find good agreement with experiments performed in the group of Klaus Sengstock at Hamburg University, using ultracold Potassium-40. We begin by investigating the effect of the harmonic trap on a collisionless system. We find a dynamical mechanism for spin-segregation, the mean-field driven creation of two domains of opposite magnetization in phase-space. The effect finds a transparent explanation when introducing the concept of dynamically induced long-range interactions, occurring when the fast phase-space rotation induced by a strong parabolic trap effectively smears out the contact interactions. Further results in this thesis have been achieved in collaboration with the experimental group in Hamburg. In the first project, we study the collective excitations of a trapped four-component Fermi gas. Long wavelength spin waves are excited by using a magnetic field gradient to wind up a spin spiral. During the subsequent dynamics, the spin components oscillate in the trap, while the total density remains constant. The dynamics can be understood quantitatively by disentangling it into dipolar, nematic and octupolar configurations. In a further experiment with spin-9/2 fermions, it was found that spin-changing interactions can lead to collective and coherent oscillations of the spin state of the whole Fermi sea with long lifetimes. It is found theoretically, that these giant oscillations are protected from spatial dephasing by dynamically induced long-range interactions. We identify the suppression of such oscillations in the high-density regime as the consequence of incoherent non-forward scattering. In the last project, we study collision processes in ultracold Potassium in greater detail. We find that they can be arranged in 3 categories: Spin-changing vs. spin-conserving collisions, processes depending on density vs. processes depending on density gradients and forward vs. lateral scattering. With this categorization, as well as the exact dependence of each process on scattering lengths and momenta, we can explain and simulate not only the coherent mean-field driven oscillations, but also relaxation effects that appear to be incoherent on the single-particle levelGases atómicos ultrafríos han establecido como sistemas cuánticos limpias que ofrecen un alto grado de control sobre parámetros cruciales. Están bien aisladas de su entorno y por eso ofrecen la posibilidad de estudiar la dinámica coherente de muchos cuerpos. En esta tesis, estudiamos la dinámica de fermiones ultrafríos con spin largo. Gases espinoriales fermiónicos difieren de la situación típica en la física de materia condensada por la presencia de la trampa y la posibilidad de tener un spin largo (> 1/2). En comparación con el caso de spin 1/2, fermiones de espín largo deben tener una de dos posibles propiedades nuevas. Obedecen a una simetría ampliada SU(N), o muestran colisiones spin-cambiante y un efecto Zeeman cuadrático. Aqui tratamos el segundo caso. En el escenario de interacciónes débiles, hay tres regímenes diferentes. Para interacciones muy débiles, el sistema está en el régimen sin colisiones e interacciones se puede describir en un nivel de campo medio. Para interacciones fuertes, las colisiones garantizan el equilibrio local y el sistema es descrito por ecuaciones hidrodinámicas. Para el régimen intermedio, no hay una descripción sencilla. Ademas, la sección transversa de dispersión para colisiones spin-cambiantes y de spin-conservación puede ser diferente para fermiones de espín largo. Encontramos una situación, donde el sistema es hidrodinámico con respecto a un proceso, pero no a la otra. En esta tesis desarrollamos una ecuación de Boltzmann semi-clásica, que permite interpolar el régimen intermedio, en presencia de la trampa y para espín largo. Este enfoque trata la dinámica de un cuerpo como un sistema abierto, acoplado a un entorno determinado por todas las atomos demás. Encontramos un buen acuerdo con experimentos realizados en el grupo de Klaus Sengstock en la Universidad de Hamburgo, hechos con potasio-40 ultrafrío. Comenzamos investigando el efecto de la trampa armónica en un sistema sin colisiones. Encontramos un mecanismo dinámico par la segregación de spin, la creación de dos dominios de magnetización opuesta en el espacio fásico, impulsada por el campo medio. Encontramos una explicación transparente de este efecto con la introducción del concepto de interacciones de largo alcance inducidos dinámicamente, que se forma cuando una fuerte trampa parabólica desenfoque eficazmente las interacciones de contacto. Otros resultados de esta tesis han sido realizados en colaboración con el grupo experimental en Hamburgo. En el primer proyecto, estudiamos las excitaciones colectivas de un gas de Fermi atrapada, con cuatro componentes de spin. Ondas de spin con larga longitud de onda se excitan mediante un gradiente de campo magnético. Durante la dinámica siguiente, los componentes de spin oscilan en la trampa, mientras que la densidad total permanece constante. Podemos entender esta dinámica cuantitativamente desligandola en configuraciones dipolares, nemáticos y octupolares de espín. En un experimento siguiente con fermiones de spin 9/2, se encontró que las interacciones spin-cambiando pueden activar oscilaciones colectivas y coherentes del estado de spin de todo el mar de Fermi con duración larga. Descubrimos teóricamente, que estas oscilaciones gigantes están protegidos de desfase espacial por las interacciones de largo alcance inducidos dinámicamente. Identificamos la supresión de tales oscilaciones en el régimen de alta densidad como la consecuencia de la dispersión incoherente lateral. En el último proyecto, estudiamos los procesos de colisión en potasio ultrafrío en mas detalle. Podemos organizarlos en tres categorías: Colisiones spin-cambiante vs. spin-conservación, procesos dependiente de la densidad vs. gradientes de densidad y colisiones hacia adelante vs. laterales. Con esta clasificación y la dependencia en la longitud de dispersión y momentos, podemos explicar y simular no sólo las oscilaciones coherentes impulsados por el campo medio, sino también efectos de relajació

References, Appendices & All Parts Merged

Includes: Appendix MA: Selected Mathematical Formulas; Appendix CA: Selected Physical Constants; References; EGP merged file (all parts, appendices, and references)https://commons.library.stonybrook.edu/egp/1007/thumbnail.jp