Dynamics of the Gross-Pitaevskii Equation and Shortcuts to Adiabaticity

Procedures which vary the parameters of a model in an adiabatic way have applications in many areas of quantum technology. However, explicitly employing adiabatic evolution often leads to decoherence issues due to systems interacting with their environment. For this reason, there has been much interest in developing shortcuts to adiabaticity in which the target final state is reached in a finite duration change of parameter. In this thesis, we design and study a shortcut to adiabaticity in an interacting Bose-Einstein condensate. In particular, we study the response induced by ramps in the interaction strength of such a system. We determine the power law decay exponents of the induced excitations as well as the characteristic frequency with which these excitations oscillate with respect to the duration and mean values of the ramps

Multi-Symplectic Integrators for Nonlinear Wave Equations

Symplectic (area-preserving) integrators for Hamiltonian ordinary differential equations have shown to be robust, efficient and accurate in long-term calculations. In this thesis, we show how symplectic integrators have a natural generalization to Hamiltonian PDEs by introducing the concept of multi-symplectic partial differential equations (PDEs). In particular, we show that multi-symplectic PDEs have an underlying spatio-temporal multi-symplectic structure characterized by a multi-symplectic conservation law MSCL). Then multi-symplectic integrators (MSIs) are numerical schemes that preserve exactly the MSCL. Remarkably, we demonstrate that, although not designed to do so, MSIs preserve very well other associated local conservation laws and global invariants, such as the energy and the momentum, for very long periods of time. We develop two types of MSIs, based on finite differences and Fourier spectral approximations, and illustrate their superior performance over traditional integrators by deriving new numerical schemes to the well known 1D nonlinear Schrödinger and sine-Gordon equations and the 2D Gross-Pitaevskii equation. In sensitive regimes, the spectral MSIs are not only more accurate but are better at capturing the spatial features of the solutions. In particular, for the sine-Gordon equation we show that its phase space, as measured by the nonlinear spectrum associated with it, is better preserved by spectral MSIs than by spectral non-symplectic Runge-Kutta integrators. Finally, to further understand the improved performance of MSIs, we develop a backward error analysis of the multi-symplectic centered-cell discretization for the nonlinear Schrödinger equation. We verify that the numerical solution satisfies to higher order a nearby modified multi-symplectic PDE and its modified multi-symplectic energy conservation law. This implies, that although the numerical solution is an approximation, it retains the key feature of the original PDE, namely its multi-symplectic structure

A two level approach for simulating Bose-Einstein condensates by localized orthogonal decomposition

In this work, we consider the numerical computation of ground states and dynamics of single-component Bose-Einstein condensates (BECs). The corresponding models are spatially discretized with a multiscale finite element approach known as Localized Orthogonal Decomposition (LOD). Despite the outstanding approximation properties of such a discretization in the context of BECs, taking full advantage of it without creating severe computational bottlenecks can be tricky. In this paper, we therefore present two fully-discrete numerical approaches that are formulated in such a way that they take special account of the structure of the LOD spaces. One approach is devoted to the computation of ground states and another one for the computation of dynamics. A central focus of this paper is also the discussion of implementation aspects that are very important for the practical realization of the methods. In particular, we discuss the use of suitable data structures that keep the memory costs economical. The paper concludes with various numerical experiments in 1d, 2d and 3d that investigate convergence rates and approximation properties of the methods and which demonstrate their performance and computational efficiency, also in comparison to spectral and standard finite element approaches

Exact nonlinear dynamics of Spinor BECs applied to nematic quenches

In this thesis we study the nonlinear dynamics of spin-1 and spin-2 Bose-Einstein condensates, with particular application to antiferromagnetic systems exhibiting nematic (beyond magnetic) order. Firstly, we give a derivation of the spinor energy functionals with a focus on the connections between the nonlinear terms. We derive a hierarchy of nonlinear irreducible multipole observables sensitive to different levels of nematic order, and explore the various nematic states in terms of their multipolar order, representations of their symmetries, and topological defects. We then develop an exact solution to the nonlinear dynamics of spinor Bose-Einstein condensates. We use this solution to construct efficient and accurate numerical algorithms to evolve the spinor Gross-Pitaevskii equation in time. We demonstrate the advantages of our algorithms with several 1D numerical test problems, comparing with existing methods in the literature. We apply our numerical methods to simulating quenches of the condensate between various antiferromagnetic phases for spin-1 and spin-2. For spin-1, we carry out quenches for a theoretical uniform system in 2D, and then specialize to the parameters used in a recent harmonically trapped experiment in 3D. We connect the long-time coarsening growth law of the relevant order parameter to the decay of half-quantum vortices, which are the relevant topological defects of the ground state. For the spin-2 system, we investigate a novel quench from two different quadrupolar-nematic phases to an octupolar-nematic “cyclic” phase which supports 1/3 fractional vortices. We develop appropriate order parameter observables which couple to the spin and superfluid currents generated by these defects, and show that a new growth law appears with exponent 1/3

Phase Transitions of Repulsive Two-Component Fermi Gases in Two Dimensions

We predict the phase separations of two-dimensional Fermi gases with repulsive contact-type interactions between two spin components. Using density-potential functional theory with systematic semiclassical approximations, we address the long-standing problem of itinerant ferromagnetism in realistic settings. We reveal a universal transition from the paramagnetic state at small repulsive interactions towards ferromagnetic density profiles at large interaction strengths, with intricate particle-number dependent phases in between. Building on quantum Monte Carlo results for uniform systems, we benchmark our simulations against Hartree-Fock calculations for a small number of trapped fermions. We thereby demonstrate that our employed corrections to the mean-field interaction energy and especially to the Thomas-Fermi kinetic energy functional are necessary for reliably predicting properties of trapped mesoscopic Fermi gases. The density patterns of the ground state survive at low finite temperatures and confirm the Stoner-type polarization behavior across a universal interaction parameter, albeit with substantial quantitative differences that originate in the trapping potential and the quantum-corrected kinetic energy. We also uncover a zoo of metastable configurations that are energetically comparable to the ground-state density profiles and are thus likely to be observed in experiments. We argue that our density-functional approach can be easily applied to interacting multi-component Fermi gases in general.Comment: 23 pages, 8 figure

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

Lattice methods for strongly interacting many-body systems

Lattice field theory methods, usually associated with non-perturbative studies of quantum chromodynamics, are becoming increasingly common in the calculation of ground-state and thermal properties of strongly interacting non-relativistic few- and many-body systems, blurring the interfaces between condensed matter, atomic and low-energy nuclear physics. While some of these techniques have been in use in the area of condensed matter physics for a long time, others, such as hybrid Monte Carlo and improved effective actions, have only recently found their way across areas. With this topical review, we aim to provide a modest overview and a status update on a few notable recent developments. For the sake of brevity we focus on zero-temperature, non-relativistic problems. After a short introduction, we lay out some general considerations and proceed to discuss sampling algorithms, observables, and systematic effects. We show selected results on ground- and excited-state properties of fermions in the limit of unitarity. The appendix contains details on group theory on the lattice.Comment: 64 pages, 32 figures; topical review for J. Phys. G; replaced with published versio