Numerical Investigation of Strongly Interacting Bosons at Zero Temperature

We review some numerical works carried out within the department for Quantum Optics and Statistics at the University of Freiburg’s Institute of Physics, between September 2016 and June 2018. Our activities focus on quantum properties of matter at zero temperature, i.e., a regime where the thermal energy kBT is negligible with respect to the other energy scales of the considered system. This area of research, related to ultracold gases, has attracted a great deal of interest, both experimentally and theoretically, since the first realization of a Bose-Einstein condensate in 1995. In a context where the theoretical understanding of these systems still remains challenging, the growing power of computers offers a unique and efficient way to tackle such challenges. In our theory group, we particularly use powerful numerical methods that give exact results, in contrast to other theoretical approaches based on an a priori assumption, e.g., mean field theory. To illustrate it, we focus on few typical results that would not be available other than by using high performance computing. These results have been obtained by using three numerical methods: quantum Monte Carlo (QMC), Gutzwiller Monte Carlo (GMC), and the Multiconfigurational Time-dependent Hartree method for bosons (MCTDHX)

Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double-Well

We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behaviour of the particles launched either at the single particle ground state or saddle point energy, in a time-independent potential. We complement these results by a characterisation of the cross-over from diabatic to quasi-adiabatic evolution under finite-time switching of the potential barrier, via the associated time-evolution of a single particle's von Neumann entropy. This is achieved with the help of the multiconfigurational time-dependent Hartree method for indistinguishable particles (\textsc{Mctdh-x}) -- which also allows us to extrapolate our results for increasing particle numbers.Comment: 20 pages, 14 figure

BOSONS DE SPIN-1/2 ET 1 SUR RÉSEAUX OPTIQUES EN UNE ET DEUX DIMENSIONS

Optical trapping of atoms on optical lattices allows the study of their behavior in the range of ultra low temperature, at the nanokelvin scale, without freezing their angular momentum degrees of freedom. Those recent trapping methods offer the possibility to analyse the quantum magnetism spontaneously adopted by the atoms. In this thesis we study numerically bosons with two internal effective degrees of freedom, referred to as spin-1/2 bosons, and also bosons with three degrees of freedom, spin-1 bosons, with two conceptually different methods: a simplified one using mean field approximation and an exact method, the Quantum Monte Carlo method. Beyond the study of these two systems, we compare a mean field method, sometimes excessively used, to the Quantum Monte Carlo method. The thorough investigation of the spin-1/2 bosons system in one and two dimensions in the zero temperature limit shows the influence of dimensionality on the physical properties of this system. The thermal effects, still present experimentally, are also analysed. Lastly, the spin-1 boson system trapped into a two dimensionnal lattice, a richer and more complicated system than the previous one, is investigated in the zero temperature limit. The study of these two systems reveals different magnetic organization in Mott insulators and superfluid phases, such as a ferromagnetic superfluid. First order phase transitions and coherent anticorrelated movements, present in the Mott phases, are also observed.Le piégeage optique d'atomes sur réseaux optiques permet d'étudier leur comportement dans un régime de très basse température, de l'ordre du nanokelvin, sans geler le degré de liberté de leur moment cinétique. Ces méthodes récentes de piégeage offrent la possibilité d'analyser le magnétisme des phases quantiques spontanément adopté par les atomes. Dans cette thèse, nous étudions numériquement des bosons à deux états effectifs de spin, ou bosons de spin-1/2, ainsi que des bosons à trois états de spin, ou bosons de spin-1, avec deux méthodes conceptuellement différentes: une approche simplifiée en champ moyen et une méthode exacte, la méthode de Monte Carlo Quantique. Au delà de l'étude de ces deux systèmes, nous comparons une méthode en champ moyen, parfois abusivement employée, à la méthode de Monte Carlo Quantique. L'étude approfondie du système de bosons de spin-1/2 en une et deux dimensions dans la limite de température nulle montre l'influence de la dimension sur la physique de ce système. Les effets thermiques, non négligeables expérimentalement, sont aussi étudiés. Enfin, le système de bosons de spin-1 piégés sur un réseau bidimensionnel, système plus riche et plus complexe à étudier que le précédent, est étudié dans la limite de température nulle. L'étude de ces deux systèmes révèle différentes organisations magnétiques dans les phases isolantes de Mott ainsi que dans la phase superfluide, telles qu'un superfluide ferromagnétique. Des transitions de phases du premier ordre et des mouvements cohérents anticorrélés, présents dans les phases de Mott, sont aussi observés

Magnetic phase transition in the ground-state phase diagram of binary bose gases in optical lattices

We investigate the ground-state phase diagram of interacting binary Bose gases trapped in two-dimensional optical lattices by means of quantum Monte Carlo simulations. Our simulations reveal a magnetic phase transition from a $x{\text{-}}y$ ferromagnetic-order to a spin insulator inside the Mott insulating phase with two particles per site for quasi-balanced on-site inter- and intra-particle interactions, i.e., $U_{\uparrow \downarrow} \lesssim U$ . This 3D-XY transition is characterized by the establishment of a finite local magnetic moment along the z-axis, ferromagnetic correlations in the $x{\text{-}}y$ plane and by counterflow superfluidity inside the Mott phase. When decreasing $U_{\uparrow \downarrow}/U$ , this transition merges with the Mott-superfluid transition and becomes first order. The merging of the two transitions is investigated with respect to the $U_{\uparrow \downarrow}/U$ parameter

Anderson Tower of States and Nematic Order of Spin-1 Bosonic Atoms on a 2D Lattice

We investigate the structure of the spectrum of antiferromagnetically coupled spin-1 bosons on a square lattice using degenerate perturbation theory and exact diagonalizations of finite clusters. We show that the superfluid phase develops an Anderson tower of states typical of nematic long-range order with broken SU(2) symmetry. We further show that this order persists into the Mott-insulating phase down to zero hopping for one boson per site and down to a critical hopping for two bosons per site, in agreement with mean-field and quantum Monte Carlo results. The connection with the transition between a fragmented condensate and a polar one in a single trap is briefly discussed