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Exploring the Effects of Quantum Fluctuations in the Waveguide Expansion of a Bose-Einstein Condensate by means of Numerical Simulations

Maia Aguirre Pascual de Zulueta

Published Dec 15, 2020• 0 citations
OpenAlex

Abstract

[EN] In this work, we have studied a dilute and ultracold bosonic gas of weakly interacting atoms, \ni.e. a Bose-Einstein Condensate (BEC) by means of the Modi ed Gross-Pitaevskii equation \n(MGPE) taking into account beyond-mean- eld corrections due to quantum \nuctuations. We \nhave considered the case where the cloud of atoms is strongly con ned transversally (within a \ncigar-shaped trap) and therefore, the three-dimensional (3D) MGPE is reduced to an e ective \none-dimensional (1D) form by averaging over the transverse coordinates. Based on this approach, \nwe have performed numerical simulations for analyzing the behavior of the BEC under the \nin \nuence of quantum \nuctuations, which give rise to an additional nonlinear quartic term. After \nreviewing the mathematical theory of Bose-Einstein condensation based on the GPE and the \nnumerical methods used to integrate it we discuss the computation of the ground state and the \nsimulation of its dynamic evolution in a waveguide (free expansion).

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[EN] In this work, we have studied a dilute and ultracold bosonic gas of weakly interacting atoms, \ni.e. a Bose-Einstein Condensate (BEC) by means of the Modi ed Gross-Pitaevskii equation \n(MGPE) taking into account beyond-mean- eld corrections due to quantum \nuctuations. We \nhave considered the case where the cloud of atoms is strongly con ned transversally (within a \ncigar-shaped trap) and therefore, the three-dimensional (3D) MGPE is reduced to an e ective \none-dimensional (1D) form by averaging over the transverse coordinates. Based on this approach, \nwe have performed numerical simulations for analyzing the behavior of the BEC under the \nin \nuence of quantum \nuctuations, which give rise to an additional nonlinear quartic term. After \nreviewing the mathematical theory of Bose-Einstein condensation based on the GPE and the \nnumerical methods used to integrate it we discuss the computation of the ground state and the \nsimulation of its dynamic evolution in a waveguide (free expansion).
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
In this paper, we mainly review recent results on mathematical theory andnumerical methods for Bose-Einstein condensation (BEC),based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are thencarried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models forspin-1 BEC, Bogoliubov excitation and BEC at finite temperature.
We explore atom-laser-like transport processes of ultracold Bose-condensed atomic vapors in mesoscopic waveguide structures beyond the Gross-Pitaevskii mean-field theory. Based on a microscopic description of the transport process in the presence of a coherent source that models the outcoupling from a reservoir of perfectly Bose-Einstein condensed atoms, we derive a system of coupled quantum evolution equations that describe the dynamics of a dilute condensed Bose gas in the framework of the Hartree-Fock--Bogoliubov approximation. We apply this method to study the transport of dilute Bose gases through an atomic quantum dot and through waveguides with disorder. Our numerical simulations reveal that the onset of an explicitly time-dependent flow corresponds to the appearance of strong depletion of the condensate on the microscopic level and leads to a loss of global phase coherence.
In the mean-field approximation, the number density $\ensuremath{\rho}(\mathbf{r})$ for the ground state of a Bose-Einstein condensate trapped by an external potential V(r)V(\mathbf{r})V(r) satisfies a classical field equation called the Gross-Pitaevskii equation. We show that quantum corrections to $\ensuremath{\rho}$ are dominated by quantum fluctuations with wavelengths of order $1/\sqrt{\ensuremath{\rho}a}$, where aaa is the SSS-wave scattering length. By expanding the equations for the Hartree-Fock approximation to second order in the gradient expansion, we derive local correction terms to the Gross-Pitaevskii equation for $\ensuremath{\rho}$ that take into account the dominant effects of quantum fluctuations. We also show that the gradient expansions for the density and for the condensate break down at fourth order and at second order, respectively.
A variational technique is applied to solve the time-dependent nonlinear Schr\"odinger equation (Gross-Pitaevskii equation) with the goal to model the dynamics of dilute ultracold atom clouds in the Bose-Einstein condensed phase. We derive analytical predictions for the collapse, equilibrium widths, and evolution laws of the condensate parameters and find them to be in very good agreement with our numerical simulations of the nonlinear Schr\"odinger equation. It is found that not only the number of particles, but also both the initial width of the condensate and the effect of different perturbations to the condensate may play a crucial role in the collapse dynamics. The results are applicable when the shape of the condensate is sufficiently simple.
We model the three-dimensional (3D) trapped Bose-Einstein condensate with the Gross-Pitaevskii equation (GPE) incorporating higher-order nonlinear interaction effects. Based on the F-expansion method and self-similar approach, we analytically derive the exact soliton solutions of the 3D GPE, identifying the oscillatory mode under certain experimental settings. The oscillatory kinetic parameter values we obtained coincide very well with those reported in previous work, which is based on the variational approach, showing the applicability of the theoretical treatment presented in this work.