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arXiv:1509.08328 · Communications in Contemporary Mathematics

Interface layer of a two-component Bose–Einstein condensate

Amandine Aftalion, Christos Sourdis

Published Jun 14, 2016• 21 citations
arXivHTMLPDF

Abstract

This paper deals with the study of the behavior of the wave functions of a two-component Bose–Einstein condensate near the interface, in the case of strong segregation. This yields a system of two coupled ordinary differential equations for which we want to have estimates on the asymptotic behavior, as the strength of the coupling tends to infinity. As in phase separation models, the leading order profile is a hyperbolic tangent. We construct an approximate solution and use the properties of the associated linearized operator to perturb it into a genuine solution for which we have an asymptotic expansion. We prove that the constructed heteroclinic solutions are linearly nondegenerate, in the natural sense, and that there is a spectral gap, independent of the large interaction parameter, between the zero eigenvalue (due to translations) at the bottom of the spectrum and the rest of the spectrum. Moreover, we prove a uniqueness result which implies that, in fact, the constructed heteroclinic is the unique minimizer (modulo translations) of the associated energy, for which we provide an expansion.

DOI
10.1142/s0219199716500528

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This paper deals with the study of the behavior of the wave functions of a two-component Bose–Einstein condensate near the interface, in the case of strong segregation. This yields a system of two coupled ordinary differential equations for which we want to have estimates on the asymptotic behavior, as the strength of the coupling tends to infinity. As in phase separation models, the leading order profile is a hyperbolic tangent. We construct an approximate solution and use the properties of the associated linearized operator to perturb it into a genuine solution for which we have an asymptotic expansion. We prove that the constructed heteroclinic solutions are linearly nondegenerate, in the natural sense, and that there is a spectral gap, independent of the large interaction parameter, between the zero eigenvalue (due to translations) at the bottom of the spectrum and the rest of the spectrum. Moreover, we prove a uniqueness result which implies that, in fact, the constructed heteroclinic is the unique minimizer (modulo translations) of the associated energy, for which we provide an expansion.
Recently, coupled systems of nonlinear Schrödinger equations have been used extensively to describe mixtures Bose–Einstein condensates. In this paper, we study the distribution of two different hyperfine spin states of a binary mixture of three dimensional Bose–Einstein condensates. In a double condensate, an interface may occur due to large intraspecies and interspecies scattering lengths. We prove that there is an asymptotic separation of different phases in the strong coupling (Thomas–Fermi) limit.
In this Thesis we study steady state solutions and dynamical evolutions of two– component atomic Bose–Einstein Condensates. We initially investigate the equilibrium properties of condensate mixtures in harmonic trapping potentials at zero temperature. Subsequently we simulate the coupled growth of these condensates by inclusion of damping terms. Finally, we investigate the evolution of coupled Bose gases via the so-called classical–field method. A recent experiment [D. J. McCarron et al., Phys. Rev. A, 84, 011603(R) (2011)] achieved Bose–Einstein Condensation of a two–species 87Rb–133Cs phase segregated mixture in harmonic trapping potentials. Depending on relative atom numbers of the two species, three distinct regimes of density distributions were observed. For these experimental parameters, we investigate the corresponding time–independent ground state solutions through numerical simulations of the coupled Gross–Pitaevskii equations. By including experimentally relevant shifts between the traps, we observe a range of structures including ‘ball and shell’ formations and axially/radially separated states. These are found to be very sensitive to the trap shifts. For all three experimental regimes, our numerical simulations reveal good qualitative agreement. The observed experimental profiles cannot be guaranteed to be fully equilibrated. This, coupled with the rapid sympathetic cooling of the experimental system, leads to a situation where growth may play a determining factor in the density structures formed. To investigate this further, we introduce phenomenological damping to describe the associated condensate growth/decay, revealing a range of transient structures. However, such a model always predicts the predominance of one condensate species over longer evolution times. Work undertaken by collaborators with the more elaborate Stochastic Projected Gross–Pitaevskii equations, which can describe condensate formation by coupling to a heat bath, predicts the spontaneous formation of dark–bright solitons. Motivated by this, we show how the presence of solitons can affect the condensate distribution, thus highlighting the overall dynamical role in the emerging patterns. Finally, we use classical field methods to analyse the evolution of non trapped Bose gases from strongly nonequilibrium initial distributions. The contrast between miscible (overlapping) and immiscible (phase segregated) components gives rise to important distinctions for condensate fractions and the formation of domains and vortices. In addition, splitting the particles of a single component thermalised state into two components is investigated. We then study the effects of suddenly quenching the strength of the interspecies interactions. Under suitable conditions, this quench generates isotropic vortex tangles. While this tangle subsequently decays over time, we propose how a repeat sequence of quenches at regular intervals could be employed to drive the tangle, thereby potentially providing a novel route to the generation of quantum turbulence.
We investigate the expansion dynamics of a Bose–Einstein condensate that consists of two components and is initially confined in a quasi-one-dimensional trap. We classify the possible initial states of the two-component condensate by taking into account the nonuniformity of the distributions of its components and construct the corresponding phase diagram in the plane of nonlinear interaction constants. The differential equations that describe the condensate evolution are derived by assuming that the condensate density and velocity depend on the spatial coordinate quadratically and linearly, respectively, which reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas–Fermi approximation. We have obtained self-similar solutions of these differential equations for several important special cases and write out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms may be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves is considered. We compare the numerical solutions of the Gross–Pitaevskii equations with their approximate analytical solutions and numerically study the situations where the analytical method being used admits no exact solutions.
In this thesis, we consider a Gross–Pitaevskii (GP) energy functional as a model for rotating one component and two components Bose–Einstein condensates (BEC) in two dimensions. This model can be non-dimensionalized to highlight a strong confinement regime with strong interaction between the two components. We introduce a new discretization of this energy, featuring both finite difference and fast Fourier transform approaches, in a bounded domain of R 2 using Dirichlet boundary conditions. We develop an explicit gradient method algorithm with adaptive step and projection (EPG) over the constraints manifold for the minimization of the discrete energy. This method allows for the derivation of a stopping criterion. Moreover, we propose two post processing algorithms for the numerical minimizers. One is aimed for single vortices while the other is aimed for vortex sheets. Both algorithms detect these structures and compute their indices. In a recent article [6], the authors study the behaviour of a segregated two components BEC set into rotation. They are able to prove that for large rotation, the interface between the components gets long, conjecturing the possibility towards vortex sheets. They also study the vortex structures of BEC in a segregated regime. In this thesis, we produce numerical simulations using EPG, validating these recent theoretical results, supporting conjectures, and covering different physical cases (the cases of one component and two components in coexistence regime [45, 39]). We also illustrate the efficiency of EPG compared to the well-known GPELab method [11] which requires to solve a linearly implicit system at each step. Finally, we adapt a few theorems found in the literature to our discrete setting. We prove the existence of a global minimizer of the GP energy functional for the finite difference scheme and study some of its properties. We also work on symmetrical problems we encountered in some of the numerical simulations.
Nambu-Goldstone modes in immiscible two-component Bose-Einstein condensates are studied theoretically. In a uniform system, a flat domain wall is stabilized and then the translational invariance normal to the wall is spontaneously broken in addition to the breaking of two U(1) symmetries in the presence of two complex order parameters. We clarify the properties of the low-energy excitations and identify that there exist two Nambu-Goldstone modes: an in-phase phonon with a linear dispersion and a ripplon with a fractional dispersion. The signature of the characteristic dispersion can be verified in segregated condensates in a harmonic potential.