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Physical Review A

Pattern formations in miscellaneous mixtures of Bose-Einstein condensates and the higher-dimensional time-gated Manakov system

Victor K. Kuetche, Thomas Bouetou Bouetou, Timoléon Crépin Kofané, A. B. Moubissi, Kuppuswamy Porsezian

Published Nov 18, 2010• 18 citations
Publisher

Abstract

In this article, we investigate the structure and dynamics of miscellaneous mixtures of Bose-Einstein condensates confined within a time-independent anisotropic parabolic trap potential. In the zero-temperature mean-field approximation leading to coupled Gross-Pitaevskii equations for the macroscopic wave functions of the condensates, we show that these equations can be mapped onto the higher-dimensional time-gated Manakov system up to a first-order of accuracy. Paying particular attention to two-species mixtures and looking forward deriving a panel of miscellaneous excitations to the above equations, we analyze the singularity structure of the system by means of Weiss et al.'s [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 24, 522 (1983); 25, 13 (1984).] methodology and provide its general Lax representation. As a result, we unearth a typical spectrum of localized and periodic coherent patterns while depicting elastic and nonelastic interactions among such structures alongside the splitting and resonance phenomena occurring during their motion.

DOI
10.1103/physreva.82.053619

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In this article, we investigate the structure and dynamics of miscellaneous mixtures of Bose-Einstein condensates confined within a time-independent anisotropic parabolic trap potential. In the zero-temperature mean-field approximation leading to coupled Gross-Pitaevskii equations for the macroscopic wave functions of the condensates, we show that these equations can be mapped onto the higher-dimensional time-gated Manakov system up to a first-order of accuracy. Paying particular attention to two-species mixtures and looking forward deriving a panel of miscellaneous excitations to the above equations, we analyze the singularity structure of the system by means of Weiss et al.'s [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 24, 522 (1983); 25, 13 (1984).] methodology and provide its general Lax representation. As a result, we unearth a typical spectrum of localized and periodic coherent patterns while depicting elastic and nonelastic interactions among such structures alongside the splitting and resonance phenomena occurring during their motion.
We assume the macroscopic wave function of a Bose-Einstein condensate as a superposition of Gaussian wave packets, with time-dependent complex width parameters, insert it into the mean-field energy functional corresponding to the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational principle. In this way the GPE is mapped onto a system of coupled equations of motion for the complex width parameters, which can be analyzed using the methods of nonlinear dynamics. We perform a stability analysis of the fixed points of the nonlinear system, and demonstrate that the eigenvalues of the Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation spectrum of a condensate in an axisymmetric trap.
Quantum phenomena appear in a macroscopic scale in Bose-Einstein condensates. The Gross-Pitaevskii (GP) equation describes the dynamics of the weakly interacting Bose-Einstein condensates. The GP equation has a form of the Schroedinger equation with self-interaction. The coupled Gross-Pitaevskii equations are used to describe some mixtures of Bose-Einstein condensates. In this paper, we will show some numerical results of coupled Gross-Pitaevskii equations without self-interaction, which has a form of nonlinearly-coupled Schroedinger equations. We demonstrate that the macroscopic oscillation and the interference of two quantum wave packets decay in time owing to mutual interaction, which is analogous to the decoherence in quantum mechanics of many particles.
We consider a quasi-one-dimensional Bose-Einstein condensate with contact and long-range dipolar interactions, under the action of the time-periodic modulation applied to the harmonic-oscillator and optical-lattice trapping potentials. The modulation results in generation of a variety of harmonics in oscillations of the condensate's width and centre-of-mass coordinate. These include multiple and combinational harmonics, represented by sharp peaks in the system's spectra. Approximate analytical results are produced by the variational method, which are verified by systematic simulations of the underlying Gross-Pitaevskii equation. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.
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 study the interaction of dark–bright solitons in two-component Bose–Einstein condensates by suitably tailoring the trap potential, atomic scattering length and atom gain or loss. We show that the coupled Gross–Pitaevskii equation can be mapped onto the Manakov model. An interesting class of matter wave solitons and their interaction are identified with time-independent and periodically modulated trap potentials, which can be experimentally realized in two-component condensates. These include periodic collapse and revival of solitons, and snake-like matter wave solitons as well as different kinds of two-soliton interactions.
We study the motion of a vortex dipole in a Bose--Einstein condensate confined to an anisotropic trap. We focus on a system of ODEs describing the vortices' motion, which is in turn a reduced model of the Gross--Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. We uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals in the overall direction of rotation of the dipole. Near the separatrix orbit in the isotropic system, we find other families of periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the guiding center orbits, we derive an explicit iterated map that simplifies the problem further. Numerical calculations are used to illustrate the phenomena discovered through the analysis. Using the results from the reduced system, we are able to construct complex periodic orbits in the original, PDE, mean-field model for Bose--Einstein condensates, which corroborates the phenomenology observed in the reduced dynamical equations.