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Physical Review A

Spatially localized structures and oscillons in atomic Bose-Einstein condensates confined in optical lattices

M. V. Charukhchyan, Evgeny S. Sedov, Sergei M Arakelyan, A. P. Alodjants

Published Jun 30, 2014• 16 citations
Publisher

Abstract

We consider the problem of formation of small-amplitude spatially localized oscillatory structures for atomic Bose-Einstein condensates confined in two- and three-dimensional optical lattices, respectively. Our approach is based on applying the regions with different signs of atomic effective masses where an atomic system exhibits effective hyperbolic dispersion within the first Brillouin zone. By using the $\mathbf{kp}$ method we have demonstrated mapping of the initial Gross-Pitaevskii equation on nonlinear Klein-Gordon and/or Ginzburg-Landau-Higgs equations, which is inherent in matter fields within ${\ensuremath{\phi}}^{4}$-field theories. Formation of breatherlike oscillating localized states---atomic oscillons---as well as kink-shaped states have been predicted in this case. Apart from classical field theories atomic field oscillons occurring in finite lattice structures possess a critical number of particles for their formation. The obtained results pave the way to simulating some analogues of fundamental cosmological processes occurring during our Universe's evolution and to modeling nonlinear hyperbolic metamaterials with condensed matter (atomic) systems.

DOI
10.1103/physreva.89.063624

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Abstract. We show that the phenomenon of modulational instability in arrays of Bose-Einstein condensates confined to optical lattices gives rise to coherent spatial structures of localized excitations. These excitations represent thin disks in 1D, narrow tubes in 2D, and small hollows in 3D arrays, filled in with condensed atoms of much greater density compared to surrounding array sites. Aspects of the developed pattern depend on the initial distribution function of the condensate over the optical lattice, corresponding to particular points of the Brillouin zone. The long-time behavior of the spatial structures emerging due to modulational instability is characterized by the periodic recurrence to the initial low-density state in a finite optical lattice. We propose a simple way to retain the localized spatial structures with high atomic concentration, which may be of interest for applications. Theoretical model, based on the multiple scale expansion, describes the basic features of the phenomenon. Results of numerical simulations confirm the analytical predictions.
Long-lived, spatially localized, and temporally oscillating nonlinear excitations are predicted by numerical simulation of coupled Gross-Pitaevskii equations. These oscillons closely resemble the time-periodic breather solutions of the sine-Gordon equation but decay slowly by radiating Bogoliubov phonons. Their time-dependent profile is closely matched with solutions of the sine-Gordon equation, which emerges as an effective field theory for the relative phase of two linearly coupled Bose fields in the weak-coupling limit. For strong coupling the long-lived oscillons persist and involve both relative and total phase fields. The oscillons decay via Bogoliubov phonon radiation that is increasingly suppressed for decreasing oscillon amplitude. Possibilities for creating oscillons are addressed in atomic gas experiments by collision of oppositely charged Bose-Josephson vortices and direct phase imprinting.
We show that the phenomenon of modulational instability in arrays of Bose–Einstein condensates confined to optical lattices gives rise to coherent spatial structures of localized excitations. These excitations represent thin discs in 1D, narrow tubes in 2D, and small hollows in 3D arrays, filled in with condensed atoms of much greater density compared to surrounding array sites. Aspects of the developed pattern depend on the initial distribution function of the condensate over the optical lattice, corresponding to particular points of the Brillouin zone. The long-time behaviour of the spatial structures emerging due to modulational instability is characterized by the periodic recurrence to the initial low-density state in a finite optical lattice. We propose a simple way to retain the localized spatial structures with high atomic concentration, which may be of interest for applications. A theoretical model, based on the multiple-scale expansion, describes the basic features of the phenomenon. Results of numerical simulations confirm the analytical predictions.
Long-lived, spatially localized, and temporally oscillating nonlinear excitations are predicted by numerical simulation of coupled Gross-Pitaevskii equations. These oscillons closely resemble the time-periodic breather solutions of the sine-Gordon equation but decay slowly by radiating Bogoliubov phonons. Their time-dependent profile is closely matched with solutions of the sine-Gordon equation, which emerges as an effective field theory for the relative phase of two linearly coupled Bose fields in the weak-coupling limit. For strong coupling the long-lived oscillons persist and involve both relative and total phase fields. The oscillons decay via Bogoliubov phonon radiation that is increasingly suppressed for decreasing oscillon amplitude. Possibilities for creating oscillons are addressed in atomic gas experiments by collision of oppositely charged Bose-Josephson vortices and direct phase imprinting.
We investigate the spatial profiles of periodic localized modes in attractive Bose-Einstein condensates, by solving the mean-field Gross-Pitaevskii equation in the presence of elliptic-type periodic potential. By considering a one-dimensional time independent linearized Gross-Pitaevskii equation, we obtained three bound state solutions and energies emanating from the first order Lamé equation. When the nonlinearity induced by the two body inter-atomic interactions are fully activated, spatially localized trivial phase periodic solutions of the attractive condensates are analytically obtained using the ansatz technique coupled with the direct integral method. Results of numerical simulation depicts trivial phase solutions, which are uniform train of spatially localized modes that are insensitive to variation of the elliptic modulus. However in the non-trivial phase regime, the spatially localized trains of soliton become very structurally unstable. This work underscores the spontaneous generation of periodic potential by the condensate wave function, determine the band structure of the lattice and basic properties of periodic matter waves under linear conditions, and highlight various spatial nonlinear periodic modes in the condensate. Finally, our investigation provides a solid theoretical framework that finds potential application in the fabrication of atomic lasers, periodic matter-wave gratings and quantum logic gates.