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Nonlinear Guided Waves and Their Applications

Subdiffractive Solitons in Bose-Einstein Condensates

Kęstutis Staliūnas, R. Herrero, Germán J. de Valcárcel

Published Jan 1, 2005• 1 citation
Publisher

Abstract

We predict the disappearance of diffraction (equivalently the increase of the effective mass) of Bose-Einstein condensates in counter-moving periodic potentials. The kinetic energy term of the corresponding Gross-Pitaevskii equation transforms into one proportional to the fourth power of the momentum. We demonstrate subdiffractive stable solitons (stable droplets of the condensate) in the vicinity of this zero diffraction point.

DOI
10.1364/nlgw.2005.wd29

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We predict the disappearance of diffraction (equivalently the increase of the effective mass) of Bose-Einstein condensates in counter-moving periodic potentials. The kinetic energy term of the corresponding Gross-Pitaevskii equation transforms into one proportional to the fourth power of the momentum. We demonstrate subdiffractive stable solitons (stable droplets of the condensate) in the vicinity of this zero diffraction point.
A type of matter wave diffraction management is presented that leads to subdiffractive solitonlike structures. The proposed management technique uses two counter-moving, identical periodic potentials (e.g., optical lattices). For suitable lattice parameters, a different type of atomic bandgap structure appears in which the effective atomic mass becomes infinite at the lowest edge of an energy band. This way, normal matter-wave diffraction (proportional to the square of the atomic momentum) is replaced by fourth-order diffraction, and hence the evolution of the system becomes subdiffractive.
The existence and stability of solitons in Bose-Einstein condensates with attractive interatomic interactions, described by the Gross-Pitaevskii equation with a three-dimensional (3D) periodic potential, are investigated in a systematic form. We find a one-parameter family of stable 3D solitons in a certain interval of values of their norm, provided that the strength of the potential exceeds a threshold value. The minimum number of 7Li^{7}\mathrm{Li}7Li atoms in the stable solitons is 60, and the energy of the soliton at the stability threshold is $\ensuremath{\approx}6$ recoil energies in the lattice. The respective energy versus norm diagram features two cuspidal points, resulting in a typical swallowtail pattern, which is a generic feature of 3D solitons supported by quasi-two-dimensional or fully dimensional lattice potentials.
We investigate in detail the effects of nonlinearity on optical diffraction of Bose-Einstein condensates (BECs). By directly integrating the optically coupled two-component Gross-Pitaevskii equation in real space-time, comprehensive analyses of BEC optical diffraction phenomena are done under various conditions of light-pulse irradiation, total number of BEC atoms, etc., without using the adiabatic elimination approximation for an atomic excited state. Calculation results for the optical diffraction of 87Rb{}^{87}\text{Rb}87Rb BECs revealed that (1) the effect of nonlinearity on the atomic states causes the ``nonkinetic'' nonlinear effect in the Raman-Nath regime of diffraction, while the dynamics of BEC atoms due to the nonlinearity-induced repulsive forces works dominantly to produce the ``kinetic'' nonlinear effect in the Bragg regime of diffraction; (2) nonlinearity reduces the amplitude and frequency of the two-photon Rabi oscillation between BEC stationary and moving states, suggesting limitations in implementing the BEC Mach-Zehnder interferometer; and (3) the observed nonlinear effects are free from kinetic effects of the atomic excited state and not responsible for the optical transition process.
The dissipative dynamic stability is investigated of dark solitons in elongated Bose–Einstein condensates that can be described by the Gross–Pitaevskii equation including an additional term. Based on the direct perturbation theory for the nonlinear Schrodinger equation, the dependence of the soliton velocity on time is explicitly given, and the shape of dark solitons remaining unchanged under the dissipative condition is confirmed theoretically for the first time. It is found that the dynamically stable dark solitons turn out to be thermodynamically unstable.
We study the motion of bright matter-wave solitons in nonlinear potentials, produced by periodic or random spatial variations of the atomic scattering length. We obtain analytical results for the soliton motion, the radiation of matter wave, and the radiative soliton decay in such configurations of the Bose-Einstein condensate. The stable regimes of propagation are analyzed. The results are in remarkable agreement with the numerical simulations of the Gross-Pitaevskii equation with periodic or random spatial variations of the mean field interactions.
Using the variational approximation and direct simulations in real and imaginary time, we find stable two-dimensional (2D) and 3D solitons in the self-attractive Gross-Pitaevskii equation (GPE) with a potential which is uniform in one direction (z)(z)(z) and periodic in the others (however, the quasi-1D potentials cannot stabilize 3D solitons). The family of solitons includes single- and multiple-peaked ones. The results apply to Bose-Einstein condensates (BEC's) in optical lattices (OL's) and to spatial or spatiotemporal solitons in layered optical media. This is the first prediction of mobile 2D and 3D solitons in BEC's, as they keep mobility along zzz. Head-on collisions of in-phase solitons lead to their fusion into a collapsing pulse. Slow collisions between two multiple-peaked solitons whose main peaks are separated by an intermediate channel end up with their fusion into one single-peaked soliton in the middle channel, $\ensuremath{\simeq}1∕3$ of the original number of atoms being shed off. Stable localized states in the self-repulsive GPE with the low-dimensional OL combined with a parabolic trap are found too. Two such pulses in one channel perform recurrent elastic collisions, periodically featuring sharp interference patterns in the strong-overlap state.