arXiv:cond-mat/0702154 · Physical Review A

Nearly-one-dimensional self-attractive Bose-Einstein condensates in optical lattices

Luca Salasnich, Alberto Cetoli, Boris A. Malomed, Flavio Toigo

Published Mar 30, 2007• 30 citations

Abstract

Within the framework of a mean-field description, we investigate atomic Bose-Einstein condensates, with attraction between atoms, under the action of a strong transverse confinement and periodic [optical-lattice (OL)] axial potential. Using a combination of the variational approximation, one-dimensional (1D) nonpolynomial Schr\"odinger equation, and direct numerical solutions of the underlying 3D Gross-Pitaevskii equation, we show that the ground state of the condensate is a soliton belonging to the semi-infinite band gap of the periodic potential. The soliton may be confined to a single cell of the lattice or extended to several cells, depending on the effective self-attraction strength $g$ (which is proportional to the number of atoms bound in the soliton) and depth of the potential, ${V}_{0}$, the increase of ${V}_{0}$ leading to strong compression of the soliton. We demonstrate that the OL is an effective tool to control the soliton's shape. It is found that, due to the 3D character of the underlying setting, the ground-state soliton collapses at a critical value of the strength, $g={g}_{c}$, which gradually decreases with the increase of ${V}_{0}$; under typical experimental conditions, the corresponding maximum number of $^{7}\mathrm{Li}$ atoms in the soliton, ${N}_{\mathrm{max}}$, ranges between 8000 and 4000. Examples of stable multipeaked solitons are also found in the first finite band gap of the lattice spectrum. The respective critical value ${g}_{c}$ again slowly decreases with the increase of ${V}_{0}$, corresponding to ${N}_{\mathrm{max}}\ensuremath{\simeq}5000$.

DOI: 10.1103/physreva.75.033622

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2 Quasi-one-dimensional Bose-Einstein condensates in nonlinear lattices

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arXiv:cond-mat/0702154 · Physical Review A

Nearly-one-dimensional self-attractive Bose-Einstein condensates in optical lattices

Luca Salasnich, Alberto Cetoli, Boris A. Malomed, Flavio Toigo

Published Mar 30, 2007• 30 citations

arXiv HTML PDF

Abstract

g

(which is proportional to the number of atoms bound in the soliton) and depth of the potential,

{V}_{0}

, the increase of

{V}_{0}

leading to strong compression of the soliton. We demonstrate that the OL is an effective tool to control the soliton's shape. It is found that, due to the 3D character of the underlying setting, the ground-state soliton collapses at a critical value of the strength,

g={g}_{c}

, which gradually decreases with the increase of

{V}_{0}

; under typical experimental conditions, the corresponding maximum number of

^{7}\mathrm{Li}

atoms in the soliton,

{N}_{\mathrm{max}}

, ranges between 8000 and 4000. Examples of stable multipeaked solitons are also found in the first finite band gap of the lattice spectrum. The respective critical value

{g}_{c}

again slowly decreases with the increase of

{V}_{0}

, corresponding to

{N}_{\mathrm{max}}\ensuremath{\simeq}5000

DOI: 10.1103/physreva.75.033622

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2 Quasi-one-dimensional Bose-Einstein condensates in nonlinear lattices

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We consider the three-dimensional (3D) mean-field model for the Bose–Einstein condensate, with a one-dimensional (1D) nonlinear lattice (NL), which periodically changes the sign of the nonlinearity along the axial direction, and the harmonic-oscillator trapping potential applied in the transverse plane. The lattice can be created as an optical or magnetic one, by means of available experimental techniques. The objective is to identify stable 3D solitons supported by the setting. Two methods are developed for this purpose: the variational approximation, formulated in the framework of the 3D Gross–Pitaevskii equation, and the 1D nonpolynomial Schrödinger equation (NPSE) in the axial direction, which allows one to predict the collapse in the framework of the 1D description. Results are summarized in the form of a stability region for the solitons in the plane of the NL strength and wavenumber. Both methods produce a similar form of the stability region. Unlike their counterparts supported by the NL in the 1D model with the cubic nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick may help to stabilize them against the collapse, by causing the solitons to shed the excess norm. A dynamical effect specific to the NL is found in the form of freely propagating small-amplitude wave packets emitted by perturbed solitons.We consider the three-dimensional (3D) mean-field model for the Bose–Einstein condensate, with a one-dimensional (1D) nonlinear lattice (NL), which periodically changes the sign of the nonlinearity along the axial direction, and the harmonic-oscillator trapping potential applied in the transverse plane. The lattice can be created as an optical or magnetic one, by means of available experimental techniques. The objective is to identify stable 3D solitons supported by the setting. Two methods are developed for this purpose: the variational approximation, formulated in the framework of the 3D Gross–Pitaevskii equation, and the 1D nonpolynomial Schrödinger equation (NPSE) in the axial direction, which allows one to predict the collapse in the framework of the 1D description. Results are summarized in the form of a stability region for the solitons in the plane of the NL strength and wavenumber. Both methods produce a similar form of the stability region. Unlike their counterparts supported by the NL in the 1D model with the cubic nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick may help to stabilize them against the collapse, by causing the solitons to shed the excess norm. A dynamical effect specific to the NL is found in the form of freely propagating small-amplitude wave packets emitted by perturbed solitons.

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arXiv:quant-ph/0410235Europhysics Letters (EPL)Mar 8, 200534 citations

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DUMITRU BOGDAN MIHALACHE, Dumitru Mazilu, F. Lederer +4 more

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 $^{7}\mathrm{Li}$ 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.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 $^{7}\mathrm{Li}$ 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.

arXiv:nlin/0507007Physical Review AAug 2, 200540 citations

g

(which is proportional to the number of atoms bound in the soliton) and depth of the potential,

{V}_{0}

, the increase of

{V}_{0}

g={g}_{c}

, which gradually decreases with the increase of

{V}_{0}

; under typical experimental conditions, the corresponding maximum number of

^{7}\mathrm{Li}

atoms in the soliton,

{N}_{\mathrm{max}}

, ranges between 8000 and 4000. Examples of stable multipeaked solitons are also found in the first finite band gap of the lattice spectrum. The respective critical value

{g}_{c}

again slowly decreases with the increase of

{V}_{0}

, corresponding to

{N}_{\mathrm{max}}\ensuremath{\simeq}5000

^{7}\mathrm{Li}

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.