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arXiv:cond-mat/0501250 · Journal of Physics B Atomic Molecular and Optical Physics

Bound states of attractive Bose–Einstein condensates in shallow traps in two and three dimensions

Sadhan K Adhikari

Published Feb 22, 2005• 12 citations
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Abstract

Using variational and numerical solutions of the mean-field Gross-Pitaevskii equation for attractive interaction (with cubic or Kerr nonlinearity) we show that a stable bound state can appear in a Bose-Einstein condensate (BEC) in a localized exponentially-screened radially-symmetric harmonic potential well in two and three dimensions. We also consider an axially-symmetric configuration with zero axial trap and a exponentially-screened radial trap so that the resulting bound state can freely move along the axial direction like a soliton. The binding of the present states in shallow wells is mostly due to the nonlinear interaction with the trap playing a minor role. Hence these BEC states are more suitable to study the effect of the nonlinear force on the dynamics. We illustrate the highly nonlinear nature of breathing oscillation of these states. Such bound states could be created in BECs and studied in the laboratory with present knowhow.

DOI
10.1088/0953-4075/38/5/009

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Using variational and numerical solutions of the mean-field Gross-Pitaevskii equation for attractive interaction (with cubic or Kerr nonlinearity) we show that a stable bound state can appear in a Bose-Einstein condensate (BEC) in a localized exponentially-screened radially-symmetric harmonic potential well in two and three dimensions. We also consider an axially-symmetric configuration with zero axial trap and a exponentially-screened radial trap so that the resulting bound state can freely move along the axial direction like a soliton. The binding of the present states in shallow wells is mostly due to the nonlinear interaction with the trap playing a minor role. Hence these BEC states are more suitable to study the effect of the nonlinear force on the dynamics. We illustrate the highly nonlinear nature of breathing oscillation of these states. Such bound states could be created in BECs and studied in the laboratory with present knowhow.
The dynamics of weakly interacting three-dimensional Bose-Einstein condensates (BECs), trapped in external axially symmetric plus anharmonic distortion potential are studied. Within a variational approach and time-dependent Gross-Pitaevskii equation, the coupled condensate width equations are derived. By modulating anharmonic distortion of the trapping potential, nonlinear features are studied numerically and illustrated analytically, such as mode coupling of oscillation modes, and resonances. Furthermore, the stability of attractive interaction BEC in both repulsive and attractive anharmonic distortion is examined. We demonstrate that a small repulsive and attractive anharmonic distortion is effective in reducing (extending) the condensate stability region since it decreases (increases) the critical number of atoms in the trapping potential.
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.
The stability of an attractive Bose–Einstein condensate on a joint one-dimensional optical lattice and an axially symmetrical harmonic trap is studied using the numerical solution of the time-dependent mean-field Gross–Pitaevskii equation and the critical number of atoms for a stable condensate is calculated. We also calculate this critical number of atoms in a double-well potential which is always greater than that in an axially symmetrical harmonic trap. The critical number of atoms in an optical trap can be made smaller or larger than the corresponding number in the absence of the optical trap by moving a node of the optical lattice potential in the axial direction of the harmonic trap. This variation of the critical number of atoms can be observed experimentally and compared with the present calculations.
The stability of Bose–Einstein condensates (BECs) loaded into a two-dimensional shallow harmonic potential well is studied. By using the variational method, the ground state properties for interacting BECs in the shallow trap are discussed. It is shown that the possible stable bound state can exist. The depth of the shallow well plays an important role in stabilizing the BECs. The stability of BECs in the shallow trap with the periodic modulating of atom interaction by using the Feshbach resonance is also discussed. The results show that the collapse and diffusion of BECs in a shallow trap can be controlled by the temporal modulation of the scattering length.
It is shown using the Gross–Pitaevskii equation that resonance states of Bose–Einstein condensates with attractive interactions can be stabilized into true bound states. A semiclassical variational approximation and an independent quantum variational numerical method are used to calculate the energies (chemical potentials) and linewidths of resonances of the time-independent Gross–Pitaevskii equation; both methods produce similar results. Borders between the regimes of resonances, bound states and, in two and three dimensions, collapse are identified.