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arXiv:1105.5537 · Modern Physics Letters B

BRIGHT-LIKE SOLITON SOLUTION IN QUASI-ONE-DIMENSIONAL BEC IN THIRD ORDER BY INTERACTION RADIUS

Pavel A. Andreev, L. S. KUZ'MENKOV

Published Aug 13, 2012• 17 citations
arXivHTMLPDF

Abstract

Nonlinear Schrödinger equations and corresponding quantum hydrodynamic (QHD) equations are widely used in studying of ultracold boson–fermion mixtures and superconductors. In this article, we show that more exact account of interaction in Bose–Einstein condensate (BEC), in comparison with the Gross–Pitaevskii (GP) approximation, leads to the existence of a new type of solitons. We use a set of QHD equations in the third order by the interaction radius (TOIR), which corresponds to the GP equation in the first order by the interaction radius. We analytically obtain a soliton solution which is an area of increased atom concentration. The conditions for existence of the soliton are studied. It is shown what solution exists if the interaction between the particles is repulsive. Particle concentration has been achieved experimentally for the BEC is of order of 1012–1014 cm-3. In this case the solution exists if the scattering length is of the order of 1 μm, which can be reached using the Feshbach resonance. It is one of the limit case of existence of the solution. The corresponding scattering length decrease with the increasing of concentration of particles. We have shown that account of interaction up to TOIR approximation leads to new effects. The investigation of effects in the TOIR approximation gives a more detail information on interaction potentials between the atoms and can be used for a more detail investigation of the interatomic potential structure.

DOI
10.1142/s0217984912501527

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Nonlinear Schrödinger equations and corresponding quantum hydrodynamic (QHD) equations are widely used in studying of ultracold boson–fermion mixtures and superconductors. In this article, we show that more exact account of interaction in Bose–Einstein condensate (BEC), in comparison with the Gross–Pitaevskii (GP) approximation, leads to the existence of a new type of solitons. We use a set of QHD equations in the third order by the interaction radius (TOIR), which corresponds to the GP equation in the first order by the interaction radius. We analytically obtain a soliton solution which is an area of increased atom concentration. The conditions for existence of the soliton are studied. It is shown what solution exists if the interaction between the particles is repulsive. Particle concentration has been achieved experimentally for the BEC is of order of 1012–1014 cm-3. In this case the solution exists if the scattering length is of the order of 1 μm, which can be reached using the Feshbach resonance. It is one of the limit case of existence of the solution. The corresponding scattering length decrease with the increasing of concentration of particles. We have shown that account of interaction up to TOIR approximation leads to new effects. The investigation of effects in the TOIR approximation gives a more detail information on interaction potentials between the atoms and can be used for a more detail investigation of the interatomic potential structure.
Abstract A quantum hydrodynamic model is developed for axial symmetric anisotropic short-range interactions. The quantum stress tensor represents the interaction and is derived up to the third order by the interaction radius. The first order by the interaction radius contains the isotropic part only. This leads to interaction in the Gross–Pitaevskii approximation. Terms existing in the third order by the interaction radius are caused by the isotropic and nonisotropic parts of the interaction, and each of them introduces an interaction constant. Therefore, three interaction constants are involved in the model. Atoms, except for alkali and alkali-earth atoms, can have an anisotropic potential of interaction (this has particularly been demonstrated for the lanthanides). Short-wavelength instability caused by nonlocal terms appears in the Bogoliubov spectrum, and the conditions for stable and unstable behavior are described. Bright solitons in a repulsive Bose–Einstein condensate (BEC) are studied under the influence of anisotropic short-range interaction in the BEC of one species. The area of existence of bright solitons corresponds to the area of instability of the Bogoliubov spectrum. Approximate reduction of nonlocal nonlinearity to quintic nonlinearity in the bright soliton regime is also demonstrated.
Abstract Repulsive Bose–Einstein condensates, where short-range interaction is included up to the third order by the interaction radius, demonstrate the existence of bright solitons in a narrow interval of parameters. These solitons are studied here for the boson–fermion mixture, where spin-1/2 fermions are considered in the regime of full-spin polarization. The influence of fermions on bosonic bright solitons via the boson–fermion interaction is considered up to the third order by the interaction radius. Fermions themselves are considered within the hydrodynamic model, which includes the kinetic pressure-evolution equation. The interactions between fermions are also considered. The first order by the interaction radius makes a zero contribution to the Euler equation and the kinetic pressure evolution equation for fermions, but the third order by the interaction radius provides nonzero contributions to both equations. The repulsive (attractive) boson–fermion interaction leads to bright (dark) fermionic solitons.
The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE. In a dilute gas of N Bosons at zero temperature practically all particles are in the same single-particle state of the density matrix ρ(r,r ′;t). 1), 2) This macroscopically occupied single-particle state is called Bose-Einstein condensate (BEC) and it is well described by a complex classical field ψ(r,t) (order parameter or macroscopic
We analyze the dynamics of a bright soliton in Bose–Einstein condensates (BECs) with time-dependent atomic scattering length in an expulsive parabolic potential. Under a safe range of parameters in which the Gross–Pitaevskii (GP) equation is effective in one dimension, our results show that, the dynamics of the bright soliton can be classed into two phases, depending on the value of the scattering length. Meanwhile, there exists a critical value of the absolute value of the atomic scattering length, below which, the dynamics of the bright soliton is very regular. Those phenomena can be useful for developing concrete applications of the nonlinear matter waves. We also obtain the orbital equation of the bright soliton and get some interesting data which may be useful for the experimental observation of the bright soliton and the application of the atom laser with manipulated intensity.
The analytical dark and bright soliton solutions of the one-dimensional Gross-Pitaevskii equation with a confining potential are obtained. For the bright soliton, the recent experimental finding is studied, and the particle number of the soliton and the window of the particle numbers for the bright soliton to occur are estimated analytically and in good agreement with the experimental data. The existence of dark soliton for the attractive interaction and bright soliton for the repulsive interaction is predicted under proper conditions.
A one-dimensional (1D) Gross-Pitaevskii (GP) equation which governs the dynamics of Bose-Einstein condensates (BECs) with time-dependent scattering length in a harmonic potential is considered. Through a modified phase imprint transformation into the macroscopic order parameter, we suitably generate a derivative cubic-quintic Gross-Pitaevskii (DCQ-GP) equation that describes the dynamics of condensates in the harmonic potential for both twoand three-body interatomic interactions in the presence of nonlinear dispersion. Employing a modified lens-type transformation, we reduce the DCQ-GP equation into a derivative cubic-quintic Ginzburg-Landau (DCQ-GL) equation that allows us to investigate the modulational instability (MI) of the system and study, through the analytical exact solutions of the considered 1D GP equation, the effects of the three-body interactions and nonlinear dispersion on the dynamics of the BECs. We observe that for BECs with repulsive two-body interatomic interactions, the attractive three-body interactions and the nonlinear dispersion have opposite effects on the condensate: (i) The attractive three-body interactions enhance the instability, while the nonlinear dispersion softens the instability of the system; (ii) the attractive three-body interactions contributes to the ejections of atoms from the condensates while the nonlinear dispersion contributes to the accumulation of atoms in the condensates; (iii) the attractive three-body interactions contributes to the broadening of the matter wave solitons, while the nonlinear dispersion contributes to the compression of the matter wave solitons.