Two routes to the one-dimensional discrete nonpolynomial Schrödinger equation

Abstract

The Bose-Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the onedimensional (1D) discrete nonpolynomial Schrödinger equation (NPSE). Both models are derived from the three-dimensional Gross-Pitaevskii equation (3D GPE). To produce "model 1" (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. "Model 2", that was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction of the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2 -in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC. PACS numbers: 03.75.Lm; 05.45.YvThe dynamics of a dilute quantum gas which forms the Bose-Einstein condensate (BEC) is very accurately described by the three-dimensional Gross-Pitaevskii equation (3D GPE). This equation treats effects of collisions between atoms in the condensate in the mean-field approximation. In experimentally relevant settings, the BEC is always confined by a trapping potential. In many cases, the trap is designed to have the "cigar-shaped" form, allowing an effective reduction of the dimension from 3 to 1. In turn, the 1D dynamics of the trapped condensate may be controlled by means of an additional periodic potential, induced by an optical lattice (OL), which acts along the axis of the "cigar". If the OL potential is sufficiently strong, the eventual dynamical model reduces to a 1D discrete equation. In both the continual and discrete versions of the 1D description, a crucially important feature is the form of the nonlinearity in the respective equations. In the limit of low density, the nonlinearity is cubic -the same as in the underlying 3D GPE. In the general case, a consistent derivation, which starts from the cubic nonlinearity in 3D, leads to 1D equations with a nonpolynomial nonlinearity, the respective model being called the "nonpolynomial Schrödinger equation" (NPSE). The discrete lim...

Discrete Bright Solitons in Bose-Einstein Condensates and Dimensional Reduction in Quantum Field Theory

“…that is the 1D discrete nonpolynomial Schrödinger equation, describing the BEC under a transverse anisotropic harmonic confinement and an axial optical lattice [12,13].…”

“…The three-dimensional (3D) Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equations which accurately describes a Bose-Einstein condensate (BEC) made of dilute and ultracold atoms [9], is usually analyzed in the case of repulsive interaction strength which corresponds to a positive inter-atomic s-wave scattering length [2]. Indeed, a negative s-wave scattering length implies an attractive interaction strength which may bring to the collapse [9] due to the shrink of the transverse width of a realistic quasi-1D bosonic cloud [12,13,19]. Nevertheless, in certain regimes of interaction both continuous and discrete 3D Gross-Pitaevskii equations predict the existence of meta-stable configurations which are usually called continuous and discrete bright solitons [12,13,19].…”

“…Indeed, a negative s-wave scattering length implies an attractive interaction strength which may bring to the collapse [9] due to the shrink of the transverse width of a realistic quasi-1D bosonic cloud [12,13,19]. Nevertheless, in certain regimes of interaction both continuous and discrete 3D Gross-Pitaevskii equations predict the existence of meta-stable configurations which are usually called continuous and discrete bright solitons [12,13,19]. We remark that continuous bright solitons have been observed in various experiments [5,14,17,24] involving attractive bosons of 7 Li and 85 Rb vapors.…”

“…In the first part we discuss an effective one-dimensional discrete nonpolynomial Schrödinger equation obtained from the continuous 3D Gross-Pitaevskii equation with transverse harmonic confinement and axial periodic potential [12,13]. We show that this 1D discrete nonpolynomial Schrödinger equation reduces to the 1D discrete Gross-Pitaevskii equation only in the weak-coupling regime and we compare the bright soliton of the discrete nonpolynomial Schrödinger equation bright solitons with the bright solitons of the discrete Gross-Pitaevskii equation.…”

One-dimensional reduction of the three-dimenstional Gross-Pitaevskii equation with two- and three-body interactions

“…Other recent works have considered confinement patterns for several distinct geometries [22][23][24][25][26][27][28]. For instance, in [22] the authors derived an effective NPSE for quasi-1D trap with periodic modulation and in [23] an effective nonlinear Schrödinger equation (NLSE) for cigar-shaped and disk-shaped Fermi superfluid was suggested.…”