We analyze the existence and stability of spatially extended (Bloch-type) and localized states of a Bose-Einstein condensate loaded into an optical lattice. In the framework of the Gross-Pitaevskii equation with a periodic potential, we study the band-gap structure of the matter-wave spectrum in both the linear and nonlinear regimes. We demonstrate the existence of families of spatially localized matter-wave gap solitons, and analyze their stability in different band gaps, for both repulsive and attractive atomic interactions.
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,
V0, the increase of
V0 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=gc, which gradually decreases with the increase of
V0; under typical experimental conditions, the corresponding maximum number of
7Li atoms in the soliton,
Nmax, 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
gc again slowly decreases with the increase of
V0, corresponding to ${N}_{\mathrm{max}}\ensuremath{\simeq}5000$.
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 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 the spatial profiles of periodic localized modes in attractive Bose-Einstein condensates, by solving the mean-field Gross-Pitaevskii equation in the presence of elliptic-type periodic potential. By considering a one-dimensional time independent linearized Gross-Pitaevskii equation, we obtained three bound state solutions and energies emanating from the first order Lamé equation. When the nonlinearity induced by the two body inter-atomic interactions are fully activated, spatially localized trivial phase periodic solutions of the attractive condensates are analytically obtained using the ansatz technique coupled with the direct integral method. Results of numerical simulation depicts trivial phase solutions, which are uniform train of spatially localized modes that are insensitive to variation of the elliptic modulus. However in the non-trivial phase regime, the spatially localized trains of soliton become very structurally unstable. This work underscores the spontaneous generation of periodic potential by the condensate wave function, determine the band structure of the lattice and basic properties of periodic matter waves under linear conditions, and highlight various spatial nonlinear periodic modes in the condensate. Finally, our investigation provides a solid theoretical framework that finds potential application in the fabrication of atomic lasers, periodic matter-wave gratings and quantum logic gates.
The cubic nonlinear Schrodinger equation with a lattice potential is used to model a periodic dilute gas Bose-Einstein condensate. Both two- and three-dimensional condensates are considered, for atomic species with either repulsive or attractive interactions. A family of exact solutions and corresponding potential is presented in terms of elliptic functions. The dynamical stability of these exact solutions is examined using both analytical and numerical methods. For condensates with repulsive atomic interactions, all stable, trivial-phase solutions are off-set from the zero level. For condensates with attractive atomic interactions, no stable solutions are found, in contrast to the one-dimensional case.
Using the variational approximation and numerical simulations, we study one-dimensional gap solitons in a binary Bose-Einstein condensate trapped in an optical-lattice potential. We consider the case of interspecies repulsion, while the intraspecies interaction may be either repulsive or attractive. Several types of gap solitons are found: symmetric or asymmetric; unsplit or split, if centers of the components coincide or separate; intragap (with both chemical potentials falling into a single band gap) or intergap, otherwise. In the case of the intraspecies attraction, a smooth transition takes place between solitons in the semi-infinite gap, those in the first finite band gap, and semigap solitons (with one component in a band gap and the other in the semi-infinite gap).