We investigate the ground state (GS) of a collisionless Bose-Einstein condensate (BEC) trapped in a soft one-dimensional optical lattice (OL), which is formed by two counterpropagating optical beams perturbed by the BEC density profile through the local-field effect (LFE). We show that LFE gives rise to an envelope-deformation potential, a nonlocal potential resulting from the phase deformation, and an effective self-interaction of the condensate. As a result, stable photon-atomic (polaritonic) lattice solitons, including an optical component, in the form of the deformation of the soft OL, in a combination with a localized matter-wave component, are generated in the blue-detuned setting, without any direct interaction between atoms. These self-trapped modes, which realize the system's GS, are essentially different from the gap solitons supported by the interplay of the OL potential and collisional interactions between atoms. A transition to tightly bound modes from loosely bound ones occurs with the increase of the number of atoms in the BEC.
We investigate the ground state (GS) of a collisionless Bose-Einstein condensate (BEC) trapped in a soft one-dimensional optical lattice (OL), which is formed by two counterpropagating optical beams perturbed by the BEC density profile through the local-field effect (LFE). We show that LFE gives rise to an envelope-deformation potential, a nonlocal potential resulting from the phase deformation, and an effective self-interaction of the condensate. As a result, stable photon-atomic (polaritonic) lattice solitons, including an optical component, in the form of the deformation of the soft OL, in a combination with a localized matter-wave component, are generated in the blue-detuned setting, without any direct interaction between atoms. These self-trapped modes, which realize the system's GS, are essentially different from the gap solitons supported by the interplay of the OL potential and collisional interactions between atoms. A transition to tightly bound modes from loosely bound ones occurs with the increase of the number of atoms in the BEC.
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 ${N}_{\mathrm{max}}\ensuremath{\simeq}5000$.
We study the dynamics of nonlinear localized excitations (‘solitons’) in two-dimensional (2D) Bose–Einstein condensates (BECs) with repulsive interactions, loaded into an optical lattice (OL), which is combined with an external parabolic potential. First, we demonstrate analytically that a broad (‘loosely bound’, LB) soliton state, based on a 2D Bloch function near the edge of the Brillouin zone (BZ), has a negative effective mass (while the mass of a localized state is positive near the BZ centre). The negative-mass soliton cannot be held by the usual trap, but it is safely confined by an inverted parabolic potential ( anti-trap ). Direct simulations demonstrate that the LB solitons (including those with intrinsic vorticity) are stable and can freely move on top of the OL. The frequency of the elliptic motion of the LB-soliton's centre in the anti-trapping potential is very close to the analytical prediction which treats the solition as a quasi-particle. In addition, the LB soliton of the vortex type features real rotation around its centre. We also find an abrupt transition, which occurs with the increase of the number of atoms, from the negative-mass LB states to tightly bound (TB) solitons. An estimate demonstrates that for the zero-vorticity states, the transition occurs when the number of atoms attains a critical number N cr ∼ 10 3 , while for the vortex the transition takes place at N cr ∼ 5 × 10 3 atoms. The positive-mass LB states constructed near the BZ centre (including vortices) can also move freely. The effects predicted for BECs also apply to optical spatial solitons in bulk photonic crystals.
We study the dynamics of one-dimensional solitons in attractive and repulsive Bose–Einstein condensates (BECs) loaded into an optical lattice (OL), which is combined with an external parabolic potential. First, we demonstrate analytically that, in the repulsive BEC, where the soliton is of the gap type, its effective mass is negative . This gives rise to a prediction for the experiment: such a soliton cannot be held by the usual parabolic trap, but it can be captured (performing slow harmonic oscillations, with a period that is estimated to be ∼0.01 s in realistic experimental conditions) by an anti-trapping inverted parabolic potential. We also study the motion of the soliton in a long system, concluding that, in the cases of both the positive and negative mass, it moves freely, provided that its amplitude is below a certain critical value; above it, the soliton's velocity decreases due to interaction with the OL. At a later stage, the damped motion becomes chaotic. We also investigate the evolution of a two-soliton pulse in the attractive model. The pulse generates a persistent breather, if its amplitude is not too large; otherwise, fusion into a single fundamental soliton takes place. Collisions between two solitons captured in the parabolic trap or anti-trap are considered too. Depending on their amplitudes and phase difference, the solitons either perform stable oscillations, colliding indefinitely many times, or merge into a single soliton. Effects reported in this work for BECs can also be formulated for optical solitons in nonlinear photonic crystals. In particular, the capture of the negative-mass soliton in the anti-trap implies that a bright optical soliton in a self-defocusing medium with a periodic structure of the refractive index may be stable in an anti-waveguide .
We study the dynamics of one-dimensional solitons in the attractive and repulsive Bose-Einstein condensates (BECs) loaded into an optical lattice (OL), which is combined with an external parabolic potential. First, we demonstrate analytically that, in the repulsive BEC, where the soliton is of the gap type, its effective mass is negative. This gives rise to a prediction for the experiment: such a soliton cannot be not held by the usual parabolic trap, but it can be captured (performing harmonic oscillations) by an anti-trapping inverted parabolic potential. We also study the motion of the soliton a in long system, concluding that, in the cases of both the positive and negative mass, it moves freely, provided that its amplitude is below a certain critical value; above it, the soliton’s velocity decreases due to the interaction with the OL. At a late stage, the damped motion becomes chaotic. We also investigate the evolution of a two-soliton pulse in the attractive model. The pulse generates a persistent breather, if its amplitude is not too large; otherwise, fusion into a single fundamental soliton takes place. Collisions between two solitons captured in the parabolic trap or anti-trap are considered too. Depending on their amplitudes and phase difference, the solitons either perform stable oscillations, colliding indefinitely many times, or merge into a single soliton. Effects reported in this work for BECs can also be formulated for optical solitons in nonlinear photonic crystals. In particular, the capture of the negative-mass soliton in the anti-trap implies that a bright optical soliton in a self-defocusing medium with a periodic structure of the refractive index may be stable in an anti-waveguide. 1 1
We demonstrate a possibility to generate localized states in effectively one-dimensional Bose-Einstein condensates with a negative scattering length in the form of a dark soliton in the presence of an optical lattice (OL) and/or a parabolic magnetic trap. We connect such structures with twisted localized modes (TLMs) that were previously found in the discrete nonlinear Schrödinger equation. Families of these structures are found as functions of the OL strength, tightness of the magnetic trap and chemical potential, and their stability regions are identified. Stable bound states of two TLMs are also found. In the case when the TLMs are unstable, their evolution is investigated by means of direct simulations, demonstrating that they transform into large-amplitude fundamental solitons. An analytical approach is also developed, showing that two or several fundamental solitons, with the phase shift π between adjacent ones, may form stable bound states, with parameters quite close to those of the TLMs revealed by simulations. TLM structures are also found numerically and explained analytically in the case when the OL is absent, the condensate being confined only by the magnetic trap.
We study dipolar Bose–Einstein condensate (BEC) solitons in a one-dimensional optical lattice with the combined effect of local and nonlocal nonlinearities. The local nonlinearity is imposed by a magnetic or optical field via the Feshbach (FB) resonance. In contrast, the nonlocal nonlinearity is created by the long-range dipole–dipole interaction among the condensates. The orientations of the dipoles are directed by a rotatable uniform external field, which gives rise to a controllable nonlocal nonlinearity resulting from the angle θ between the direction of the dipole and the elongation of the lattice. If the lattice is sufficiently deep, this model can be described by the Gross–Pitaevskii equation (GPE) with tunable local and nonlocal nonlinear strengths (g and θ respectively). The formation, motion, and collision of the solitons in this system are studied by numerical simulations. The combined effect of the local and nonlocal nonlinearities gives a controllable scheme for all these characteristics via the relation between g and θ.