We analyze gap solitons in trapped Bose-Einstein condensates (BECs) in optical lattice potentials under Feshbach resonance management. Starting with an averaged Gross-Pitaevsky equation with a periodic potential, we employ an envelope-wave approximation to derive coupled-mode equations describing the slow BEC dynamics in the first spectral gap of the optical lattice. We construct exact analytical formulas describing gap soliton solutions and examine their spectral stability using the Chebyshev interpolation method. We show that these gap solitons are unstable far from the threshold of local bifurcation and that the instability results in the distortion of their shape. We also predict the threshold of the power of gap solitons near the local bifurcation limit.
We analyze gap solitons in trapped Bose-Einstein condensates (BECs) in optical lattice potentials under Feshbach resonance management. Starting with an averaged Gross-Pitaevsky equation with a periodic potential, we employ an envelope-wave approximation to derive coupled-mode equations describing the slow BEC dynamics in the first spectral gap of the optical lattice. We construct exact analytical formulas describing gap soliton solutions and examine their spectral stability using the Chebyshev interpolation method. We show that these gap solitons are unstable far from the threshold of local bifurcation and that the instability results in the distortion of their shape. We also predict the threshold of the power of gap solitons near the local bifurcation limit.
We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.
Optical lattices and Feshbach resonance management are two mighty and widely used techniques for studying, both experimentally and theoretically, various physical issues and the underlying nonlinear dynamics of Bose−Einstein condensates. Both techniques show great power for studying solitons of different types, and particularly, the former technique realizes the creation of 1D bright matter‐wave gap solitons, whose formation and properties in multidimensional systems, however, are much less known. Herein, both techniques and study are combined, theoretically and numerically, the formation and dynamics of 3D nonlinear localized gap modes, including fundamental gap solitons and their higher‐order ones as soliton clusters, as well as gap vortices with topological charge s = 1. The stability regions of all the localized gap modes are identified by means of direct perturbed numerical simulations, revealing important insights into soliton physics in multidimensional space.
We propose a scheme for making a Bose-Einstein condensate (BEC) of molecules from a BEC of atoms in a strongly confining two-dimensional optical lattice and a weak one-dimensional optical lattice in the third dimension. The stable solutions obtained for the order parameters take the form of a different type of gap soliton, with both atomic and molecular BECs, and also standard gap solitons with only a molecular BEC. The strongly confining dimensions of the lattice stabilize the BEC against inelastic energy transfer in atom-molecule collisions. The solitons with atoms and molecules may be obtained by starting with an atomic BEC, and gradually tuning the resonance by changing the external magnetic-field strength until the desired atom-molecule soliton is obtained. A gap soliton of a BEC of only molecules may be obtained nonadiabatically by starting from an atom-only gap soliton, far from a Feshbach resonance and adjusting the magnetic field to near Feshbach resonance. After a period of time in which the dimer field grows, change the magnetic field such that the detuning is large and negative and Feshbach effects wash out, turn off the optical lattice in phase with the atomic BEC, and turn on an optical lattice in phase with the molecules. The atoms disperse, leaving a gap soliton composed of a molecular BEC. Regarding instabilities in the dimension of the weak optical lattice, the solitons which are comprised of both atoms and molecules are sometimes stable and sometimes unstable---we present numerically obtained results. Gap solitons comprised of only molecules have the same stability properties as the standard gap solitons: stable from frequencies slightly below the middle of the band gap to the top, and unstable below that point. Instabilities are only weakly affected by the soliton velocities, and all instabilities are oscillatory.
We consider a model of Bose-Einstein condensates which combines a stationary optical lattice (OL) and periodic change of the sign of the scattering length (SL) due to the Feshbach-resonance management. Ordinary solitons and ones of the gap type being possible, respectively, in the model with constant negative and positive SL, an issue of interest is to find solitons alternating, in the case of the low-frequency modulation, between shapes of both types, across the zero-SL point. We find such \textit{alternate solitons} and identify their stability regions in the 2D and 1D models. Three types of the dynamical regimes are distinguished: stable, unstable, and semi-stable. In the latter case, the soliton sheds off a conspicuous part of its initial norm before relaxing to a stable regime. In the 2D case, the threshold (minimum number of atoms) necessary for the existence of the alternate solitons is essentially higher than its counterparts for the ordinary and gap solitons in the static model. In the 1D model, the alternate solitons are also found only above a certain threshold, while the static 1D models have no threshold. In the 1D case, stable antisymmetric alternate solitons are found too. Additionally, we consider a possibility to apply the variational approximation (VA) to the description of stationary gap solitons, in the case of constant positive SL. It predicts the solitons in the first finite bandgap very accurately, and does it reasonably well in the second gap too. In higher bands, the VA predicts a border between tightly and loosely bound solitons.
We investigate stability of gap solitons (GSs) in the first two band gaps in the framework of the one-dimensional Gross-Pitaevskii equation, combining the repulsive nonlinearity and a moderately strong optical lattice (OL), which is subjected to ``management,'' in the form of time-periodic modulation of its depth. The analysis is performed for parameters relevant to the experiment, characteristic values of the modulation frequency being $\ensuremath{\omega}\ensuremath{\sim}2\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}20\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$. First, we present several GS species in the two band gaps in the absence of the management. These include fundamental solitons and their bound states, as well as a subfundamental soliton in the second gap, featuring two peaks of opposite signs in a single well of the periodic potential. This soliton is always unstable, and quickly transforms into a fundamental GS, losing a considerable part of its norm. In the first band gap, (stable) bound states of two fundamental GSs are possible solely with opposite signs, if they are separated by an empty site. Under the periodic modulation of the OL depth, we identify stability regions for various GS species, in terms of $\ensuremath{\omega}$ and modulation amplitude, at fixed values of the soliton's norm, N. In either band gap, the GS species with smallest N has a largest stability area; in the first and second gaps, they are, respectively, the fundamental GS proper, or the one spontaneously generated from the subfundamental soliton. However, with the increase of N, the stability region of every species expands in the first gap, and shrinks in the second one. The outcome of the instability development is also different in the two band gaps: it is destruction of the GS in the first gap, and generation of extra side lobes by unstable GSs in the second one.
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.