We construct families of incoherent matter-wave solitons in a repulsive degenerate Bose gas trapped in an optical lattice (OL), i.e., gap solitons, and investigate their stability at zero and finite temperature, using the Hartree-Fock-Bogoliubov equations. The gap solitons are composed of a coherent condensate, and normal and anomalous densities of incoherent vapor cotrapped with the condensate. Both intragap and intergap solitons are constructed, with chemical potentials of the components falling in one or different band gaps in the OL-induced spectrum. Solitons change gradually with temperature. Families of intragap solitons are completely stable (both in direct simulations and in terms of eigenvalues of perturbation modes), while the intergap family may have a very small unstable eigenvalue (nevertheless, they feature no instability in direct simulations). Stable higher-order (multihumped) solitons and bound complexes of fundamental solitons are found too.
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).
We introduce two- and one-dimensional (1D) models of a binary BEC (Bose-Einstein condensate) in a periodic potential, with repulsive interactions. We chiefly consider the most fundamental case of the interspecies repulsion with zero intraspecies interactions. The same system may also model a mixture of two mutually repulsive fermionic species. Existence and stability regions for gap solitons (GSs) supported by the interplay of the interspecies repulsion and periodic potential are identified. Two-component GSs are constructed by means of the variational approximation (VA) and in a numerical form. The VA provides an accurate description for the GS which is a compound state of two tightly bound components, each essentially trapped in one cell of the periodic potential. GSs of this type dominate in the case of intragap solitons, with both components belonging to the first finite bandgap of the linear spectrum (only this type of solitons is possible in a weak lattice). Intergap solitons, with one component residing in the second bandgap, and intragap solitons which have both components in the second gap, are possible in a deeper periodic potential, with the strength essentially exceeding the recoil energy of the atoms. Intergap solitons are, typically, bound states of one tightly and one loosely bound component. In this case, results are obtained in a numerical form. The number of atoms in experimentally relevant situations is estimated to be $\ensuremath{\sim}5000$ in the 2D intragap soliton, and $\ensuremath{\sim}25\phantom{\rule{0.2em}{0ex}}000$ in its intergap counterpart; in 1D solitons, it may be up to
105. For 2D solitons, the stability is identified in direct simulations, while in the 1D case it is done via eigenfrequencies of small perturbations, and then verified by simulations. In the latter case, if the intragap soliton in the first bandgap is weakly unstable, it evolves into a stable breather, while unstable solitons of other types (in particular, intergap solitons) get completely destroyed. The intragap 2D solitons in the first bandgap are less robust, and in some cases they are completely destroyed by the instability. Addition of intraspecies repulsion to the repulsion between the components leads to further stabilization of the GSs.
Using coupled equations for the bosonic and fermionic order parameters, we construct families of gap solitons (GSs) in a nearly one-dimensional Bose-Fermi mixture trapped in a periodic optical-lattice (OL) potential, the boson and fermion components being in the states of the Bose-Einstein condensation and Bardeen-Cooper-Schrieffer superfluid, respectively. Fundamental GSs are compact states trapped, essentially, in a single cell of the lattice. Full families of such solutions are constructed in the first two band gaps of the OL-induced spectrum, by means of variational and numerical methods, which are found to be in good agreement. The families include both intragap and intergap solitons, with the chemical potentials of the boson and fermion components falling in the same or different band gaps, respectively. Nonfundamental states, extended over several lattice cells, are constructed too. The GSs are stable against strong perturbations.
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.
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.
In past years, optical lattices have been demonstrated as an excellent platform for making, understanding, and controlling quantum matters at nonlinear and fundamental quantum levels. Shrinking experimental observations include matter-wave gap solitons created in ultracold quantum degenerate gases, such as Bose–Einstein condensates with repulsive interaction. In this paper, we theoretically and numerically study the formation of one-dimensional gap soliton molecules and clusters in ultracold coherent atom ensembles under electromagnetically induced transparency conditions and trapped by an optical lattice. In numerics, both linear stability analysis and direct perturbed simulations are combined to identify the stability and instability of the localized gap modes, stressing the wide stability region within the first finite gap. The results predicted here may be confirmed in ultracold atom experiments, providing detailed insight into the higher-order localized gap modes of ultracold bosonic atoms under the quantum coherent effect called electromagnetically induced transparency.