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arXiv:cond-mat/0512227 · Physical Review A

Two-component gap solitons in two- and one-dimensional Bose-Einstein condensates

Arthur Gubeskys, Boris A. Malomed, Ilya M. Merhasin

Published Feb 9, 2006• 46 citations
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Abstract

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 ${10}^{5}$. 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.

DOI
10.1103/physreva.73.023607

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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∼5000\ensuremath{\sim}5000\ensuremath∼5000 in the 2D intragap soliton, and \ensuremath∼25000\ensuremath{\sim}25\phantom{\rule{0.2em}{0ex}}000\ensuremath∼25000 in its intergap counterpart; in 1D solitons, it may be up to 105{10}^{5}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 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).
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 consider effects of interspecies attraction on two-component gap solitons (GSs) in the binary BEC with intraspecies repulsion, trapped in the one-dimensional optical lattice (OL). Systematic simulations of the coupled Gross-Pitaevskii equations corroborate an assumption that, because the effective mass of GSs is negative, the interspecies attraction may split the two-component soliton. Two critical values, ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$, of the OL strength $(\ensuremath{\kappa})$ are identified. Two-species GSs with fully overlapping wave functions are stable in strong lattices $(\ensuremath{\kappa}>{\ensuremath{\kappa}}_{1})$. In an intermediate region, ${\ensuremath{\kappa}}_{1}>\ensuremath{\kappa}>{\ensuremath{\kappa}}_{2}$, the soliton splits into a double-humped state with separated components. Finally, in weak lattices $(\ensuremath{\kappa}<{\ensuremath{\kappa}}_{2})$, the splitting generates a pair of freely moving single-species GSs. We present and explain the dependence of ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ on the number of atoms (total norm), and on the relative strength of the competing interspecies attraction and intraspecies repulsion. The splitting of asymmetric solitons, with unequal norms of the two species, is briefly considered too. It is found and explained that the splitting threshold grows with the increase of the asymmetry.
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.
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, NNN. In either band gap, the GS species with smallest NNN 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 NNN, 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 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.