Hybrid matter-wave - microwave solitons on the lattice

We introduce a two-component system which models a pseudospinor Bose-Einstein condensate (BEC), with a microwave field coupling its two components. The feedback of BEC of the field (the local-field effect) is taken into account by dint of the respective Poisson equation, which is solved using the Green's function. This gives rise to an effective long-range self-trapping interaction, which may act alone, or be combined with the contact cubic nonlinearity. The system is made discrete by loading the BEC into a deep optical-lattice potential. Numerical solutions demonstrate that onsite-centered fundamental solitons are stable in the cases of attractive or zero contact interactions, while offsite-centered solitons are unstable. In the case of the repulsive onsite nonlinearity, offsite solitons are stable, while their onsite-centered counterparts are stable only at sufficiently small values of the norm, where bistability between the off- and onsite-centered mode takes place. The shape of the onsite-centered solitons is very accurately predicted by a variational approximation (which includes essential technical novelties). Spatially-antisymmetric (\textquotedblleft twisted") solitons are stable at small values of the norm, being unstable at larger norms. In the strongly asymmetric version of the two-component system, which includes the Zeeman splitting, the system is reduced to a single discrete Gross-Pitaevskii equation, by eliminating the small higher-energy component.