Nonlinear tunneling in two-dimensional lattices

We present a thorough analysis of the nonlinear tunneling of Bose-Einstein condensates in static and accelerating two-dimensional lattices within the framework of the mean-field approximation. We deal with nonseparable lattices, considering different initial atomic distributions in highly symmetric states. For an analytical description of the condensate before instabilities develop, we derive several few-mode models, analyzing essentially both nonlinear and quasilinear regimes of tunneling. By direct numerical simulations, we show that two-mode models provide an accurate description of tunneling when either initially two states are populated or tunneling occurs between two stable states. Otherwise, a two-mode model may give only useful qualitative hints for understanding tunneling, but does not reproduce many features of the phenomenon. This reflects the crucial role of instabilities developed due to two-body interactions resulting in a non-negligible population of the higher bands. This effect becomes even more pronounced in the case of accelerating lattices. In the latter case we show that the direction of the acceleration is a relevant physical parameter which affects the tunneling by changing the atomic rates at different symmetric states and by changing the numbers of bands involved in the atomic transfer.