Internal Josephson phenomena in a tuned linear-nonlinear two-state Bose-Einstein condensate
We discuss the coherent oscillations between two coupled quantum states of a Bose-Einstein condensate in two-dimensional space at zero temperature. In the system we consider, weak interparticle repulsive interactions occur between the particles in state one only, while the state two particles remain non-interacting. The system is described by the coupled Gross-Pitaevskii (non-linear) and Schr\''{o}dinger (linear) equations for macroscopic wave functions of the two states. Analytical as well as numerical solution reveals a variety of regimes of dynamics of the relative quantum phase and population imbalance between the two states of the condensate, such as harmonic and anharmonic Rabi oscillations, internal a.c. Josephson effect, and macroscopic quantum self-trapping. We show that there is a value of energy detuning between the states at which oscillations are fully suppressed.