Nonlinear from linear states in two-component Bose-Einstein condensates

In this work, we consider the extension of linear eigenmodes of the harmonic oscillator potential into nonlinear states, for the case of two-component Gross-Pitaevskii equations with a parabolic potential, motivated by the context of two interacting hyperfine states of Rb-87 in Bose-Einstein condensates. In particular, we establish that nonlinear continuations of various eigenmode combinations are possible and corroborate this analytical finding with numerical computations for the lowest few eigenmode combinations involving the ground state and the first two excited states. A multitude of nonlinear states can be constructed in this way, some of which spontaneously deform, as the interactions become stronger, into previously obtained nonlinear eigenstates. The Bogolyubov-de Gennes analysis of the excitations on top of such states illustrates that some of them may become unstable beyond a critical threshold (of the chemical potentials associated with the states), while others may be stable within the entire range of chemical potentials considered herein. When the modes are found to be unstable, their evolution is followed, leading to interesting dynamical effects such as spontaneous symmetry breaking or oscillatory growth.