Eigenstates and excitations of the simple atom-molecule Bose-Einstein condensate

We model an atom-molecule Bose-Einstein condensate (AMBEC) using simplified set of coupled Gross-Pitaevskii equations (GPE), where we neglect the background (elastic) scattering length of the atoms. We analyze the ground state numerically and analytically, and construct its twin state through transformation $\delta\rightarrow-\delta, K\rightarrow-K$. We find that the ground state is a collection of three local eigenstates: all-atom state, mixed (atom-molecule) state, and all-molecule state, while the twin state comprises of mixed state with tunable fraction of atoms including unity but excluding all-molecules. We find the analytic boundaries of the local eigenstates from the stability analysis of the underlying all-molecule and all-atom eigenstate. In the ground state we find either regular oscillations in size and fraction of both condensates, or shrinking of the atomic condensate that resembles the collapse. In the twin state we find rapid irregular disintegration of both condensates. We contrast the properties of the mean-field parametric excitations found in AMBEC with the experimentally observed excitations of the BEC. We discuss some enticing possibilities for creation of atomic BEC of controllable half-width by performing the parameter sweep around the boundary.