Entanglement Properties of a Three-Mode Atom-Molecule Bose-Einstein Condensates System Considering the Interactions due to the s-wave Intramodal Elastic Scattering

Ultracold atoms in the atomic Bose-Einstein condensate (ABEC) state can form molecular Bose-Einstein condensate (MBEC) through photoassociation. In the atom-molecule Bose-Einstein condensates (BECs), two or more atoms in the ABEC can combine to form a molecule in the MBEC and again a molecule from a MBEC can decompose to atoms in the ABEC. The Bose-stimulated Raman adiabatic passage is an efficient mechanism for conversion of an atomic BEC to a molecular BEC. A three-mode atom-molecule Bose-Einstein condensates system can be prepared through the photoassociative Bose-stimulated Raman adiabatic passage. In our system, three modes are one ABEC, one excited MBEC, and one stable MBEC. The intramodal interactions due to the χ(3) nonlinearity is present in all three BEC modes along with ABEC-excited MBEC and excited MBEC-stable MBEC intermodal couplings. The quantum mechanical Hamiltonian of the system is constructed considering all three intraspecies interactions and intermodal couplings among the modes. The Hamiltonian of the system is solved analytically using a special intuitive approach which is more general and gives more accurate result than the well-known short time approximation method. The correctness of the solution is verified through the equal time commutation relation. Staring from a three-mode composite coherent state we compute the time evolution of the field annihilation operators of all three modes in presence of all possible interactions and couplings. Using these solutions, we investigate the quantum entanglement properties of the system for all three two-mode combinations. Entanglement is found for two combinations of modes, where as one combination is always separable. Also, we study the dependence of the entanglement properties of the system with the interaction and coupling constants.