Coherent Superposition States of Atoms and Molecules in a Bose-Einstein Condensate with Exactly Balanced Photo-Associations and Photo-Dissociations

We show that there exist a series of coherent superposition states of atoms and molecules in a dilute Bose-Einstein condensate with exactly balanced photo-associations and photo-dissociations, and their analytical expressions are explicitly given. They also correspond to the coherent superposition states of two kinds of photons in optical second harmonic generation processes, which shows exactly balanced down- and up-conversions.