Quantum state exchange between indirectly coupled modes

The problem of quantum state exchange between three coupled quantum harmonic oscillators (which can represent interacting modes of the electromagnetic field in nonlinear crystal or interaction between laser beams and vibrational modes of ions in traps) is considered. We review different definitions of state exchange and find conditions of its possibility for arbitrary initial states in a chain of three oscillators with bilinear resonance coupling (corresponding to the rotating wave approximation), analyzing the structure of integrals of motion and propagators of the Schroedinger equation, as well as the Wigner functions. The influence of different parameters (such as detunings between coupling constants and temperatures of the intermediate oscillator and 'receiver') on the fidelity of exchange is analyzed