Classical and quantum targeted energy transfer between nonlinear oscillators

Complete energy exchange between two weakly coupled classical Klein–Gordon anharmonic oscillators is investigated. According to the recently introduced theory of targeted energy transfer [Physica B 296 (2001) 222; Phys. Rev. Lett. 87 (2001) 165501], the condition for a specific amount of energy initially on the donor to be completely transferred to the acceptor is that the detuning function, defined as the variation of the oscillators energy during a transfer conserving the total action, is bounded by a certain function determined by the coupling. We show that, for given donor with given initial energy, the potential of the acceptor can be numerically tuned so that the detuning function becomes exactly zero. Then, complete energy transfer occurs in the limit of zero coupling. We also show that it is sufficient to fit the coefficient of the nonlinearity of the acceptor at lowest order for obtaining a small, although nonzero, detuning function. Then, total energy transfer can be obtained at coupling larger than a nonzero but small critical value. Quantum targeted energy transfer is also investigated in the discrete nonlinear Schrödinger dimer and in the Klein–Gordon dimer. In both cases, it is found that a quantum wave packet on the donor at the classical energy for targeted transfer is transferred to the acceptor, as in the classical case. The wave packet oscillates back and forth between donor and acceptor but loses its coherence after some number of oscillations which tends to be large when approaching the classical limit. In the Klein–Gordon dimer the energy transfer cannot be complete, since the zero point energy of the donor cannot be removed. Quantum targeted transfer appears to be related to the existence of a quantum pathway of almost degenerate eigenenergies the variation of which becomes the detuning function in the classical limit.