Selftrapping and Quantum Fluctuations in the Discrete Nonlinear   Schrodinger Equation
  We study the robustness of the selftrapping phenomenon exhibited by the Discrete Nonlinear Schrodinger (DNLS) equation against the effects of nonadiabaticity and quantum fluctuations in a two-site system (dimer). To test for nonadiabatic effects (in a semiclassical context), we consider the dynamics of an electron (or excitation) in a dimer system and coupled to the vibrational degrees of freedom, modeled here as classical Einstein oscillators of mass M. For relaxed (coherent state) oscillators initial condition, the DNLS selftrapping transition persists for a wide range of M spanning 5 decades. When undisplaced initial conditions are used, the selftrapping transition is destroyed for masses greater than 0.02 . To test for the effects of quantum fluctuations, we performed a first-principles numerical calculation for the fully quantum version of the above system: the two-site Holstein model. We computed the long-time averaged probability for finding the electron at the initial site as a function of the asymmetry and nonlinearity parameters, defined in terms of the electron-phonon coupling strength and the oscillator frequency. Substantial departures from the usual DNLS system are found: A complex landscape in asymmetry-nonlinearity phase space, which is crisscrossed with narrow "channels", where the average electronic probability on the initial site remains very close to 1/2, being substantially larger outside. In the adiabatic case, there are also low-probability "pockets" where the average electronic probability is substantially smaller than 1/2. 
C. A. Bustamante and M. I. Molina