Well dynamics within the quantum Kramers model is studied. The validity of the approach based on linear-response theory for the bath of normal modes is explored by including the second-order terms in the classical equations of motion (EOM) for the stable modes. The effect of nonsecular terms describing the linear interaction between the stable modes vanishes upon averaging over the distribution of initial positions and velocities of stable modes. By accounting for the secular terms, the classical EOM’s for the stable modes are reduced to those of forced oscillator with time-dependent frequency. An exact expression for the transition probability kernel is derived using the known quantum-mechanical solution for the model. The time-reversal symmetry of EOM ensures that the kernel satisfies the detailed-balance condition. The corrections to the linear-response-theory results for the kernel and for the average energy loss and its dispersion are exponentially small for the high-frequency modes due to the slow (adiabatic) variation of the frequency. They are important in the case of the continuum spectrum of the bath, when the dominant contribution is due to the low-frequency modes. The results demonstrate the validity of the perturbative approach in the study of well dynamics.

• Received 14 May 1990

DOI:https://doi.org/10.1103/PhysRevA.42.4427