Fresnel integrals and irreversible energy transfer in an oscillatory system with time-dependent parameters

We demonstrate that in significant limiting cases the problem of irreversible energy transfer in an oscillatory system with time-dependent parameters can be efficiently solved in terms of the Fresnel integrals. For definiteness, we consider a system of two weakly coupled linear oscillators in which the first oscillator with constant parameters is excited by an initial impulse, whereas the coupled oscillator with a slowly varying frequency is initially at rest but then acts as an energy trap. We show that the evolution equations of the slow passage through resonance are identical to the equations of the Landau-Zener tunneling problem, and therefore, the suggested asymptotic solution of the classical problem provides a simple analytic description of the quantum Landau-Zener tunneling with arbitrary initial conditions over a finite time interval. A correctness of approximations is confirmed by numerical simulations.