A Classical Analog of Landau-zener Tunneling as a New Type of the Mechanical Energy Trap

A detailed analytical study of irreversible energy transfer in an oscillatory system with time-dependent parameters has not been addressed thus far in the literature. This paper demonstrates a closed-form asymptotic solution of this problem for a system of two weakly coupled linear oscillators; the first oscillator with constant parameters is excited by an initial impulse, whereas the coupled oscillator with a time-dependent frequency is initially at rest but then acts as an energy trap. It is shown that in physically meaningful limiting cases the problem of irreversible energy transfer from the excited oscillator to the trap is reduced to a first-order equation with the solution in the form of the Fresnel integrals. In view of a mathematical analogy between energy transfer in a classical oscillatory system with variable parameters and non-adiabatic quantum Landau-Zener transition, the results of this paper, in addition to providing an analytical framework for understanding the transient dynamics of coupled oscillators, suggest an approximate procedure for solving the linear LandauZener problem with arbitrary initial conditions over a finite time-interval. A correctness of approximations is confirmed by numerical simulations.