Effects of resonance states in barrier region on non-exponential decay of wave-packets scattered by rounded-rectangular potentials

Abstract

The decay processes of wave-packets scattered by periodically perturbed and unperturbed rounded-rectangular potentials are studied numerically and theoretically, when the widths of the potentials L are very large. For the case of the unperturbed potentials, four different stages successively arise in the decay process of the wave in the potential region: two pre-exponential decays, namely power–law decay of t −3 and oscillating power–law decay, exponential decay and post-exponential decay, which is also power–law decay of t −3. The post-exponential decay is usually extremely small in magnitude. The characteristics of the pre-exponential and exponential decays are explained with the properties of resonance states, i.e. the Gamow states, for the unperturbed system. The rate of the exponential decay is determined by the imaginary part of the eigenenergy of the first resonance state. For the two pre-exponential decays, the ending time of the t −3 decay is a linear function of L and that of the oscillating power-law decay is proportional to L 3. In the limit of L → ∞, the t −3 decay is observed persistently, namely the decay for the rounded-step potential. For the perturbed potentials, even if the average energy of an initial wave-packet is relatively smaller than the oscillating potential, the noninstanton tunnelling, i.e. the multi-quanta absorption tunnelling, raises the tunnelling wave component up to the oscillating top of the rounded-rectangular potential, and the tunnelling probability rapidly increases with the perturbation strength. The properties of the resonance states are almost the same as those of the Gamow states because of the flatness of the potential top. As a result, the decay process after the tunnelling is almost the same as that for the unperturbed system. It is suggested that the tunnelling amplitude and tunnelling time, namely the amplitude and period of the pre-exponential decay, can be controlled by the perturbation strength and the potential width, respectively.

Quantum Tunneling and Complex Dynamics in the Suris’s Integrable Map

“…On the other hand, one can find nonlinear integrable Hamiltonian flow systems, while nonlinear integrable maps are rather rare. The instanton is a path describing the tunneling transition in continuous Hamiltonian flow systems, and it is sometimes used as a reference when comparing the behavior of tunneling in integrable and nonintegrable systems [30][31][32]. In contrast, to the authors' knowledge, there has not yet been an analysis of the tunneling effect for the map that is obtained by perturbing an integrable map.…”