Semiclassical reproducibility of sawtooth structure observed for a periodically perturbed rounded-rectangular potential.

In the previous work [Takahashi and Ikeda, Phys. Rev. E 109, 044203 (2024)2470-004510.1103/PhysRevE.109.044203], we found that tunneling probabilities for a periodically perturbed rounded rectangular potential form a sawtoothlike structure as a function of either the Planck constant ℏ or the angular frequency of the perturbation ω owing to multiquanta absorption tunneling. The replacement of the dominant harmonic channel with the change of either ℏ or ω occurs in every transition region of the sawtooth structure, which causes a sudden change in the tunneling probability. The tunneling probability in the potential region forms a resonance peak reflecting the fundamental resonance scattering state at each edge of the sawtooth structure. In this paper, we explore the underlying mechanism of the sawtooth structure in terms of semiclassics. The semiclassical method reproduces the sawtooth structure except for narrow transition regions accompanied by resonance peaks. The sawtooth structure is constructed by superpositioning a sufficiently large number of complex branches as an analogy of the Fourier decomposition of a sawtoothlike wave. The baseline of tunneling probability, i.e., the average line of the sawtooth structure, is well reproduced by the Melnikov method, i.e., the semiclassical weight estimated based on the theory of stable-unstable manifold guided tunneling. When ℏ and ω are fixed, the average line changes as exponentiation with the base ε, i.e., ∝ε^{α}, where α is roughly estimated as ∼c/ℏω.