Semiclassical mechanism of sawtooth structure tunneling

We investigate the tunneling mechanism in a rounded-step potential under a time-periodic perturbation, where instanton tunneling is substantially prohibited. The tunneling probability exhibits a characteristic sawtoothlike structure as a function of the inverse Planck's constant 1/ℏ. This structure arises from the multiquanta absorption tunneling mechanism, specifically owing to the replacement of the dominant harmonic channel at every edge. Indeed, the replacement causes a sudden change in tunneling probability. Furthermore, the interference between the dominant and subdominant harmonic channels plays a crucial role in forming detailed properties, specifically probability oscillations superposed on the sawtooth structure. By employing a semiclassical approach, we show that the sawtooth structure is reproduced by the superposition of many complex branches, i.e., complex classical trajectories over many periods of the perturbation. The semiclassical decomposition of the wave operator to many complex branches is analogous to frequency decomposition and reproduces the sawtooth structure. We analyze the convergence of the semiclassical results to the quantum calculations as the number of time periods increases and elucidate the underlying mechanism of constructing the sawtooth structure from the viewpoint of semiclassics. Furthermore, the semiclassical method reproduces the probability oscillations induced by the interference between dominant and subdominant harmonic channels.