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.
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.
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.
Sawtooth structures are observed in tunneling probabilities with changing Planck's constant for a periodically perturbed rounded-rectangular potential with a sufficiently wide width for which instanton tunneling is substantially prohibited. The sawtooth structure is a manifestation of the essential nature of multiquanta absorption tunneling. Namely, the periodic perturbation creates an energy ladder of harmonic channels at E_{n}=E_{I}+nℏω, where E_{I} is an incident energy and ω is an angular frequency of the perturbation. The harmonic channel that absorbs the minimum amount of quanta of n=n[over ¯], such that V_{0}<E_{n[over ¯]}≤V_{0}+ℏω, makes a dominant contribution to the tunneling process, where V_{0} is the height of the static potential and V_{0}-E_{I}≫ℏω. At each steep slope part of the sawtooth structure, replacement of the dominant harmonic channel, i.e., E_{n[over ¯]}→E_{n[over ¯]+1}, occurs and the tunneling probability suddenly drops with increasing 1/ℏ. Due to the flatness of the potential top, resonance eigenstates exist just above the potential and the first resonance state appears as the peak of each edge of the sawtooth structure for the tunneling probability in the potential region. Sawtooth structures are also observed with changing the perturbation frequency. We introduce an effective formula to characterize the basic profile of those sawtooth structures.
Tunneling essentially different from instanton-type tunneling, say noninstanton tunneling, is studied from both semiclassical and quantum viewpoints. Taking a periodically perturbed rounded-off step potential for which the instanton-type tunneling is substantially prohibited, we analyze change of the tunneling probability with change of the perturbation frequency based on the stable-unstable manifold-guided tunneling (SUMGT) theory, which we have recently introduced. In the large and small limits of the frequency, the tunneling rate rapidly decays, but it is markedly enhanced in an intermediate range. We will also make a quantum interpretation of the noninstanton tunneling by using an exactly solvable model--a periodically perturbed right-angled step potential. Analysis with this model shows that SUMGT is considered as a sort of photoassisted tunneling through a large energy gap induced with absorbing a huge number quanta, which is completely different from the instanton-type tunneling. Both approaches from the semiclassical and quantum viewpoints complement each other to cause a better understanding of noninstanton tunneling.
In recent papers we demonstrated a new class of barrier tunneling phenomena which can be observed in periodically driven one dimensional barrier systems (i.e., 1.5D barrier tunneling systems). In such systems the tunneling component is in general accompanied by remarkable interference fringe patterns as the perturbation strength exceeds a certain characteristic value. Such a phenomenon is quite general and is not restricted to periodically driven 1D systems, and it may be also observed in intrinsic two dimensional autonomous systems. The complex-domain semiclassical theory developed by ourselves was applied to the problem, and we succeeded in reconstructing every detail of the fringed tunneling wavefunction only by using the complexified classical tunneling trajectories. This fact greatly motivates us to investigate the mechanism of the tunneling in terms of “classical mechanics”. The later part of the present paper is devoted to asserting the generality of the “classical mechanism” of the fringed tunneling. We develop detailed semiclassical analyses in the complex domain and demonstrate that the key trajectories contributing to the fringed tunneling are quite different from the instanton-like trajectories, which works only in unperturbed or weakly-perturbed regimes. Indeed, such trajectories are traveling in the complex phase space being guided by the complexified stable and unstable manifolds which exhibit heteroclinic-like entanglement with the incoming wave manifolds.
In multidimensional barrier tunneling, there exist two different types of tunneling mechanisms, instanton-type tunneling and noninstanton tunneling. In this paper we investigate transitions between the two tunneling mechanisms from the semiclassical and quantum viewpoints taking two simple models: a periodically perturbed Eckart barrier for the semiclassical analysis and a periodically perturbed rectangular barrier for the quantum analysis. As a result, similar transitions are observed with change of the perturbation frequency ω for both systems, and we obtain a comprehensive scenario from both semiclassical and quantum viewpoints for them. In the middle range of ω, in which the plateau spectrum is observed, noninstanton tunneling dominates the tunneling process, and the tunneling amplitude takes the maximum value. Noninstanton tunneling explained by stable-unstable manifold guided tunneling (SUMGT) from the semiclassical viewpoint is interpreted as multiphoton-assisted tunneling from the quantum viewpoint. However, in the limit ω→0, instanton-type tunneling takes the place of noninstanton tunneling, and the tunneling amplitude converges on a constant value depending on the perturbation strength. The spectrum localized around the input energy is observed, and there is a scaling law with respect to the width of the spectrum envelope, i.e., the width ∝ℏω. In the limit ω→∞, the tunneling amplitude converges on that of the unperturbed system, i.e., the instanton of the unperturbed system.
The fully complex domain semiclassical theory based upon the complexified stable-unstable manifold theory, which we have developed in our recent studies, is successfully applied to explain anomalous tunneling phenomena numerically observed in a periodically modulated round-off-step potential. Numerical experiments show that tunneling through the oscillating step potential is characterized by a spatially nondecaying tunneling tail and an anomalously slow relaxation. The key is the existence of a critical trajectory exhibiting singular behavior, and the analysis of neighboring trajectories around it reproduces the essence of such anomalous phenomena.