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.
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.
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.
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 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.
Sawtooth potentials-piecewise-linear potentials defined by alternating rising and falling slopes-have garnered interest for their roles in classical Brownian ratchets, quantum transport, and flat bands in ultracold-atom lattices.However, a general solution of the time-independent Schrödinger equation for an arbitrary asymmetric sawtooth potential and a study of its band structure remain absent from the literature.We derive the exact analytical eigenstates in terms of Airy functions, but due to numerical difficulty, we solved for the eigenstates using a piecewise constant "staircase approximation."We find localized low-energy states confined within individual wells and plane waves for high energies.We apply the obtained solutions to investigate the relationship between bandwidth and parameters of the sawtooth potentials, which could offer design principles for realizing flatbands in condensed-matter and ultracold-atom research.We find that the bandwidth of the lowest energy band is entirely independent of the sawtooth's asymmetry, while it remains inversely correlated with potential height and cell size.The methodologies presented here provide a toolkit for further exploration of sawtooth potentials.
This study reports tunneling through a one-dimensional (1D) square potential barrier (SPB) under fluctuations in an observer’s frame of reference (OFR). To date, tunneling through an SPB has been studied under the assumption that the OFR remains constant throughout the tunneling measurements; therefore, the change of the tunneling probability when the OFR is assumed to fluctuate remains unanswered. In this paper, a 1D SPB is considered under fluctuations of an OFR. The average transmission probability of a particle through an SBP for two types of OFR fluctuations (periodic-square-wave and periodic-sawtooth-wave fluctuations) is formulated in time representations. Under these types of fluctuations, the average transmission probability gradually increases with a particle’s energy, which is saturated to the transmission probability in the case of the stationary OFR at a much greater energy than the amplitude of the fluctuations. The average transmission probability is much higher at the amplitude of the fluctuations in the case of periodic-square-wave fluctuations. Therefore, the average transmission probability with a particle’s energy has the potential to reveal the distribution of OFR fluctuations.
The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.