# Half-Spectrum Suppression in Dynamic Resonant Tunneling
## 1 Introduction  
The subject of tunneling suppression has been studied for decades. It has been shown that the tunneling current between two wells can be totally suppressed [1; 2]. This finding was surprising not only due to its counter-intuitiveness, but also since it contradicts previously reported behavior of a similar system [3; 4]. Later on, it was presented in a simple two-level solvable system [5; 6]. At about the same time, it was shown that an oscillating delta function potential at certain energies shows zero transmission [7]. A similar phenomenon was found in more complex systems, such as in Dynamic Resonant Tunneling (DRT) processes (see Figure 1); recently, it was shown that zero transmission appears under certain conditions in dynamic tunneling processes [8]. The current paper shows that zero transmission can only appear when half of the spectrum is completely suppressed. Again, this was counter-intuitive, since it was anticipated that due to the high sensitivity of resonant tunneling current on the system's parameters, any spectrum widening would improve the probability of penetrating the opaque barrier. Indeed, many studies showed both experimentally and theoretically that oscillations and potential variations increase tunneling probability [9; 10; 11; 12; 13; 14; 15; 16; 17]. However, when the resonant level varies in time, at certain incoming particle energies, which obey a quantization rule, the tunneling current is highly suppressed [18; 19; 20; 21; 22; 23]. Despite the fact that these quantization rules show a high agreement with exact numerical analysis, since these criteria are approximate analytical expressions, it is not clear whether they predict a criterion for complete suppression, i.e., zero current, or if they predict significant suppression, albeit where the current does not vanish. Recently, it has been shown that DRT can be analyzed similarly to a stationary RT process [22]. In an RT process, the particle tunnels via a quasi-bound state (QBS), which is created due to the presence of a well inside the barrier. When the incoming particle's energy (\(\Omega\)) matches the resonance energy \(\Omega=\Omega_{R}\), transmission is high, while when there is no match the tunneling current is suppressed drastically. In a DRT process, the well oscillates, and therefore the QBS turns into a Quasi-Bound-Super-State (QBSS), which consists of an infinite array of coupled sub-states. Therefore, in accordance with the stationary model, when the incoming particle energy agrees with that of the sub-state energies, whose spectral component is high, the entire QBSS is excited and the total tunneling current is high, while when there is no match, the current is suppressed. That is, if \(\left|\Phi_{\rm QB}\right>\) represents the QBSS,then therefore the probability of an incoming particle with energy \(\Omega\) tunneling out with energy \(\Omega_{out}\) is  
\[p_{\Omega_{in},\Omega_{out}}\sim\left|\left\langle\Omega\right|\Phi_{QB}\right\rangle \right|^{2}\left|\left\langle\Phi_{QB}\right|\Omega_{out}\right\rangle\right|^{2} \tag{1}\]  
When the well oscillates harmonically, the incoming particle can gain or lose energy quanta. Consequently, the spectrum of the particle consists only of discrete energy levels. Therefore, this model is useful when there is an agreement between \(\left\langle\Omega_{R}\right\rangle\) (the mean value of the varying resonance energy level) and one of the particle's discrete spectral components. In this case, the QBSS \(\left|\Phi_{QB}\right\rangle\) is well-defined, and its spectrum consists of the discrete levels  
\[E_{n}=\left\langle\Omega_{R}\right\rangle+n\omega \tag{2}\]  
where \(\omega\) is the well's oscillating frequency, \(n\) is an integer and \(\left\langle\Omega_{R}\right\rangle\equiv\left(\omega/2\pi\right)\int_{0}^{2 \pi/\omega}\Omega_{R}(t)dt\) is the mean resonance energy level. In which case, if the component \(\left|\left\langle\Omega\right|\Phi_{QB}\right\rangle\right|\) is high, the current is high, and when it is low the current is low as well.