# Half-Spectrum Suppression in Dynamic Resonant Tunneling
## 1 Introduction
If a high-current criterion is valid in the \(t_{1}<t<t_{2}\) domain, i.e.,  
\[\frac{1}{2}\int\limits_{t_{1}}^{t_{2}}\big{|}\Omega-\Omega_{R}\big{(}t^{\prime }\big{)}\big{|}dt^{\prime}=\pi\bigg{(}n_{1}+\frac{1}{4}\bigg{)} \tag{8}\]  
then in the case where \(\Omega=E_{n}\) a high current criterion is valid in the \(t_{2}<t<t_{1}+2\pi/\omega\) domain as well, i.e.,  
\[\frac{1}{2}\int\limits_{t_{2}}^{t_{1}+2\pi/\omega}\big{|}\Omega-\Omega_{R} \big{(}t^{\prime}\big{)}\big{|}dt^{\prime}=\pi\bigg{(}n_{2}+\frac{1}{4}\bigg{)} \tag{9}\]  
\[E_{n}=\left\langle\Omega_{R}\right\rangle+n\omega \tag{2}\]0 Temporal variations of the resonance energy level. **Left panel**: The resonance level varies like a Gaussian perturbation and crosses the particle energy twice (only one extremum). **Right panel**: The resonance level varies periodically. In every period there are two extrema. Therefore, when \(\Omega=E_{n}\) there is a consistency between the two parts of the period. Consequently, when \(\Omega=E_{n}\), the QBSS in DRT behaves like a QBS in a stationary RT process--there is a clear criterion for maximum current and a clear criterion for current suppression. It is clear that (2) and (8) are the two required criteria to achieve the maximum current. However, it is also clear that despite the fact that (2) and (3) are criteria for current suppression, they cannot be sufficient for zero current (Fano resonances [8; 24; 25; 26; 27; 28; 29; 30; 31]). The reason is that like the stationary scenario, the transmission at the minima is not zero. Even in off-resonance cases some particles penetrate the barrier into the well and escape from the well across the barrier. To generate a scenario of zero transmission, i.e., zero current, one must demand suppression criterion in only one part of the period. In this case, the entire spectrum is suppressed but half of it can go to zero. If, without loss of generality, we again take \(\Omega>\Omega_{R}(t)\) for \(t_{2}<t<t_{1}+2\pi/\omega\) (and \(\Omega<\Omega_{R}(t)\) for \(t_{1}<t<t_{2}\)) and we further take the suppression criterion for \(t_{1}<t<t_{2}\) and current-enhancement criterion for \(t_{2}<t<t_{1}+2\pi/\omega\), then both (5) and (9) must be valid simultaneously. In this case,  
\[ \begin{split}\frac{\pi}{\omega}(\Omega-\langle\Omega_{R}\rangle)& =\frac{1}{2}\int\limits_{t_{1}}^{t_{1}+2\pi/\omega}[\Omega-\Omega _{R}(t^{\prime})]dt^{\prime}\\ &=-\frac{1}{2}\int\limits_{t_{1}}^{t_{2}}|\Omega-\Omega_{R}(t^{ \prime})|dt^{\prime}+\frac{1}{2}\int\limits_{t_{2}}^{t_{1}+2\pi/\omega}|\Omega -\Omega_{R}(t^{\prime})|dt^{\prime}=\pi\Big{(}(n_{2}-n_{1})+\frac{1}{2}\Big{)} \end{split}  \tag{10}\]  
Therefore, this specific scenario can occur only between the QBSS energy levels, i.e., when  
\[\Omega=E_{n+1/2}=\langle\Omega_{R}\rangle+\bigg{(}n+\frac{1}{2}\bigg{)}\omega \tag{11}\]  
where in this case \(n=n_{2}-n_{1}\). Clearly, the same conclusion appears when the suppression criterion is applied for the \(t_{2}<t<t_{1}+2\pi/\omega\) part, and current enhancement for the \(t_{1}<t<t_{2}\) part. Below, we will investigate these cases. We will show that in the unique scenarios where the incoming particle energy is exactly between the QBSS sub levels, i.e., \(\Omega=E_{n+1/2}=\langle\Omega_{R}\rangle+(n+1/2)\omega\), and (5) or (6) (but not both) holds, then the spectrum is split into two. Both are suppressed, but half is suppressed to zero (see Figure 3). Consequently, if the suppressed part is the high-energy one, and the incoming energy is lower than the oscillating frequency, then the current reaches zero. Figure 3: Upper panel (**a**): the system fulfills the criterion which completely suppresses the lower part of the spectrum. Lower panel (**b**): the higher part of the spectrum is completely suppressed.