# Multiple Fano Resonances in Dynamic Resonant Tunneling Processes
## 7 Strong Oscillations and Opaque Barrier Regime  
In the wide barrier regime, i.e., when the barrier is opaque, the dependence of the resonance energy on the barrier's width \(L\) is exponentially small and can therefore be neglected. Furthermore, in this regime, the particle is approximately trapped in the well and the QBSS is well defined. It has been shown that the transmission via this state is extremely sensitive to the incoming particle's energy (\(\Omega\)). For some values of \(\Omega\), which are consistent with the spectral component of the QBSS, the QBSS is highly excited and, consequently, the tunneling current is very high. Similarly, for other energy values, which are inconsistent with the spectral components of the QBSS, the particle cannot be trapped and the tunneling  
Figure 12: The current as a function of the particle’s incoming energy \(\Omega\) and the barrier’s width \(L\). The black dashed curves on the lower panel are the analytical solutions of Equation (27) for \(m=2\), \(3\). These are the same parameters as in Figure 11.  
current is highly suppressed. These scenarios have been well investigated [10; 14; 18]. In particular, it has been found that current suppression occurs for the energies:  
\[\int\limits_{t_{1}}^{t_{2}}\bigl{(}\Omega-\Omega_{R}\bigl{(}t^{\prime}\bigr{)} \bigr{)}dt^{\prime}=\biggl{(}m-\frac{1}{4}\biggr{)}\pi \tag{31}\]  
where, for the case under study, i.e., Equation (7),  
\[\Omega_{R}(t)=U-\frac{(\alpha+\beta\cos(\omega t))^{2}}{4} \tag{32}\]  
is the instantaneous resonance energy of the well and \(t_{1}\) and \(t_{2}\) are the solutions of \(\Omega=\Omega_{R}(t)\)[10; 14; 18], i.e., these are the instances in which the incoming particle's energy coincides with the instantaneous resonant state's energy. The solutions of (31) for the case (32) are:  
\[\Omega_{m}\cong U-\biggl{(}\frac{\alpha^{2}}{4}+\frac{\beta^{2}}{8}\biggr{)}- \biggl{[}\frac{|\alpha|\beta}{\pi}-2\omega\biggl{(}m-\frac{1}{4}\biggr{)} \biggr{]} \tag{33}\]  
Similarly, for a given incoming energy, the oscillating amplitudes, which correspond to the current suppression, are:  
\[\beta_{m}=-\frac{4|\alpha|}{\pi}+\sqrt{\frac{16\alpha^{2}}{\pi^{2}}+8\biggl{(} U-\Omega-\frac{\alpha^{2}}{4}+2\omega\biggl{(}m-\frac{1}{4}\biggr{)}\biggr{)}} \tag{34}\]  
However, although this expression has been found to be in good agreement with numerical simulations, it only predicts current suppression. It does not predict _zero_ current. The reason for this is that, as has been explained in refs. [10; 11], at the suppression energies, only the upper half of the spectrum (\(\Omega_{out}\geq\Omega_{in}\)) is suppressed. Therefore, in a case where the incoming energy is higher than the oscillating frequency \(\Omega>\omega\), the lower part of the spectrum still includes propagating modes, since the particle can lose a quantum \(\omega\) and still escape from the well. To prevent this, the existence of a Fano resonance (and zero transmission) requires the additional condition (additional to (31)) that \(\Omega<\omega\), in which case, the output spectrum consists of only negative energies. Therefore, positive energies cannot propagate and the incoming particles are completely reflected. This required additional condition is consistent with the explanation that follows Equation (23). The difference between these two kinds of suppression (low current vs. zero current) is illustrated in Figure 13, where the current, as a function of the incoming energy \(\Omega\) and a function of the oscillation amplitude \(\beta\), is presented. When the incoming energies are lower than the oscillation frequency \(\Omega<\omega\), a total reflection appears for appropriate amplitudes. In the lower panel, the analytical results of Equation (33) are presented by dashed black curves above the numerical results. The analytical results represent suppression scenarios; however, the suppressions' characteristics are fundamentally different in the regions \(\Omega<\omega\) and \(\Omega>\omega\).