# Half-Spectrum Suppression in Dynamic Resonant Tunneling
## 5 A Formal Derivation in the Quasi-Classical Regime
The solutions of Equation (48), which are related to current suppression, are presented by red lines (solid and dashed), while the solutions of Equation (49), which are related to the current enhancement, are presented by blue lines (solid and dashed). Figure 4: The logarithm of \(|s_{n}|^{2}\) (solution of Equation (27)) as a function of the incoming energy and the output mode number \(n\). Dark color designates high amplitude. FSE, FSS and HSS stand for Full Spectrum Excitation, Full Spectrum Suppression and Half Spectrum Suppression, respectively. The parameters are \(LUl^{1/2}=7\), \(\alpha l^{-1/2}=-1.6\), \(\beta U^{-1/2}=0.08\) and \(\omega/U=0.01009\) and the vectors (28) consist of 71 components, of which only 51 are presented. Figure 5: (a): The current as a function of the incoming energy and the oscillation amplitude. Dark color designates high current. The larger the amplitude the wider the QBSS spectrum, which supports a wider range of incoming energies. (b): An enlargement of the blue area on the left panel but with the analytical solutions of Equations (48) (red curves) and (49) (blue curves) above the numerical results. The parameters are \(LUl^{1/2}=7\), \(\alpha l^{-1/2}=-1.6\) and \(\omega/U=0.002\). Despite the fact that the equations in Section 4 and Equations (48) and (49) were derived, for simplicity, in the high amplitude regime, Figure 5 shows that these equations show high agreement with numerical analysis even in the low amplitude regime. The main results can be clearly seen: the difference between adjacent destructive and constructive solutions is \(\omega\). Two blue lines intersect when \(\frac{\beta\alpha}{\pi}=-2\omega\left(m_{1}+\frac{1}{4}\right)\) and two red lines intersect when \(\frac{\beta\alpha}{\pi}=-2\omega\left(m_{2}-\frac{1}{4}\right)\). On the other hand, a red line intersects a blue line when Equations (48) and (49) are valid, simultaneously with the opposite signs; hence, a red line intersects a blue line when \(\frac{\alpha\beta}{\pi}=2\omega m\), and the gap between adjacent similar scenarios is \(\alpha\Delta\beta=2\pi\omega\). However, Equations (48) and (49) can be valid simultaneously both when Equation (48) has a '\(+\)' sign and Equation (49) has a '\(-\)' sign, or when Equation (48) has a '\(-\)' sign and Equation (49) has a '\(+\)' sign. Both scenarios represent half-spectrum suppressions; however, the first scenario (Equation (48) with a '\(-\)' sign) represents the total suppression of the lower energies, and the second scenario (Equation (48) with a '\(+\)' sign) represents the suppression of the higher energies. Therefore, the gap between the two scenarios is \(\alpha\Delta\beta=\pi\omega\). In Figure 5, two red lines' intersection occurs only when the current is suppressed (see the red curves in Figures 6 and 7). Two blue line intersections occur when the current is enhanced (see the blue curves in Figures 6 and 7). As can be seen from Figures 5 and 6, in both cases the spectrum is symmetric with respect to the central energy of the spectrum, i.e., the spectrum has a similar shape for \(\Omega-\langle\Omega_{R}\rangle=\omega\) and for \(\Omega-\langle\Omega_{R}\rangle=-\omega\). At the intersections of blue and red lines in Figure 5, the current is suppressed. In these cases, the spectrum is anti-symmetric with respect to \(\langle\Omega_{R}\rangle\). As can be seen in Figure 8, the spectrum is flipped: for \(\Omega-\langle\Omega_{R}\rangle=\omega/2\), the lower part of the spectrum vanishes, while for \(\Omega-\langle\Omega_{R}\rangle=-\omega/2\) the higher part of the spectrum vanishes. Figure 6: The same parameters as in Figure 5 but with the amplitude \(\beta U^{-1/2}=0.04909\). The output spectrum is presented for different incoming energies. (**a**) and (**c**): The current suppression criterion: Equation (48) for \(\Omega=\langle\Omega_{R}\rangle\pm\omega\) (the intersection between the red curves in Figure 5b, i.e., destructive interference in both directions). The scenarios (**a**) and (**c**) are enlarged in (**d**) and (**e**), respectively.