# Half-Spectrum Suppression in Dynamic Resonant Tunneling
## 5 A Formal Derivation in the Quasi-Classical Regime  
\(\chi_{n}\) of Equation (24) in the quasi-classical approximation, i.e., \(\omega<<U-\Omega\), reads  
\[\chi_{n}\cong-2\sqrt{U-\Omega_{R}-\Omega+\Omega_{R}-n\omega}\cong-2\rho_{0}+( \Omega-\Omega_{R}+n\omega)/\rho_{0} \tag{37}\]where \(\rho_{0}=\sqrt{U-\Omega_{R}}\) and  
\[\frac{4}{\beta}\rho_{0}\delta(n)=2\frac{\omega}{\rho_{0}}\bigg{(}\frac{\big{[} \rho_{0}(\alpha+2\rho_{0})-\Omega+\Omega_{R}\big{]}/\omega-n}{\beta}\bigg{)}s_{n }+(s_{n+1}+s_{n-1}), \tag{38}\]  
respectively. Therefore, using the notations  
\[\Delta=[\rho_{0}(\alpha+2\rho_{0})-\Omega+\Omega_{R}]/\omega\text{ and }z\equiv\beta\rho_{0}/\omega \tag{39}\]  
the solution of (38) can be written as follows:  
\[s_{n}=\left\{ \begin{array}{ll}AJ_{n-\Delta}(\beta\rho_{0}/\omega)&n>0\\ BJ_{-n+\Delta}(\beta\rho_{0}/\omega)&n<0\end{array} \right. \tag{40}\]  
where \(A\) and \(B\) are coefficients, and \(J_{\
u}(z)\) are the Bessel function (see, for example, Ref. [34]). Matching the solution at \(n=0\) yields  
\[AJ_{-\Delta}(\beta\rho_{0}/\omega)=BJ_{+\Delta}(\beta\rho_{0}/\omega) \tag{41}\]  
and  
\[\frac{4}{\beta}\rho_{0}=(AJ_{-\Delta+1}(\beta\rho_{0}/\omega)-BJ_{+\Delta+1}( \beta\rho_{0}/\omega)) \tag{42}\]  
After solving Equations (41) and (42), the final solution reads  
\[s_{n}=\left\{ \begin{array}{ll}WJ_{+\Delta}(\beta\rho_{0}/\omega)J_{n-\Delta} (\beta\rho_{0}/\omega)&n>0\\ WJ_{-\Delta}(\beta\rho_{0}/\omega)J_{-n+\Delta}(\beta\rho_{0}/\omega)&n<0\end{array} \right. \tag{43}\]  
where  
\[W=\frac{4\rho_{0}/\beta}{J_{+\Delta}(\beta\rho_{0}/\omega)J_{-\Delta+1}(\beta \rho_{0}/\omega)-J_{+\Delta+1}(\beta\rho_{0}/\omega)J_{-\Delta}(\beta\rho_{0} /\omega)} \tag{44}\]  
Therefore, to suppress the high energy part of the spectrum, one requires  
\[J_{+\Delta}(\beta\rho_{0}/\omega)=0 \tag{45}\]  
and to suppress the lower part of the spectrum,  
\[J_{-\Delta}(\beta\rho_{0}/\omega)=0 \tag{46}\]  
Clearly, Equations (45) and (46) suppress the half spectrum to zero. In the regime where \(z\equiv\beta\rho_{0}/\omega>>1\), i.e., with slowly varying approximation and large oscillating amplitudes, one can use the approximation [34]\(J_{\
u}(z)\sim\sqrt{2/(\pi z)}\cos\!\left(z-\frac{1}{2}v\pi-\frac{1}{4}\pi\right)\) to obtain equivalent expressions to (34) and (36), i.e.,  
\[\frac{\beta\rho_{0}}{\pi}\pm\frac{1}{2}\big{[}\rho_{0}(\alpha+2\rho_{0})- \Omega+\langle\Omega_{R}\rangle\big{]}=\omega\bigg{(}m-\frac{1}{4}\bigg{)} \tag{47}\]  
And again, since \(2\sqrt{U-\langle\Omega_{R}\rangle}-|\alpha|<<\beta\) and \(\beta<<|\alpha|\), then up to the second order of \(\beta\)  
\[\Omega=\langle\Omega_{R}\rangle\pm\left[\frac{\beta\alpha}{\pi}+2\omega\bigg{(} m-\frac{1}{4}\bigg{)}\right] \tag{48}\]  
Therefore, the upper sign (+) leads to Equation (34) and the lower sign (-) leads to Equation (36). Similarly, an enhancement criterion leads to  
\[\Omega=\langle\Omega_{R}\rangle\pm\left[\frac{\beta\alpha}{\pi}+2\omega\bigg{(} m+\frac{1}{4}\bigg{)}\right] \tag{49}\]where, again, the upper sign (+) leads to Equation (34) and the lower sign (-) leads to Equation (36), as should be expected. In Figure 4, the logarithm of the coefficient \(|s_{n}|^{2}\) is presented as a function of both the incoming particle's energy \(\Omega\) and the spectral mode \(n\). As can be seen, all three patterns appear. When \(\Omega=\langle\Omega_{R}\rangle+n\omega\), then when (49) is valid, Full Spectrum Excitation (FSE) is achieved. When (48) is valid, then Full Spectrum Suppression (FSS) is reached. On the other hand, when \(\Omega=\langle\Omega_{R}\rangle+(n+0.5)\omega\), then when (48) is valid the Half Spectrum Suppression (HSS) is reached. In Figure 5, the current is presented as a function of the incoming particle's energy and the oscillation amplitude. Therefore, it presents the dependence of the current spectrum on the oscillations' amplitude. As should be expected, the spectrum widens with the corresponding increase in the amplitude. On the right panel of Figure 5, the solutions of Equations (48) and (49) are presented.