# Half-Spectrum Suppression in Dynamic Resonant Tunneling
## 4 A Solution in the Slowly Varying Approximation  
When the slowly varying approximation of the Sisyphus effect (Equation (16)) is applied to Equation (14), i.e., to the harmonically oscillating case, we find  
\[\begin{split}\Big{(}\Omega-U+\frac{2\alpha^{2}+\beta^{2}}{8} \Big{)}t_{1}+\frac{\beta^{2}}{16\omega}\Big{[}\sin(2\omega t_{1})+8\frac{ \alpha}{\beta}\sin(\omega t_{1})\Big{]}=\pi\Big{(}n_{1}\mp\frac{1}{4}\Big{)} \\ \text{where}\ t_{1}=\frac{1}{\omega}\arccos\Big{(}\frac{2\sqrt{U- \Omega}-\alpha}{\beta}\Big{)}\end{split} \tag{29}\]where the \(-/+\) sign corresponds to Equations (3) and (8), respectively, i.e., the minus corresponds to current suppression and the plus corresponds to current enhancement. After substituting \(t_{1}\) in Equation (29), the equation can be rewritten:  
\[\begin{array}{l}\pi\Big{(}n_{1}\mp\frac{1}{4}\Big{)}=\\ \frac{1}{\omega}\left[\frac{\beta}{8}\Big{(}3\alpha+2\sqrt{U-\Omega}\Big{)} \sqrt{1-\frac{\left(\alpha-2\sqrt{U-\Omega}\right)^{2}}{\beta^{2}}}+(\Omega- \langle\Omega_{R}\rangle)acos\Big{(}\frac{2\sqrt{U-\Omega}-\alpha}{\beta} \Big{)}\right]\end{array} \tag{30}\]  
since  
\[\frac{1}{2}\int\limits_{-\pi/\omega}^{\pi/\omega}\Big{[}\Omega-U+\frac{1}{4}( \alpha+\beta\cos\omega t)^{2}\Big{]}dt=\left(\frac{\alpha^{2}+\beta^{2}/2}{4} +(\Omega-U)\right)\frac{\pi}{\omega}=(\Omega-\langle\Omega_{R}\rangle)\frac{ \pi}{\omega} \tag{31}\]  
Equivalently, the same criterion applied in the second part of the period leads to  
\[\begin{array}{l}\pi\Big{(}n_{2}\mp\frac{1}{4}\Big{)}=\\ \frac{1}{\omega}\Bigg{[}\frac{\beta}{8}\Big{(}3\alpha+2\sqrt{U-\Omega}\Big{)} \sqrt{1-\frac{\left(\alpha-2\sqrt{U-\Omega}\right)^{2}}{\beta^{2}}}+(\Omega- \langle\Omega_{R}\rangle)\Big{[}acos\Big{(}\frac{2\sqrt{U-\Omega}-\alpha}{ \beta}\Big{)}-\pi\Big{]}\Bigg{]}\end{array} \tag{32}\]  
To simplify the expressions, let us focus on the large oscillating amplitude regime (\(2\sqrt{U-\Omega}-\alpha<<\beta\)), since \(\Omega-\langle\Omega_{R}\rangle<<\alpha^{2}\) and \(|\beta|<<|\alpha|\) (note that \(\alpha\) is negative); then, Equation (30) reduces to  
\[(\Omega-\langle\Omega_{R}\rangle)\frac{\pi}{2\omega}-\frac{\alpha\beta}{2 \omega}=\pi\bigg{(}n_{1}\mp\frac{1}{4}\bigg{)} \tag{33}\]  
and therefore, the energy must be  
\[\Omega=\langle\Omega_{R}\rangle+\frac{\alpha\beta}{\pi}+2\omega\bigg{(}n_{1} \mp\frac{1}{4}\bigg{)} \tag{34}\]  
Similarly, the suppression/enhancement criterion applied in the second part of the period (following Equation (30)) leads to  
\[(\Omega-\langle\Omega_{R}\rangle)\frac{\pi}{2\omega}+\frac{\alpha\beta}{2 \omega}=-\pi\bigg{(}n_{2}\mp\frac{1}{4}\bigg{)} \tag{35}\]  
and therefore the energy must be  
\[\Omega=\langle\Omega_{R}\rangle-\frac{\alpha\beta}{\pi}-2\omega\bigg{(}n_{2} \mp\frac{1}{4}\bigg{)} \tag{36}\]  
Therefore, (34) and (36) can both be valid with the same sign, provided that \(\Omega=\Omega_{m}=\langle\Omega_{R}\rangle+m\omega\) and \(m=n_{1}-n_{2}\). Therefore, as was shown above, when the energy coincides with one of the energies \(\Omega_{m}\) then the entire spectrum is suppressed or is excited.  
On the other hand, when \(\Omega=\Omega_{m+1/2}=\langle\Omega_{R}\rangle+(m+1/2)\omega\), then if one part of the spectrum is suppressed (say the lower one), then the other side (say the higher one) is excited and vice versa.