# Multiple Fano Resonances in Dynamic Resonant Tunneling Processes
## 3 Total Reflection via Narrow Barriers  
From the stationary problem above, we learn that barrier narrowing shifts the resonance to lower energies and even negative ones. Clearly, in this stationary scenario, it is not very useful, since there is no access to the bound state and the resonance states are too spectrally wide. However, when the well oscillates, the particle can lose oscillation quanta and be transferred, at least temporally, to the bound state. Consequently, a Fano resonance can be generated. In the dynamic scenario, an oscillating component is added, and the Schrodinger equation reads:  
\[-\frac{\partial^{2}}{\partial x^{2}}\psi+U(x)\psi+(\alpha+\beta\cos\omega t) \delta(x)\psi=i\frac{\partial\psi}{\partial t} \tag{7}\]  
The dynamic system is illustrated in Figure 3.  
The solution of Equation (7) can be written as a superposition of discrete energy expressions:  
\[\psi(x,t)=\left\{ \begin{array}{cc}\varphi_{0}^{+}\exp(-i\Omega_{0}t)+\sum \limits_{n=-\infty}^{\infty}r_{n}\varphi_{n}^{-}\exp(-i\Omega_{n}t)&x<0\\ \sum\limits_{n=-\infty}^{\infty}t_{n}\varphi_{n}^{+}\exp(-i\Omega_{n}t)&x>0 \end{array} \right. \tag{8}\]  
Figure 2: Contour plot of the transmission coefficient of the barrier as a function of both the particle’s energy \(\Omega\) and the barrier’s width \(L\) (for the case \(\alpha U^{-1/2}=-1.9\)). The darker the color, the higher the transmission coefficient. The dashed red curve represents Equation (5). In the positive energy regime it represents a resonant state and in the negative energy regime it represents a bound state. where \(\varphi_{n}^{\pm}\) are the homogeneous solutions of waves that propagate from left to right \((\varphi_{n}^{+})\) and from right to left \((\varphi_{n}^{-})\), i.e.,  
\[\varphi_{n}^{+}\rightarrow\tau_{\Omega_{n}}\exp[ik_{n}x-i\Omega_{n}t]\ (for\,x \rightarrow\infty) \tag{9}\]  
\[\varphi_{n}^{-}\rightarrow\tau_{\Omega_{n}}\exp[-ik_{n}x-i\Omega_{n}t]\ (for\,x \rightarrow-\infty) \tag{10}\]  
where \(k_{n}=\sqrt{\Omega_{n}}\), \(\Omega_{n}\equiv\Omega+n\omega\). \(|\tau_{\Omega}|^{2}\) is the probability of penetrating the stationary barrier with energy \(\Omega\), and finally, \(r_{n}\) and \(t_{n}\) are the modes' coefficients. It is worth noting the difference between the coefficients of the modes, i.e., \(r_{n}\) and \(t_{n}\), which need to be found, and the transmission coefficient of a single mode \(\tau_{\Omega_{n}}\), which is obtained by solving the stationary case. Using the boundary conditions at \(x=0\):  
\[\psi_{l}(0)=\psi_{r}(0) \tag{11}\]  
\[\psi^{\prime}{}_{l}(0)-\psi^{\prime}{}_{r}(0)=-(\alpha+\beta\cos(\omega t)) \psi_{r}(0) \tag{12}\]  
where the tags stand for spatial derivations. The problem reduces to the following difference equation:  
\[-\chi_{n}\delta(n)=(\alpha-\chi_{n})s_{n}+\frac{\beta}{2}(s_{n+1}+s_{n-1}) \tag{13}\]  
where \(s_{n}\equiv t_{n}\big{(}\varphi_{n}^{+}/\varphi_{0}^{+}\big{)}\),  
\[\chi_{n}\equiv\frac{\varphi_{n}^{+}}{\varphi_{n}^{+}}-\frac{\varphi_{n}^{-}}{ \varphi_{n}^{-}}=\frac{1}{G_{n}^{+}(0)}\equiv-2\rho_{n}\text{tanh}[\rho_{n}L- i\text{arctan}(k_{n}/\rho_{n})] \tag{14}\]  
\(G_{n}^{+}(0)\equiv G_{\Omega_{n}}^{+}(0)\) is the outgoing Green function of Equation (7) at \(x=0\) and \(\rho_{n}=\sqrt{U-\Omega_{n}}\). Figure 3: Incoming particles propagate through a potential barrier with an oscillating part at its center. **Upper** panel: the particles are transmitted through the barrier, where the spectrum is split into multiple discrete energies. **Lower** panel: when the barrier is narrower, then for specific barrier widths, Fano resonances appear and the particles are totally reflected (with 100% probability). Finally, the current through the barrier is:  
\[j=2\Re\sum_{n=-\infty}^{\infty}|t_{n}|^{2}|\tau_{n}|^{2}k_{n} \tag{15}\]  
where \(\Re\) represents the real part. The current Equation (15), as a function of the barrier's width \(L\), is presented in Figure 4 for a given incoming energy \(\Omega\).