# Multiple Fano Resonances in Dynamic Resonant Tunneling Processes
## 8 High Precision Interferometry  
When an object is measured with a wave beam, then usually, the shorter the wavelength, the better the resolution. In the case of a simple rectangular barrier, the interference between the reflections from its boundaries determines the dynamic range of the measure  
Figure 14: Comparison between the two types of suppressions: the current as a function of the oscillation amplitude is presented for two adjacent incoming energies, \(\Omega/\omega=1.01\) with a thick blue dotted curve and \(\Omega/\omega=0.99\) with a black curve. The vertical red lines are the solution of \(\beta_{m}\) (Equation (34)) for \(m=1,2\ldots 12\). These are the same parameters as in Figure 13.  
Figure 13: The current as a function of the incoming energy and the oscillation amplitudes. On the lower panel, the black dashed curves represent the analytical expressions (Equation (33)) and the horizontal line represents the \(\Omega=\omega\) boundary. The parameters are \(LU^{1/2}=5\), \(\alpha U^{-1/2}=-1.9\) and \(\omega/U=0.1\). ment. Therefore, if a small object with dimensions \(\sim\Delta L\) is measured, then, unless the particle's frequency is of the order \(\sim\Delta L^{-2}\), the dynamic range would be minuscule and of the order of \((\Delta L/\lambda)^{2}\ll 1\), where \(\lambda\) is the particle's de-Broglie wavelength. Ref. [18] suggested fabricating a quantum device like the one presented in Equation (7) and Figure 3 as a frequency-controlled quantum transistor. The oscillations can be confined in space using well-known technology (see, for example, ref. [35]). The present work teaches that the dynamic range of these devices can be increased substantially. The Fano nature of these resonances shows that the dynamic range is almost 100%, regardless of the barrier's width and oscillating frequency. In the case where the oscillating part (width \(\varepsilon\) and height \(V\)) and measured object (width \(L\) and height \(U\)) are very narrow in comparison to the incoming particle's wavelength, Equation (30) can be simplified to (since now \(\alpha_{0}\cong V\varepsilon\)):  
\[\Omega\cong m\omega-\frac{(V\varepsilon+UL_{m})^{2}}{4}-\frac{\beta^{2}}{8} \tag{35}\]  
This equation can be rewritten as:  
\[\Omega\cong\left[-\frac{(V\varepsilon+UL_{m})^{2}}{4}-\frac{\beta^{2}}{8} \right]\text{mod}\omega \tag{36}\]  
where mod represents the modulo operation. In the case where \(\alpha_{0}>>UL_{m}\), then the widths \(L_{m}\), for which zero transmission is reached, must maintain:  
\[\Omega\cong m\omega-\frac{\alpha_{0}^{2}+2\alpha_{0}UL_{m}}{4}-\frac{\beta^{2 }}{8} \tag{37}\]  
Therefore, the difference between the adjacent widths is \(L_{m}-L_{m+1}=\Delta L\cong\frac{2\omega}{\alpha_{0}U}\). Or, equivalently, a change \(\delta L\) in the barrier's width will change the zero-transmission energy by:  
\[\Delta\Omega\cong\frac{V\varepsilon U}{2}\delta L \tag{38}\]  
In real physical units, Equation (38) can be written as \(\Delta\Omega\cong\frac{m}{\hbar^{3}}VU\varepsilon\delta L\), where \(m\) represents the effective mass in the material, which, for GaAs (where the effective mass is 0.067 times the mass of the free electron [35]), Equation (38) can therefore be written as:  
\[\Delta\Omega[\text{Hz}]\cong 1.5\times 10^{15}VU\varepsilon\delta L \tag{39}\]  
In Equation (39), the potential heights \(V\) and \(U\) are measured in eV and their widths \(\varepsilon\) and \(\delta L\) are measured in nanometers. The change in the frequency is measured in Hz. Therefore, if the oscillating part and measured object have a potential of \(\sim\)0.01 eV and a width of \(\sim\)1 nm, then every change of 0.01 nm (0.1 A ) varies the zero-transmission frequency by more than 1.5 GHz. Clearly, this result is valid, provided that the oscillating frequency is higher than this value. These values are easily achievable using contemporary electronics. Since the frequency can be measured with great accuracy, this device can exhibit an extremely high precision.