## 3 Experimental Results
### Coupled Three-Rod Waveguides, \(N=3\)
To create such a nonseparable elastic state experimentally, we identify a isofrequency state of 33.25 kHz (see insets of Figure 5a,b). At this frequency, there is substantial transmission of \(e_{2}\) and \(e_{3}\) eigen modes with negligible transmission of \(e_{1}\). We drive the system at 33.25 kHz by  
\[\left( \begin{array}{c}F_{1}\sin(\omega t)\\ F_{2}\sin(\omega t+\pi)\\ 0\end{array} \right), \tag{14}\]  
where \(F_{1},F_{2}\) are the excitation amplitudes of rod 1 and 2, respectively. We also define \(r=|F_{1}/F_{2}|\), the excitation amplitude ratio. Figure 6a shows the transmission amplitudes of each rod, and Figure 6b shows the phase difference, \(\phi_{ij}\); \(i,j=1,2,3\) and \(i\
eq j\), between the transmission of rods as a function of \(r\). From Figure 6b, we see that by manipulating only one parameter \(r\), the excitation amplitude ratio, the eigen modes superposition can be experimentally tuned. From Figure 6, we also see that, for a particular excitation amplitude ratio \(r=0.4356\) (dashed line in Figure 6), the transmitted amplitudes of rod 1 and 2 are almost equal and for rod 3 is small (close to zero). In addition, phase difference  
Figure 5: (**a**) Transmission spectrum of the coupled three-rod waveguides (\(N=3\)) for the three spatial eigen modes of (1 1 1), (1 0 -1), and (1 -2 1). The transmission amplitude is in arbitrary units. (**b**) Band structure determined and calculated from (**a**). The asterisks in (**b**) are obtained from the resonances of (**a**), and the solid line associated with the asterisks is a fit to the low frequency resonances. The open and solid circles in (**b**) correspond to resonant modes associated with the \(e_{2}\) and \(e_{3}\) spatial eigen modes. The solid lines are a fit to the identifiable resonances using the dispersion relations with cutoff frequencies 14.23 kHz for \(e_{2}\) and 24.27 kHz for \(e_{3}\). The inset on (**a**) and the dashed line in the inset of (**b**) focus on a frequency interval showing nearly overlapping (1 0 -1) and (1 -2 1) resonances and a trough between two resonances in the (1 1 1) transmission around 33.25 kHz. between rods 2 and 3 (\(\phi_{23}\)), as well as 1 and 3 (\(\phi_{13}\)), are \(\pi/2\) and between 1 and 2 (\(\phi_{12}\)) is close to \(\pi\). This relation signifies that though the driving amplitude of rods 1, 2, and 3 are \(F_{0}(0.4356,-1,0)\), the transmitted amplitudes are \(A_{0}(1,-1,\epsilon)\), where \(\epsilon\ll 1\) and \(F_{0}\) and \(A_{0}\) are constants. To accommodate these conditions using Equation (8), the Bell state coefficients, \(A\) and \(B\), must be complex. We now determine \(A\) and \(B\), the Bell state (complex) coefficients, for the excitation amplitude ratio \(r=0.4356\). The Bell state is constructed as a superposition of elastic waves, each a product of a plane wave part and a spatial mode part:  
\[u_{3\times 1}=\left(Ae_{2}e^{ik_{2}x}+Beg^{ik_{3}x}\right)e^{i\omega t}. \tag{15}\]  
For the case of isofrequency state at \(\omega=33.25\) kHz, from Figure 5b, we have \(k_{2}=\frac{2\pi}{2L}8\) and \(k_{3}=\frac{2\pi}{2L}6\) (in the finite waveguide, the wave numbers are integer multiples of \(\pi/L\)). At that frequency and at \(x=L\), Equation (15) reduces to:  
\[\frac{A}{\sqrt{2}}\left( \begin{array}{c}1\\ 0\\ -1\end{array} \right)+\frac{B}{\sqrt{6}}\left( \begin{array}{c}1\\ -2\\ 1\end{array} \right)=\left( \begin{array}{c}C_{1}e^{i\phi_{1}}\\ C_{2}e^{i\phi_{2}}\\ C_{3}e^{i\phi_{3}}\end{array} \right). \tag{16}\]  
In Equation (16), \(C_{ij}\); \(i=1,\ 2,\ 3\) are the maximum displacement amplitudes of each rod recorded at the detecting end of the waveguide, and \(\phi_{i}\); \(i=1,\ 2,\ 3\) are the corresponding phases. Equation (16) can be simplified in terms of the phase differences, \(\phi_{ij};i,j=1,2,3\) and \(i\
eq j\), between the transmission of each rods as:  
\[\left( \begin{array}{c}C_{1}e^{i\phi_{12}}\\ C_{2}\\ C_{3}e^{-i\phi_{23}}\end{array}