## 1 Introduction
For this, we have developed a numerical model of externally driven elastically coupled waveguides including dissipation. This model is fitted to some of our experimental data which it reproduces with good fidelity. This model is then used to explore the different ways of manipulating the Bell states complex amplitudes. This model shows that the driving frequency, the relative driving amplitudes and phases applied to the waveguides, are essential parameters for controlling the elastic Bell states. Finally, the model highlights the importance of the resonant frequencies of the bands associated with spatial modes and the damping coefficient of the elastic system in achieving Bell states with complex amplitudes. Let us consider an elastic system composed of \(N\) identical 1D waveguides coupled elastically along their length and arranged in parallel array fashion (cf. Figure 1). In the continuum limit, the propagation of longitudinal modes is characterized by [43]:  
\[\left\{H\cdot I_{N\times N}+\alpha^{2}M_{N\times N}\right\}u_{N\times 1}=0 \tag{1}\]  
where the dynamical differential operator,  
\[H=\frac{\partial^{2}}{\partial t^{2}}-\beta^{2}\frac{\partial^{2}}{\partial x ^{2}} \tag{2}\]  
models the propagation of elastic waves in the direction \(x\) along the waveguides. The parameter \(\beta\) is proportional to the speed of sound in the waveguides. The parameter \(\alpha\) measures the strength of the elastic coupling between the waveguides and we assume that \(\alpha\) is the same for all coupled waveguides. \(u_{N\times 1}\) is a vector whose components, \(u_{i},\ i=1,2,\ldots,N\), represent the displacement of the ith waveguide? \(I_{N\times N}\) is the \(N\times N\) identity matrix and the coupling matrix operator \(M_{N\times N}\) describes the elastic coupling between waveguides? [PERSON] factorizing the Klein-Gordon (KG) Equation (1) leads to:  
\[\left\{U_{N\times N}\otimes\sigma_{1}\frac{\partial}{\partial t}+\beta U_{N\times N }\otimes(-i\sigma_{2})\frac{\partial}{\partial x}\pm i\alpha U_{2N\times 2N} \sqrt{M_{N\times N}}\otimes\sigma_{1}\right\}\Psi_{2N\times 1}=0 \tag{3}\]  
The KG equation is used to describe relativistic particles in quantum mechanics. [PERSON] factorized the KG equation and established the theoretical foundation for the positron. The mathematical formulation of the Dirac equation naturally reveals the spin of relativistic particles. The factorization of the KG Equation (1) into a Dirac-like Equation (3) leads to non-trivial eigen modes that have spinor character and therefore non-conventional topology in wave number space [18; 25]. Moreover, from Equation (3) we see that the equation does not satisfy time-reversal symmetry \((t\rightarrow-t)\), T-symmetry, nor parity symmetry \((x\rightarrow-x)\), P-symmetry, separately. That is, one does not recover Equation (3) when the sign of time and position are changed individually, though, they obey time and parity (PT) symmetry when applied simultaneously. In Equation (3), \(U_{N\times N}\) and \(U_{2N\times 2N}\) are antidiagonal matrices with unit elements, and  
\[\sigma_{1}=\left( \begin{array}{cc}0&1\\ 1&0\end{array} \right)\text{and }\sigma_{2}=\left( \begin{array}{cc}0&i\\ -i&0\end{array} \right) \tag{4}\]  
are two of the Pauli matrices. \(\Psi_{2N\times 1}\) is a \(2N\)-dimensional vector which represents the modes of vibration of the \(N\) waveguides projected in the two possible directions of propagation (forward and backward). \(\sqrt{M_{N\times N}}\) is the square root of the coupling matrix. The square root of a matrix is not unique since the eigen vectors of \(\sqrt{M_{N\times N}}\) are also the eigen vectors of the coupling matrix \(M_{N\times N}\) itself; the non-uniqueness of \(\sqrt{M_{N\times N}}\) does not introduce difficulties in determining the elastic modes of the coupled system in the Dirac representation.