Non-classical behaviour of coherent states for systems constructed using exceptional orthogonal polynomials

We construct the coherent states and Schr\"odinger cat states associated with new types of ladder operators for a particular case of a rationally extended harmonic oscillator involving type III Hermite exceptional orthogonal polynomials. In addition to the coherent states of the annihilation operator, $c$, we form the linearised version, \tilde{c}, and obtain its coherent states. We find that while the coherent states defined as eigenvectors of the annihilation operator $c$ display only quantum behaviour, those of the linearised version, \tilde{c}, have position probability densities displaying distinct wavepackets oscillating and colliding in the potential. The collisions are certainly quantum, as interference fringes are produced, but the remaining evolution indicates a classical analogue.