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We construct the coherent states and Schr\"odinger cat states associated withnew types of ladder operators for a particular case of a rationally extendedharmonic oscillator involving type III Hermite exceptional orthogonalpolynomials. 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 theannihilation operator $c$ display only quantum behaviour, those of thelinearised version, \tilde{c}, have position probability densities displayingdistinct wavepackets oscillating and colliding in the potential. The collisionsare certainly quantum, as interference fringes are produced, but the remainingevolution indicates a classical analogue.

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