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We construct the coherent states of general order, $m$ for the ladderoperators, $c(m)$ and $c^\dagger(m)$, which act on rational deformations of theharmonic oscillator. The position wavefunctions of the eigenvectors involvetype III Hermite exceptional orthogonal polynomials. We plot energyexpectations, time-dependent position probability densities for the coherentstates and for the even and odd cat states, Wigner functions, and Heisenberguncertainty relations. We find generally non-classical behaviour, with oneexception: there is a regime of large magnitude of the coherent stateparameter, $z$, where the otherwise indistinct position probability densityseparates into $m+1$ distinct wavepackets oscillating and colliding in thepotential, forming interference fringes when they collide. The Mandel $Q$parameter is calculated to find sub-Poissonian statistics, another indicator ofnon-classical behaviour. We plot the position standard deviation and findsqueezing in many of the cases. We calculate the two-photon-number probabilitydensity for the output state when the $m=4$, $\mu=-5$ coherent states (where$\mu$ labels the lowest weight in the superposition) are placed on one arm of abeamsplitter. We find that it does not factorize, again indicatingnon-classical behaviour. Calculation of the linear entropy for thisbeamsplitter output state shows significant entanglement, another non-classicalfeature. We also construct linearized versions, $\tilde c(m)$, of theannihilation operators and their coherent states and calculate the sameproperties that we investigate for the coherent states. For these we findsimilar behaviour to the $c(m)$ coherent states, at much smaller magnitudes of$z$, but comparable average energies.

Coherent states for ladder operators of general order related to exceptional orthogonal polynomials