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We construct ladder operators, $\tilde{C}$ and $\tilde{C^\dagger}$, for amulti-step rational extension of the harmonic oscillator on the half plane,$x\ge0$. These ladder operators connect all states of the spectrum in onlyinfinite-dimensional representations of their polynomial Heisenberg algebra.For comparison, we also construct two different classes of ladder operatoracting on this system that form finite-dimensional as well asinfinite-dimensional representations of their respective polynomial Heisenbergalgebras. For the rational extension, we construct the position wavefunctionsin terms of exceptional orthogonal polynomials. For a particular choice ofparameters, we construct the coherent states, eigenvectors of $\tilde{C}$ withgenerally complex eigenvalues, $z$, as superpositions of a subset of the energyeigenvectors. Then we calculate the properties of these coherent states,looking for classical or non-classical behaviour. We calculate the energyexpectation as a function of $|z|$. We plot position probability densities forthe coherent states and for the even and odd cat states formed from thesecoherent states. We plot the Wigner function for a particular choice of $z$.For these coherent states on one arm of a beamsplitter, we calculate the twoexcitation number distribution and the linear entropy of the output state. Weplot the standard deviations in $x$ and $p$ and find no squeezing in the regimeconsidered. By plotting the Mandel $Q$ parameter for the coherent states as afunction of $|z|$, we find that the number statistics is sub-Poissonian.

Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator