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

Abstract

We construct ladder operators, $\tilde{C}$ and $\tilde{C^\dagger}$, for a multi-step rational extension of the harmonic oscillator on the half plane, $x\ge0$. These ladder operators connect all states of the spectrum in only infinite-dimensional representations of their polynomial Heisenberg algebra. For comparison, we also construct two different classes of ladder operator acting on this system that form finite-dimensional as well as infinite-dimensional representations of their respective polynomial Heisenberg algebras. For the rational extension, we construct the position wavefunctions in terms of exceptional orthogonal polynomials. For a particular choice of parameters, we construct the coherent states, eigenvectors of $\tilde{C}$ with generally complex eigenvalues, $z$, as superpositions of a subset of the energy eigenvectors. Then we calculate the properties of these coherent states, looking for classical or non-classical behaviour. We calculate the energy expectation as a function of $|z|$. We plot position probability densities for the coherent states and for the even and odd cat states formed from these coherent 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 two excitation number distribution and the linear entropy of the output state. We plot the standard deviations in $x$ and $p$ and find no squeezing in the regime considered. By plotting the Mandel $Q$ parameter for the coherent states as a function of $|z|$, we find that the number statistics is sub-Poissonian.