{
  "id": "1811.09338",
  "version": "v1",
  "published": "2018-11-23T01:24:15.000Z",
  "updated": "2018-11-23T01:24:15.000Z",
  "title": "Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator",
  "authors": [
    "Scott E. Hoffmann",
    "Véronique Hussin",
    "Ian Marquette",
    "Yao-Zhong Zhang"
  ],
  "comment": "16 pages, 11 figures",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-23T01:24:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "coherent states",
      "multi-step supersymmetric rational extensions",
      "ladder operator",
      "truncated oscillator",
      "polynomial heisenberg algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}