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

Abstract

We construct ladder operators,C andC † , for a multi-step rational extension of the harmonic oscillator on the half plane, x ≥ 0. 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 finitedimensional 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 ofC 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. * scott.hoffmann@uqconnect.edu.au 2

Time-dependent rational extensions of the parametric oscillator: quantum invariants and the factorization method

“…It is just required to compute the last integral (68), although, it has to be done once we specify the form of F ptq. an important role since those lead to a new family of Hamiltonians with eigenfunctions that belong to the class of exceptional Hermite polynomials [38,39,50]. In this section we explore the construction presented in previous sections to construct the time-dependent counterparts of rational extensions associated with the parametric oscillator.…”

Ladder operators and coherent states for the Rosen-Morse system and its rational extensions

“…Let us shortly exhibit the three cases for further referring. More details can be found in [27,29,30,31].…”

“…In some cases (see, e.g. [48,31]), it is possible to construct ladder operators separately in the state-added subspace. But since in our case this subspace is spanned by { ψ(0; x)} and contains only one energy level, the idea of a consistent ladder operator is not well-defined.…”

Coherent states and scattering for exactly solvable quantum systems

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