  Ladder operators for the simplest version of a rationally extended quantum
harmonic oscillator (REQHO) are constructed by applying a Darboux
transformation to the quantum harmonic oscillator system. It is shown that the
physical spectrum of the REQHO carries a direct sum of a trivial and an
infinite-dimensional irreducible representation of the polynomially deformed
bosonized osp(1|2) superalgebra. In correspondence with this the ground state
of the system is isolated from other physical states but can be reached by
ladder operators via non-physical energy eigenstates, which belong to either an
infinite chain of similar eigenstates or to the chains with generalized Jordan
states. We show that the discrete chains of the states generated by ladder
operators and associated with physical energy levels include six basic
generalized Jordan states, in comparison with the two basic Jordan states
entering in analogous discrete chains for the quantum harmonic oscillator.
