Exactly solvable non - Hermitian Jaynes - Cummings - type Hamiltonian admitting entirely real spectra from supersymmetry

Abstract It is shown that for a given hermitian Hamiltonian possessing supersymmetry, there is always a non-hermitian Jaynes-Cummings-type Hamiltonian(JCTH) admitting entirely real spectra. The parent supersymmetric Hamiltonian and the corresponding non-hermitian JCTH are simultaneously diagonalizable. The exact eigenstates of these non-hermitian Hamiltonians are constructed algebraically for certain shape-invariant potentials, including a non-hermitian version of the standard Jaynes-Cummings(JC) model for which the parent supersymmetric Hamiltonian is the superoscillator. It is also shown that a non-hermitian version of the several physically motivated generalizations of the JC model admits entirely real spectra. The positive-definite metric operator in the Hilbert space is constructed explicitly along with the introduction of a new inner product structure, so that the eigenstates form a complete set of orthonormal vectors and the time-evolution is unitary.