Derivation of multi-photon anharmonic oscillator model from its classical counterpart

Dolan Krishna Bayen, Swapan Mandal

Dolan Krishna Bayen, Swapan Kumar Mandal

Christopher C. Gerry

Davood Afshar, Mojtaba Jafarpour

We present a new method to study the eigenvalues and eigenfunctions of anharmonic oscillators. It involves a new class of Bogoliubov transformations and leads to the introduction of k-photon coherent states. We consider the Hamiltonians for the simple harmonic and anharmonic oscillators as the two generators of a Lie algebra, whose other generators may be found exactly, or up to any desired order of the perturbation parameter involved. An element of this Lie group, turning out to be the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one, thus facilitating the calculation of the eigenvalues and eigenfunctions of the former. The transformation of the ordinary annihilation and creation operators leads to generalized ones, corresponding to generalized oscillation modes, and also helps us out to introduce multi-photon coherent states. We specifically consider four-photon coherent states in detail and study time dependent position and momentum uncertainties in these states.

Christopher C. Gerry

Chi‐Fai Lo, K.L. Liu

Recently, Sharma et al. [Phys. Rev. A 56, 694 (1997)] proposed a q-deformed oscillator trap model to investigate the time evolution of a single two-level ion in an anharmonic oscillator trap, with the anharmonicity being parametrized by the deformation parameter q. The model is obtained by simply replacing the usual bosonic creation and annihilation operators in the harmonic trap model by their q-deformed counterparts. In the present paper we have reexamined the validity of the q-deformed oscillator trap model, and our analysis indicates that such a naive generalization actually does not exist. This is accomplished by demonstrating that the eigenstates of this q-deformed oscillator trap model are not normalizable.