Pseudo-invariant Eigen-operator＂ Method for Deriving Energy-Gap of an Atom-Cavity Jaynes-Cummings Hamiltonian with Atomic Centre-of-Mass Motion

Using the ＂pseudo-invariant eigen-operator＂ method we find the energy-gap of the Jaynes-Cummings Hamiltonian model of an atom-cavity system. This model takes the atomic centre-of-mass motion into account. The supersymmetric structure is involved in the Hamiltonian of an atom-cavity system. By selecting suitable supersymmettic generators and using supersymmetrie transformation the Hamiltonian is diagonalized and energy eigenvectors are obtained.