Nonlinear model for a trapped ion interacting with a standing wave via gauge-like transformations

We treat the interaction of a trapped ion with a standing-wave light field relaxing the previous condition of the trapping ion being on the node of th e standing wave. In addition our treatment is valid to all orders in the La mb-Dicke parameter, eta. Only a simple transformation to the "dressed basis " (subject to the constraints of the Pauli algebra) is required in order to show that the ion interacting with a standing light-wave may be described by the Jaynes-Cummings like Hamiltonian to first order in the parameter eta . Going beyond the leading order in eta leads one naturally to a Nonlinear Jaynes-Cummings type Hamiltonian. Nonlinear Jaynes-Cummings dynamics of a t rapped ion was recently discussed by Vogel and Filho. However our approach gives/specifies an ansatz on how to arrive at the nonlinearity. The ansatz of leaving invariant the Pauli algebra after transformations amounts to a l inear transformation in the su(2) algebra describing the two-level system. The outcome is a Jaynes-Cummings like Hamiltonian or a Nonlinear Jaynes-Cum mings like Hamiltonian (if non-linear terms are kept) with the effective co upling dependence on the parameter chi aimed to figure the trapping locatio n in the standing wave. The main idea, namely leaving the Hamiltonian form invariant while changing the physical parameters, can pave the way to a new understanding of gauge-like transformations in the present context.