Nonlinear Model for a Trapped Ion Interacting with a Standing Wave via Gauge-Like Transformations

S. Alam ； C. Bentley

《Chinese Journal of Physics》 37卷1期 (1999/02) Pp. 44-50

並列摘要

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 the standing wave. In addition our treatment is valid to all orders in the Lamb-Dicke parameter, η. 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 η. Going beyond the leading order in η leads one naturally to a Nonlinear Jaynes- Cummings type Hamiltonian. Nonlinear Jaynes-Cummings dynamics of a trapped 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 linear transformation in the su(2) algebra describing the two-level system. The outcome is a Jaynes-Cummings like Hamiltonian or a Nonlinear Jaynes-Cummings like Hamiltonian (if non-linear terms are kept) with the effective coupling dependence on the parameter X aimed to figure the trapping location 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.