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arXiv:0704.0117 · Journal of Physics B Atomic Molecular and Optical Physics

Lower ground state due to counter-rotating wave interaction in a trapped ion system

T. Liu, K. L. Wang, Mang Feng

Published May 18, 2007• 16 citations
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Abstract

We consider a single ion confined in a trap under the radiation of two travelling waves of lasers. In the strong-excitation regime and without the restriction of the Lamb–Dicke limit, the Hamiltonian of the system is similar to a driving Jaynes–Cummings model without the rotating wave approximation (RWA). The approach we developed enables us to present complete eigensolutions, which makes it possible to compare with the solutions under the RWA. We find that the ground state in our non-RWA solution is energetically lower than the counterpart under the RWA. If we have the ion in the ground state, it is equivalent to a spin dependent force on the trapped ion. Discussion is given for the difference between the solutions with and without the RWA, and for the relevant experimental test, as well as for the possible application to quantum information processing.

DOI
10.1088/0953-4075/40/11/002

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We consider a single ion confined in a trap under the radiation of two travelling waves of lasers. In the strong-excitation regime and without the restriction of the Lamb–Dicke limit, the Hamiltonian of the system is similar to a driving Jaynes–Cummings model without the rotating wave approximation (RWA). The approach we developed enables us to present complete eigensolutions, which makes it possible to compare with the solutions under the RWA. We find that the ground state in our non-RWA solution is energetically lower than the counterpart under the RWA. If we have the ion in the ground state, it is equivalent to a spin dependent force on the trapped ion. Discussion is given for the difference between the solutions with and without the RWA, and for the relevant experimental test, as well as for the possible application to quantum information processing.
Raman interaction of a trapped ultracold ion with two travelling wave lasers has been used extensively in ion-trap experiments. We solve this interaction in the absence of a rotating wave approximation (RWA) by a continued fraction, without considering the restriction of the Lamb-Dicke limit and the weak excitation regime. Some interesting characteristics of the ion-trap system, particularly for the ion outside the weak excitation regime are found. Finally, a comparison of our results with the solution under the RWA is made.
We present a second-order perturbative analysis of the model describing a two-level trapped ion interacting with a travelling laser field, in the Lamb–Dicke regime. Unlike the customary approach, based on the interaction picture and the rotating wave approximation, we reduce the original Hamiltonian to a time-independent model, unitarily equivalent to a spin-boson model, which features a form well suited to apply the perturbation theory. We determine the first- and second-order corrections of eigenstates and eigenvalues. By varying the interaction parameters we identify four regimes characterized by different eigenvalue distributions (possibly including level doublets) and suggest the presence of an intermediate regime where the spectrum undergoes macroscopic changes. In this regime, where the perturbation approach does not hold, our reduced Hamiltonian is shown to reproduce the Jaynes–Cummings model.
We present a family of generalized unitary transformations that simplifies the Hamiltonian for a harmonically trapped two-level atom (or ion) interacting with a plane wave laser field. Near-resonant single as well as double vibrational phonon dynamical regimes are found. The validity condition of the often used motional rotating-wave approximation (MRWA) is examined both numerically and analytically. Large errors are found within typical regimes of the MRWA with respect to the motional degrees of freedom. The effects of the MRWA in trapped ion systems are shown to be opposite to that of the rotating-wave approximation in the usual Jaynes-Cummings model. Our study points to a more restrictive condition on the particle localization (Lamb-Dicke) parameter for the validity of the MRWA in the single phonon dynamical regime. It also sheds light on quantum information storage and processing with trapped atoms.
The eigenenergy spectrum of the Jaynes–Cummings (JC) model with and without the rotating-wave approximation (RWA) is investigated. The numerical analysis indicates that the non-RWA spectrum can only be approximated by the RWA in the range of sufficiently small coupling constant and detuning. In other region, the counter-rotating terms remarkably change the nature of the RWA energy spectrum. A simple expression with high accuracy for ground eigenenergy and eigenvector for non-RWA is available. The ground eigenvector for non-RWA shows that the ground state is not a dark state and very different from that of RWA which is a dark state.