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arXiv:quant-ph/0012085 · Journal of Physics B Atomic Molecular and Optical Physics

Continued fraction solution to the Raman interaction of a trapped ultracold ion with two travelling wave lasers

Mang Feng

Published Jan 26, 2001• 9 citations
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Abstract

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.

DOI
10.1088/0953-4075/34/3/320

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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.
By means of the Bloch–Maxwell equation, the Raman interaction of a trapped ultracold ion with two traveling wave lasers is treated semiclassically. As the model works without limitation of the Lamb–Dicke limit and the weak excitation regime, we study chaotic behavior of the system in the wide range of the parameters. It is shown that the chaotic behavior is more and more pronounced with the increase of the Lamb–Dicke parameter and Rabi frequency, and can be exhibited experimentally by using quantum jump technique.
In coherent state representation and in the absence of rotating wave approximation, the interaction between a standing wave laser and a trapped ultracold ion within the Lamb-Dicke limit, expanded to the second-order term of the Lamb-Dicke parameter, can be solved analytically by series expansion technique. It can be seen that, for some particular detunings, the eigenenergies of the system can be exactly obtained and the eigenfunctions, presented as series forms, are a superposition of displaced Fock states and displaced squeezed states. Making use of our solution, some particular nonclassical states could be prepared.
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.