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Chinese Physics Letters

Nonlinear Jaynes-Cummings Dynamics of a Trapped Ion Under a Dressed-State Description

Xiaoxue Yang, Ying Wu, Xue-li Luo, Kelin Gao, Yi Xiao

Published Mar 1, 1998• 10 citations
Publisher

Abstract

Introducing a dressed-state description, we show that a trapped ion in a standing wave field can be described by a nonlinear Jaynes-Cummings model different from the counterpart of Vogel and Matos Filho [ Phys. Rev. A 52 (1995) 4214 ] in several aspects, which may carry important implications for the preparation and reconstruction of the nonclassical states of a trapped ion.

DOI
10.1088/0256-307x/15/3/012

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Introducing a dressed-state description, we show that a trapped ion in a standing wave field can be described by a nonlinear Jaynes-Cummings model different from the counterpart of Vogel and Matos Filho [ Phys. Rev. A 52 (1995) 4214 ] in several aspects, which may carry important implications for the preparation and reconstruction of the nonclassical states of a trapped ion.
Recently [F. Krumm and W. Vogel, Phys. Rev. A 97, 043806 (2018)], the detuned and nonlinear Jaynes-Cummings model describing the quantized motion of a trapped ion was introduced and its corresponding dynamics was solved via considering the driving laser in a quantized manner. For the dynamics with a classical driving laser an approximation in terms of neglected time ordering was considered to obtain analytical expressions and compared to numerically obtained exact dynamics, which revealed the insufficiency of this approximation. In the present work we reconsider this model and show that it can likewise be solved analytically, i.e., including time ordering, with a classical driving laser field. Using the analytical solution, we investigate the quantum time-ordering effects of the system with respect to nonclassicality and nonsimulatability of the motional states of the ion. Furthermore, we use the Magnus expansion to analyze the impact of certain orders of the time ordering. We derive an exact radius of convergence for these time-ordering corrections beyond the established and only sufficient criterion. Finally, the differences between the solution derived here and the previously found one using a quantized pump are discussed.
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 we work to second order in the Lamb-Dicke parameter ηηη, the extension to higher orders in ηηη is also indicated. Only a simple transformation to the ``dressed basis'' [subject to the constraint of Pauli algebra] is required in order to show that the ion interacting with a standing light-wave may be described by the Jaynes-Cummings Hamiltonian.
A single ion trapped in a harmonic potential and laser cooled at the node of a standing wave can be described in terms of the Jaynes-Cummings Hamiltonian known in quantum optics. Similar to nonclassical states of light we propose to prepare nonclassical states of the ion motion in a harmonic trap. In particular, several schemes for the preparation of Fock states are presented, and techniques for the preparation of squeezed states and coherent states of the ion motion are indicated. For an observation of these nonclassical states we propose to measure quantum collapse and revival in the population inversion of a single trapped ion.
Starting from the Jaynes-Cummings (J-C) model describing the interaction of a trapped ion with a standing-wave laser field, we derive the interaction Hamiltonian on the nonlinear J-C model of the trapped ion excitated resonantly with 1-quantum vibrational mode. After solving the time-dependent states of this model, we study the evolution of population inversion for the ion numerically. It is shown that the period of the quantum collapse and revival of the population inversionis related with the Lamb-Dick parameter and the position of the ion in the standing-wave laser field. With the increasing of the parameter, it will decrease the period of the collapse and revival. When moving the ion from the node of the standing-wave laser to the antinode, the period of the collapse and revival becomes longer.