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Journal of Optics B Quantum and Semiclassical Optics

Nonlinear vibronic dynamics of three laser-driven ions in a linear trap

S Bendel, Th. Richter, W. Vogel

Published Mar 1, 2004• 0 citations
Publisher

Abstract

The nonlinear interaction of three ions in a linear trap with laser light is studied by taking into account the three vibrational modes within a rotating-wave approximation. It is demonstrated that, for localization conditions far from the Lamb-Dicke limit, all three modes have a significant impact on the dynamics of the system. Preparations of vibrational Fock states can be used to realize two different regimes with an exactly solvable three-atom dynamics. The first one corresponds to a nonlinear version of the Tavis-Cummings coupling, whereas the Tavis-Cummings type interactions are suppressed in the second one.

DOI
10.1088/1464-4266/6/3/012

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The nonlinear interaction of three ions in a linear trap with laser light is studied by taking into account the three vibrational modes within a rotating-wave approximation. It is demonstrated that, for localization conditions far from the Lamb-Dicke limit, all three modes have a significant impact on the dynamics of the system. Preparations of vibrational Fock states can be used to realize two different regimes with an exactly solvable three-atom dynamics. The first one corresponds to a nonlinear version of the Tavis-Cummings coupling, whereas the Tavis-Cummings type interactions are suppressed in the second one.
Raman sideband excitations driven by counterpropagating Raman beams are analyzed for the trapped ion under the localization conditions in and beyond the Lamb-Dicke regime. Multiquantum vibrational couplings are demonstrated to produce significant effects on the Raman sideband transitions and the relevant Raman excitations of the trapped ion beyond the Lamb-Dicke regime. The quantum features of the vibrational states attained by exciting the motional ground state are closely associated with the nonlinear vibronic couplings, Raman excitation, and localization conditions. Quantum entanglement and quantum interference are discussed for the trapped ion under various Raman excitation and localization conditions. The results are compared with the recent experiments. Further experimental attempts are proposed to use the single trapped 25Mg+{}^{25}{\mathrm{Mg}}^{+}25Mg+ ion in a linear ion trap.
The dynamics of the interaction of a quantized cavity field and the vibronic degrees of freedom of a trapped ion is studied under realistic conditions by including cavity losses, spontaneous electronic transitions, and atomic nonlinearities. As long as spontaneous electronic transitions are negligible, analytical results are derived for describing the interaction of the trapped ion and the damped cavity field in the secular approximation. Under more general conditions, when the secular approximation breaks down and spontaneous emission effects become important, the dynamics of the system is studied by quantum-trajectory methods. As an example we demonstrate that, by exploiting the nonlinearities inherent in the dynamics of the system, one may prepare correlated vibronic states and vibrational Fock states. Moreover, we study the generation of an entangled Greenberger-Horne-Zeilinger state. Taking into account the decoherence due to electronic and cavity losses, the best fidelity is obtained under conditions far from the Lamb-Dicke regime.
The first sideband Raman excitations driven by counter-propagating Raman beams are analysed for the trapped ion under the localization condition in or beyond the Lamb-Dicke regime. It is demonstrated that multiquantum processes in the vibrational couplings produce significant effects on the the first sideband Raman transitions and the relevant Raman displacements for the trapped ion beyond the Lamb-Dicke regime. The quantum features of the vibrational states attained by displacing the motional ground state are demonstrated to be closely associated with the nonlinear vibronic couplings. Quantum entanglement, and quantum interference as well, are discussed for ion traps with various Raman excitation and localization conditions. The results are compared with the recent experiment reported by Monroe C et al in 1996 Science 272 1131-5.
We showed that in Lamb-Dicke regime and under rotating wave approximation, the dynamical behavior of two trapped ions interacting with a laser beam resonant to the first red side-band of center-of-mass mode can be described by Jaynes-Cummings Model. An exact analytic solution for this kind of Jaynes-Cummings model is presented. The results showed that quantum collapses and revivals for the occupation of two atoms, and squeezing for vibratic motion of center-of-mass mode existed in both two different types of initial conditions. The maximum momentum squeezing for center-of-mass mode in these two types of conditions are found to be 42.4% and 43.8% respectively. The coherence, in the first type of initial conditions can keeps long times, and in the second type of initial conditions, a concrete form of coherent state is obtained, when the initial average number is very small.
Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.