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Journal of Modern Optics

Quantum-mechanical localization of an ion in a Paul trap

M. El Ghafar, E. Mayr, В. И. Савичев, Päivi Törmä, A. Zeiler, W. P. Schleich

Published Oct 1, 1997• 5 citations
Publisher

Abstract

We show that dynamical localization of vibrational states occurs for an ion stored in a Paul trap and interacting with a standing laser field [1A]. The relation between the classical diffusion and the localization length is found to be of the same form as for the paradigm of dynamical localization, the kicked rotator.
DOI
10.1080/09500349708231860

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The phenomenon of localization manifests itself in many quantum mechanical systems ranging from the localization of light in a random medium via Anderson localization of an electronic wave to the motion of atoms in time-dependent laser fields. In all these cases the underlying classical system is chaotic and shows diffusion as a function of time. In contrast, the quantum mechanical counterpart has a localized wave function whose width is governed by the classical diffusion and Planck’s constant. In this paper we show that there exists an additional quantum parameter that controls the localization length. In the system of a two-level ion stored in a Paul trap and interacting with a standing wave it is the detuning between the transition frequency and the laser field. We also discuss the effect of decoherence in form of spontaneous emission. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
When an ion confined in a linear ion trap interacts with a coherent laser field, the internal degrees of freedom, related to the electron transitions, couple to the vibrational degree of freedom of the ion. As a result of this interaction, the quantum dynamics of the vibrational degree of freedom becomes complicated, and in some ranges of parameters even chaotic. We analyze the vibrational ion dynamics using a formal analogy with the solid-state problem of electron localization. In particular, we show how the resonant approximation used in analysis of the ion dynamics, leads to a transition from a two-dimensional (2D) to a one-dimensional problem (1D) of electron localization. The localization length in the solid-state problem is estimated in cases of weak and strong interaction between the sites of the 2D cell by using the methods of resonance perturbation theory, common in analysis of 1D time-dependent dynamical systems. We show that the localization length can be used as an indicator of the effective temperature of the trapped ion, which can be experimentally measured.
A classically chaotic system consisting of a Paul trapped ion and a sequences of standing laser pulses is treated quantum-mechanically. Under the circumstance of time-dependence, we derive the transition probability from the ion's motional state to ', and find, in the first-order approximation, the classically chaotic character disappears. Theoretical analysis and numerical calculations show that by regulating the phase parameter we can control the transition probability. When reaches some specific values, the transition from the state to ' is forbidden and, for some laser periods, resonance occurs, which leads to the corresponding transitions between different motional states. The time-evolution of an initial motional state just over one period is also studied in detail.