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arXiv:1002.3748 · New Journal of Physics

The localization of phonons in ion traps with controlled quantum disorder

A. Bermúdez, Miguel Angel Martin-Delgado, Diego Porras

Published Dec 9, 2010• 34 citations
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Abstract

We show that the vibrations of a chain of trapped ions offer an interesting route to explore the physics of disordered quantum systems. By preparing the internal state of the ions in a quantum superposition, we show how the local vibrational energy becomes a stochastic variable, its statistical properties inherited from the underlying quantum parallelism of the internal state. We describe a minimally perturbing measurement of the resonance fluorescence, which allows us to study effects such as Anderson localization without the need for ground-state cooling or individual addressing and thus paves the way for high-temperature ion experiments.

DOI
10.1088/1367-2630/12/12/123016

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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.
Trapped ions offer an excellent platform to investigate fascinating phenomena in quantum physics. Here we propose an experimental scheme to achieve Anderson localization of phonons using an ion chain under irradiation of a laser Bessel beam. The quasiperiodicity comes from the characteristic of the Bessel beam, which is only relevant to the positions of the individual ions with respect to the standing wave of the Bessel beam, rather than the internal degrees of freedom of the ions. We demonstrate, by the inverse participation ratio, the delocalization-localization transition, which is sensitive to the strength and the phase of the Bessel beam laser. Our model involving long-range phonon hopping could present richer phenomena than the Aubry-Andr\'e model including only nearest-neighbor coupling.
A single trapped and cooled ion interacting with light is the realization of a fundamental system of quantum optics. Relevant viewpoints include the light scattering and its relation to the preparation of the ion, the dynamics of the ion's internal motion, and the dynamics of its vibrational motion. As cases in point, recent results are reviewed that illustrate those viewpoints: (i) The preparation of the ion, by spontaneous emission, in a superposition of energy eigenstates of quasi-permanent lifetime. (ii) The dispersive effect of quantum jumps of the ion, to and from a metastable state, that gives rise to internally self-locking the ion in either the light-scattering bright or the non-scattering dark state. (iii) The stochastic cooling of the ion by repeated null detection of its resonance scattering. These counter-intuitive phenomena characterize a physical system, where quantum mechanics models ensembles of observations.
The detailed characterization of non-trivial coherence properties of composite quantum systems of increasing size is an indispensable prerequisite for scalable quantum computation, as well as for understanding non-equilibrium many-body physics. Here, we show how autocorrelation functions in an interacting system of phonons as well as the quantum discord between distinct degrees of freedoms can be extracted from a small controllable part of the system. As a benchmark, we show this in chains of up to 42 trapped ions, by tracing a single phonon excitation through interferometric measurements of only a single ion in the chain. We observe the spreading and partial refocusing of the excitation in the chain, even on a background of thermal excitations. We further show how this local observable reflects the dynamical evolution of quantum discord between the electronic state and the vibrational degrees of freedom of the probe ion.