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arXiv:quant-ph/0305125 · Physical Review A

Cooperative fluorescence effects for dipole-dipole interacting systems with experimentally relevant level configurations

Volker Hannstein, Gerhard C. Hegerfeldt

Published Oct 29, 2003• 6 citations
arXivHTMLPDF

Abstract

The mutual dipole-dipole interaction of atoms in a trap can affect their fluorescence. Extremely large effects were reported for double jumps between different intensity periods in experiments with two and three ${\mathrm{Ba}}^{+}$ ions for distances in the range of about ten wave lengths of the strong transition while no effects were observed for ${\mathrm{Hg}}^{+}$ at 15 wavelengths. In this theoretical paper we study this question for configurations with three and four levels, which model those of ${\mathrm{Hg}}^{+}$ and ${\mathrm{Ba}}^{+},$ respectively. For two systems in the ${\mathrm{Hg}}^{+}$ configuration we find cooperative effects of up to 30% for distances around one or two wavelengths, about 5% around ten wavelengths, and, for larger distances in agreement with experiments, practically none. This is similar for two V systems. However, for two four-level configurations, which model two ${\mathrm{Ba}}^{+}$ ions, cooperative effects are practically absent, and this latter result is at odds with the experimental findings for ${\mathrm{Ba}}^{+}.$

DOI
10.1103/physreva.68.043826

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The mutual dipole-dipole interaction of atoms in a trap can affect their fluorescence. Extremely large effects were reported for double jumps between different intensity periods in experiments with two and three Ba+{\mathrm{Ba}}^{+}Ba+ ions for distances in the range of about ten wave lengths of the strong transition while no effects were observed for Hg+{\mathrm{Hg}}^{+}Hg+ at 15 wavelengths. In this theoretical paper we study this question for configurations with three and four levels, which model those of Hg+{\mathrm{Hg}}^{+}Hg+ and Ba+,{\mathrm{Ba}}^{+},Ba+, respectively. For two systems in the Hg+{\mathrm{Hg}}^{+}Hg+ configuration we find cooperative effects of up to 30% for distances around one or two wavelengths, about 5% around ten wavelengths, and, for larger distances in agreement with experiments, practically none. This is similar for two V systems. However, for two four-level configurations, which model two Ba+{\mathrm{Ba}}^{+}Ba+ ions, cooperative effects are practically absent, and this latter result is at odds with the experimental findings for Ba+.{\mathrm{Ba}}^{+}.Ba+.
We investigate the effect of the dipole-dipole interaction on the quantum jump statistics of three atoms. This is done for three-level systems in a V\mathsf{V}V configuration and in what may be called a D\mathsf{D}D configuration. The transition rates between the four different intensity periods are calculated in closed form. Cooperative effects are shown to increase by a factor of 2 compared to two of either three-level systems. This results in transition rates that are, for distances of about one wavelength of the strong transition, up to 100% higher than for independent systems. In addition the double and triple jump rates are calculated from the transition rates. In this case cooperative effects of up to 170% for distances of about one wavelength and still up to 15% around 10 wavelengths are found. Nevertheless, for the parameters of an experiment with Hg+{\mathrm{Hg}}^{+}Hg+ ions the effects are negligible, in agreement with the experimental data. For three Ba+{\mathrm{Ba}}^{+}Ba+ ions this seems to indicate that the large cooperative effects observed experimentally cannot be explained by the dipole-dipole interaction.
A simplified scheme for the investigation of cooperative effects in the quantum jump statistics of small numbers of fluorescing atoms and ions in a trap is presented. It allows the analytic treatment of three dipole-dipole interacting four-level systems which model the relevant level scheme of Ba+ ions. For the latter, a huge rate of double and triple jumps was reported in a former experiment and the huge rate was attributed to the dipole-dipole interaction. Our theoretical results show that the effect of the dipole-dipole interaction on these rates is at most 5% and that for the parameter values of the experiment there is practically no effect. Consequently it seems that the dipole-dipole interaction can be ruled out as a possible explanation for the huge rates reported in the experiment.
The control and manipulation of the atomic translational and internal degrees of freedom via atom-laser interaction is an essential tool in modern quantum optical experiments which allows for a large variety of applications. The present work is concerned with theoretical investigations which may shed some light on certain experiments in this field.In the first part of this work, cooperative effects in the fluorescence of dipole-dipole interacting atoms in a trap are studied. The famous macroscopic quantum jumps can occur for a single atom with a metastable state when illuminated by light. Then periods of fluorescence of fixed intensity alternate stochastically with dark periods. If one has two or three such atoms in a trap, there are dark periods and periods of single and multiple intensity, from overlap. The statistics of these periods is influenced by the dipole-dipole interaction between the atoms in a sensitive manner. This work is concerned with a theoretical description of these effects, including a discussion of the seemingly contradictory experimental results which have been obtained in this context. The main result indicates that the huge rate of double and triple jumps reported for three trapped Ba+ ions can in fact not be attributed to the dipole-dipole interaction.In the second part, the possible use of an evanescent wave atomic mirror as a measuring device for the properties of an atomic wave packet is investigated. The idea is taken from an experiment, in which such a mirror was used for the temporal diffraction of a cloud of laser cooled atoms dropping out of a magneto-optical trap. A model for the investigation of the atomic wave function with such a mirror is given, and the outcomes of different set-ups are investigated. The main result shows that with such a mirror one can sample the mod-squared of a spread-out wave function, and this opens the way for a realistic possibility for measuring this otherwise not easily accessible quantity.
We consider the effect of dipole-dipole interaction on the quantum jumps in a system of two three-level atoms in a ssV configuration interacting with coherent fields. We show that the intensity-intensity correlation for the strong transition shows additional fluctuations for some interatomic separations between the atoms signifying three levels of fluorescence. It is argued that the intermediate level of fluorescence arises from the coupling between the symmetric and antisymmetric states caused by the dipole-dipole interaction.
If an atom is able to exhibit macroscopic dark periods, or electron shelving, then a driven system of two atoms has three types of fluorescence periods (dark, single, and double intensity). We propose to use the average durations of these fluorescence types as a simple and easily accessible indicator of cooperative effects. As an example, we study two dipole-interacting V systems by simulation techniques. We show that the durations of the two types of light periods exhibit marked separation-dependent oscillations and that they vary in phase with the real part of the dipole-dipole coupling constant.