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

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 Ba ϩ ions for distances in the range of about ten wave lengths of the strong transition while no effects were observed for Hg ϩ at 15 wavelengths. In this theoretical paper we study this question for configurations with three and four levels, which model those of Hg ϩ and Ba ϩ , respectively. For two systems in the Hg ϩ 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 ϩ ions, cooperative effects are practically absent, and this latter result is at odds with the experimental findings for Ba ϩ .

Simplified approach to double jumps for fluorescing dipole-dipole interacting atoms

“…However, the systems used in the experimental setups of references [3,12,19] were not V systems so that a direct comparison between theory and experiment was not possible. For this reason the present authors extended their investigation to two other systems [20], namely a D shaped system modeling the Hg + ions used in reference [12] and a four-level system (see Figure 1) modeling the Ba + ions of references [3,4]. For two D systems cooperative effects in the same order of magnitude as for the V systems were found for ion-distances of a few wavelengths of the laser-driven transition.…”

“…kl becomes the C j of reference [20]. The quasi-steady states are already known from the calculations for three three-level systems.…”

“…In terms of these transition rates the double and triple jump rate, i.e. the rate of two or three subsequent jumps within a short time window T W , are then given by [20] In Fig. 4 a plot of n DJ for the experimental parameter values of reference [19] is shown.…”

“…This leads to [20] ρ(t + ∆t, The last term can be neglected and the coefficient α ij can therefore be interpreted as transition rate p ij from intensity period I i to period I j . They can be calculated by means of the dual eigenstates for eigenvalue 0 of L 0 [20]. For a single D system the quasi-steady states are given by…”

“…This process is characterized by the transition rates between the different intensity periods. In references [18,20,22] they were calculated for different model level systems and different numbers of atoms using a perturbation approach based on the Bloch equation of the corresponding systems. This approach will be illustrated in the following by applying it to the simple case of a single three level system in a D-type configuration as depicted in figure 2.…”