Dipole-dipole interaction between orthogonal dipole moments in time-dependent geometries

In two nearby atoms, the dipole-dipole interaction can couple transitions with orthogonal dipole moments. This orthogonal coupling accounts for a number of interesting effects, but strongly depends on the geometry of the setup. Here, we discuss several setups of interest where the geometry is not fixed, such as particles in a trap or gases, by averaging over different sets of geometries. Two averaging methods are compared. In the first method, it is assumed that the internal electronic evolution is much faster than the change of geometry, whereas in the second, it is vice versa. We find that the orthogonal coupling typically survives even extensive averaging over different geometries, albeit with qualitatively different results for the two averaging methods. Typically, one- and two-dimensional averaging ranges modeling, e.g., low-dimensional gases, turn out to be the most promising model systems.