Transition cancellations of $^{87}$Rb and $^{85}$Rb atoms in a magnetic field setting new standards

Abstract

We have analyzed the magnetic field dependences of intensities of all the optical transitions between magnetic sublevels of hyperfine levels, excited with $\sigma^+$, $\pi$ and $\sigma^-$ polarized light, for the $D_1$ and $D_2$ lines of $^{87}$Rb and $^{85}$Rb atoms. Depending on the type of transition and the quantum numbers of involved levels, the Hamiltonian matrices are of $1\times 1$, $2\times 2$, $3\times 3$ or $4\times 4$ dimension. As an example, analytical expressions are presented for the case of $2\times 2$ dimension matrices for $D_1$ line of both isotopes. Eigenvalues and eigenkets are given, and the expression for the transition intensity as a function of $B$ has been determined. It is found that some $\pi$ transitions of $^{87}$Rb and $^{85}$Rb get completely canceled for certain, extremely precise, values of $B$. No cancellation occurs for $\sigma^+$ or $\sigma^-$ transitions of $D_1$ line. For matrices with size over $2\times 2$, analytical formulas are heavy, and we have performed numerical calculations. All the $B$ values cancelling $\sigma^+$, $\pi$ and $\sigma^-$ transitions of $D_1$ and $D_2$ lines of $^{87}$Rb and $^{85}$Rb are calculated, with an accuracy limited by the precision of the involved physical quantities. We believe our modeling can serve as a tool for determination of standardized values of magnetic field. The experimental implementation feasibility and its possible outcome are addressed. We believe the experimental realization will allow to increase precision of the physical quantities involved, in particular the upper state atomic levels energy.