# Hyperfine Structure: Breit-Rabi Diagram

The energy-level shifts of a system with hyperfine structure in response to a magnetic field have a different character depending on whether the shifts are smaller or larger than the hyperfine splitting. When the Larmor frequency is much smaller than the hyperfine splitting, the hyperfine sublevels behave as if they were isolated, and the shift of each Zeeman sublevel is proportional to M_{F}. For large magnetic fields, for which the Larmor frequency is larger than the hyperfine splitting, the hyperfine sublevels are strongly mixed with each other, and the system is best represented by the uncoupled _{J}I M_{I}〉 J M basis. The energy spectrum then consists of manifolds of levels with the same M_{J} and Zeeman shift proportional to M_{I}. Here we analyze this situation for the ground state of ^{87}Rb.

Define the atomic system using values from the AtomicData database. |

The eigenvectors are the eigenstates in the presence of the magnetic field, written as a linear combination of the original sublevels. However, they are not listed in the order corresponding to the original sublevels. Here we reorder the eigenvalues and eigenvectors so that they match the unperturbed quantities when the magnetic field approaches zero.

We can use the correctly ordered energies to plot the level diagram for the modified energy eigenstates in the presence of the magnetic field.

For a small field, the states are best described in the coupled basis , with three sublevels (with ), and five sublevels (with ). For a large field, the states are best described in the uncoupled basis _{J}I M_{I}〉 J M, with four M_{J}=1/2 states (with ), and four M_{J}=-1/2 states.