Dynamics of a two-state system through a real level crossing

The dynamics of a two-state system whose energies undergo a real crossing at some instant of time is studied. At this instant, both the coupling and the detuning vanish simultaneously, which leads to an exact degeneracy of the eigenenergies of the system. It is found that the dynamics of the system is primarily determined by the manner in which the degeneracy occurs. This interesting behavior is reminiscent of a symmetry-breaking process, since the totally symmetric situation occurring at the crossing is significantly altered by infinitesimal quantities, which remove the degeneracy, with very important dynamical implications from there on. A very simple analytical formula is derived, which is found to describe the population changes very accurately.