  Classically, the dynamics of the chiral oscillator (CO) may be described by
the Landau model (LM) through a well established mathematical procedure known
as duality mapping. In this letter we show how this duality is broken in
quantum mechanics due to the presence of a $Z_2$-anomaly in the CO. We give the
theoretical basis for an experimental setup displaying the possibility to
measure this global anomaly using cold Rydberg atoms.
