Observation of Topological and Parity-dependent Phase of $m=0$ Spin States

A Ramsey interrogation scheme was used to measure the phase shift of laser-cooled ${}^{87}$Rb clock-transition pseudospins arising as a result of a reversal of a bias magnetic field, i.e., $\mathbf{\text{B}}\to -\mathbf{\text{B}}$, during the interrogation. While no phase shift occurred when the reversal was sudden, the Ramsey fringes were shifted by a factor of $\pi$ when the reversal was adiabatic. We thus verified the prediction that the spin states $\mathrm{\mid }F,m=0>$ acquire a purely topological and parity-dependent phase factor of $(-1{)}^F$ as a result of $\mathbf{\text{B}}\to -\mathbf{\text{B}}$.