Clock transition ($\partial {\omega }_{21}/\partial B=0$)

A transition whose frequency is stationary to first order in a specified perturbation at an operating point. For magnetic sensitivity, the derivative shown vanishes, but second and higher derivatives can remain nonzero. A clock transition need not have zero individual energy slopes: equal slopes suffice. Optical light shifts and other perturbations require their own sensitivity analysis.

For the two $m=0$ branches of the Breit-Rabi formula, their splitting is $\sqrt{{\Delta }_{\mathrm{hf}}^2+[(g_J-g_I){\mu }_{B}B{]}^2}$. Expanding at small field gives the quadratic frequency shift displayed above. Vanishing linear frequency sensitivity at zero field therefore does not eliminate field-gradient dephasing at a finite bias field.

The ground-hyperfine magnetic-dipole transition between $F=1,m_F=0$ and $F=2,m_F=0$ has vanishing first-order Zeeman shift near zero field. It obeys $\Delta F=1,\Delta m_F=0$. Its residual quadratic shift follows from the Breit-Rabi formula, rather than from assigning both states permanently zero magnetic dipole moment.