Experimental study of magnetically insensitive transitions in ultracold Fermi gas of $^{40}$K

This paper presents an experimental study of microwave single-photon transitions that are magnetic-field-insensitive in degenerate Fermi gases of $^{40}$K. This contrasts with microwave single-photon clock transitions for 0-0 magnetic-field-insensitive states and two-photon clock transitions for non 0-0 magnetic-field-insensitive states in bosonic alkali metal atoms. We show that there are two sets of special transitions between two different hyperfine ground states ($|F$=9/2, $m_{F}$=1/2$\rangle$ $\Leftrightarrow$ $|$7/2, -1/2$\rangle$ and $|$9/2, -1/2$\rangle$ $\Leftrightarrow$ $|$7/2, 1/2$\rangle$), whose microwave single-photon transition frequency is insensitive to low magnetic fields, as the first-order Zeeman shift is almost completely canceled. By using the microwave spectrum and Ramsey interference fringes, we demonstrate the long-time stability of the coherent transition under magnetic field fluctuations. These magnetic-field-insensitive microwave hyperfine transitions in ultracold $^{40}$K Fermi gases offer promising applications in quantum information and precision measurements.