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

Abstract

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.

Figures (9)

• Figure 1: (a) Scheme of the energy levels and transitions among the lowest hyperfine Zeeman states of $^{40}$K. The gray lines indicate the energy levels at zero magnetic field, while the black lines represent the Zeeman-split energy levels at 2.8 G. Nine sets of microwave transitions between the Zeeman sublevels are considerred in our current study, involving $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$, and $e_{i}$ transitions, where $i$=1, 2, 3 corresponds to $\sigma^{+}, \pi, \sigma^{-}$ transitions. (b) Differential Zeeman shifts as a function of magnetic field. Two magnetically insensitive transitions, $a_{3}$ and $b_{1}$, are identified, which exhibit minimal dependence on the magnetic field. The inset shows an enlarged view of the differential Zeeman shifts for the $a_{3}$ and $b_{1}$ transitons, with the transition frequency difference being approximately 1.38 kHz near a magnetic field of 2.8 G.
• Figure 2: Experimental setup. The atoms are confined in a crossed optical dipole trap formed by 1064-nm lasers. The control laser for the ac-Stark shift propagates along the $\hat{x}$ direction and is collimated with a Gaussian beam waist of 2 mm. A magnetic field applied along the $\hat{z}$ direction induces Zeeman splitting and serves as the quantization axis. A microwave antenna generates a signal along the $\hat{z}$ direction to drive MW transitions.
• Figure 3: Microwave spectra of the hyperfine transitions: (a) $e_1$, (b) $d_1$, (c) $c_1$, and (d) $b_1$. The measurements were performed using a Gaussian-shaped $\pi$ pulse with a duration of 0.11 ms and an optimized amplitude under a bias magnetic field of 2.8 G. The spectra are quantified by the population ratio $\eta = N_2 / (N_1 + N_2)$, where $N_1$ and $N_2$ denote the atom numbers of the hyperfine Zeeman states for each transition. The full width at half maximum (FWHM) of the observed spectra is $2\pi \times \{7.18, 7.18, 7.02, 7.14\}$ kHz for the respective transitions, extracted by fitting the experimental data with the Gaussian pulse formula $\eta_1(\delta) = \eta_0 \exp\left(-\delta^2 / 2\sigma^2\right)$. The center transition frequency $\omega_0$ is determined by $\omega_0 = \Delta E_{hf}/\hbar + \Delta E(B)/\hbar$, where $\Delta E(B)/\hbar$ is the magnetic-field-induced frequency shift, with $\Delta E(B)/\hbar = 2\pi \times (-5.219)$ MHz for $e_1$, $2\pi \times (-3.469)$ MHz for $d_1$, $2\pi \times (-1.722)$ MHz for $c_1$, and $2\pi \times (0.023)$ MHz for $b_1$. The symbol points represent experimental data, and the solid lines indicate the Gaussian fit of the data. Error bars represent the standard deviation from five repeated measurements.
• Figure 4: Microwave spectra of the hyperfine transition $b_1$ for different durations of Gaussian $\pi$ pulses. The spectra correspond to pulse durations of 0.11 ms (black circles), 0.3 ms (orange inverted triangles), 1.2 ms (green triangles), and 35 ms (red pentagons), with respective FWHMs of $2\pi \times \{7140, 2600, 1380, 62\}$ Hz. The inset highlights the spectrum for the 35 ms pulse, showing a FWHM of $2\pi \times 62$ Hz. Different marker shapes represent experimental data, while the colored lines indicate Gaussian fits to the data.
• Figure 5: Microwave spectra of two sets of magnetically insensitive transitions, $b_1$ and $a_3$. The measurements are performed using a Gaussian-shaped $\pi$ pulse with a duration of 10.5 ms and optimized amplitude under a bias magnetic field of 2.8 G. The FWHMs for transitions $b_1$ and $a_3$ are $2\pi \times$172 Hz and $2\pi \times$175 Hz, respectively. The measured transition frequency difference between the two magnetically insensitive transitions is 1.34 kHz, which is consistent with the theoretical calculation shown in the inset of Fig. 1(b). Different shaped markers represent the experimental data, while colored lines indicate the results of Gaussian fits.