Degenerate Rabi spectroscopy of the Floquet engineered optical lattice clock

Simulating physics with large $\mathrm{SU}\left(N\right)$ symmetry is one of the unique advantages of alkaline-earth-metal atoms. Introducing periodical driving modes to the system may provide richer $\mathrm{SU}\left(N\right)$ physics that the static one cannot reach. However, a discussion of whether or not the driving modes will break the SU($N$) symmetry is still lacking. Here we experimentally study a Floquet engineered degenerate ${}^{87}\mathrm{Sr}$ optical lattice clock (OLC) by periodically modulating the frequency of the lattice laser. With the help of Rabi spectroscopy, we find that the atoms at different Zeeman sublevels are tuned by the same driven function. Meanwhile, our experimental results suggest that the uniform distribution among the sublevels does not change despite the driving. Our experimental demonstrations may pave the way for implementation of Floquet engineering on tailoring the $\mathrm{SU}\left(N\right)$ physics in the OLC system.

Figure 1

Floquet engineered degenerate optical lattice clock. In the absence of magnetic field, both clock states have 10 degenerate sublevels (denoted by different colors [11]). (a) Without driving, the atoms follow the Boltzmann distribution in the external states. After the $\pi$ clock transitions, the atoms can hop between two clock states without changing their Zeeman sublevels $\mathrm{\Delta }{m}_F=0$. (b) Under the lattice driving pattern, the carrier peak in Rabi spectroscopy can split into several Floquet sidebands. Each Floquet sideband corresponds to the $\pi$ clock transitions in each Floquet energy level which can still retain the $\mathrm{SU}\left(N\right)$ symmetry.

Figure 2

The Zeeman spectrum in the presence of a 460-mG magnetic field at 150-ms clock-laser interrogation time. These peaks, from left to right, correspond to the $\pi$ transition for the $m_F=-9/2$ to $m_F=+9/2$ sublevels, respectively.

Figure 3

Rabi oscillations of different Zeeman sublevels at (a) $|m_F|=9/2$, (b) $|m_F|=7/2$, (c) $|m_F|=5/2$, and (d) $|m_F|=3/2$. The experimental parameters are the same as in Fig. 2.

Figure 4

The degenerate narrow spectrum at 150-ms clock-laser interrogation time. The blue diamonds are the experimental data. The solid and dashed lines are the theoretical results with magnetic field $B=6$ mG and 0, respectively.

Figure 5

Rabi oscillations of (a) the zeroth Floquet sideband at $\overline{V}=1.5\text{V}$ and (b) the first Floquet sideband at $\overline{V}=2.5\text{V}$. The insets show the corresponding Rabi spectroscopy.

Figure 6

The Floquet degenerate spectrum at 150-ms clock-laser interrogation time, driving frequency ${\omega }_s/2\pi =200$ Hz, and residual magnetic field $B=6$ mG. The experimental data (blue diamonds) are compared to the theoretical results (red solid lines) at the driving voltages (a) $\overline{V}=0.5\text{V}$, (b) $\overline{V}=1.0\text{V}$, (c) $\overline{V}=1.5\text{V}$, (d) $\overline{V}=2.0\text{V}$, (e) $\overline{V}=2.5\text{V}$, and (f) $\overline{V}=3.0\text{V}$.

Figure 7

The relationship between the renormalized driving amplitude $A$ and driving voltage $\overline{V}$ added to the PZT in the Floquet degenerate system (red diamonds) and the Floquet polarized system (blue circles, from Ref. [39]). The dashed lines are linear fitting results, and the red region shows the $1\sigma$ deviation from the fitting line of the Floquet degenerate results.

Figure 8

The atomic distribution at different sublevels obtained by fitting the experimental data with Eq. (3) from Floquet theory. The voltages added to the PZT are (a) $\overline{V}=0.5\text{V}$, (b) $\overline{V}=1.0\text{V}$, (c) $\overline{V}=1.5\text{V}$, (d) $\overline{V}=2.0\text{V}$, (e) $\overline{V}=2.5\text{V}$, and (f) $\overline{V}=3.0V$. The dashed lines indicate the weighted mean, and the red region shapes the $1\sigma$ standard deviation. The atomic distribution at sublevel $m_F=1/2$ labeled with a star is calculated by subtracting the other sublevels from 1.

Figure 9

Determination of the experimental parameters in the nondriven case. (a) The experimental data for the motional sideband spectrum in the $z$ direction. (b) The experimental data for the motional sideband spectrum in the $r$ direction. (c) Theoretical fitting (red line) with experimental data (blue diamonds) for the blue sideband in the $z$ direction.