Floquet Engineering Hz-Level Rabi Spectra in Shallow Optical Lattice Clock

Quantum metrology with ultrahigh precision usually requires atoms prepared in an ultrastable environment with well-defined quantum states. Thus, in optical lattice clock systems deep lattice potentials are used to trap ultracold atoms. However, decoherence, induced by Raman scattering and higher order light shifts, can significantly be reduced if atomic clocks are realized in shallow optical lattices. On the other hand, in such lattices, tunneling among different sites can cause additional dephasing and strongly broadening of the Rabi spectrum. Here, in our experiment, we periodically drive a shallow ${}^{87}\mathrm{Sr}$ optical lattice clock. Counterintuitively, shaking the system can deform the wide broad spectral line into a sharp peak with 5.4 Hz linewidth. With careful comparison between the theory and experiment, we demonstrate that the Rabi frequency and the Bloch bands can be tuned, simultaneously and independently. Our work not only provides a different idea for quantum metrology, such as building shallow optical lattice clock in outer space, but also paves the way for quantum simulation of new phases of matter by engineering exotic spin orbit couplings.

Figure 1

The Rabi spectra of the excited state population at $U_z=9.1\mathrm{Er}$, $g_0=21\mathrm{Hz}$ with probing time $t=150\mathrm{ms}$ is measured. (a) In the undriven case, the spectrum is weak and broad ($\sim \text{kilohertz}$). (b) In the driven case with frequency excursion of the modulation ${\nu }_a=13.7644\mathrm{kHz}$ and driving frequency ${\nu }_s=600\mathrm{Hz}$, a strong and narrow spectral line emerges with 5.4 Hz linewidth estimated by Lorentz fitting function (dashed line). (c) Experimental setup: The ${}^{87}\mathrm{Sr}$ atoms are trapped in one dimensional optical lattice formed with two counterpropagating lattice laser beams which are tuned via an AOM. While the frequency of the stronger lattice laser is locked to the magic wavelength, the FE is implemented on the weaker one. The AOM aligned on the clock laser is used for band preparation and measurement. The inset presents the effective triangular velocity $v(t)$ of atoms and the effective force $F(t)$ that they feel.

Figure 2

(a) The Rabi oscillation of the zeroth Floquet sideband at different $K$’s. The markers are experimental data, and the solid lines are theoretical results [see Eq. (15) in the SM] with some fitting parameters [17, 20]. (b) The ratio between the effective Rabi frequency of the Floquet sidebands and its bare value, namely $g_{\mathrm{eff}}/g_0$, for both the zeroth and the first order, versus the modulation index $K$ (dots line). The effective Rabi frequency is extracted from the corresponding Rabi oscillations. The markers are the experimental data and the solid lines are the modulation function ${\mathcal{R}}^m(K)$.

Figure 3

(a) The Rabi spectrum of the atoms prepared at the internal state $|g⟩$ of $n_z=0$ Bloch band. The experimental parameters are $U_z=9E_r$, $g_0=104\mathrm{Hz}$, the probe time $t=6\mathrm{ms}$, and the driving frequency is fixed at ${\nu }_s=2.5\mathrm{kHz}$. (b) The Rabi spectrum of the atoms prepared at the internal state $|e⟩$ of $n_z=1$ Bloch band. The experimental parameters are $U_z=27.2E_r$, $g_0=196.7\mathrm{Hz}$, the probe time $t=3\mathrm{ms}$, and the driving frequency is fixed at ${\nu }_s=2.5\mathrm{kHz}$. The theoretical results (red solid lines), see Eq. (13) in SM, are calculated by taking the same fitting parameters as undriven case except one free fitting parameter radial temperature $T_r$ (see details in [17, 20]).

Figure 4

The VHS splitting ratio with ${\nu }_a$ for both $n_z=0$ and $n_z=1$, which is consistent with modification function ${\mathcal{F}}_1({\nu }_a)$.

Figure 5

The experimental data of Rabi oscillations (left) and spectra (right) are measured at ${\nu }_a=13.7644\mathrm{kHz}$, driving frequency ${\nu }_s=600\mathrm{Hz}$ and bare Rabi frequency $g_0=21\mathrm{Hz}$ with different lattice potentials and probe times: (a) $U_z=23E_r$, $t=135\mathrm{ms}$; (b) $U_z=15.1E_r$, $t=135\mathrm{ms}$; (c) $U_z=9.1E_r$, $t=150\mathrm{ms}$; and (d) $U_z=7.2E_r$, $t=150\mathrm{ms}$. The theoretical results [see Eq. (13) in SM] of Rabi oscillations (left) and Rabi spectrum (right) are red solid lines. The FWHM of carrier peaks in (a), (b), and (c) are obtained by Lorentz function fitting. The FWHM for $U_z=7.2E_r$ is not given, because its Rabi oscillation is strongly deviated from the theoretical prediction (${\chi }^2\approx 1.04$ vs $<0.09$ in other cases).