Imaging Optical Frequencies with $100\mu \mathrm{Hz}$ Precision and $1.1\mu \mathrm{m}$ Resolution

We implement imaging spectroscopy of the optical clock transition of lattice-trapped degenerate fermionic Sr in the Mott-insulating regime, combining micron spatial resolution with submillihertz spectral precision. We use these tools to demonstrate atomic coherence for up to 15 s on the clock transition and reach a record frequency precision of $2.5\times {10}^{-19}$. We perform the most rapid evaluation of trapping light shifts and record a 150 mHz linewidth, the narrowest Rabi line shape observed on a coherent optical transition. The important emerging capability of combining high-resolution imaging and spectroscopy will improve the clock precision, and provide a path towards measuring many-body interactions and testing fundamental physics.

Figure 1

Imaging spectroscopy experimental sequence. (a) A spin mixture of $|{{}^{1}S}_0,m_F=\pm 1/2⟩$ atoms is loaded into a 3D optical lattice. A $\pi$ pulse drives singly occupied sites of one spin state to the excited state, after which all other atoms are removed. The remaining atoms are placed in a superposition state and then read out with absorption imaging. Images are processed to yield the column density of (b) the ground state ${\tilde{n}}_g$, (c) the excited state ${\tilde{n}}_e$, and (d) the excitation fraction ${\tilde{p}}_e$. Small spatial frequency shifts can be measured through changes in ${\tilde{p}}_e$. (e) The normalized autocorrelation function of the central $15\mu \mathrm{m}\times 10\mu \mathrm{m}$ region of ${\tilde{p}}_e$ (black points) demonstrates correlations induced by the imaging system that correspond to a $1.1\mu \mathrm{m}$ imaging resolution (red line).

Figure 2

Imaging the local excitation fraction allows for the determination of small frequency shifts within the lattice-trapped sample. (a) A map of the excitation fraction of atoms after a Ramsey sequence ($T=5\mathrm{s}$) is split into two separate regions. (b) Parametric plots of $P_1$ against $P_2$ (black points) show ellipses, created by a reproducible phase shift between the two regions. A maximum likelihood estimator determines the ellipse properties (red line). The phase shift increases with $T$ for a fixed frequency difference $f_1-f_2$, while the contrast decays. (c) The measured uncertainty $\delta (f_1-f_2)$ for 900 s averaging time (blue points) closely follows the expected QPN limit for ellipse fitting (blue line). The measurements remain QPN-limited for 1000 experimental repetitions (red square and line), reaching a fractional uncertainty of $2.5\times {10}^{-19}$, with (inset) a total Allan deviation that averages with a slope of $3.6\times {10}^{-17}/\sqrt{\mathrm{Hz}}$. This uncertainty would correspond to a 2.3 mm gravitational redshift on the Earth. The frequencies are normalized to the clock frequency $f_0\approx 429\mathrm{THz}$ of ${}^{87}\mathrm{Sr}$.

Figure 3

Determining the lattice magic frequency through imaging spectroscopy. (a) When the lattice frequency is higher than the magic frequency, $e$ atoms (red line) experience a deeper potential than $g$ atoms (blue line, not to scale). (b) The mismatch in lattice potentials results in a spatially dependent clock frequency that we measure via ${\tilde{p}}_e$ after a Ramsey sequence, with a population that varies from $\frac{1}{2}+C$ to $\frac{1}{2}-C$, where $C$ is the contrast. An image of ${\tilde{p}}_e$ from a single cycle of the experiment is shown in (c). (d) A fit of the distance between the imaged rings gives a single-shot estimate of the differential ac Stark shift, which we determine separately for the $m_F=1/2$ (purple circles) and $-1/2$ (orange squares) states. Detuning of the lattice light is from the scalar magic frequency 368.554 725 THz [38]. (e) The uncertainty in the differential ac Stark shift averages down rapidly (black circles, fit is black line), with a QPN limit (dashed black line) 10 times better than the state-of-the-art Ref. [3].

Figure 4

Synchronous Rabi and Ramsey spectroscopy of an ultrastable laser. An applied magnetic field gradient creates a spatially dependent clock frequency. This effectively “disperses” the signal by converting the frequency response into a spatial pattern, a multiplexed measurement of the laser noise and atomic response. (a) In Ramsey spectroscopy, nearly all laser noise is common mode. “Ramsey fringes” are imaged as a spatially sinusoidal excitation fraction that measures the linearly increasing frequency shift across the sample, with $T=6\mathrm{s}$. (b) In Rabi spectroscopy, the entire lattice is illuminated by a clock pulse for 8 s. Only atoms in a narrow spatial region can be excited. Here, the measured linewidth is 150 mHz.