Clock precision beyond the Standard Quantum Limit at $10^{-18}$ level

Abstract

Optical atomic clocks with unrivaled precision and accuracy have advanced the frontier of precision measurement science and opened new avenues for exploring fundamental physics. A fundamental limitation on clock precision is the Standard Quantum Limit (SQL), which stems from the uncorrelated projection noise of each atom. State-of-the-art optical lattice clocks interrogate large ensembles to minimize the SQL, but density-dependent frequency shifts pose challenges to scaling the atom number. The SQL can be surpassed, however, by leveraging entanglement, though it remains an open problem to achieve quantum advantage from spin squeezing at state-of-the-art stability levels. Here we demonstrate clock performance beyond the SQL, achieving a fractional frequency precision of 1.1 $\times 10^{-18}$ for a single spin-squeezed clock. With cavity-based quantum nondemolition (QND) measurements, we prepare two spin-squeezed ensembles of $\sim$30,000 strontium atoms confined in a two-dimensional optical lattice. A synchronous clock comparison with an interrogation time of 61 ms achieves a metrological improvement of 2.0(2) dB beyond the SQL, after correcting for state preparation and measurement errors. These results establish the most precise entanglement-enhanced clock to date and offer a powerful platform for exploring the interplay of gravity and quantum entanglement.

Figures (4)

• Figure 1: Enhanced control of atomic motion. (a) Overview of the setup. Atoms are trapped in a 2D optical lattice formed by a movable lattice along the vertical ($z$) direction and a transverse lattice along $y$. Relative detuning of the movable lattice beams enables transport of atoms along $z$ direction, allowing independent cavity-based QND measurements of subensembles A (red tubes) and B (blue tubes). The transverse lattice enhances confinement and atom-cavity coupling homogeneity. A clock laser propagating along $z$ globally drives the clock transition. Combined with sideband cooling along $z$ and Doppler cooling in the $x-y$ plane, atomic temperatures below 0.5 $\upmu$K are achieved. (b) Two-ensemble Ramsey contrast as a function of dark time, demonstrating an atom-atom coherence time of $4.5(1)$ s. The inset shows the parametric plot of the excitation fraction $P_{\text{A}}$ and $P_{\text{B}}$ for subensembles A and B at 500 ms dark time. (c) Contrast decay as a function of the number of lattice transport roundtrips, measured at a dark time of 141 ms.
• Figure 2: Characterization of QND measurements. (a) The cavity resonance is detuned from the ${}^1S_0 \rightarrow {}^3P_1$ transition by $\delta_{\text{c}}/(2\pi)=-4\ \text{MHz}$. The probe laser for cavity-based QND measurements induces inhomogeneous light shifts $\delta_{\text{QND}}$ on the ${}^1S_0 \rightarrow {}^3P_0$ clock transition due to non-uniform atom-cavity coupling. (b) Measured QND-induced light shifts $\delta_{\text{QND}}$ using Rabi spectroscopy in two lattice configurations: 1D (movable lattice only, 19 $E_{\text{r}}$ depth) and 2D (movable lattice at 19 $E_{\text{r}}$ plus transverse lattice at 12 $E_{\text{r}}$). The inset shows a broadened, distorted line shape resulting from light shifts. (c,d) Inhomogeneous light shifts lead to loss of coherence and phase shift in Ramsey spectroscopy, which can be greatly suppressed with spin echo. Ramsey contrast with and without echo sequence is compared in two lattice configurations, showing that enhanced control of atomic motion in the 2D lattice improves spin-echo performance. The phase shift induced by QND probing with spin echo is negligible.
• Figure 3: Spin-noise reduction and metrological enhancement. (a) Measurement sequence. The "Pre" $J_z$ measurements project the CSSs into conditionally spin-squeezed states. The clock interrogation sequence consists of two $\pi/2$ pulses separated by an interrogation time $T$. "Final" $J_z$ measurements readout the spin-squeezed states after interrogation. (b) Correlations among successive $J_z$ measurements. Dashed circles with radius $2 \Delta J_{z, \text{A(B)}}^{\text{QPN}}$ indicate the QPN-limited scatter, independently calibrated from atom-cavity coupling. (c) Metrological enhancement as a function of probe photon number. Upper panel: the squared effective contrast $C^2=C_{\text{f}}^2/C_{\text{i}}$ versus photon number. Lower panel: spin noise reduction $R$ and metrological gain $\xi^2 = R/C^2$. A maximum spin-noise reduction of $R = -7.2(1.0)$ dB is observed. At the optimal photon number, a metrological enhancement $\xi^2=-5.1(1.0)$ dB is achieved.
• Figure 4: Synchronous clock comparison. With the movable lattice, ensembles A and B are alternately transported into the cavity for QND measurements, enabling direct synchronous clock comparison. Since the clock laser addresses two ensembles globally, common-mode clock laser noise is largely rejected. The measured frequency instability of CSS-CSS comparison (green points) agree with the theoretical QPN limit. In the meantime, the SSS-SSS comparison (blue points) not only achieves a 3.3(2) dB enhancement relative to the CSS-CSS case, but also demonstrates a 2.0(2) dB metrological advantage beyond the Standard Quantum Limit (SQL), after correcting for SPAM errors (initial contrast $C_{\text{i}} = 82(1)\%$ corresponds to a 0.9(1) dB correction to SQL). The synchronous clock comparison achieves a fractional frequency instability of $8.0(2)\times10^{-17}/\sqrt{\tau}$, resulting in a comparison precision of $1.6\times10^{-18}$ for full measurement time. Allan deviations of the frequency difference between two ensembles are plotted after subtracting a linear frequency drift of 1.6 $\upmu$Hz/s. Error bars represent $1\sigma$ statistical confidence interval.