Near-Unitary Spin Squeezing in ${}^{171}\mathrm{Yb}$

Spin squeezing can improve atomic precision measurements beyond the standard quantum limit (SQL), and unitary spin squeezing is essential for improving atomic clocks. We report substantial and nearly unitary spin squeezing in ${}^{171}\mathrm{Yb}$, an optical lattice clock atom. The collective nuclear spin of $\sim {10}^3$ atoms is squeezed by cavity feedback, using light detuned from the system’s resonances to attain unitarity. The observed precision gain over the SQL is limited by state readout to 6.5(4) dB, while the generated states offer a gain of 12.9(6) dB, limited by the curvature of the Bloch sphere. Using a squeezed state within 30% of unitarity, we demonstrate an interferometer that improves the averaging time over the SQL by a factor of 3.7(2). In the future, the squeezing can be simply transferred onto the optical-clock transition of ${}^{171}\mathrm{Yb}$.

Figure 1

(a) Experimental setup. A one-dimensional optical lattice at ${\lambda }_t=759\mathrm{nm}$ (red) traps the atoms. Light at $\lambda =556\mathrm{nm}$ (green), whose transmission is detected using a dichroic mirror ($D$) and a single-photon counter (SPC), is used for squeezing and probing. (b) Relevant energy levels of ${}^{171}\mathrm{Yb}$ with ground state $|{{}^{1}S}_0,I=\frac{1}{2}⟩$ and excited state $|{{}^{3}P}_1,F=\frac{3}{2}⟩$. The Zeeman splitting in $|{{}^{3}P}_1,F=\frac{3}{2}⟩$ is ${\mathrm{\Delta }}_z/(2\pi )=18.5\mathrm{MHz}$ for a magnetic field $B_z=13.6\mathrm{G}$ along the cavity axis. (c) Cavity transmission spectrum showing vacuum Rabi splitting for $N\eta =1800$ as well as the two squeezing light pulses ${\omega }_{l1}$ and ${\omega }_{l2}$. (d) Simplified representation of the squeezing and measurement sequence. (e) Quasiprobability distributions for a CSS (red) and SSS (blue).

Figure 2

Measured normalized spin noise ${\sigma }^2(\alpha )$, as a function of the state rotation angle $\alpha$, for shearing strengths $Q=0.3$ (blue circles), $Q=2.2$ (green squares), $Q=4.5$ (yellow triangles), and $Q=6.3$ (red diamonds). For visualization, the data measured at $\alpha =0$ are displayed at $\alpha =0.02\mathrm{rad}$. The solid lines are theoretical fits. States in the violet region below the dashed line at ${\sigma }^2={\sigma }_d^2=0.11$ (detection limit) cannot be directly observed. Inset: ${\sigma }_{st}^2$ of SSS after subtracting measurement noise ${\sigma }_d^2$ for the same parameters.

Figure 3

Shearing strength $Q$ (filled blue squares) and excess broadening factor $F$ (red circles) plotted versus the number $p_t$ of transmitted photons. The open red circles correspond to the directly measured data with the theoretical model without free parameters (dotted red line), and the solid circles are after subtraction of measurement noise ${\sigma }_d^2$ with the parameter-free model without (solid red line) and with (dashed red line) Bloch-sphere-curvature-induced broadening. The theoretical predictions are given by Eqs. (S7) and (S8) of Supplemental Material [32].

Figure 4

(a) Wineland metrological gain ${\xi }_W^2$ (blue), measured spin noise reduction ${\xi }_{-}^2$ (green), and inferred SSS noise ${\xi }_{st}^2$ (red) as a function of shearing strength $Q$. ${\xi }_{-}^2$ is limited by the measurement resolution and ${\xi }_{st}^2$ by the curvature of the Bloch sphere. (b) Ramsey contrast as a function of $Q$ with initial contrast as the only fitting parameter (solid line). (c) Allan deviation of a phase measurement for a CSS (black squares) with SQL (dashed line) and for an SSS with $Q=3.8(p_t=130)$, $F=0.8$ (red data). The red solid line is fit to the first three data points. The reduction in the measurement time over the SQL is a factor of 3.7(2), represented by the orange star in (a).