States of an Ensemble of Two-Level Atoms with Reduced Quantum Uncertainty

We generate entangled states of an ensemble of $5\times {10}^4$ ${}^{87}\mathrm{Rb}$ atoms by optical quantum nondemolition measurement. The resonator-enhanced measurement leaves the atomic ensemble, prepared in a superposition of hyperfine clock levels, in a squeezed spin state. By comparing the resulting reduction of quantum projection noise [up to 8.8(8) dB] with the concomitant reduction of coherence, we demonstrate a clock input state with spectroscopic sensitivity 3.0(8) dB beyond the standard quantum limit.

Figure 1

(a) Experimental setup. (b) Atomic level structure. (c) Experimental sequence. Timing of probe pulses (solid line) and microwave pulses (dashed line) in preparation and readout of a squeezed state. $\tilde{\pi }$ designates a composite $\pi$ pulse [25]. Various procedures are inserted at $X$, as described in the text, to measure the CSS variance, measure the noise of a spin component other than $S_z$, or operate a clock. $\mathbf{A}–\mathbf{C}$ illustrate semiclassical probability distribution functions for the Gaussian states discussed in the text.

Figure 2

Spin noise measurements (see text): $y_1=4\mathrm{Var}(M_1)$ for a single CSS preparation (open triangles), $y_2=2\mathrm{Var}({\tilde{M}}_1-{\tilde{M}}_2)$ for two independently prepared CSSs (open circles), and $4\Delta S_{z\mathrm{meas}}^2=2\mathrm{Var}(M_1-M_2)$ for two measurements after a single CSS preparation (solid diamonds), all in units of atom number. Vertical error bars are statistical; horizontal error bars indicate standard deviation of measured atom numbers. The solid (dashed) line corresponds to $a_1=1$, $a_2=9\times {10}^{-6}$ ($a_2=0$). Inset: Variance $\Delta S_{\alpha }^2$ of $S_z$ after rotating a squeezed state by an angle $\alpha$ about $⟨\mathbf{S}⟩$, with parameter-free theory curve.

Figure 3

Measured data with fits of simple models [25] for normalized spin noise ${\sigma }^2=[\Delta S_z^2{]}_{M_1}/\Delta S_{z\mathrm{CSS}}^2$ (open diamonds), contrast $\mathcal{C}$ (open squares), and metrological squeezing parameter ${\zeta }_m$ (solid triangles).