Magnetic Sensitivity Beyond the Projection Noise Limit by Spin Squeezing

We report the generation of spin squeezing and entanglement in a magnetically sensitive atomic ensemble, and entanglement-enhanced field measurements with this system. A maximal $m_f=\pm 1$ Raman coherence is prepared in an ensemble of $8.5\times {10}^5$ laser-cooled ${}^{87}\mathrm{Rb}$ atoms in the $f=1$ hyperfine ground state, and the collective spin is squeezed by synthesized optical quantum nondemolition measurement. This prepares a state with large spin alignment and noise below the projection-noise level in a mixed alignment-orientation variable. 3.2 dB of noise reduction is observed and 2.0 dB of squeezing by the Wineland criterion, implying both entanglement and metrological advantage. Enhanced sensitivity is demonstrated in field measurements using alignment-to-orientation conversion.

Figure 1

(a) Experimental geometry. PD: photodiode; L: lens; WP: wave plate; BS: beam splitter; PBS: polarizing beam splitter. (b) Measurement pulse sequence for (i) dispersive measurement of Raman coherence and (ii) QND measurement and alignment-to-orientation conversion. (c) Entanglement-enhanced field measurement: (i) a two-pulse QN measurement prepares a $T_x$-aligned state with reduced uncertainty in an alignment-orientation variable with a mixing angle $\theta$ that (ii) rotates into $T_y$ due to Zeeman shifts and (iii) is then read out by a single pulse, giving an integrated signal proportional to the Zeeman shift. See text for details.

Figure 2

Joint probability distribution of successive QND measurements with (a) no atoms in the trap, i.e., readout noise, (b) independent CSS preparations, and (c) a single CSS preparation. Solid curves indicate $2\sigma$ radii for Gaussian fits. Dashed blue circle in (c) reproduces solid circle in (b) indicating the standard quantum limit for the input CSS. Note: For presentation purposes, a small mean offset has been subtracted from the data.

Figure 3

Noise scaling of the QND measurement of the CSS and spin-squeezed state. Shaded area represents region of increased metrological sensitivity due to spin squeezing. Curves are plotted from independently measured experimental parameters unless noted; see text for details. Horizontal and vertical error bars represent $\pm 1\sigma$ statistical errors. Read-out noise has been subtracted from each data set.

Figure 4

Log-log plot of alignment-to-orientation measurement sensitivity with an input coherent spin state (blue circles) and spin-squeezed state (red diamonds). Curves are plotted from independently measured experimental parameters. Solid curves are the predicted measurement sensitivity with an input CSS (blue), and spin-squeezed state (red) with atomic projection noise reduced by a factor $1/(1+\zeta )$. Broken curves represent the light shot noise (dotted line), atomic projection-noise (dashed line), and spin-squeezed projection-noise (dot-dashed line) contributions to the sensitivity. Horizontal and vertical error bars represent $\pm 1\sigma$ statistical errors.