Ultrasensitive Atomic Spin Measurements with a Nonlinear Interferometer

We study nonlinear interferometry applied to a measurement of atomic spin and demonstrate a sensitivity that cannot be achieved by any linear-optical measurement with the same experimental resources. We use alignment-to-orientation conversion, a nonlinear-optical technique from optical magnetometry, to perform a nondestructive measurement of the spin alignment of a cold ${}^{87}\mathrm{Rb}$ atomic ensemble. We observe state-of-the-art spin sensitivity in a single-pass measurement, in good agreement with covariance-matrix theory. Taking the degree of measurement-induced spin squeezing as a figure of merit, we find that the nonlinear technique’s experimental performance surpasses the theoretical performance of any linear-optical measurement on the same system, including optimization of probe strength and tuning. The results confirm the central prediction of nonlinear metrology, that superior scaling can lead to superior absolute sensitivity.

Popular Summary

Interferometric measurements form the basis of the most sensitive instruments, including gravitational wave detectors, atomic clocks, and optical magnetometers. The quantum theory of linear interferometers has been studied in detail, and quantum enhancements such as spin squeezing have been demonstrated in demanding, real-world applications. On the other hand, nonlinear interferometers, a class of devices that includes both optical magnetometers and Bose-Einstein condensates, have only recently begun to be understood at the quantum level. We demonstrate an ultrasensitive, nondestructive nonlinear measurement of atomic spin that achieves state-of-the-art sensitivity and, moreover, surpasses the best possible linear measurement of the same spin component.

A key feature of nonlinear interferometers is their so-called super-Heisenberg scaling, in which the sensitivity improves with particle number faster than even the best quantum-enhanced linear measurement. Better scaling suggests a fundamentally better approach, guaranteed to surpass linear interferometry for sufficient particle number. This suggestion has been the subject of much recent debate and, to date, is unconfirmed by experiment. We nondestructively measure the spin alignment of $6\times {10}^5$ laser-cooled ${}^{87}\mathrm{Rb}$ atoms in an optical dipole trap using 2-$\mu \mathrm{s}$-long pulses of light. We show how the atomic spin-alignment component scales with photon number, in agreement with covariance-matrix theory. Our nonlinear measurements become more sensitive than linear measurements when the photon number increases above $3\times {10}^7$, confirming the theoretical prediction that improved scaling should lead to greater sensitivity. We furthermore reproduce spin squeezing (i.e., better metrological sensitivity), which would not be induced for linear measurements given our experimental parameters.

The ability of nonlinear measurements to detect spin alignment is an improvement over previous analyses, which could only infer spin orientation. Furthermore, we expect that nonlinear measurements will contribute to the detection and preparation of exotic quantum phases in ensembles of ultracold atoms.

Figure 1

Alignment-to-orientation conversion measurement of atomic spins. (a) An unknown field $B_z$ rotates an initially ${\widehat{J}}_x$-polarized state in the ${\widehat{J}}_x-{\widehat{J}}_y$ plane. The ${\widehat{J}}_y$ component is detected using an ${\widehat{S}}_x$-polarized probe, which produces a rotation of ${\widehat{J}}_y$ toward ${\widehat{J}}_z$ by an angle ${\theta }_{\mathrm{AOC}}={\kappa }_2{\widehat{S}}_x/2$. (b) Simultaneously, paramagnetic Faraday rotation produces a rotation of ${\widehat{S}}_x$ toward ${\widehat{S}}_y$. The net effect is a rotation ${\mathrm{\Phi }}_{\mathrm{AOC}}={\kappa }_1{\kappa }_2{N}_L{\widehat{J}}_y/4$, which is observed by detecting ${\widehat{S}}_y$. In an alternative strategy, the linear-to-elliptical rotation of ${\widehat{S}}_x$ toward ${\widehat{S}}_z$ by the angle ${\mathrm{\Phi }}_{\mathrm{LTE}}={\kappa }_2{\widehat{J}}_y$ can be observed by detecting ${\widehat{S}}_z$. (c) Experimental geometry. Near-resonant probe pulses pass through a cold cloud of ${}^{87}\mathrm{Rb}$ atoms and experience a Faraday rotation by an angle proportional to the on-axis collective spin ${\widehat{J}}_z$. Atoms are prepared in a coherent spin state ${\widehat{J}}_x$ via optical pumping. The pulses are initially polarized with maximal Stokes operator ${\widehat{S}}_x$, measured at the input by a photodiode (${\mathrm{PD}}_3$). Rotation toward ${\widehat{S}}_y$ is detected by a balanced polarimeter consisting of a wave plate (WP), polarizing beam splitter (PBS), and photodiodes (${\mathrm{PD}}_{1,2}$).

Figure 2

Alignment-to-orientation conversion measurement of $J_y$. In the main frame, we plot the signal $S_y=⟨{\widehat{S}}_y^{(\text{out})}⟩$ of the AOC measurement as a function of $S_x$ for various $J_x$. We find the measured signal $J_y$ from fit to data using the function $S_y=({\kappa }_1{\kappa }_2/2)J_y{S}_x^2$ (solid lines). Error bars represent $\pm 1\sigma$ statistical errors. Inset: Measured $J_y$ versus $J_x$. For small rotation angles, $J_y\simeq 2{\omega }_L{B}_{z}t{J}_x$.

Figure 3

Log-log plot of the uncertainty $\mathrm{\Delta }{J}_y$ of the AOC measurement versus number of photons $N_L$. Blue diamonds indicate the measured sensitivity. Nonlinear enhanced scaling of the sensitivity is observed over more than an order of magnitude in $N_L$. A fit to the data yields $\mathrm{\Delta }{J}_y\propto N_L^k$ with $k=-1.46\pm 0.04$. The best observed sensitivity is $\mathrm{\Delta }{J}_y=1290\pm 90$ spins with $N_L=2\times 10^8\mathrm{photons}$. For reference, we also plot the data (light blue circles) and theory (dotted curve) for the measurement of ${\widehat{J}}_z$ via nonlinear Faraday rotation reported by Napolitano et al. [8]. The solid blue curve represents theory given by Eq. (3) with no free parameters, plus the independently measured electronic noise contribution. The dashed green curve shows the theoretical prediction describing an ideal LTE measurement of $J_y$ without technical or electronic noise contributions. The nonlinear measurement sensitivity surpasses an ideal LTE measurement with $N_L=3\times 10^7\mathrm{photons}$. Error bars for standard errors would be smaller than the symbols and are not shown. Inset: Observed metrologically significant spin squeezing ${\xi }_m^2$ as a function of photon number. The dashed line is a guide to the eye. Error bars indicate $\pm 1\sigma$ standard errors.

Figure 4

Theoretical comparison of AOC (solid blue curves) and LTE (dashed green curves) measurement sensitivity. (a) Number of photons $N_L$ needed to achieve projection-noise-limited sensitivity $(\mathrm{\Delta }{\widehat{J}}_y{)}_{\mathrm{AOC}}^2=(\mathrm{\Delta }{\widehat{J}}_y{)}_{\mathrm{LTE}}^2=N_A/4$ as a function of detuning $\mathrm{\Delta }$. The gray line indicates $(\mathrm{\Delta }{\widehat{J}}_y{)}_{\mathrm{AOC}}^2=(\mathrm{\Delta }{\widehat{J}}_y{)}_{\mathrm{LTE}}^2$, so that the AOC (LTE) strategy is more sensitive in the shaded (white) region. Magenta curves represent damage ${\eta }_{\mathrm{sc}}=0.1$ (dot–dashed curve) and 0.5 (dotted curve) to the atomic state due to spontaneous emission. (b) Estimated metrologically significant spin squeezing ${\xi }_m^2$, optimized as a function of $N_L$, versus probe detuning.

Figure 5

Theoretical calculation of spin squeezing as a function of optical depth and probe detuning for (a) the AOC strategy and (b) the LTE strategy with our experimental parameters. Contours indicate the minimum achievable metrologically significant squeezing ${\xi }_{m,\min }^2$ with respect to $N_L$, with values indicated, as a function of detuning $\mathrm{\Delta }$ and OD. (c) The blue diamonds represent the observed spin squeezing ${\xi }_m$ as a function of optical depth for the AOC measurement with $N_L=2\times 10^8\mathrm{photons}$. The solid curves show the theoretically predicted minimum achievable squeezing, optimized with respect to both $N_L$ and $\mathrm{\Delta }$, for the AOC measurement (solid blue line) and the LTE measurement (solid green line). The scaling of each curve is roughly ${\xi }^2\propto {\mathrm{OD}}^{-1/2}$, so the advantage for AOC continues also to large OD.