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) = ± 1 Raman coherence is prepared in an ensemble of 8.5 × 10(5) laser-cooled (87)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.

Quantum metrology studies the use of entanglement and other quantum resources to improve precision measurement 1 . An interferometer using N independent particles to measure a parameter X can achieve at best the "standard quantum limit" (SQL) of sensitivity δX ∝ N −1/2 . The same interferometer 2 using N entangled particles can achieve in principle the "Heisenberg limit" δX ∝ N −1 , using exotic states 3 . Recent theoretical work argues that interactions among particles may be a valuable resource for quantum metrology, allowing scaling beyond the Heisenberg limit [4][5][6] . Specifically, a k-particle interaction will produce sensitivity δX ∝ N −k with appropriate entangled states and δX ∝ N −(k−1/2) even without entanglement 7 . Here we demonstrate this "super-Heisenberg" scaling in a nonlinear, nondestructive 8, 9 measurement of the magnetisation 10, 11 of an atomic ensemble 12 . We use fast optical nonlinearities to generate a pairwise photon-photon interaction 13 (k = 2) while preserving quantum-noise-limited performance 7,14 , to produce δX ∝ N −3/2 . We observe superHeisenberg scaling over two orders of magnitude in N , limited at large N by higher-order 1 arXiv:1012.5787v1 [quant-ph] 28 Dec 2010 nonlinear effects, in good agreement with theory 13 . For a measurement of limited duration, super-Heisenberg scaling allows the nonlinear measurement to overtake in sensitivity a comparable linear measurement with the same number of photons. In other scenarios, however, higher-order nonlinearities prevent this crossover from occurring, reflecting the subtle relationship of scaling to sensitivity in nonlinear systems. This work shows that inter-particle interactions can improve sensitivity in a quantum-limited measurement, and introduces a fundamentally new resource for quantum metrology.The best instruments are interferometric in nature, and operate according to the laws of quantum mechanics. A collection of particles, e.g., photons or atoms, is prepared in a superposition state, allowed to evolve under the action of a Hamiltonian containing an unknown parameter X , and measured in agreement with quantum measurement theory. The complementarity of quantum measurements 15 determines the ultimate sensitivity of these instruments.Here we describe polarisation interferometry, used for example in optical magnetometry to detect atomic magnetisation 11,16,17 ; similar theory describes other interferometers 2 . A collection of N photons, with circular plus/minus polarisations |+ , |− is described by single-photon Stokeswhere the σ i are the Pauli matrices and σ 0 is the identity.In traditional quantum metrology, a Hamiltonian of the formĤ = X N j=1ŝ (j) z uniformly and independently couples the photons to X , the parameter to be measured 1 . If the input state consists of independent photons, the possible precision scales as δX ∝ N −1/2 , the shot-noise or standard quantum limit (SQL). The N −1/2 factor reflects the statistical averaging of independent results. 2In contrast, entangled states can be highly, even perfectly...

We report the generation of a macroscopic singlet state in a cold atomic sample via quantum nondemolition measurement-induced spin squeezing. We observe 3 dB of spin squeezing and detect entanglement with 5σ statistical significance using a generalized spin-squeezing inequality. The degree of squeezing implies at least 50% of the atoms have formed singlets.

Measurement of spin precession is central to extreme sensing in physics,1,2 geophysics,3 chemistry,4 nanotechnology5 and neuroscience,6 and underlies powerful magnetic resonance spectroscopies.7 Because there is no spin-angle operator, any measurement of spin precession is necessarily indirect, e.g., inferred from spin projectors Fα at different times. Such projectors do not commute, and thus quantum measurement back-action (QMBA) necessarily enters the spin measurement record, introducing errors and limiting sensitivity. Here we show how to reduce this disturbance below δFα∼N, the classical limit for N spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a planar squeezed state8 which, because spins obey non-Heisenberg uncertainty relations,9,10 allows simultaneous precise knowledge of spin angle and amplitude. We use high-dynamic-range optical quantum non-demolition measurements11–13 applied to a precessing magnetic spin ensemble, to demonstrate spin tracking with steady-state angular sensitivity 2.9 dB beyond the standard quantum limit, simultaneous with amplitude sensitivity 7.0 dB beyond Poisson statistics.14 This method for the first time surpasses classical limits in non-commuting observables, and enables orders-of-magnitude sensitivity boosts for state-of-the-art sensing15–18 and spectroscopy.19,20

We extend the covariance matrix description of atom-light quantum interfaces, originally developed for real and effective spin-1/2 atoms, to include 'spin alignment' degrees of freedom. This allows accurate modelling of optically probed spin-1 ensembles in arbitrary magnetic fields. We also include technical noise terms that are very common in experimental situations. These include magnetic field noise, variable atom number and the effect of magnetic field inhomogeneities. We demonstrate the validity of our extended model by comparing numerical simulations to a free-induction decay measurement of polarized 87 Rb atoms in the f = 1 ground state. We qualitatively and quantitatively reproduce experimental results with no free parameters. The model can be easily extended to larger spin systems, and adapted to more complicated experimental situations. 4 These authors contributed equally to this work.