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Interaction-based quantum metrology showing scaling beyond the Heisenberg limit


Quantum metrology aims to use entanglement and other quantum resources to improve precision measurement1. An interferometer using N independent particles to measure a parameter can achieve at best the standard quantum limit of sensitivity, δN−1/2. However, using N entangled particles and exotic states2, such an interferometer3 can in principle achieve the Heisenberg limit, δN−1. Recent theoretical work4,5,6 has argued that interactions among particles may be a valuable resource for quantum metrology, allowing scaling beyond the Heisenberg limit. Specifically, a k-particle interaction will produce sensitivity δNk with appropriate entangled states and δN−(k−1/2) even without entanglement7. Here we demonstrate ‘super-Heisenberg’ scaling of δN−3/2 in a nonlinear, non-destructive8,9 measurement of the magnetization10,11 of an atomic ensemble12. We use fast optical nonlinearities to generate a pairwise photon–photon interaction13 (corresponding to k = 2) while preserving quantum-noise-limited performance7,14. We observe super-Heisenberg scaling over two orders of magnitude in N, limited at large numbers by higher-order nonlinear effects, in good agreement with theory13. 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 situations, however, higher-order nonlinearities prevent this crossover from occurring, reflecting the subtle relationship between scaling and sensitivity in nonlinear systems. Our work shows that interparticle interactions can improve sensitivity in a quantum-limited measurement, and experimentally demonstrates a new resource for quantum metrology.

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Figure 1: Atom–light interface.
Figure 2: Calibration of nonlinear Faraday rotation.
Figure 3: Super-Heisenberg scaling.


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We thank I. H. Deutsch and F. Illuminati for comments. We thank C. M. Caves and A. D. Codorníu for inspiration. This work was supported by the Spanish Ministry of Science and Innovation through the Consolider-Ingenio 2010 project QOIT, the Ingenio-Explora project OCHO (ref. FIS2009-07676-E/FIS) and project ILUMA (ref. FIS2008-01051), by the Marie-Curie RTN EMALI, and by Fundacio CELLEX Barcelona.

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Correspondence to M. Napolitano.

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Napolitano, M., Koschorreck, M., Dubost, B. et al. Interaction-based quantum metrology showing scaling beyond the Heisenberg limit. Nature 471, 486–489 (2011).

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