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, δ  ∝ N^−1/2. However, using N entangled particles and exotic states², such an interferometer³ can in principle achieve the Heisenberg limit, δ  ∝ N⁻¹. Recent theoretical work^{4, 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 δ  ∝ N^−k with appropriate entangled states and δ  ∝ N^−(k−1/2) even without entanglement⁷. Here we demonstrate ‘super-Heisenberg’ scaling of δ  ∝ N^−3/2 in a nonlinear, non-destructive^{8, 9} measurement of the magnetization^{10, 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 performance^{7, 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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