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Quantum metrology studies the use of entanglement and other quantum resourcesto improve precision measurement. An interferometer using N independentparticles to measure a parameter X can achieve at best the "standard quantumlimit" (SQL) of sensitivity {\delta}X \propto N^{-1/2}. The same interferometerusing N entangled particles can achieve in principle the "Heisenberg limit"{\delta}X \propto N^{-1}, using exotic states. Recent theoretical work arguesthat interactions among particles may be a valuable resource for quantummetrology, allowing scaling beyond the Heisenberg limit. Specifically, ak-particle interaction will produce sensitivity {\delta}X \propto N^{-k} withappropriate entangled states and {\delta}X \propto N^{-(k-1/2)} even withoutentanglement. Here we demonstrate this "super-Heisenberg" scaling in anonlinear, non-destructive measurement of the magnetisation of an atomicensemble. We use fast optical nonlinearities to generate a pairwisephoton-photon interaction (k = 2) while preserving quantum-noise-limitedperformance, to produce {\delta}X \propto N^{-3/2}. We observe super-Heisenbergscaling over two orders of magnitude in N, limited at large N by higher-ordernonlinear effects, in good agreement with theory. For a measurement of limitedduration, super-Heisenberg scaling allows the nonlinear measurement to overtakein sensitivity a comparable linear measurement with the same number of photons.In other scenarios, however, higher-order nonlinearities prevent this crossoverfrom occurring, reflecting the subtle relationship of scaling to sensitivity innonlinear systems. This work shows that inter-particle interactions can improvesensitivity in a quantum-limited measurement, and introduces a fundamentallynew resource for quantum metrology.

Interaction-based quantum metrology showing scaling beyond the Heisenberg limit