Interaction-based quantum metrology showing scaling beyond the Heisenberg limit
Abstract
Quantum metrology studies the use of entanglement and other quantum resources to improve precision measurement. 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 using N entangled particles can achieve in principle the "Heisenberg limit" δX N-1, using exotic states. Recent theoretical work argues 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 δX N-k with appropriate entangled states and δX N-(k-1/2) even without entanglement. Here we demonstrate this "super-Heisenberg" scaling in a nonlinear, non-destructive measurement of the magnetisation of an atomic ensemble. We use fast optical nonlinearities to generate a pairwise photon-photon interaction (k = 2) while preserving quantum-noise-limited performance, to produce δX N-3/2. We observe super-Heisenberg scaling over two orders of magnitude in N, limited at large N by higher-order nonlinear effects, in good agreement with theory. 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.