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 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...

Quantum metrology for a general Hamiltonian parameter

“…Moreover, when there is interaction among the N entangled qubits, the measurement precision can be further increased to beyond the Heisenberg limit [11][12][13][14][15][16].…”

Nonlocality in many-body quantum systems detected with two-body correlators

“…Last, but not least, one should consider cold and ultracold ensembles (for an excellent review see [106]), in which, by employing quantum Faraday effect, one can reach unprecendented degeees of squeezing of the total atomic spin (cf. [107,108], and unprecendented degrees of precision of quantum magnetometry (cf. [109]).…”

Precision bounds for noisy nonlinear quantum metrology

“…For example, in quantum optics we usually have G =n, wheren is the photon number operator. However, more generally, the parameter shift φ can be generated by an optical nonlinearity [3,4]. In such a case, the generator G is a nonlinear function of the photon number operatorn such as G =n q , where q denotes the order of the nonlinearity.…”