Θ MLE ( μ ) is the absolute maximum (dashed line) of the likelihood function P ( μ | φ ) (solid line) corresponding to the observed sequence of results μ. Schematic representation of different phase estimation protocols.
Here N is the total number of particles or, in the presence of fluctuations, the mean total. Every symbol is accompanied by a number (in chronological order) corresponding to the reference reported in the side table. Filled (open) circles indicate results obtained (inferred) without (with) subtraction of technical and/or imaging noise. Circles are expected gains based on a characterization of the quantum state, e.g., calculated as Δ θ = ξ R / N, where ξ R is the spin-squeezing parameter or as Δ θ = 1 / F Q, where F is the Fisher information. Stars refer to directly measured phase sensitivity gains, performing a full phase estimation experiment. The solid thick line is the Heisenberg limit Δ θ HL = 1 / N. The gain is shown on logarithmic and linear scales. Gain of phase sensitivity over the standard quantum limit Δ θ SQL = 1 / N achieved experimentally with trapped ions (black symbols), Bose-Einstein condensates (red), and cold thermal ensembles (blue).
The goal of this article is to review and illustrate the theory and the experiments with atomic ensembles that have demonstrated many-particle entanglement and quantum-enhanced metrology. All these factors call for generating nonclassical atomic states designed for phase estimation in atomic clocks and atom interferometers, exploiting many-body entanglement to increase the sensitivity of precision measurements. Moreover, atoms can strongly couple to external forces and fields, which makes them ideal for ultraprecise sensing and time keeping. Thanks to the large and tunable nonlinearities and the well-developed techniques for trapping, controlling, and counting, many groundbreaking experiments have demonstrated the generation of entangled states of trapped ions, cold, and ultracold gases of neutral atoms.
Progress has been particularly rapid for atoms. This has stimulated the research in different areas of physics to engineer and manipulate fragile many-particle entangled states. Quantum technologies exploit entanglement to revolutionize computing, measurements, and communications.
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