All-order evaluation of weak measurements: --- The cases of an operator $\mathbf{A}$ which satisfies the property ${\mathbf{A}}^2=1$ ---

Abstract

Some exact formulae of the expectation values and probability densities in a weak measurement for an operator $\mathbf{A}$ which satisfies the property ${\mathbf{A}}^2=1$ are derived. These formulae include all-order effects of the unitary evolution due to the von-Neumann interaction. These are valid not only in the weak measurement regime but also in the strong measurement regime and tell us the connection between these two regime. Using these formulae, arguments of the optimization of the signal amplification and the signal to noise ratio are developed in two typical experimental setups.

17 pages, 10 figures (v1); Fig.3 and some typos are corrected (v2); Comments and references are added and some typos are corrected (v3)

References in corpus (7)

• Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification
• Experimental joint weak measurement on a photon pair as a probe of Hardy's Paradox
• Complex weak values in quantum measurement
• Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair
• Optimizing the Signal to Noise Ratio of a Beam Deflection Measurement with Interferometric Weak Values
• Quantum Averages of Weak Values
• Weak measurement: Effect of the detector dynamics

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