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

TL;DR

This paper derives exact all-order formulas for expectation values and probability densities in weak measurements involving operators with ${f A}^2=1$, bridging weak and strong measurement regimes and optimizing signal amplification.

Contribution

It provides the first comprehensive all-order formulas for such measurements, including strong interaction effects, and explores optimization strategies in experimental setups.

Findings

Derived exact all-order expectation values and probability densities.

Established connection between weak and strong measurement regimes.

Analyzed optimization of signal amplification and SNR in experiments.

Abstract

Some exact formulae of the expectation values and probability densities in a weak measurement for an operator ${\bf A}$ which satisfies the property ${\bf A}^{2}=1$ 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.