On Amplification by Weak Measurement

TL;DR

This paper provides a comprehensive analysis of weak measurement amplification, revealing that the traditional proportionality to the weak value breaks down at all coupling strengths, and highlights the importance of nonlinear effects for high amplification regimes.

Contribution

It derives the exact relationship between measured displacement and weak value, showing the limits of amplification and introducing the importance of nonlinear effects in experiments.

Findings

Measured displacement is not proportional to the weak value.

Maximum amplification depends on optimal pre- and post-selected states.

Nonlinear effects become significant at high amplification levels.

Abstract

We analyze the amplification by the Aharonov-Albert-Vaidman weak quantum measurement on a Sagnac interferometer [P. B. Dixon et al., Phys. Rev. Lett. 102, 173601 (2009)] up to all orders of the coupling strength between the measured system and the measuring device. The amplifier transforms a small tilt of a mirror into a large transverse displacement of the laser beam. The conventional analysis has shown that the measured value is proportional to the weak value, so that the amplification can be made arbitrarily large in the cost of decreasing output laser intensity. It is shown that the measured displacement and the amplification factor are in fact not proportional to the weak value and rather vanish in the limit of infinitesimal output intensity. We derive the optimal overlap of the pre- and post-selected states with which the amplification become maximum. We also show that the nonlinear effects begin to arise in the performed experiments so that any improvements in the experiment, typically with an amplification greater than 100, should require the nonlinear theory in translating the observed value to the original displacement.