Geometrical interpretation of quantum weak measurement

Abstract

We revisit the quantum weak measurement (QWM) by sketching the polarisation state dynamics on the Poincaré sphere and the associated geometric phase, which is considered the soul of QWM. Our experimental arrangement comprises a coherent laser beam as a source of pure state, two polarizers corresponding to pre- and post-selection and a tilted wave plate placed in-between them to introduce weak interaction. The pre-selected state is a linearly polarised beam that can be represented as a point on the Poincaré sphere equator and the weak interaction with the wave plate results in a small spread in S3 axis. Now, an orthogonal projection of this state leads two different geodesics on the surface of the sphere through its poles, starting from both sides of the spread to the orthogonal point. The post-selection is made by moving the projection point slightly away from the orthogonal position so that both geodesics shift to the equator which results in a rapid accumulation of geometric phase in the beam crosssection. A gradient in the linear momentum is developed as a consequence which gives amplified shift in the beam position.