Weak Measurement of Berry's Phase

TL;DR

This paper explores how weak quantum measurements relate to Berry's phase, revealing that weak values encode geometric phase information and can be directly measured using specific measurement sequences.

Contribution

It establishes a direct connection between weak values and Berry's phase, providing a method to measure geometric phases through weak measurement techniques.

Findings

Weak values relate to third order Bargmann invariants.

Berry's phase corresponds to the argument of weak values.

Alternating strong and weak measurements can determine Berry's phase.

Abstract

Quantum measurements can be generalized to include complex quantities. It is possible to relate the quantum weak values of projection operators to the third order Bargmann invariants. The argument of the weak value becomes, up to a sign, equal to the Berry's phase associated with the three state vectors. In case of symmetric informationally complete, positive operator valued measures (SIC-POVMs), this relation takes a particularly simple form. Alternating strong and weak measurements can be used to determine Berry's phase directly, which demonstrates that not only their real and imaginary parts but also moduli and arguments of weak values have a physical significance. For an arbitrary projection operator, weak value is real when the projector, pre- and post-selected states lie on a so-called null phase curve which includes the geodesic containing the three states as a special case.