Dynamical discontinuities in repeated weak measurements revealed by complex weak values

Abstract

This work demonstrates that repeated weak measurements together with post-selection can produce sharp dynamical discontinuities in meter observables, even in minimal quantum systems. The discontinuous behavior is governed by the polar angle of the post selected state, which serves as a continuous control parameter. As this angle is varied, the expectation value of the meter observables changes abruptly at the point where the imaginary part of the associated weak value, a complex quantity that arises in weak measurements with post-selection, becomes zero. Such non-analytic behavior emerges only when the weak value is genuinely complex for a range of post-selection angles. If the weak value remains purely real for all angles, the dynamics remain smooth. The discontinuity originates from an exchange of stability between fixed points of the non-unitary Kraus operator governing the meter's evolution. Remarkably, despite the absence of a thermodynamic limit, the relaxation time in the vicinity of the discontinuity exhibit universal critical behavior characterized by a critical exponent equal to $1$, independent of system parameters. These results establish the weak value as a tunable control parameter capable of inducing non-analytic dynamical responses and reshaping the stability structure of measurement-induced quantum dynamics.