Experimental probe of weak-value amplification and geometric phase through the complex zeros of the response function

We experimentally demonstrate a fundamental relationship between the weak value of an observable and the complex zero of the response function of a system by employing weak measurement on spin Hall shift of a Gaussian light beam. Using this relationship, we show that extremely large weak values far beyond its upper bound in the conventional linear-response regime can be experimentally obtained from the position of the minima of the pointer intensity profile corresponding to the real part of the complex zero of the response function. The imaginary part of the complex zero, on the other hand, is related to the spatial gradient of the geometric phase of light, which in this particular scenario evolves due to the weak interaction and the pre- and postselections of the polarization states. These relationships between the weak value and the complex zeros of the response function may provide new insights on weak measurements in a wide class of physical systems. Extraction of large weak values outside the usual domain of its validity and quantification of small interaction parameters using a physically meaningful and experimentally accessible system property such as the response function may open up a new paradigm of weak measurement.