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The extraordinary concept of weak value amplification has attractedconsiderable attention for addressing foundational questions in quantummechanics and for metrological applications in high precision measurement ofsmall physical parameters. Here, we experimentally demonstrate a fundamentalrelationship between the weak value of an observable and complex zero of theresponse function of a system by employing weak value amplification of spinHall shift of a Gaussian light beam. Using this relationship, we show thatarbitrarily large weak value amplification far beyond the conventional weakmeasurement limit can be experimentally obtained from the position of theminima of the pointer intensity profile corresponding to the real part of thecomplex zero of the response function. The imaginary part of the complex zero,on the other hand, is related to the spatial gradient of geometric phase oflight, which in this particular scenario evolves due to the weak interactionand the pre and post selections of the polarization states. These universalrelationships between the weak values and the complex zeros of the responsefunction may provide new insights on weak measurements in a wide class ofphysical systems. Extremely large weak value amplification and extraction ofsystem parameters outside the region of validity of conventional weakmeasurements may open up a new paradigm of weak measurement enhancing itsmetrological applications.

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