  The extraordinary concept of weak value amplification has attracted
considerable attention for addressing foundational questions in quantum
mechanics and for metrological applications in high precision measurement of
small physical parameters. Here, we experimentally demonstrate a fundamental
relationship between the weak value of an observable and complex zero of the
response function of a system by employing weak value amplification of spin
Hall shift of a Gaussian light beam. Using this relationship, we show that
arbitrarily large weak value amplification far beyond the conventional weak
measurement limit 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 geometric phase of
light, which in this particular scenario evolves due to the weak interaction
and the pre and post selections of the polarization states. These universal
relationships between the weak values and the complex zeros of the response
function may provide new insights on weak measurements in a wide class of
physical systems. Extremely large weak value amplification and extraction of
system parameters outside the region of validity of conventional weak
measurements may open up a new paradigm of weak measurement enhancing its
metrological applications.
