Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization

We experimentally acquire the disordered wave functions from the conical second-harmonic generation to explore the variation of weak localization from extended to prelocalized states. We numerically verify that the experimental density distributions with different extents of weak localization can be excellently analyzed with a reduced version of the nonlinear σ model (RV-NLS model). Moreover, we demonstrate that the χ-square distributions with fractional degrees of freedom are practically equivalent to the density distributions of the RV-NLS model. Our finding indicates that the concept of fractional degrees of freedom can be applied to the statistical properties of disordered wave functions.