Quantum Phase Extraction in Isospectral Electronic Nanostructures
  Quantum phase is not a direct observable and is usually determined by interferometric methods. We present a method to map complete electron wave functions, including internal quantum phase information, from measured single-state probability densities. We harness the mathematical discovery of drum-like manifolds bearing different shapes but identical resonances, and construct quantum isospectral nanostructures possessing matching electronic structure but divergent physical structure. Quantum measurement (scanning tunneling microscopy) of these "quantum drums" [degenerate two-dimensional electron states on the Cu(111) surface confined by individually positioned CO molecules] reveals that isospectrality provides an extra topological degree of freedom enabling robust quantum state transplantation and phase extraction. 
Christopher R. Moon, Laila S. Mattos, Brian K. Foster, Gabriel   Zeltzer, Wonhee Ko, Hari C. Manoharan (Stanford University)