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.Comment: Published 8 February 2008 in Science; 13 page manuscript (including 4
figures) + 13 page supplement (including 6 figures); supplementary movies
available at http://mota.stanford.ed
The ability of the scanning tunnelling microscope to manipulate single atoms and molecules has allowed a single bit of information to be represented by a single atom or molecule. Although such information densities remain far beyond the reach of real-world devices, it has been assumed that the finite spacing between atoms in condensed-matter systems sets a rigid upper limit on information density. Here, we show that it is possible to exceed this limit with a holographic method that is based on electron wavefunctions rather than free-space optical waves. Scanning tunnelling microscopy and holograms comprised of individually manipulated molecules are used to create and detect electronically projected objects with features as small as approximately 0.3 nm, and to achieve information densities in excess of 20 bits nm-2. Our electronic quantum encoding scheme involves placing tens of bits of information into a single fermionic state.
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