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

Fourier transform infrared (FT-IR) spectroscopic imaging has been widely tested as a tool for stainless digital histology of biomedical specimens, including for the identification of infiltration and fibrosis in endomyocardial biopsy samples to assess transplant rejection. A major barrier in clinical translation has been the slow speed of imaging. To address this need, we tested and report here the viability of using high speed discrete frequency infrared (DFIR) imaging to obtain stain-free biochemical imaging in cardiovascular samples collected from patients. Images obtained by this method were classified with high accuracy by a Bayesian classification algorithm trained on FT-IR imaging data as well as on DFIR data. A single spectral feature correlated with instances of fibrosis, as identified by the pathologist, highlights the advantage of the DFIR imaging approach for rapid detection. The speed of digital pathologic recognition was at least 16 times faster than the fastest FT-IR imaging instrument. These results indicate that a fast, on-site identification of fibrosis using IR imaging has potential for real time assistance during surgeries. Further, the work describes development and applications of supervised classifiers on DFIR imaging data, comparing classifiers developed on FT-IR and DFIR imaging modalities and identifying specific spectral features for accurate identification of fibrosis. This addresses a topic of much debate on the use of training data and cross-modality validity of IR measurements. Together, the work is a step toward addressing a clinical diagnostic need at acquisition time scales that make IR imaging technology practical for medical use.

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.

Progress in quantum nanoscience has engendered a physically diverse array of controllable solid-state quantum systems [4][5][6] . The prototypical quantum system consists of two wavefunctions that can be coherently combined into superpositions. In this work, we create and study superpositions of electron wavefunctions in nanoassembled quantum corrals where we can finely tune the geometry. We engineered an elliptical resonator to harbour degenerate wavefunctions whose superpositions could be manipulated. The solutions to the Schrödinger equation in a hard-walled ellipse possess two quantum numbers: n, the number of nodes crossing the minor axis, and l, half the number of nodal intersections along the perimeter (these map to the radial and angular momentum quantum numbers in a circle We assembled our designed resonator using a home-built scanning tunnelling microscope (STM) operating in ultrahigh vacuum. The single-crystal Cu(111) substrate was prepared, cooled to ~4 K, and dosed with ~15 Co atoms per (100 Å) 2 . We individually manipulated 2 44 Co adatoms to bound the corral. With spectroscopy, we verified that modes 41 and 42 occurred within a few mV of one another (Supp. Fig. 1). A constant-current ( I ) topograph of the finished structure is shown in Fig 1c. To confirm that the wavefunctions ψ closely describe this system, we used them to calculate (see Methods) a theoretical topograph ( Fig. 1d) that reproduces the data -4/13 -without any fitting parameters. Figure 1e displays the calculated contributions j c of the significant modes composing the topograph ( ) z r , such thatNext, we added a nanoscopic gate: a single cobalt atom. While moving the adatom across the ellipse-effectively sweeping a local electrostatic potential across the eigenstates-we measured topographs (Fig. 2, first column) and simultaneously acquired / dI dV image maps. By subtracting the / dI dV map of the empty ellipse, we created / dI dV difference maps (Fig. 2, second column). We began by placing the gate atom at one of the maxima of the calculated 2, 7 state. The resultant difference map (Fig. 2e) strongly resembles the 2, 7 state. Surprisingly, however, when the Co atom was moved rightward to one of the strong maxima of the state 4, 4 , the image produced ( Fig. 2g) was manifestly different from either of the two eigenstates.We will show that our / dI dV Δ maps are images of superpositions: phase- Electrons in quantum corrals are well modelled by particle-in-a-box solutions to the Schrödinger equation because the surface state wavelength [30 Å in Cu(111) is much larger than the spacing between the wall atoms 1,3,10,[15][16][17][18] . As a first clue to the underlying physics, the original report of the quantum mirage 3 pointed out the similarity between the solitary eigenfunction closest to E F and the spatial fine structure around the (Fig. 3a) comprise an overall 3-dimensional space; the crossover between planes in this space can be inferred from the structure of the wavefunctions (Fig. 3b-d).To generate arbitrary superpositio...