Quantum holographic encoding in atwo-dimensional electron gasChristopher R. Moon1, Laila S. Mattos1, Brian K. Foster2, Gabriel Zeltzer3 and Hari C. Manoharan1*

The ability of the scanning tunnelling microscope to manipulate single atoms and molecules has allowed a single bit ofinformation to be represented by a single atom or molecule. Although such information densities remain far beyond thereach of real-world devices, it has been assumed that the finite spacing between atoms in condensed-matter systems setsa rigid upper limit on information density. Here, we show that it is possible to exceed this limit with a holographic methodthat is based on electron wavefunctions rather than free-space optical waves. Scanning tunnelling microscopy andholograms comprised of individually manipulated molecules are used to create and detect electronically projected objectswith features as small as �0.3 nm, and to achieve information densities in excess of 20 bits nm22. Our electronicquantum encoding scheme involves placing tens of bits of information into a single fermionic state.

We use atomic manipulation1 to first construct open nano-structures—molecular holograms—which can concentrateinformation into a medium that is free of lattice con-

straints: the quantum states of a two-dimensional gas of electrons.Electron waves scattered from the potential imposed by themolecules propagate into an atomically uniform area of a coppersurface. Here, through coherent electron interference, the infor-mation inherent in the molecule positions is densely projectedinto two spatial degrees of freedom and one energy degree offreedom. Similar to optical volume holography2–4, thisrequires precise amplitude and phase engineering5,6 of electronwavefunctions to store ‘pages’ of information volumetrically.These data are read out by mapping the energy-resolved electrondensity of states (DOS) with a scanning tunnelling microscope(STM). As the projection and readout are both extremely near-field,and because we use native quantum states rather than an externalbeam7, we are not limited by lensing or collimation. This allowsus to encode information at length scales smaller than the atomicspacing limit, thus heralding a new horizon for attainableinformation density.

In conventional holography, a three-dimensional (3D) object isstored in a two-dimensional (2D) hologram by interfering twocoherent waves: an object beam carrying information and asimple reference beam. When the resulting interference pattern islater illuminated with a replica of the reference beam, the object isprojected and the information can be retrieved (Fig. 1a). The inter-ference pattern can also be simulated digitally on a computer,which then sends the hologram for a given projection to a spatiallight modulator. New developments in this arena include opticalholographic data storage2–4 (projecting data into a 3D storagemedium), holographic optical tweezers8,9 (creating precise 3Doptical trapping potentials) and holovideo10 (producing dynamicholographic projections in real time). Here, we use a conjugate tech-nique (Fig. 1b): we directly construct a scattering environmentmade from individual molecules that is permanently illuminatedby a continuous spectrum of electrons confined to a metalsurface. The scattering geometry is arranged at the atomic scalesuch that a target holographic projection results. However, this pro-jection is not into three spatial dimensions, as in traditional optical

or electron-beam holography, but rather into 3D position–energy(r–E) space, where r ¼ (x, y) is a planar position.

This new form of holography is achievable in any quantum 2Delectron system where the local potential can be sufficiently con-trolled, such as with fixed or scanned metallic gates, or with theaddition of atoms or molecules as local gates. In our implemen-tation, we began with the nearly free gas of 2D surface state electronshosted by the (111) face of a copper crystal11. Atomic manipulationenables confinement of these electrons into quantum corrals12, per-mitting the study of lifetime effects13, Kondo physics14,15,single-atom gating6 and quantum phase5. Curiously, however,limited attention has been paid to applications of electron waves

Illumination

Hologram

Projection

Detection

x1 nmy

E

Quantumelectrons

x

y

z

Classicalphotons

1 cm

Figure 1 | Electronic quantum holography concept. a, In traditional optical

holography, light shone on a 2D hologram projects a 3D object viewable by

eye. b, In this work, two-dimensional quantum electrons illuminate coplanar

holograms assembled with atomic manipulation. The analogue of the 3D

object (density of mass) that can be seen by the eye is the electron density

of states, which is observed by means of scanning tunnelling microscopy.

Dense holographic encoding is made possible by projecting information into

one energy and two spatial dimensions.

1Department of Physics, Stanford University, Stanford, California 94305, USA, 2Department of Electrical Engineering, Stanford University, Stanford, California94305, USA, 3Department of Applied Physics, Stanford University, Stanford, California 94305, USA; *e-mail: manoharan@stanford.edu

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in open geometries, other than the recognition that they cantheoretically possess very small features16. Additionally motivatedby existing subquantum detection schemes for both electroniccharge17 and magnetic flux18, and new storage technology trendsmaking use of a third dimension19, we sought to use open, coherentquantum structures to encode data in a fashion scalable below thespatial limit of atomic condensed matter.

Molecular hologramsOur molecular holograms are created by positioning COmolecules on the Cu(111) surface using a STM (Fig. 1b).Illuminated by the resident 2D electron gas (which is analogousto white light), these strictly planar holograms project electronicstates into 3D position-energy space, as electron waves scatterfrom the molecules and coherently interfere. Data can be encodedinto this volumetric projection in various ways; here we project2D pages of information into a readout area of the surface devoidof molecules. Page i is a pattern of bits written at energy Ei at pos-itions r in the electronic local density of states (LDOS), r(r, Ei). Each

bit is defined by whether r(r, Ei) is above or below a constantthreshold value. A page is read with the STM as horizontal slicesthrough the full 3D electronic object (Fig. 1b). In essence, infor-mation in the discrete molecule positions is carried by the electrongas and concentrated into the central readout region, where weattempt to maximize information density. Holography is ideal forthis task as it allows multiple pages to be stacked within the samephysical space.

Coherent electron scattering in molecular holograms is not assimple as in closed quantum corrals, where electrons are welldescribed by particle-in-a-box wavefunctions6,12. Fortunately, theDOS can still be predicted in a straightforward manner with amultiple-scattering Green’s function method20,21. In sharp contrast,engineering a specific pattern in r(r, E)—the encoding process—requires solving the difficult inverse problem. We used simulatedannealing (see Methods) to optimize the molecular hologram geo-metry. Related techniques have recently been applied22 to find thetheoretical shape of a quantum corral maximizing the quantummirage14. Apart from the molecular positions, the free parameters

2 nm 1 nm

−80 −40 0Height (pm)

Low HighdI/dV (a.u.)

Figure 2 | Holographic page encoding and readout. a,c,e, Topographs of molecular holograms constructed with CO on the Cu(111) surface. b,d,f, High-

resolution dI/dV maps (VAC¼ 4 mVr.m.s.; I ¼ 1 nA) of the resulting DOS at particular energies. Rightmost column: encoded (top, binary image) and retrieved

(bottom, dI/dV at pixel centres) information. S page (a,b; topo: 17� 17 nm2, V¼ 10 mV; dI/dV map: 5� 5 nm2, 217.5 mV). U page (c,d; topo: 17� 17 nm2,

210 mV; dI/dV map: 5� 5 nm2, 210 mV). Spatially offset S and U pages (e,f; topo: 25� 25 nm2, 210 mV; dI/dV map: 8 � 8 nm2, 210 mV). Central boxes on

topographs mark the readout regions in which binary target data were encoded into the electronic DOS. The pointwise readout of the DOS retrieves the template

images exactly.

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are size and energy Ei for an encoded page. Initially, we used electronsnear the Fermi energy EF (at Fermi wavevector kF ¼

p(2m*EF/h� 2)

and wavelength lF ¼ 2p/kF) so they could be imaged at lowsample bias V. The wavelength of 2kF Friedel oscillations is lF/2 �1.5 nm on Cu(111), where EF ¼ 0.45 eV(corresponding to V ¼ 0)and the effective mass m* ¼ 0.38 electron masses. For our baselineholography, we aimed to encode data with a linewidth of half ofthis wavelength, or lF/4 � 0.75 nm. The quantum nature of theelectrons provides a distinct advantage over holographic techniquesbased on free electrons or photons, because the Pauli exclusionprinciple guarantees a relatively large baseline energy, the Fermienergy, which is essentially built into the sample and accessible atzero bias.

Baseline quantum holographyWithout loss of generality, we chose to encode pages with areadout (either high or low LDOS) that would resemble letters ofthe alphabet. Starting with random molecular positions, we opti-mized the match between the theoretical r(r, E) and specifictarget bit patterns (Fig. 2, black-and-white images in right panels)for each page in the 2.9 � 4.3 nm2 central readout area. Then, oper-ating in ultrahigh vacuum at 4 K, we positioned the COmolecules according to the annealed designs, the first of whichwas encoded for the letter S. A constant-current (I) topograph of

the assembled molecular hologram is shown in Fig. 2a. The centreof the topograph does not resemble the template image (Fig. 2b,upper right), because this topograph was acquired at V ¼ 10 mV,whereas the target was 210 mV, demonstrating the energy sensi-tivity of the readout.

To decode the projection more precisely, we acquiredconstant-height dI/dV maps (conductance maps; see Methods) ofthe image area. These are direct images of the electronic LDOS atspecific energies. At V ¼ 217.5 mV, a clear S was revealed(Fig. 2b). If r was measured solely at the intended bit locations ofthe target image, the encoded information was retrieved withperfect fidelity (Fig. 2b, lower right). This binary image persistedthroughout a 20 mV range centred at V ¼ 217.5 mV; that is, theS is projected into every electronic state in a 2.9 nm � 4.3 nm �20 mV region of r–E space. The template image was 12 � 8pixels, so that the S would image with fairly constant linewidth.However, this inter-pixel spacing (�0.4 nm) was too small foreach pixel to be independently settable. Our experiments and simu-lations indicate that a 7 � 5 pixel array of this size allows full bitcontrol, corresponding to a spacing of �lF /4 (0.75 nm).

Thus, as a conservative estimate this S page r(r, ES) ; S(r) con-tained 35 bits of information. As the essential area required toretrieve these bits is only 2.9 � 4.3 nm2, the areal informationdensity was 2.8 bits nm22. Naturally, this excludes the 17� 17 nm2

10

−45

−18

−80

5 nm

Electronic object projection

xy

10

−45

−18

−80

Target page correlation

xy

0.6 nm

Sam

ple

volta

ge (m

V)

Sam

ple

volta

ge (m

V)

1 nm1 nm 1 nm 1 nm

Figure 3 | Volumetric quantum holography. a, Topograph (13.5 � 13.5 nm2; I¼ 1 nA; V¼ 260 mV) of a molecular hologram that encodes two pages of data

at different energies in the same region of space. b, A dI/dV map taken at V¼ 218 mV shows the S page. VAC ¼ 4 mV r.m.s. c, Measuring dI/dV at

V¼ 245 mV retrieves the U page. The molecules are unchanged. Colour scales are the same as those in Fig. 2. d, By mapping dI/dV at many voltages

between 280 and 10 mV, r(r, E) was measured throughout the readout region. A translucent surface of constant DOS is shown. Slices through this space at

the appropriate energies reveal the S and U pages. e, The normalized cross-correlation of r(r, E) with each template image (insets) was computed as a

function of r and E. Surfaces of constant correlation (at 98% of the global maximum) are shown for each page, confirming their locations in the information

cube. Contours show the maximum correlation projected in each dimension (95–99.9%).

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overhead surface area occupied by the molecules, which we have notattempted to minimize. We emphasize that we are not aiming to tilethese devices and create large-scale STM-written memories, whichwould be plainly impractical. Our focus here is the ultimatereduction of the fundamental area required to measure a setof information on a surface, breaking conventional limits onboth local information density and the feature sizes ofengineered electrons.

Using our nanoscale writing and encoding scheme, we alsocreated a U page at the same energy, with different molecular pos-itions (Fig. 2c). The corresponding page r(r, EU) ; U(r) is shown inFig. 2d. We then constructed a hologram (Fig. 2e) projecting the Sand U pages simultaneously. The result (Fig. 2f ) is a single pageSU2D ¼ S(r1) þ U(r2), containing both individual pages at thesame energy but spatially offset by jr1 2 r2j ¼ 3.6 nm. (As inoptical holography, even though the projection can be decomposedin this fashion, the SU hologram is unrelated to the individual S andU holograms.) Each letter measures �3.0 � 4.7 nm2, and combinedthey require twice the surface area as the S or U alone; the infor-mation density is essentially unchanged.

Volumetric quantum holographyMoving beyond this baseline result, we encoded the S and U pagessimultaneously in the same spatial area by embedding them

volumetrically—a feat impossible with traditional surface writing. Atopograph of the molecular hologram that accomplishes this isshown in Fig. 3a. Because it was acquired at V ¼ 260 mV, thecentre of the image reflects the integration of the LDOS containingboth pages. Conductance maps at energies ES ¼ EF 2 18 mV andEU ¼ EF 2 45 mV separate the DOS into the S and U (Fig. 3b,c).We mapped the full electronic object by acquiring dense dI/dVmaps of the central readout region at many voltages. Figure 3d dis-plays a translucent surface of constant dI/dV showing how thedetected r evolves through r–E space. Slices through the four-dimensional dataset reveal that the information is contained in atruly volumetric function SU3D¼ S(r, ES) þ U(r, EU), superposingthe individual pages in the same space but offset in energy.(See Supplementary Information, Video 1, for a full readout of thisdata cube.)

To quantify the positions of the encoded pages, we computed thenormalized cross-correlation between each dI/dV map and eachtemplate image. Figure 3e shows surfaces of constant correlationsurrounding the points where the correlation of each letter is great-est. This demonstrates that the S and U exist in the same region ofspace (localized to within 0.1 nm) but are separated by �27 mV.Stacking the second page doubled the areal information density to5.6 bits nm22, and highlights a novel scaling strategy for multiplex-ing information using volumetric holography.

Fourier amplitude (a.u.)Low High

16

−16

−16 16 −16 16κx (nm−1) κx (nm−1) κ (nm−1)

Atomic limit

10 mV1.91 V

200

Am

plitu

de (a

.u.)

κ y (n

m−1

)16

−16

κ y (n

m−1

)

10 mV 1.91 V

1 nm1 nm

Height (pm)0 40 80

Height (pm)−80 0−40

1 nm

dI/dV (a.u.)Low High

Figure 4 | Electronic versus atomic writing. a,b, 2D Fourier transforms of electron standing waves (not shown) measured at V¼ 10 mV (a) and 1.91 V (b).

c, Their radial averages are each strongly peaked at a single wavevector k that increases with energy, allowing smaller features in the DOS. The weight

surrounding the d.c. peak at k¼ 0 is not plotted. d, Topograph of a molecular hologram (10� 10 nm2, I¼ 1 nA, V¼ 10 mV) encoding an S. e, A dI/dV map

(2.5� 2.5 nm2, VAC¼ 50 mV r.m.s., centred around the dashed readout region in d) taken at V¼ 1.91 V, a much higher energy than that of maps shown

previously. The upper inset shows the encoded data and the bottom inset shows its retrieval. f, Topograph (same scale, I¼ 1 nA, V¼ 10 mV) of the smallest

S that can be written with atoms. Dots mark the centres of the eight atoms. The electronic S is significantly smaller, surpassing the ultimate scale of atoms

on surfaces. The wavevector k corresponding to the minimum spacing between two atomic lines is shown as the green dotted line in c.

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Subatomic quantum holographyThe written features shown thus far have a characteristic size of�0.8 nm, already much smaller than the wavelength of thesynchrotron-generated photons now used in state-of-the-art X-rayholography23. This can be even further reduced by using electronsat higher energies. An accurate measure of the wavevector contentof electrons at a given energy is the spatial Fourier transform of r(Fig. 4a,b). The result is a circular ring whose radial average(Fig. 4c) is peaked at wavevector k(E). As the voltage increasesfrom V ¼ 10 mV to 1.91 V, k grows by a factor of 2.8. In thisway, electronic writing can be realized that surpasses the spatiallimits imposed by the coarseness of atoms.

Traditional writing scaled to its ultimate limit can be no smallerthan atoms registered to a discrete underlying surface lattice1. Underthis rubric, using eight copper atoms manipulated with the STM tip,we created (Fig. 4f ) the smallest atomic S possible on Cu(111).Atoms must be spaced by two lattice constants (0.51 nm); atcloser separations metal atoms bond to form inseparable clustersand cannot be placed arbitrarily. Electrons are subject to no suchconstraint. We created a significantly smaller S(r) (Fig. 4e) using amolecular hologram (Fig. 4d) projecting into V ¼ 1.91 V. Asmeasured from the atom centres and the corresponding points inthe DOS, the electronic S is �1.24 nm in height, has linewidth�0.3 nm, and occupies 50% of the area of its atomic counterpart.(See Supplementary Information, Fig. S1, for a comparison withother nanoscale writing technologies and milestones.) The high-energy S page contains over 20 bits nm22, significantly exceedingthe maximum atomic density of �9 bits nm22. Note that thiswriting is even smaller than proposed24 (presently technologicallyimpossible) writing on a single highly excited Rydberg atom25,which would yield letters over 50 times taller than the V ¼ 1.91 Vprojected S.

ConclusionsAccording to theory, a quantum state can encode any amount ofinformation (at zero temperature), requiring only sufficiently highbandwidth and time in which to read it out26. In practice, onlyrecently has progress been made towards encoding several bitsinto the shapes of bosonic single-photon wavefunctions27, whichhas applications in quantum key distribution28. We have experimen-tally demonstrated that 35 bits can be permanently encoded into atime-independent fermionic state, and that two such states can besimultaneously prepared in the same area of space. We have simu-lated hundreds of stacked pairs of random 7 � 5-pixel arrays as wellas various ideas for pathological bit patterns, and in every case theinformation was theoretically encodable. In all experimentalattempts, extending down to the subatomic regime, the encodingwas successful and the data were retrieved at 100% fidelity. Webelieve the limitations on bit size are �l/4, but surprisingly theinformation density can be significantly boosted by using higher-energy electrons and stacking multiple pages holographically.Determining the full theoretical and practical limits of this tech-nique—the trade-offs between information content (the numberof pages and bits per page), contrast (the number of measurementsrequired per bit to overcome noise), and the number of atoms in thehologram—will involve further work.

MethodsForward scattering. The multiple-scattering Green’s function theory20,21 was usedto predict the LDOS and hence our holographic projections. It requires that thescattering phase shift for a molecule, i1, matches the data well at low biases andthe surface state dispersion relation (which is non-parabolic at high energies29).For V � 1 V, it was necessary to add a phase relaxation length L to the free-electronGreen’s function, such that G0(r, r0 , E) ¼ G2D(r, r0 , E)exp(2jr2r0j/L), where G2D isthe undamped propagator for a particle in two dimensions. For V ¼ 1.91 V, we usedL ¼ 5 nm, which lessened but did not eliminate some mismatch to the data, andwhich is in agreement with the coherence length deduced from themomentum-space peak width (Fig. 4c).

Inverse scattering. CO molecules bond above the copper atoms in the (111) surface,which form a triangular lattice with spacing a ¼ 0.255 nm. A typical hologramrequires optimizing the positions of molecules restricted to a 15 � 15 nm2 area,reserving the inner 4 � 5 nm2 for the page readout area. The total area containingthe molecules is flexible and is chosen to allow a reasonable spacing betweenmolecules and to limit the number of possible sites. As in conventional holography,any subset of the hologram is responsible for projecting the entire image and wetherefore include enough total hologram area for sufficient contrast for the particularinformation content. The readout area is selected such that the bit spacing fora target image is �l/4. For 80 molecules, there are �3,500 possible sites and1 � 10167 ensembles. We used simulated annealing to optimize the molecularpositions, which were initialized randomly. At each iteration, a random moleculewas moved to an unoccupied site, obeying design rules: sites having two occupiednearest neighbours were forbidden because they are physically unstable30. Athigher voltages (V � 1 V) we found that all dimers became unstable, so thesewere then forbidden as well.

The merit function for a projection (its match to the target bits) that weattempted to maximize was f ¼ [min(W)�max(B)]þ b[W � B], where W and Bare the sets of DOS computed in white and black pixels, respectively, and W denotesthe mean of W. The first term represents the lowest contrast between high and lowbits and the second term, the average contrast, encourages growth; we typically choseb ¼ 0.15 2 0.3. In the simulated annealing for SU3D holography, we optimized thecombined merit function F ¼ min( fS;V1 ;

fU;V2), where the two f terms represent the

S at V1 and the U at V2. As a fictitious temperature T was decreased, a change wasaccepted with probability exp(2Df/T). We used a linear cooling schedule, starting ata T that essentially randomized the molecular positions.

Conductance mapping. The dI/dV maps were acquired open loop: the tip was heldat constant height (above the previously determined sample plane) and voltage as itwas scanned. At each point, the response to a small a.c. voltage modulation (typicallyVr.m.s. ¼ 4 mV at 865 Hz) was recorded. By keeping the feedback loop disengaged,the dI/dV signal was not scaled by changes in tip height due to the apparenttopography; these maps faithfully reflect r(r, E) at any E. Topographs, which reflectthe integration of r from E ¼ EF to EF þ eV, can only retrieve encoded patterns inr if the patterns exist in the majority of this energy window.

Received 10 November 2008; accepted 8 December 2008;published online 25 January 2009

References1. Eigler, D. M. & Schweizer, E. K. Positioning single atoms with a scanning

tunnelling microscope. Nature 344, 524–526 (1990).2. Heanue, J. F., Bashaw, M. C. & Hesselink, L. Volume holographic storage and

retrieval of digital data. Science 265, 749–752 (1994).3. Coufal, H. J., Sincerbox, G. T. & Psaltis, D. Holographic Data Storage

(Springer-Verlag, 2000).4. Ashley, J. et al. Holographic data storage. IBM J. Res. Develop. 44,

341–368 (2000).5. Moon, C. R. et al. Quantum phase extraction in isospectral electronic

nanostructures. Science 319, 782–787 (2008).6. Moon, C. R., Lutz, C. P. & Manoharan, H. C. Single-atom gating of

quantum-state superpositions. Nature Phys. 4, 454–458 (2008).7. Xu, S. Y. et al. Nanometer-scale modification and welding of silicon and metallic

nanowires with a high-intensity electron beam. Small 1, 1221–1229 (2005).8. Curtis, J. E., Koss, B. A. & Grier, D. G. Dynamic holographic optical tweezers.

Opt. Commun. 207, 169–175 (2002).9. Grier, D. G. A revolution in optical manipulation. Nature 424, 810–816 (2003).10. Lucente, M. Interactive three-dimensional holographic displays: seeing the

future in depth. ACM SIGGRAPH Comput. Graphics 31, 63–67 (1997).11. Crommie, M. F., Lutz, C. P. & Eigler, D. M. Imaging standing waves in a

two-dimensional electron gas. Nature 363, 524–527 (1993).12. Crommie, M. F., Lutz, C. P. & Eigler, D. M. Confinement of electrons to

quantum corrals on a metal surface. Science 262, 218–220 (1993).13. Braun, K. F. & Rieder, K. H. Engineering electronic lifetimes in artificial atomic

structures. Phys. Rev. Lett. 88, 096801 (2002).14. Manoharan, H. C., Lutz, C. P. & Eigler, D. M. Quantum mirages formed by

coherent projection of electronic structure. Nature 403, 512–515 (2000).15. Fiete, G. A. et al. Scattering theory of Kondo mirages and observation of single

Kondo atom phase shift. Phys. Rev. Lett. 86, 2392–2395 (2001).16. Sentef, M., Kampf, A. P., Hembacher, S. & Mannhart, J. Focusing quantum states

on surfaces: a route towards the design of ultrasmall electronic devices. Phys. Rev.B 74, 153407 (2006).

17. Aassime, A., Johansson, G., Wendin, G., Schoelkopf, R. J. & Delsing, P.Radio-frequency single-electron transistor as readout device for qubits: chargesensitivity and backaction. Phys. Rev. Lett. 86, 3376–3379 (2001).

18. Awschalom, D. D. et al. Low-noise modular microsusceptometer using nearlyquantum limited dc SQUIDs. Appl. Phys. Lett. 53, 2108–2110 (1988).

19. Parkin, S. S. P., Hayashi, M. & Thomas, L. Magnetic domain-wall racetrackmemory. Science 320, 190–194 (2008).

NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2008.415 ARTICLES

NATURE NANOTECHNOLOGY | VOL 4 | MARCH 2009 | www.nature.com/naturenanotechnology 171

© 2009 Macmillan Publishers Limited. All rights reserved.

20. Heller, E. J., Crommie, M. F., Lutz, C. P. & Eigler, D. M. Scattering andabsorption of surface electron waves in quantum corrals. Nature 369,464–466 (1994).

21. Fiete, G. A. & Heller, E. J. Colloquium: Theory of quantum corrals and quantummirages. Rev. Mod. Phys. 75, 933–948 (2003).

22. Correa, A. A., Reboredo, F. A. & Balseiro, C. A. Quantum corral wave-functionengineering. Phys. Rev. B 71, 035418 (2005).

23. Marchesini, S. et al. Massively parallel X-ray holography. Nature Photon. 2,560–563 (2008).

24. Stroud, C. R. Jr & Noel, M. W. Optics inside an atom. Opt. Photon. News 10,34–37 (April 1999).

25. Weinacht, T. C., Ahn, J. & Bucksbaum, P. H. Controlling the shape of a quantumwavefunction. Nature 397, 233–235 (1999).

26. Yamamoto, Y. & Haus, H. A. Preparation, measurement andinformation capacity of optical quantum states. Rev. Mod. Phys. 58,1001–1020 (1986).

27. Kolchin, P., Belthangady, C., Du, S., Yin, G. Y. & Harris, S. E. Electro-opticmodulation of single photons. Phys. Rev. Lett. 101, 103601 (2008).

28. Inoue, K., Waks, E. & Yamamoto, Y. Differential phase shift quantum keydistribution. Phys. Rev. Lett. 89, 037902 (2002).

29. Burgi, L., Petersen, L., Brune, H. & Kern, K. Noble metal surface states:deviations from parabolic dispersion. Surf. Sci. 447, 157–161 (2000).

30. Heinrich, A. J., Lutz, C. P., Gupta, J. A. & Eigler, D. M. Molecule cascades.Science 298, 1381–1387 (2002).

AcknowledgementsThis work was supported by US Office of Naval Research (YIP/PECASEN00014-02-1-0351), US National Science Foundation (CAREER DMR-0135122 &DMR-0804402), US Department of Energy (DE-AC02-76SF00515) and the Stanford-IBMCenter for Probing the Nanoscale. The authors acknowledge the National Defense Scienceand Engineering Graduate fellowship programme (C.R.M. and B.K.F.) and the Alfred P.Sloan Foundation (H.C.M.) for fellowship support. We thank L. Bozano, M. Brongersma,G. Burr, D. Eigler, G. Fiete, J. Kirtley, P. Kolchin, S. Harris, E. Heller, R. McGorty,V. Manoharan, J. Moon, J. Randel, S.-H. Song and Y. Yamamoto for discussions.

Additional informationSupplementary Information accompanies this paper at www.nature.com/naturenanotechnology. Reprints and permission information is available online athttp://npg.nature.com/reprintsandpermissions/. Correspondence and requests for materialsshould be addressed to H.C.M.

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