Page 1
LETTERS
Interference between two indistinguishable electrons
from independent sources
I. Neder1, N. Ofek1, Y. Chung2, M. Heiblum1, D. Mahalu1& V. Umansky1
Very much like the ubiquitous quantum interference of a single
particlewithitself1,quantuminterferenceoftwoindependent,but
indistinguishable, particles is also possible. For a single particle,
the interference is between the amplitudes of the particle’s wave-
functions, whereas the interference between two particles is a
direct result of quantum exchange statistics. Such interference is
observed only in the joint probability of finding the particles in
twoseparateddetectors,aftertheywereinjectedfromtwospatially
separated and independent sources. Experimental realizations of
two-particle interferometers have been proposed2,3; in these pro-
posals it was shown that such correlations are a direct signature of
quantumentanglement4betweenthespatialdegreesoffreedomof
thetwoparticles(‘orbitalentanglement’),eventhoughtheydonot
interact with each other. In optics, experiments using indistin-
guishable pairs of photons encountered difficulties in generating
pairs of independent photons and synchronizing their arrival
times; thus they have concentrated on detecting bunching of
photons(bosons)bycoincidencemeasurements5,6.Similarexperi-
ments with electrons are rather scarce. Cross-correlation mea-
surements between partitioned currents, emanating from one
source7–10, yielded similar information to that obtained from
auto-correlation (shot noise) measurements11,12. The proposal of
ref. 3 is an electronic analogue to the historical Hanbury Brown
and Twiss experiment with classical light13,14. It is based on the
electronicMach–Zehnderinterferometer15thatusesedgechannels
in the quantum Hall effect regime16. Here we implement such an
interferometer. We partitioned two independent and mutually
incoherent electron beams into two trajectories, so that the
combined four trajectories enclosed an Aharonov–Bohm flux.
Although individual currents and their fluctuations (shot noise
measured by auto-correlation) were found to be independent of
the Aharonov–Bohm flux, the cross-correlation between current
fluctuations at two opposite points across the device exhibited
strong Aharonov–Bohm oscillations, suggesting orbital entangle-
ment between the two electron beams.
In many ways, experiments with electrons are easier than those
with photons. Injecting electrons from an extremely cold and degen-
eratefermionicreservoirproducesahighlyorderedbeamofelectrons
that is totally noiseless17; hence, a high coincidence rate is achieved
without the need to synchronize the arrival times of the electrons. As
each electron has a definite energy (Fermi energy) and momentum
(Fermi momentum), electrons can be made indistinguishable by
injecting them from two equal voltage sources. Moreover, because
the coherence length of the electrons (‘wave packet width’ or ‘spatial
size’)isdeterminedbythesourcevoltage(atlowtemperature),avery
smallsourcevoltageensuresthepresenceofasingleelectronatatime
in the interferometer, preventing electron–electron interactions.
However, the small voltage leads to an exceedingly small electrical
currentandtominutefluctuations, makingthemeasurements extre-
mely difficult to perform.
A diagram of our experiment is shown in Fig. 1a (ref. 2). Two
independent, separated, sources of electrons (S1 and S2) inject
ordered, hence noiseless, electrons towards each other. Each stream
passes through a beam splitter (A and B), and splits into two nega-
tivelycorrelatedpartitionedstreams(ifanelectronturnsright,ahole
is injected to the left). Both sets of the two partitioned streams join
each other attwo additional beam splitters (Cand D),interfere there
and generate altogether four streams that are collected bydrains D1–
D4. Hence, every electron emitted by either S1 or S2 eventually
arrived at one of the four drains. Consider now the event where
one electron arrives at D2 and the other arrives at D4. There are
twoquantummechanicalprobabilityamplitudescontributingtothis
event:S1toD2andS2toD4;or,alternatively,S1toD4andS2toD2.
These two ‘two-particle’ events can interfere because they are indis-
tinguishable. Because in the two possible events the electrons travel
along different paths (thus accumulating different phases), the joint
probability of one arriving at D2 and the other at D4 contains the
total phase of all paths—as we show below.
The two wavefunctions, corresponding to the incoming states
from each of the two sources YSi, can be expressed in the basis of
theoutgoingstatesatthefourdrainsyDj.Assuming, as intheexperi-
ment, that every beam splitter is half reflecting and half transmitting,
its unitary scattering matrix M (that ties the input and output states)
rt
t0
r0
of the four possible paths w1,::,w4:
YS1(x)~1
2ieiw1yD1(x){eiw1yD2(x)zieiw2yD3(x)zeiw2yD4(x)
canbetakenas:M~
??
~1ffiffi
2
p
i
1
1
i
??
.Considering thephases
?
?
?ð1aÞ
?ð1bÞ
YS2(x)~1
2ieiw3yD1(x)zeiw3yD2(x)zieiw4yD3(x){eiw4yD4(x)
As, in this set-up, each electron is not allowed to interfere with itself,
only particle statistics could cause interference. Because of the ferm-
ionic property of electrons, the total two-particle wavefunction must
be the antisymmetric product of equation (1a) and equation (1b):
Ytotal(x1,x2)~1ffiffiffi
with x1 and x2 any two locations in the interferometer. Substituting
equation (1) in equation (2) leads to 24 terms, expressing the prob-
abilityamplitudeforoneelectronatx1andanotheratx2.Aswewish
to concentrate on correlations between drains, we write Ytotal using
the notation yDiDj:1ffiffi
2
p
YS1x1
½ð ÞYS2x2
ð Þ{YS2x1
ð ÞYS1x2
ð Þ?ð2Þ
2
p
yDi(x1)yDj(x2){yDj(x1)yDi(x2)
hi
for an
antisymmetric state, in which one electron heads to Di and another
1Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel.2Department of Physics, Pusan National
University, Busan 609-735, Korea.
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to Dj. The two-particle wavefunction is:
Ytotal(x1,x2)~i
2
ei(w1zw3)yD1D2{ei(w2zw4)yD3D4
?
??z
Þ{cos
zi
2e
i
2(w1zw2zw3zw4)sin
Wtotal
2
?
yD2D4{yD1D3
ð
Wtotal
2
??
yD2D3zyD1D4
ðÞ
?? ð3Þ
withWtotal~w1{w3zw4{w2,whichisexactlythetotalaccumulated
phase going anti-clockwise along the four trajectories of the two
particles.
Equation(3)describesthetwo-particleinterferenceeffect,withthe
absolute value squared of the prefactor of yDiDjthe joint probability
of having one electron at Di and one at Dj. Concentrating on the
correlation between D2 and D4, one can deduce from equation (3)
the following: (1) two electrons never arrive at the same drain (Pauli
exclusion principle); (2) the first part suggests that there is a 50%
chance for two electrons to arrive at the same ‘side’ simultaneously,
namely, at D1 and D2, or at D3 and D4, but never at D2 and D4; (3)
the second part suggests that there is a 50% chance for two electrons
to arrive at opposite ‘sides’, namely, one atD1 orD2 and the other at
D3 or D4; however, the exact correlation depends on Wtotal. When
Wtotal5p, sin2(Wtotal/2)51 and two electrons arrive at (D1, D3) or
at (D2, D4), but when Wtotal50, cos2(Wtotal/2)51 and the comple-
mentary events take place. (4) Combining all events in the two parts
of the total wavefunction, one finds for Wtotal50 a perfect anti-
correlationbetweenthearrivalofelectronsinD2andinD4;however,
for Wtotal5p there is 50% chance of anti-correlation (first part) and
50% chance of positive correlation (second part)—hence, zero
correlation. The time-averaged cross-correlated signal of the current
fluctuations in the two drains is proportional to the probability of
the correlated arrival of electrons in these drains. Varying the total
phase should result in a negative oscillating cross-correlation signal
between current fluctuations in D2 and in D4. The quantitative
estimate oftheamplitude ofthatcross-correlation signalis discussed
later.
Figure 1b describes the realization of the experiment. The two-
particle interferometer is shown split in the centre, resulting in an
upper and lower segments; each is a simple optical Mach–Zehnder
interferometer (MZI)18. An electronic version of the MZI has been
recently fabricated and studied15,19,20. A quantizing magnetic field
(,6.4T) brings the two-dimensional electron gas into the quantum
Hall effect state at filling factor one. The current is carried by a single
edge channel along the boundary of the sample16. Being a chiral one-
dimensional object, the channel is highly immune to back scattering
and dephasing. The layout of the two-particle interferometer is
described in Fig. 1c, with the scanning electron micrograph of the
actual device shown in Fig. 1d. The two MZIs can be separated from
each other with a ‘middle gate’ (MDG). When it is closed, each MZI
can be tested independently for its coherence and the Aharonov–
Bohm periodicity. A quantum point contact (QPC), formed by
metallic split gates, functions as a beam splitter while ohmic contacts
serve as sources and drains. In this configuration, the phase that is
accumulatedalongthefourtrajectoriesistheAharonov–Bohmphase
(QAB), namely, Wtotal5QAB52pBA/W0, with B the magnetic field
and A the area enclosed by the four paths (W054.14310215Tm2
is the flux quantum)21. Look, for example, at the upper MZI of the
separated two-particle interferometer (Fig. 1b). An edge channel,
emanating from ohmic contact S1, is split by QPC1 into two paths
that enclose a high magnetic flux and join again at QPC2. The phase
dependent transmission coefficient from S1 to D2 is:
??
QPCs. The visibility is defined as the ratio between the phase-
dependent and the phase-independent terms, nMZI5TQ/T0. The
Aharonov–Bohm phase was controlled by the magnetic field and
the ‘modulation gate’ (MG1 or MG2) voltage VMG, which affected
the area enclosed by the two paths.
Figure 2 displays the measured conductance of the two separated
MZIs (defined as iD/VS5TMZI(e2/h), where iDis the AC current in
thedrain,VStheapplieda.c.voltageatthesources,withe2/htheedge
channel conductance). Pinching off MDG, the QPCs were tuned to
transmission 0.5 and the AC signal was measured at D2 and D4 as a
function of VMGand magnetic field. As VMGwas scanned repeatedly
TMZI~ tQPC1tQPC2zeiQABrQPC1rQPC2
??2~T0zTQcos(QAB) ð4Þ
where t and r are the transmission and reflection amplitudes of the
D1
D3
A
B
C
D
φ1
φ2
φ4
φ3
a
S1
S2
D1
D2
D3
D4
I2
QPC2
QPC3
QPC4
MDG
Preamp
Preamp
QPC1I1
0.8 MHz
0.8 MHz
c
MG2
MG1
2 µm
MZ1
MZ2
D3
A
B
C
D
b
D1
d
S2D4S2D4
S1
D2
S1
D2
Figure 1 | The two-particle Aharonov–Bohm interferometer. a, Diagram of
the interferometer. Sources S1 and S2 inject streams of particles, which are
split by beam splitters A and B, later to recombine at beam splitters C and D.
Eachparticlecanarriveatanyoffourdifferentdrains,D1–D4.Eachofthefour
trajectories accumulates phase wi. b, By breaking the interferometer in the
centre, two separate Mach–Zehnder interferometers (MZIs) are formed. The
MZIs are the building blocks of the two-particle interferometer. c, A detailed
drawing of the interferometer. It was fabricated on a high mobility GaAs-
AlGaAs heterostructure, with a two-dimensional electron gas buried some
70nmbelowthesurface(carrierdensity2.231011cm22andlowtemperature
mobility 53106cm2V21s21). Samples were cooled to ,10mK electron
temperature. Quantum point contacts (QPCs) served as beam splitters, and
ohmic contacts as sources and drains. Tuning gates MG1 and MG2 changed
the area and thus the magnetic flux threaded through the interferometer (at
filling factor one of the integer quantum Hall effect). ‘Middle gate’ MDG
separated the interferometer into two MZIs. Metallic air bridges connected
drains D1 and D3 to the outside, where they were grounded. Currents at D2
and D4 were filtered first by an LC circuit (tuned to 0.8MHz and 60kHz
bandwidth) and then amplified by a cold preamplifier (at 4.2K). d, Scanning
electronmicrographoftheactualsample.Airbridgeswereusedtocontactthe
small ohmic contacts, the split gates of the QPCs, and the MDG.
LETTERS
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the magnetic field decayed unavoidably (as the superconducting
magnet is not ideal) at a rate of ,1.4Gh21. Hence, the interference
pattern was ‘tilted’ in the two-coordinate plane of VMGand time
(magnetic field), with two basic Aharonov–Bohm periods for each
MZI15. Apparently, the seemingly identical MZIs had different peri-
odicities: 1mV and 80min in the upper MZI, and 1.37mV and
87min in the lower MZI (the asymmetry resulted from misaligning
the QPCs and modulation gates). In the two MZIs, we found visibi-
lities 75–90%, by far the highest measured in an electron interfero-
meter. The high visibility was likely to result from the smaller size of
the MZIs15,19,20; hence, dephasing mechanisms such as flux fluctua-
tionsortemperaturesmearingwerelesseffective.Moreover,thehigh
quality two-dimensional electron gas assured a better formation of
one-dimension-like edge channels and better overlap of particle
wavefunctions.
We then discharged MDG, thus opening it fully and turning the
two MZIs into a single two-particle interferometer. The conduc-
tances at D2 and D4 were now found to be independent of the
Aharonov–Bohm flux, with a visibility smaller than the background
(,0.1%). This is expected, as each electron did not enclose an
Aharonov–Bohm flux any more.
We turn now to discuss the current fluctuations, namely, the shot
noise in D2 and in D4. Feeding a d.c. current into S1, the low fre-
quency spectral density of the shot noise in the partitioned current
(by QPC1) at D2 and at D4 (with QPC2 closed and QPC3 and MDG
open) was measured. Its expected value (neglecting here finite
temperature corrections) is SD252eIS1TQPC1(12TQPC1)50.5eIS1
(A2Hz21) for TQPC15 tQPC1
j
tions in the drain were filtered by an LC circuit, with 60kHz band-
width around a centre frequency ,0.8MHz, and then amplified by
the cold amplifier, followed by a room-temperature amplifier and a
spectrum analyser. In order to calibrate the cross-correlation mea-
surement, we performed three noise measurements: (1) noise mea-
sured at D2; (2) noise measured at D4; and (3) noise measured by
cross-correlating the current fluctuations at D2 and at D4 (by an
analogue home-made correlating circuit). Measurements (1) and
(2) both led accurately to the expected result above (they are anti-
correlated and equal signals), which were used to calibrate measure-
ments (3). An electron temperature of ,10mK was deduced from
these measurements22.
We were ready at this point to measure the two-particle cross-
correlation. All four QPCs were tuned to TQPC50.5 while the
MDG was left open, hence, turning the two MZIs into a single
two-particle interferometer. Equal DC voltages were applied to
sources S1 and S2 with two separated power supplies VS15VS25
7.8mV (IS15IS2;I50.3nA). For that voltage, there is at most a
single electron in each of the four trajectories of the interferometer
(the wave packet’s width, 15–30mm, estimated from the current and
the estimated drift velocity (,3–63106cms21), is bigger then the
interferometer’spathlength,being,8mm).Thisguaranteedastron-
ger overlap between the wavefunctions of the two electrons, and
minimizedCoulombinteractionamongtheelectrons(thuseliminat-
ing nonlinear effects in the interferometer19). The measured fluctua-
tions in D2 and D4 were averaged over some 30,000 electrons,
amplified by two separate amplification channels (each fed by its
own power supply), and finally cross-correlated. In order to verify
j250.5 (ref. 15). The current fluctua-
0
1
3
24
83% visibility
0.2
0.6
1.0
1.4
Periodicity (2π h–1)
0.2
0.6
1.0
1.4
Periodicity (2π h–1)
Periodicity (2π mV–1)
(0.75, 1.00)
Gate voltage, VMG (mV)
a
0
1
3
24
79% visibility
(0.69, 0.73)
Time (h)
b
0.2
0.6
1.0
1.4
1.8
Periodicity (2π mV–1)
0.2
0.6
1.0
1.4
1.8
16.5
17.0
17.5
18.0
18.5
19.0
19.5
Gate voltage, VMG (mV)
16.5
17.0
17.5
18.0
18.5
19.0
19.5
Figure 2 | Colour plot of the conductance of the two separate MZIs as
function of the modulation gate voltage and the magnetic field that
decayed in time. Strong Aharonov–Bohm oscillations dominate the
conductance with visibilities of ,80% each. A two-dimensional FFT in the
insetprovidestheperiodicity inmodulationgatevoltage(VMG)andintime.
a
0
0
1
1
2
2
3
3
4
4
Periodicity (2π mV–1)
Periodicity (2π h–1)
Auto-correlation
signal (10–30 A2 Hz–1)
b
0
1
2
3
0
0
1
1
2
2
3
3
4
4
Periodicity (2π mV–1)
Periodicity (2π h–1)
Auto-correlation
signal (10–30 A2 Hz–1)
0
1
2
3
Figure 3 | Analysis and two-dimensional FFT of auto-correlation (shot
noise)foranopen‘middlegate’. Panelsaandbshowtwo-dimensionalFFTs
of shot noise measurements in D2 and D4, respectively. The noise is totally
featureless, with no sign of Aharonov–Bohm oscillations above the
background.
NATURE|Vol 448|19 July 2007
LETTERS
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flux insensitivity in each drain separately, we first measured the
shot noise in D2 and in D4 as function of the magnetic flux (varying
VMGand magnetic field). The noise, with a spectral density of
S50.5eI<2.4310229A2Hz21, was found to be featureless. For
further assurance, a two-dimensional fast Fourier transform (FFT)
ofthemeasurements wascalculated,withtheresultsshowninFig.3a
and b. Again, the transforms were without any feature above our
measurement resolution of ,2310231A2Hz21, confirming the
absence of flux periodicity in the noise (as was found also in the
transmission).
We estimate now the expected magnitude of the cross-correlation
signal from equation (3). When Wtotal50, a maximum anti-
correlation signal of the current fluctuations at the drains SD2D45
DID2:DID4
h
thecross-correlationspectraldensity,fora100%visibility,isthesame
asthatofthenoiseofasingleQPC,thatis,SQPC52eITQPC(12TQPC),
or0.5eI(forTQPC50.5). AsforWtotal5pthecross-correlationsignal
is expected to vanish, we may conclude that the cross-correlation
signal should oscillate with Wtotal, SD2D4~{0:25eI 1{sinWtotal
with amplitude 1.2310229A2Hz21for I50.3nA.
Without currents in the sources, the cross-correlation signal was
featureless(thebackground),withanaverageoverthetwo-dimensional
i is expected. It can be shown that the expected value of
ðÞ,
FFT of ,2310231A2Hz21(not shown). The cross-correlation mea-
surement with I50.3nA is shown in Fig. 4. The Aharonov–Bohm
oscillations are already visible in the raw data (Fig. 4a bottom panel).
In the two-dimensional FFT (Fig. 4b), one sees a sharp peak corres-
pondingtoaperiodof0.58mVinVMG(withthesamevoltageapplied
to MG1 and MG2) and a period of 42.5min in time (being propor-
tional to the magnetic field decay). The square root of the integrated
power under the FFT peak (the amplitude of the Aharonov–Bohm
oscillations)is3.0310230A2Hz21.Aroughlysimilarmagnitudewas
observed also at a bulk filling factor of 2. Moreover, we could directly
resolve the Aharonov–Bohm oscillations as a function of VMGand
time separately by coherent time averaging. As the magnetic field
decayedintime,thusaddingcontinuouslyanAharonov–Bohmphase,
thisextraphasecouldbecompensatedforbyshiftingsubsequentscans
inVMGaccordingtothedecayratefoundinthetwo-dimensionalFFT,
leading to the negative oscillatory cross-correlation fringes shown in
the top left panel of Fig. 4a. Similarly, the oscillations as a function of
magneticfieldhavebeenextracted(toprightpanel,Fig.4a).InFig.4c
we provide the vector representation of the periodicities (inverse of
periods) of each individual MZI (from Fig. 2) and that of the two-
particle interferometer, thelast being, quite accurately, the sum of the
two.Thisisexpected,astherateofchangeoftheAharonov–Bohmflux
(1.41, 1.73, 3.00)
Periodicity (2π h–1)
0.00.40.81.21.6
(0.69, 0.73)
(0.75, 1.00)
(1.41, 1.73)
6
8
10
12
14
16
18
Gate voltage, VMG (mV)
2 2
4
6
Time (h)
8
10
12
a
2
4
6
8
10
12
6
8
10
12
14
16
18
–8
–10
–12
–14
–16
0
0
1
1
2
2
3
3
4
4
Periodicity (2π mV–1)
Periodicity (2π h–1)
Excess cross-correlation
signal (10–30 A2 Hz–1)
Cross-correlation signal
(10–30 A2 Hz–1)
–8
–10
–12
–14
–16
0
1
2
3
0.4
0.8
1.2
1.6
Periodicity (2π mV–1)
0.0
2.0
bc
Figure 4 | Cross-correlation of the current fluctuations in D2 and D4.
a, Bottom, two-dimensional colour plot of the raw data as function of VMG
and time (magnetic field). The periodicity is already visible in the raw data.
Toprightpanel,coherentaveragingofsome50tracesasfunctionofVMG,by
correcting for the added phase due to the decaying magnetic field (see text).
Strong Aharonov–Bohm oscillations are seen in the negative excess cross-
correlation (the part of the cross-correlation above the background,
resulting from an injected current of 0.3nA at each source). Note that the
mean non-oscillating part of the excess cross-correlation is
21.231029A2Hz21, as expected. Top left panel, similar averaging of the
databutatafixedVMG.Thesomewhatdifferentvisibilitiesinbothpanelsare
duetoanalysisthatmustbedoneindifferentregionsofthetwo-dimensional
plot. b, Two-dimensional FFT of the cross-correlation signal. A strong peak
is visible, with an integrated power 3.0310230A2Hz21. c, A vector
representation of the different periodicities. The two vectors starting from
the origin and ending at the blue and red crosses are the two-dimensional
periodicities of the two MZIs. The green cross is the two-dimensional
periodicityofthecross-correlationsignalofthetwo-particleinterferometer.
The vectorial sum of the periodicities of the two MZIs (black dot) agrees
excellently with the corresponding two-dimensional periodicity of the two-
particle interferometer.
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of the two-particle interferometer is the sum of the rates of the two
MZIs.
Compared with the expected amplitude of the cross-correlation
oscillations, 1.2310229A2Hz21, we measured an amplitude of 3.03
10230A2Hz21. Our results are reasonably accurate, as the measure-
ments have been repeated a few times and over long periods of integ-
rationtimes,loweringtheuncertaintytobelow10231A2Hz21.Atleast
two factors could lead to the lower cross-correlation signal. First,
although we have no theory for it, it is likely that the lower visibility
in each of the MZI’s, nMZI1andnMZI2, will lower the cross-correlation
signal by nMZI13nMZI2. Whereas the visibilities at zero applied d.c.
voltage were ,80% (Fig. 2), the visibilities at the applied DC voltage
VS57.8mV were found to be ,70% (ref. 19). Second, our finite
temperature (,10mK) will lower the shot noise by ,22%, affecting
thecross-correlationsignalsimilarly.Thesetwoeffectsalonewilllower
the expected cross-correlation signalto ,4.6310230A2Hz21, which
is about 1.5 times higher than the measured one. This discrepancy is
still not understood.
Our direct observation of interference between independent part-
icles provides a reliable scheme to entangle separate, but indistin-
guishable, quantum particles. Thepresent demonstration, done with
electrons, reproduces the original Hanbury Brown and Twiss experi-
ments13,14, which were performed with classical waves. Such experi-
ments are central to the study of the wavefunctions of multiple
particles. Our scheme has the potential to test Bell inequalities2,3,23;
however,takingintoaccountthefinitetemperature,itseemsthatthe
possibility of violating Bell inequalities in our measurements (with a
visibility of merely 25%) requires further theoretical analysis.
Received 8 February; accepted 22 May 2007.
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Acknowledgements We thank Y. Imry, U. Gavish, M. Buttiker, P. Samuelsson and
D. Rohrlich for discussions. The work was partly supported by the Israeli Science
Foundation (ISF), the Minerva foundation, the German Israeli Foundation (GIF),
the German Israeli Project cooperation (DIP), and the Ministry of Science - Korea
Program. Y.C. was supported by the Korea Research Institute of Standards and
Science (KRISS), the Korea Foundation for International Cooperation of Science
and Technology (KICOS), the Nanoscopia Center of Excellence at Hanyang
University through a grant provided by the Korean Ministry of Science and
Technology, and by the Priority Research Centers Program funded by the Korea
Research Foundation.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. The authors declare no competing financial interests.
Correspondence and requests for materials should be addressed to M.H.
(heiblum@wisemail.weizmann.ac.il).
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