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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

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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.

1.Feynman,R.P.,Leighton,R.B.&Sands,M.TheFeynmanLecturesonPhysicsVol.III,

Quantum Mechanics (Addison-Wesley, New York, 1965).

Yurke, B. & Stoler, D. Bell’s-inequality experiments using independent-particle

sources. Phys. Rev. A 46, 2229–2234 (1992).

Samuelsson, P., Sukhorukov, E. V. & Buttiker, M. Two-particle Aharonov-Bohm

effect and entanglement in the electronic Hanbury Brown-Twiss setup. Phys. Rev.

Lett. 92, 026805 (2004).

Einstein, A., Podolsky, B. & Rosen, N. Can quantum-mechanical description of

physical reality be considered complete? Phys. Rev. 47, 777–780 (1935).

Mandel, L. Quantum effects in one-photon and two-photon interference. Rev.

Mod. Phys. 71, S274–S283 (1999).

Kltenbaek, R. et al. Experimental interference of independent photons. Phys. Rev.

Lett. 96, 240502 (2006).

2.

3.

4.

5.

6.

7.Kumar, A. et al. Experimental test of the quantum shot noise reduction theory.

Phys. Rev. Lett. 76, 2778–2781 (1996).

Oliver, W. D., Kim, J., Liu, R. C. & Yamamoto, Y. Hanbury Brown and Twiss-type

experiment with electrons. Science 284, 299–301 (1999).

Henny, M. et al. The fermionic Hanbury Brown and Twiss experiment. Science

284, 296–298 (1999).

10. Klessel, H., Renz, A. & Hasselbach, F. Observation of Hanbury Brown-Twiss

anticorrelations for free electrons. Nature 418, 392–394 (2002).

11.Reznikov, M., Heiblum, M., Shtrikman, H. & Ma’halu, D. Temporal correlation of

electrons: Suppression of shot noise in a ballistic quantum point contact. Phys.

Rev. Lett. 75, 3340–3343 (1995).

12. Gavish,U.,Levinson, Y.&Imry,Y.Shot-noise intransportandbeamexperiments.

Phys. Rev. Lett. 87, 216807 (2001).

13. Hanbury Brown, R. & Twiss, R. Q. A new type of interferometer for use in radio

astronomy. Phil. Mag. 45, 663–682 (1954).

14. Hanbury Brown, R. & Twiss, R. Q. Correlation between photons in two coherent

beams of light. Nature 177, 27–29 (1956).

15. Ji, Y. et al. An electronic Mach-Zehnder interferometer. Nature 422, 415–418

(2003).

16. Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the

existence of extended states in a two-dimensional disordered potential. Phys.

Rev. B 25, 2185–2190 (1982).

17. Lesovik, B. G. Excess quantum shot noise in 2D ballistic point contacts. JETP Lett.

49, 592–594 (1989).

18. Born, M. & Wolf, E. Principles of Optics 7th edn 348–352 (Cambridge Univ. Press,

Cambridge, UK, 1999).

19. Neder, I. et al. Unexpected behavior in a two-path electron interferometer. Phys.

Rev. Lett. 96, 016804 (2006).

20. Neder, I. et al. Entanglement, dephasing and phase recovery via cross-correlation

measurements of electrons. Phys. Rev. Lett. 98, 036803 (2007).

21. Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in quantum

theory. Phys. Rev. 115, 485–491 (1959).

22. Heiblum, M. Quantum shot noise in edge channels. Phys. Status Solidi B 243,

3604–3616 (2006).

23. Bell, J. S. On the Einstein, Podolsky, Rosen paradox. Physics 1, 195–200 (1964).

8.

9.

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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