Suppression of visibility in a two-electron Mach-Zehnder interferometer

Suppression of visibility in a two-electron Mach-Zehnder interferometer

Ya. M. Blanter1,2 and Yuval Gefen2

1_{Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands} 2_{Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel}

共Received 15 October 2007; published 20 December 2007兲

We investigate the suppression of the visibility of Aharonov-Bohm oscillations in a two-electron Mach-Zehnder interferometer that leaves the single-electron current unchanged. In the case when the sources emit either spin-polarized or entangled electrons, partial distinguishability of electrons共coming from two different sources兲 suppresses the visibility. Two-particle entanglement may produce behavior similar to “dephasing” of two-particle interferometry.

DOI:10.1103/PhysRevB.76.235319 PACS number共s兲: 73.23.Ad, 73.43.⫺f, 73.90.⫹f

I. INTRODUCTION

Aharonov-Bohm 共AB兲 interferometry is a centerpiece in studies of nanoscopic electronic systems. Of special interest are Mach-Zehnder interferometers共MZIs兲:1_{The absence of}

backscattering gives rise to a large interference signal共large visibility兲 and allows measurements in strong magnetic fields.

One conceptual step beyond single-particle interferometry is a “two-particle interferometry,” tailored after the Hanbury Brown–Twiss experiment. In particular, one can utilize a two-electron MZI which features two current sources and two detectors共Fig.1兲. Such a device, suitable for the

mea-surement of current cross correlations in the detectors, has been originally proposed by Samuelson-Sukhorukov-Büttiker共SSB兲. In Ref.2, it was shown that while the current 共and the noise兲 at each particular detector is insensitive to the AB flux through the interferometer, the current cross corre-lations between the two detectors do show AB oscilcorre-lations. The latter are a direct consequence of particle indistinguish-ability: Measuring current signals at points c and d 关Fig.

1共a兲兴 may be due to an electron from a 共b兲 absorbed at c 共d兲 or at d 共c兲 . The product of the amplitudes of these two processes is flux sensitive. Had the two particles been distin-guishable, it would have been possible to conclude, for ex-ample, that the electron detected at c 共d兲 originated from a 共b兲. Only one of the above amplitudes survives, hence flux insensitivity.

Trying to design an actual measurement of a two-electron interferometry, it is important to understand which manipu-lations can render the electrons distinguishable, in practice, suppress the AB signal. Evidently, one can suppress the in-terference by introducing dephasers at the interferometer’s arms. However, this will also suppress single-particle inter-ference. We do not discuss this option in this paper.3_Below,

we focus on two different interference-suppressing scenarios. First, one can introduce a spin polarization of the sources. We show below that for opposite spin polarizations of the two sources, AB oscillations are suppressed, since electrons with opposite spin projections are distinguishable. The sec-ond scenario we consider is the entanglement of electrons in one of the sources. Intuitively, sources emitting states en-tangled with respect to electron spin might behave as “a little bit spin polarized,” and thus induce a suppression of two-particle interference.

In this paper, we calculate how the visibility in these two scenarios is reduced. In particular, our analysis reveals in-deed that entangled electrons vs nonentangled electrons act as共partially兲 distinguishable particles.

Whereas spin-polarized electron sources can be consid-ered within the existing scattering approach to electron trans-port, sources of entangled electrons present considerable dif-ficulties. First, up to now, entanglement of electrons in solid state has never been demonstrated experimentally. There is a number of theoretical proposals 共see Ref. 4 for recent re-view兲; most of them use interaction to create entangled pairs. For noninteracting electrons, the proposals include spatial separation of spin-polarized particles5,6 or, indeed, edge states in the quantum Hall effect regime.2,7_{We also note that}

proposals for manifestations of non-Abelian statistics8,9_rely,

in fact, on entanglement of the appropriate quasiparticles. A proper way to consider transport of such entangled pairs would be the development of full quantum-mechanical theory including the interaction effects. It is a major chal-lenge, and it is unlikely that such theory will be developed anytime soon. Here, we follow a different avenue. We de-velop a transport theory for two particles along the same line as single-particle transport. We show that within this

ap-a c d b F (a) 5 3 8 2 A C D B 7 4 6 1 (b) F 1

φ

4

φ

2

φ

3

φ

FIG. 1. A two-electron MZI.共a兲 Two-electron sources 共a and b兲 and two detectors 共c and d兲 connected by chiral edges. An AB flux is threading the interferometer. The sensitivity of the current cross correlation in the flux is the result of the indistinguish-ability of electrons emitted from a and b. Schematically, 具IcId典⬀兩Ab→dAa→c+ Ab→cAa→d兩2, where Ab→dis the amplitude for

transmission for b to d. For example, the combination

proach, one encounters certain technical difficulties related to the current conservation, and we show how they can be fixed by introducing a number of constraints for scattering matrices. In principle, the derivation of these constraints can come from the full scattering theory with interactions 共not available yet兲, however, we believe that our considerations are general enough and hold for an arbitrary scattering setup. In particular, we note that the present manifestation of MZI requires operating in the presence of a strong magnetic field. This means that the electrons are likely to be spin polarized, and seemingly, the physics discussed here has no practical implication. We, nevertheless, maintain that the mechanism of entangled-related suppression of two-particle interferom-etry is quite general, and the example discussed here of a spin-entangled electron pair is only a conceptually conve-nient example.

Below, we first derive general expressions for the current and noise of a generic multiterminal system, in which elec-trons in the same or different leads can be entangled. Let us consider a multiterminal system, with the leads emitting 共possibly兲 entangled electrons. A two-electron state in such a system has the form

Fˆ_␣␤† 共E,E

⬘

兲兩0典 =

兺

␴,␴⬘

g_␴␴*␣␤_⬘aˆ_␣␴† 共E兲aˆ_␤␴† _⬘共E

⬘

兲兩0典, 共1兲

where the indices␣and␤label the leads,␴and␴

⬘

label the spin projections, and the coefficient g is essentially the den-sity matrix for the entangled electron state.10_{For simplicity,}

we assume that each lead supports a single channel: The energies E and E

⬘

describe the longitudinal motion. The gen-eralization to the multichannel case is straightforward. The state is entangled if it cannot be constructed as a product of creation and annihilation operators acting on the vacuum. Below, we use as a reference state the state of electron pairs emitted by nonpolarized electrons,

关aˆ_␣↑† _{共E兲 + aˆ} ␣↓ † _{共E兲兴关aˆ} ␤↑ † _共E

_⬘

_{兲 + aˆ} ␤↓ † _共E

_⬘

_{兲兴兩0典,}

to be referred as a full product state. For the full product state, the theory must retrieve the results of a single-electron scattering approach employed to describe transport.

II. CURRENT AND NOISE

The two-particle field operator depending on the coordi-nates, times, and spin projections of both particles is

⌿ˆ␴1␴2共x1t1;x2t2兲 = 1 2␲បv

兺

_␤␥

冕

dE1dE2␾␤共x1,E1兲␾␥共x2,E2兲 ⫻exp共− iE1t1/ប − iE2t2/ប兲g␴2␴1 ␥␤ _aˆ ␤␴1aˆ␥␴2, 共2兲 where, for simplicity, we take the electron velocities in all leads equal tov, and␾_␯共x兲 is the scattering state originating

from the lead␯. The asymptotic form for this function, for

x苸␣, is

␾␯共x,E兲 =␦␣␯e−ik共E兲x+ s␣␯共E兲eik共E兲x,

where, as common in the scattering approach, the coordinate

x is counted along each lead共the origin is not important兲 and sˆ is the scattering matrix 共below, it will gain some more

indices兲. The scattering states are orthogonal, in discrete no-tations 兰dx␾_␮*共x,E兲␾_␯共x,E

⬘

兲=L␦␮␯␦EE⬘, with L being the length of the lead共it drops out of the final expressions兲.11

Now, we produce the one-particle current operator in the lead␣. It consists of the current operators produced by the “first”共Iˆ₁兲 and the “second” 共Iˆ₂兲 particles. We write

Iˆ_␣1共x1,t1兲 = − ieប 2m

冕

dx2_␴

兺

_1␴2 ⫻

冓

⌿ˆ_␴†_1␴2共x 1,t1;x2,t2兲 ⳵ ⳵x₁⌿ˆ␴1␴2共x1,t1;x2,t2兲

冔

2 + H.c. 共wrong兲,

where具...典2 means quantum-mechanical averaging over the

state of the second particle. Using12

具aˆ_␭␴† _共E兲aˆ

␯␴_⬘共E

⬘

兲典 =␦␭␯␦␴␴_⬘␦共E − E

⬘

兲f共E −␮␭兲,

where f is the equilibrium distribution function and␮_␭is the corresponding chemical potential, we obtain for the single-particle current operator,

Iˆ_␣1共x,t兲 = − e 2␲ប L 2␲បv

冕

dE1dE2e i共E₁−E2兲t/ប

兺

␤␥␦␴

兺

1␴2 ⫻A␤␣␦共E1,E2兲N␥␴2g␴1␴2 *␤␥_g ␴1␴2 ␦␥ _aˆ ␤␴1 † _共E 1兲aˆ␦␴1共E2兲N␥␴2 共still wrong兲,

where we have introduced the following combination of the scattering matrices,13

A_␤␣_␦共E1,E2兲 ⬅␦␣␤␦␣␦ei关k共E1兲−k共E2兲兴x

− s_␣␤* 共E₁兲s_␣␦共E₂兲e−i关k共E1兲−k共E2兲兴x_, 共3兲

and the quantity N_␣␴⬅兰dEf_␣␴共E兲dE, which is proportional to the “number of particles” with the spin␴ emitted by the lead␥.

The expression for Iˆ_␣1 is obviously wrong: It yields the average current proportional to the number of pairs of

par-ticles and not to the difference of the number of parpar-ticles

No. of pairs =

冕

dx1dx2dE1dE2dE3dE4

兺

␤␴1 1 共2␲បv兲2 ⫻␾␤*共x1,t1,E1兲␾␥*共x2,t2,E2兲␾␥共x2,t2,E3兲 ⫻␾␤共x1,t1,E4兲兩g␴␤␥₁␴₂兩2 ⫻具a_␤␴† _1共E 1兲a␥␴2 † _共E

2兲a␥␴₂共E3兲a␤␴₁共E4兲典

=

冉

L 2␲បv

冊

2

兺

␤␴1 兩g_␴␤␥_1␴2兩2_N ␤␴1N␥␴2,

and No. of particles=共L/2␲បv兲N_␥␴

2. Thus, including the

normalization constant, we find for the current operator of the first particle,

Iˆ_␣1共x,t兲 = − e 2␲ប

冕

dE1dE2e i共E₁−E2兲t/ប

兺

␤␥␦␴

兺

1␴2 g_␴ 1␴2 *␤␥_g ␴1␴2 ␦␥ _N ␥␴2

兺

_␯␴ 兩g_␴␴ 2 ␯␥ _兩2_N ␯␴

⫻A_␤␣_␦共E1,E2兲aˆ␤␴1

† _共E

1兲aˆ␦␴₁共E2兲N␥␴₂, 共4兲

and an identical expression for the current operator of the second particle.

Next, we use Eq.共4兲 to derive the average current,

具I␣典 = 具I␣1典 + 具I␣2典 =

− e 2␲ប

兺

_␥␴ 2

兺

␤␴1 A_␤␤␣ 共␥␴2兲兩g_␴ 1␴2 ␤␥ _兩2_N ␤␴1N␥␴2

兺

␤␴1 兩g_␴ 1␴2 ␤␥ _兩2_N ␤␴1 , 共5兲

where we assumed that the scattering matrices are energy independent, and we added to them two additional indices, as is explained below.

Equation 共4兲 also yields current noise. Averaging the

product of four creation and annihilation operators,13

具aˆ␤␴† 共E1兲aˆ␤⬘␴⬘共E1

⬘兲aˆ

␦†␴共E2兲aˆ␦⬘␴⬘共E2

⬘兲典 − 具aˆ

␤␴† 共E1兲aˆ␤⬘␴⬘共E1

⬘兲典

⫻具aˆ_␦†_␴共E

2兲aˆ␦⬘␴⬘共E2

⬘兲典

=␦␤_␦⬘␦␤_⬘␦␦␴␴_⬘␦共E₁− E₂

⬘兲

␦共E₁

⬘

− E₂兲f_␤␴共E₁兲 ⫻关1 − f␤_⬘␴共E2兲兴, we obtain S_␣␣⬘= e 2 2␲ប_␤,␤

兺

_{⬘,␥,␥⬘}

兺

␴1,␴2,␴2⬘ N_␥␴ 2

兺

␤␴1 兩g_␴ 1␴2 ␤␥ _兩2_N ␤␴1 N_␥⬘␴ 2 ⬘

兺

␤␴1 兩g_␴ 1␴2⬘ ␤␥⬘_兩2_N ␤␴1 ⫻Re关g_␴*₁␤␥_␴2g ␴1␴2 ␤⬘␥_g ␴1␴2⬘ *␤⬘␥⬘_g ␴1␴2⬘ ␤␥⬘_兴A ␤␤_⬘ ␣ _共␥ ,␴2兲A_␤␣_⬘⬘_␤共␥

⬘

,␴2

⬘兲

⫻

冕

dEf_␤␴ 1共E兲关1 − f␤⬘␴1共E兲兴 + 共␣↔␣

⬘兲.

共6兲

Equations共5兲 and 共6兲 generalize standard expressions for

the multiterminal current and noise to the case of entangled electrons. It is easy to check that if electrons are not en-tangled and are in the product state, the coefficients g are

equal to one for all values of the arguments, and these ex-pressions reduce to the standard one-particle formulas.13

The problem with Eqs.共5兲 and 共6兲 is that currents are not

automatically conserved, 兺_␣I_␣⫽0. Also, if all leads have

equal chemical potentials, nonvanishing currents can be still generated according to Eq.共5兲. Usually, requirements of

cur-rent conservation and gauge invariance 共currents are un-changed if the chemical potentials of all reservoirs are shifted simultaneously兲 are guaranteed by the unitarity of the scattering matrix. In our case, they are satisfied

automati-cally provided that we choose the scattering matrix to obey

兺

_␮ s_␮␯* 共␥,␴2兲s_␮␯⬘共␥

⬘

,␴2

⬘兲 =

␦_␯␯⬘,

兺

_␯ g_␴␴ 2 *␯␥_g ␴␴2⬘ ␯␥_⬘ s_␮␯*共␥,␴2兲s_␮⬘␯共␥

⬘

,␴2

⬘兲 =

␦␮␮_⬘g_␴␴ 2 *␮␥_g ␴␴2⬘ ␮␥_⬘ . 共7兲

III. MACH-ZEHNDER INTERFEROMETER AND VISIBILITY

Next, we specialize to the eight-terminal setup with edge states suggested by SSB.2_{We will always bias sources 2 and}

3 with the same voltage V and measure the average current through lead 5 and the cross correlation between currents at 5 and 8. We are interested in the linear in V effects. Assuming that eV is much smaller that the Fermi energy in the leads, we can, in the leading order, take all quantities N_␤␴to be the same共voltage independent兲. One has then

具I5典 = − e2V 2␲ប_␥␴

兺

₂ 1

兺

␤␴1 兩g␴1␴2 ␤␥ _兩2_␤=2,3

兺

兺

␴1 兩s5␤共␥,␴2兲兩2 共8兲 and S58= − e3兩V兩 ␲ប

兺

_␥␥⬘␴

兺

1␴2␴2⬘ 1

兺

␤␴1 兩g_␴␤␥_1␴2兩2 1

兺

␤␴1 兩g_␴ 1␴2⬘ ␤␥_⬘兩2 ⫻

冏

兺

␤=2,3 g_␴ 1␴2 *␤␥_g ␴1␴2⬘ ␤␥_⬘ s_5␤*共␥,␴2兲s_8␤共␥

⬘

,␴2

⬘兲

冏

2, 共9兲 where we have used the “unitarity” conditions关Eq. 共7兲兴.

Now, we apply these expressions to the different states emitted by the reservoirs. The idea is to investigate under which conditions the electrons emitted from sources 2 and 3 are “painted,” at least partially “red” and “blue,” respec-tively, hence suppression of the flux sensitive共coherent兲 term of the correlation.

具I5典 = − e2V 2␲ប共兩s52兩 2_+兩s 53兩2兲 = − e2V 2␲ប共TATC+ RARD兲, 共10兲 where T and R are corresponding transmission and reflection probabilities, and S58= − 2e3兩V兩 ␲ប 兩s52*s82+ s53*s83兩2 = −2e 3_兩V兩 ␲ប 兩

冑

TATCRBRCei␪+

冑

TBTDRARD兩2, 共11兲 where␪is a linear combination of the phases of transmission and reflection amplitudes and the Aharonov-Bohm phase. In accordance with SSB, the average current is not sensitive to AB effect, since not a single-electron trajectory encircles the magnetic flux. By contrast, noise is a two-particle phenom-enon, and the setup is designed in such a way that the cross correlations are sensitive to the AB flux. The visibility of the noise oscillations is

v =2

冑

TATBTCTDRARBRCRD TATCRBRC+ TBTDRARD

, 共12兲

and it becomes unity共ideal oscillations兲 provided the setup is symmetric: all transmission and reflection probabilities equal to 1/2. This is the result of SSB.

共ii兲 One source is spin polarized. Imagine that states are still not entangled but source 2 is spin polarized: It only emits and absorbs spin-up electrons. The coefficients g are all equal to one, with the exception of g_↓␴

2

2␥ _{= g} ␴1↓

␤2 _{= 0. We}

need now to choose the scattering matrices in such a way so that they obey the unitarity condition共7兲. This is easily done

intuitively. For spin-up electrons, the constraints are the same as for the full product state, and thus, the scattering for spin-up electrons is described by the matrix s. The scattering matrix for spin-down electrons, which we denote as s˜, is

constrained by the condition

兺

␯⫽2 s

˜_␮␯*˜s_␮⬘␯=␦␮␮_⬘, ˜s_␮2= 0. We obtain the average current,

具I5典 = − e2_V

2␲ប共兩s52兩

2_+兩s

53兩2+兩s˜53兩2兲, 共13兲

and current noise,

S58= − e3兩V兩

␲ប 共兩s52*s82+ s53*s83兩2+兩s˜53*˜s83兩2兲. 共14兲

The first term in the brackets, similarly to Eq.共11兲, contains

both phase-insensitive and AB terms. The second term is insensitive to the AB phase. Thus, the total visibility in the noise is reduced by the presence of this second term. The visibility depends on the choice of the scattering matrices s˜,

in particular, if the setup is symmetric, and s˜₅₃= 0共unrealistic case兲; the oscillations are still ideal 共v=1兲. This is because, in this case, a spin-down electron, originating from 3, has to go to 8 with certainty, and thus, it does not change the

cur-rent noise. On the other hand, the natural choice would be 兩s52兩=兩s53兩=兩s˜52兩. In this case, which we call optimal, the

vis-ibility equalsv = 2/3.

In the same way, we can treat a setup where both sources, 2 and 3, are polarized. If they are polarized in the same direction共spin up兲, the visibility is the same as for full prod-uct states共both current and noise are reduced by the factor of 2 since now, only spin-up states contribute兲. Provided one source is spin-up polarized and the other one is spin down, the visibility vanishes: There are no AB oscillations in this case, since one can say with certainty from what source each electron has originated.

共iii兲 Entangled electrons from one source. Pairs in triplet

sz= 0 state are emitted from lead 2, 兩2↑典兩2↓典 + 兩2↓典兩2↑典,

all other leads are in the full product state. This means that all the coefficients g are equal to one except for

g_↑↑22= g_↓↓22= 0. Now, we are obliged to choose scattering matrices which depend on the second electron in the pair—the “spouse”—otherwise the unitarity 共7兲 conditions

cannot be fulfilled. We choose

s_␣␯,␴共␥,␴2兲 =

再

s␣␯, ␥⫽ 2 or␥= 2, ␴⫽␴2

s

˜_␣␯, ␥= 2, ␴=␴2.

冎

Note that this choice共affecting the visibility兲 is arbitrary and definitely not unique. Our matrices now, in addition to what we have already seen, obey

兺

␯⫽2 s ˜_␮␯* s_␮⬘␯=

兺

␯⫽2 s_␮␯*˜s_␮⬘␯=␦␮␮_⬘. We can again calculate the average current,

具I5典 = − e2V 2␲ប

再

冉

7 8+ 1 15

冊

共兩s52兩 2_+兩s 53兩2兲 + 1 15兩s˜53兩 2

冎

_, 共15兲 and the current noise,

S₅₈= −2e 3_兩V兩 ␲ប

再

冉

142 162+ 2 14 15⫻ 16+ 1 152

冊

兩s52 *_s 82+ s53*s83兩2 +

冉

14 15⫻ 16+ 1 152

冊

共兩s˜53 *_s 83兩2+兩s53*˜s83兩2兲 + 1 152兩s˜53 *_˜s 83兩2

冎

. 共16兲 Again, phase-dependent contributions are only found in the first term in the braces. All other terms are phase insensitive and thus reduce the visibility. In particular, with the same optimal choice of scattering matrices, 兩s₅₂兩=兩s₅₃兩=兩s˜₅₂兩, the visibility is reduced to 0.93.

共iv兲 Pairs in singlet state are emitted from lead 2, 兩2↑典兩2↓典 − 兩2↓典兩2↑典.

All other leads are in the full product state. This means that all the coefficients g are equal to one except for

s_␣␯,␴共␥,␴2兲 =

冦

s_␣␯, ␥⫽ 2 or␥= 2, 共␴␴2兲 = 共↑↓兲 s ˜_␣␯, ␥= 2, ␴=␴2 s ¯_␣␯, ␥= 2, 共␴␴2兲 = 共↓↑兲

冧

. 共17兲 The matrix s¯ must then obey the following constraints:

兺

␯ s ¯_␮␯*¯s_␮⬘␯=

兺

␯ s_␮␯*¯s_␮⬘␯=

兺

␯ s ¯_␮␯* s_␮⬘␯=

兺

␯⫽2 s ¯_␮␯*˜s_␮⬘␯ =

兺

␯⫽2 s ˜_␮␯*¯s_␮⬘␯=␦␮␮_⬘.

With this choice of scattering matrices, we obtain the same

current and current noise共and consequently, the same visibil-ity兲 as for the triplet sz= 0 state. We should recall, however, that this conclusion depends on the choice of scattering ma-trices 关Eq. 共17兲兴: for instance, on the fact that s_␣␯,␴共2,␴兲 is

the same for single and triplet entangled states.

IV. CONCLUSIONS

In summary, we have defined and analyzed various sce-narios for which single-particle amplitudes maintain their co-herence共as would be manifest in a Mach-Zehnder interfer-ometry measurement兲, yet 共partial兲 distinguishability of electrons emitted from different sources suppresses the flux sensitivity of the two-particle cross-correlation function 共Table I兲. Our analysis demonstrates how two-particle

en-tanglement can give rise to a behavior akin to a “dephasing” of a two-particle interferometry.

ACKNOWLEDGMENTS

We acknowledge useful discussions with M. Büttiker, M. Heiblum, I. Neder, and P. Samuelsson. This work was sup-ported by the ISF of the Israel Academy of Sciences, by the US-Israel BSF, by the Minerva Einstein Center共BMBF兲, and by the Transnational Access Program 共RITA-CT-2003-506095兲 at the Weizmann Institute of Science.

1_{Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and H.}

Shtrikman, Nature共London兲 422, 415 共2003兲.

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Lett. 92, 026805共2004兲; see also V. S.-W. Chung, P. Samuels-son, and M. Büttiker, Phys. Rev. B 72, 125320共2005兲.

3_{Still, dephasing might be responsible for the suppression of}

vis-ibility in the experiment 共Ref. 1兲, see F. Marquardt and C.

Bruder, Phys. Rev. Lett. 92, 056805 共2004兲; I. Neder and F. Marquardt, Nev. J. Phys. 9, 112共2007兲; J. T. Chalker, Y. Gefen, and M. Y. Veillette, Phys. Rev. B 76, 085320共2007兲.

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共2005兲.

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10_{We take the coefficients g, to be energy independent. A different}

choice, for instance g⬀␦共E−E⬘兲, may yield different results, G. Burkard, D. Loss, and E. V. Sukhorukov, Phys. Rev. B 61, R16303共2000兲.

11_{It will be more natural and convenient to use continuous notations}

for the energy兰dx␾_␮*共x,E兲␾_␯共x,E⬘兲=2␲បv␦_␮␯␦共E−E⬘兲. In par-ticular, one can also write the “square” of the delta function in continuous notations␦2_{共E−E⬘兲=共L/2␲v兲␦共E−E⬘兲.}

12_{Note that as a consequence of our choice of the field operator关Eq.}

共2兲兴, the rules for the averaging imply that entangled electron

pairs are characterized by the Fermi distribution function. Whereas such a choice is plausible, the actual distribution func-tion may depend on the mechanism of entanglement. We thank M. Büttiker for useful comments to this point.

13_{M. Büttiker, Phys. Rev. B 46, 12485 共1992兲; see also Ya. M.}

Blanter and M. Büttiker, Phys. Rep. 336, 1共2000兲. TABLE I. Reduction of the visibility for different scenarios and,

otherwise, optimal conditions.

Scenario Optimal visibility

2, 3: product states 1

2: product, not polarized; 3: polarized↑ 2/3

2, 3: polarized↑ 1

2: polarized↑; 3: polarized ↓ 0

2: entangled, singlet; 3: product 0.93