Chiral-mediated entanglement in an Aharonov-Bohm ring

0.49 MB·English

by  Bruno Rizzo

Chiral-mediated entanglement in an Aharonov-Bohm ring
Bruno Rizzo,1 Liliana Arrachea,1 and Juan Pablo Paz1
1Departamento de F´ısica, FCEyN and IFIBA, Universidad de Buenos Aires,
Pabell´on 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina
(Dated: January 30, 2012)
WestudytheorbitalentanglementinabiasedAharonov-Bohmringconnectedinafour-terminal
setup. WefindthattheconcurrenceachievesamaximumwhenthemagneticfluxΦB coincideswith
2 anintegernumberofhalfafluxquantumΦ0/2. Weshowthatthisbehaviorisaconsequenceofthe
1 existenceofdegeneratestatesoftheringhavingoppositechirality. Wealso analyzethebehaviorof
0 thenoise as a function of Φ and discuss the reliability of thisquantity as an entanglement witness.
2

n
(CME) because its creation is possible by the existence
a
of intermediate states in the AB-ring which are coher-
J
entsuperpositionsoftwodifferentchiralitiesfortheelec-
7
tronic motion. Remarkably, this kind of entanglement
2 I. INTRODUCTION
can be also defined in a transport setup, which is identi-

] cal to the one that was proposed to define the electron-
ll The increasing interest in quantum information pro- holeentanglement,5,6,9,10,12,13. Infact,wealsoshowhere
a cessingisboostingthesearchformechanismstoproduce
thatthenoisecurrent-currentcorrelationfunctioncanbe
h and controlentanglement in devices of different nature.1
- a good witness for the entanglement in our setup.
s Photonicdevicesareroutinelyusedforpreparingandde- The paper is organized as follows. In Sec. II, we
e tecting entangledphotonpairs.2 The mainlimitations of
presentthemodelindetailandwesketchtheformulation
m
suchdevicesisthe non-existenceofdeterministic sources
of the theory, where we calculate the reduced density-
. as well as the difficulty in controlling the interaction be-
t matrixofatwo-particlesystem. The results ofthis work
a tween the photons. Entanglement mechanisms have also
are presented in Sec. III. The conclusions are presented
m beenproposedinsolidstatedevices,likequantumdots,3
in Sec. IV.
- and superconductors.4 The fundamental ingredient be-
d
hind all these mechanisms is a many-body interaction.
n
More recently, it was determined that entanglement is
o II. THEORETICAL TREATMENT
also possible in systems of non-interacting electrons.5,6
c
[ In particular, it was shown that electron-hole entangled
A. Model
pairs can be produced by biasing a tunneling barrier.
1
TheedgestatesofsystemsinthequantumHallregime
v We consider the AB single-channel ring with a
3 can be employed in solid state devices to produce elec-
magnetic flux. The ring is connected to four
6 tron beams with properties similar to those of a photon
terminals5,6,9,10,12 as shown in Fig. 1. Two of the ter-
8 beam in optical setups. Exploiting this analogy, several
minals, those labeled by α = 1,2, at the left side of the
5 theoretical and experimental proposals of electronic in-
1. terferometershavebeenreported.7Interestingly,theelec- ring, are at a higher voltage V with respect to the ones
at the right, labeled by α = 3,4. All terminals are or-
0 tronic counterpart of the Hanbury-Brown-Twiss device
dinary single-channel metallic leads where electrons can
2 has been analyzed in a configuration of edge Hall states
1 that do not have interfering orbits.8,9 For this reason, move either to the right or to the left. For simplicity, we
: consider spinless electrons and we describe the setup by
v Aharonov-Bohm(AB)effecttakesplaceonlyatthe level
the Hamiltonian
i of two-particle correlation functions, while the single-
X
particle AB effect is not present. This entanglement 4
r seems to be related with an asymmetry in the device H = (Hα+Hc,α)+Hring, (1)
a
whichfavorstheinternalproductionofparticle-holepairs α=1
X
andmanifestitselfin the behaviorofthe current-current where H = ε c† c are Hamiltonians of non-
correlation functions. The relevance of the current- α kα kα kα kα
interactingelectronsrepresentingthe leads. For the AB-
currentcorrelationfunctionsasentanglementwitnesshas P
ring we use a non-interacting model, where electrons
been also discussed in other fermionic systems.11
move with velocity v either clockwise (+ chirality) or
Theaimofthisworkistoestablishtheexistenceofor-
anticlockwise ( chirality). The Hamiltonian is
bital entanglement in AB systems where single-particle −
L
interference is present. Using a microscopic model, we H = dxvλΨ†(x) Ψ (x), (2)
show that when coupling the AB-ring to four leads (two ring λ Dx λ
on the right and two on the left), the post select two- λX=±Z0
particlestatesofelectronsatopposite leadsaretypically being λ the chirality, = i∂ φ, φ = Φ/(LΦ ),
x x 0
D − −
entangled. We name it ”chiral-mediated entanglement” with Φ = 2πΦ , where Φ is the magnetic flux, Φ =
B B 0
2
hc/e is the flux quantum and L is the length of the rightorintheleftleads.6,10,12Weintroducetheoperators
ring. The contacts between the leads and the ring A† c† c† ,A† c† c† ,A† c† c† ,A† c† c† ,
00 ≡ k1 k3 01 ≡ k1 k4 10 ≡ k2 k3 11 ≡ k2 k4
are modeled by tunneling terms of the form Hcα = which create one particle in one of the left leads and a
w [c† Ψ (x )+H.c.],wherex definethepo- second particle at one of the right leads. The ensuing
kα,λ=± kα kα λ α α
sitions of the ring to which the leads are attached. We the4 4densitymatrixdescribesasystemoftwoqubits
P ×
consider the leads to be at zero temperature. The chem- with elements
ical potentials enforce a bias voltage between left and
1
rµiLgh−tµlRea=dse,Vi.,ew.hiµc1hw=eaµs2su=meµtLo;bµe3ve=rysµm4a=lleµVR∼a0n.d [ρ(2)(ε)]ab,a′b′ = N0 Yα Xkα δ(ε−ǫkα)hA†abAa′b′i,
where isanormalizationfactor,whilethemeanvalue
0
B. Reduced Density Matrix and Concurrence is takeNn in the nonequilibrium state with a net current
flowingfromthe leftto the right. We remarkthatinour
In a setup as the one in Fig. 1, electrons tunnel from approachmatrixelementsofρ(2) areobtainedintermsof
the left leads to the right ones. We aim to define the ef- operators appearing in the Hamiltonian H. Expectation
fectivedensitymatrixdescribingthequantumstatepost- values of four time dependent creation and/or annihi-
selected from the total two–electron state by projecting lation operators are computed using the nonequilibrium
outthecomponentswherebothelectronsareeitherinthe Green function formalism and Wick theorem,15
∞ ∞
c† (t)c† (t)c (t)c (t) = dε dε′ G< (ε ε′)G< (ε′) G< (ε ε′)G< (ε′) , (3)
h kα kβ kλ kδ i kλ,kα − kδ,kβ − kδ,kα − kλ,kβ
Z−∞ Z−∞ h i
where G< (ε) is the Fourier transformwith respect to C. Noise
kα,kβ
t t′ of the lesser Green function
−
G< (t t′)=i c† (t′)c (t) . (4) As pointed out in Refs. 6 and 13 the concurrence can
kα,kβ − h kβ kα i be expressed in terms of correlators that quantify the
We assume that eV 0 and we are interestedin analyz- degree of violation of Bell inequalities. In transport se-
ing ε µ=(µ +µ∼)/2. We present the corresponding tups the latter caninturnbe directly relatedtocurrent-
L R
expres∼sion of ρ(2) in Appendix A. current correlation functions, which are amenable to be
From the above density matrix we compute the con- experimentally detected. We, thus, turn to analyze the
currence, which is a good measure of entanglement.14 In connection between these correlation functions and the
this case it is given by above discussed entanglement. We first outline the pro-
cedure to compute the current-current noise within our
ρ(2) =max 0, λ1 λ2 λ3 λ4 , (5) treatment. The current passing through the contact to
C − − −
the terminal α, can be expressed by the following oper-
wherhe iλi aren tphe eipgenvalupes ofp Ro := ator J (t) = (e/~) w [ic† (t)Ψ(x ,t)+h.c]. The
ρ(2)σ σ ρ(2)∗σ σ in decreasing order. It is α kα kα kα α
y y y y zero frequency shot-noise is a measure of the current-
interesting to notice that the concurrence so calculated P
current correlations in different terminals. It reads
N N
is equivalent to the one obtained by the spin–dependent
scattering formalism for a single mode conductor ,5,6 1
(0)= dτ δJ (τ),δJ (0) , (8)
α,β α β
which is S 2 h{ }i
Z
=2 τ1(1−τ1)τ2(1−τ2). (6) whereδJα =Jα−hJαi. Wecalculate(ec.(8))byevaluat-
C τ +τ τ τ ing a bubble diagram in terms of nonequilibrium Green
p 1 2− 1 2
functions. The corresponding expressions are presented
Here, τ ,τ are transmission eigenvalues of the scatter-
1 2
in Appendix IV.
ing matrices s (µ) = δ i Γ Γ GR (µ), where
α,β α,β − β α α,β
GR (µ) is the Fourier transformwith respect to t t′ of
α,β p −
the retarded Green function
III. RESULTS
GR (t t′)= iΘ(t t′) Ψ (x ,t),Ψ† (x ,t′)
α,β − − − h{ λ α λ′ β }i
λ,λ′ A. Qualitative Analysis
X
(7)
evaluatedatµ,beingxα thepositionoftheringatwhich To understand the origin of CME it is useful to be-
the wire α is attached. while Γα = 2π kαwk2αδ(µ − gin analyzing the electronic states of the isolated AB–
ε ).16 ring. TheHamiltoniancanbediagonalizedinmomentum
kα P
3
space: Defining Ψ (x) = 1/√ e−ipxc with a
λ N p p,λ N
normalization factor, p = 2πn/L and n Z we obtain
P ∈
H = ε (Φ)c† c ,withε =λv(p φ)(a
ring λ p p,λ p,λ p,λ p,λ −
cutoffinthesingle-particleenergyspectrumisassumed).
P P
The effect ofthe magnetic flux onthe energiesε (Φ) is
p,λ
illustrated in Fig. 1. Depending on the magnetic flux,
there can be zero, one or two single-particle states p,λ
| i
withagivenenergy. Stateswithdifferentchiralityλ=
±
are degenerate only when the flux is an integer multiple
of πΦ .
0 FIG. 1. (a) Scheme of an Aharonov-Bohm ring with two
Let us consider first the case where two degenerate
chiralities, attached to four leads. The chemical potentials
states with opposite chiralities exist in the ring. These aresuchthatthereisabiasvoltagebetweentheleftandright
twostatesbehavesasanintermediatequbit thatcouples leads. Electronsincomingfromtheleftcantunneltothering
to the qubits defined by the leads. We will argue that andescapethroughleads. (b)Lineardispersionrelationofan
in this case CME naturally emerges between electrons isolated AB-ring,whereelectronsmovewithvelocityv either
at the right and left leads. The N-particle states with a clockwise (+ chirality) or anticlockwise (− chirality). The
Fermienergyǫ canbeobtainedfrom 0 ,thatrepresents magnetic flux determines whenever the Fermi level is two-
F
| i degenerate or not.
the Fermi sea with N 2 particles filling the states with
−
ε (Φ) < ǫ , where v(p φ) = ǫ . Thus, we have
p,λ F F F
−
Ψ = c† c† 0 . When the ring is in contact
| ringi pF,+ pF,−| i in terms of operatorsthat are linear combinations of the
withthefourleads,particlescantunnelbetweenthering lead operators c† as H = w(f†c + H.c.), with
andthereservoirs. Forweakcouplingwecanassumethat kα eff pF,λ
eachofthe chirallevelswithpF hybridizewiththe levels f† = (1/2) 4α=1eipFxαc†kα, while there are three addi-
tional orthogonal linear combinations of these operators
oftheleadshavingthesameenergyǫ . Thisisdescribed
F P
which do not hybridize with the ring. The two eigen-
by the following effective Hamiltonian
statesofH arelinearcombinationsofasingle-particle
eff
4 state of the leads and a single-particle state of the ring.
Heff =w eiλpFxα[c†kαcpF,λ+H.c.], (9) Thus, a two-particle state of this Hamiltonian has never
α=1λ=± the two particles in the leads and therefore no orbital
X X
entanglement is possible.
where w is the effective tunneling parameter, c† creates
kα Theaboveargumentsuggeststhatbyvaryingthemag-
anelectron in the single-particlestate ofthe α-lead with
neticflux,orthechemicalpotentialoftheleads(orequiv-
energy ε = ǫ (we take ǫ = 0 without loss of gen-
kα F F alently,agatevoltageappliedatthe ring)we caninduce
erality). This Hamiltonian has four eigenstates of the
the system to switch from a situation with no orbital
form
entanglement between the leads onto another situation
4 with inter-lead entanglement. This is done by varying
ψ =[ γ c† + γ c† ]0 , n=1,...,4, Φ and/or µ in order to have degenerate chiral states of
| ni n,α kα n,λ pF,λ | i the ring at the Fermi energy. We now present a rigor-
α=1 λ=±
X X
(10) ous calculation of the entanglement for the states that
where the coefficients γ are the weights of the eigen- are relevant for a transport experiment in the coherent
n
states in the chosen base. It also has two additional de- regime and explain how the CME depends on flux and
generatestatesoftheform ψ = 4 γ c† 0 , n= chemical potential. We also discuss the way in which it
| ni α=1 n,α kα| i
5,6. The latter correspond to states that do not hy- may be detected in transport experiments.
P
bridize with the ring. When two particles are present,
two different such states must be occupied. It is simple
to show that any state of this type has a sizable projec- B. Numerical Results
tion on states of the form Λ c† c† 0 for some
α6=β α,β kα kβ| i
non-vanishingcoefficientsΛ . Thistwo-particlestateis 1. Concurrence
α,β
P
typically entangledin the orbitalindices α,β ofopposite
leads. Notice that itis alsopossible to use twoAB–rings The behavior of as a function of the mean chemical
C
with degenerate levels as a two-qubit system coupled by potential µ and the magnetic flux Φ is shown in Fig. 2.
two conducting leads (intermediate qubit). TheconcurrenceismaximalatΦ/Φ =nπandforvalues
0
On the other hand if we consider the case of a one µ close to the energy of two degenerate chiral states of
chirality ring (ǫ = v(p φ) for example) the situation thering. Thesametypeofbehaviorisobservedforother
p
−
drastically changes and no significant entanglement be- configurations of the wires, corresponding to contacts at
tween left and right leads is attained. This is can be different positions x . For this Fig. we considered wires
α
seen because the effective Hamiltonian has a different with a bandwidth W and Γ = Ω W2 µ2, where
α α α α−
level structure. In fact, H can be naturally written Ω is a constant, but the same behavior is obtained for
eff α
p
4
(a) (b)
1.0
C C
0.80
S S
0.60
0.40
0.20
0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π
Φ/Φ0 Φ/Φ0
FIG. 4. (Color online) Shot noise (squares) between left and
right terminals, as a function of Φ/Φ0 for two values of the
chemical potentials at which there are degenerate states. (a)
µ=0.3175, (b) µ=0.7866. The corresponding plots for the
concurrence are also shown for comparison (circles).
FIG. 2. (Color online) Concurrence as a function of Φ/Φ0
and mean chemical potential µ for a ring with length L=20
with wires connected at x1 =1, x2 =6,x3=11,x4=16.
2. Noise
We now turn to present results on the behavior of the
current-current correlation functions. Our aim is to an-
alyze if the signatures of entanglement found in the be-
havioroftheconcurrencecanbealsoidentifiedinthebe-
havior of the noise. Results for the total left-right noise
correlations S = (0) (equal to the self-
α=1,2,β=3,4Sα,β
correlation (0) ) are shown in Fig. 4.
− αP,β=1,2Sα,β
The left (right) panel, corresponds to a chemical poten-
P
tial µ for which there are two degenerate chiral states in
the ring for Φ/Φ = 0,(π),mod(2π). The behavior of
0
the concurrence is also plotted for comparison. In both
cases, S, along with exhibit maxima at the fluxes for
C
which the two degenerate chiral states are resonant at
the given µ. This points to the idea that noise is in-
deed a reliable witness of orbital entanglement, as dis-
cussedin the contextof other electronicsetups.5,6,9,11 In
the case of the left panel, both quantities are maximum
FIG. 3. (Color online) Concurrence as a function of Φ/Φ0
and mean chemical potential µ for a ring with only one chi- at Φ/Φ0 = 0,mod(2π). Within a range of fluxes which
rality. The ring parameters are thesame as in Fig.2. scans the width of the resonant degenerate levels of the
ring they first decrease and then increase, displaying a
dip. As the flux increases further, the behavior of these
two quantities, however, depart one another. While S
tends to vanish around Φ/Φ = π, the concurrence dis-
0
leadswithaconstantdensityofstates. Forsomechemical plays a wide plateau with height 1. Qualitatively,
C ∼
potentials exhibits maxima at Φ/Φ0 = 0, mod (2π), the same type of behavior is observedin the rightpanel.
C
whichcorrespondstotheenergyoftwodegeneratestates In this case, both quantities exhibit a sharp maxima at
ofthering,butachievesagainthemaximumvaluewithin resonance(seethepeaksaroundΦ/Φ =π). Theconcur-
0
awiderangeoffluxescenteredatΦ/Φ0 =π, mod (2π). rence displays a plateau and another (lower) maximum
This feature is analyzed below in more detail. around Φ/Φ = 0,mod(2π) while S is vanishing small.
0
On general grounds this is rather surprising, since one
Instead, if we evaluate for the Hamiltonian (2) re- couldeasilyimagine situationswith a sizablenoise with-
C
stricted to a single chirality, we find negligibly orbital out entanglement, but here we have the converse situa-
entanglement in the leads within the whole range of Φ tion.
and µ. These results are presented in Fig.3. In this case
theringbehavesasasingle-levelsystem,andpreventthe In order to further understand the relation between
formation of orbital entangled states at the leads. the behavior of and S as well as the connection to the
C
5
IV. CONCLUSION
(a) (b)
1.0
P P
0.80 r,r r,r To summarize, we introduced a new mechanism for
P P
l,l l,l
0.60 orbitalentanglement. Thistypeofentanglementisorigi-
natedinthe spectralnatureofthe AB ringandishighly
0.40
sensitive to the magnetic field. Orbitally entangled elec-
0.20 tronicpairscanbeproducedbysuitablytuningthemag-
0 netic field, the chemical potential or a voltage gate at
0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π
the ring, in order to have a degenerate pair of electronic
Φ/Φ0 Φ/Φ0 stateswithdifferentchiralitiesattheFermienergy(inter-
mediatequbit). Infact,iftherightandleftleadsareme-
FIG. 5. (Color online) Probability of findingtwo particles at diated by a single–levelsystem(single-chiralityring) the
the left and right leads as a function of magnetic flux Φ/Φ0. orbitalentanglementisnegligible. Thistypeofentangle-
Panels(a)and(b)correspondtothesameparametersofFig.
ment can be detected in transportexperiments, the shot
4 (a) and (b),respectively .
noisebeing agoodwitness. The setupofFig. 1couldbe
experimentally realized in an architecture based on the
quantum Hall regime of a 2D electron gas, by substitut-
CME, we analyze the probabilities ing each wire by a pair of incoming and outgoing edge
statesandthe ringbyapairofedgestateswithdifferent
chiralities, separated by a narrow circular wall. In such
P = δ(ε ǫ ) c† c c† c , a setupit wouldbe possible to combine the mainsystem
LL − kα h k1 k1 k2 k2i of Fig. 1 with beam splitters connected at the wires, in
Yα Xkα
order to test Bell inequalities and even perform the full
P = δ(ε ǫ ) c† c c† c (11)
RR − kα h k3 k3 k4 k4i quantum tomography following the protocol of Ref. 10.
Yα Xkα Twointerestingpossiblegeneralizationsarethecombina-
tion between static and flying qubits by combining with
quantumdots,17 aswellastheintroductionofdynamical
of finding two particles at the left (right) leads, respec-
single-particle emitters at the sources.18
tively. In Fig. 5 we show the behavior of these two
quantities for the same parameters of Fig. 4. It is
clearthat P andP areboth sizablewhen tunneling
LL RR ACKNOWLEDGMENTS
through the ring is allowed. For these chemical poten-
tials, this corresponds to the narrow window of fluxes
We thank D. Frustaglia for useful conversations. This
around Φ/Φ = 0(π), for the case of the left (right)
0
work is supported by CONICET, MINCyT and UBA-
panels, respectively, within which the degenerate levels
CYT, Argentina. LA thanks support from the J. S.
of the ring remain resonant. Beyond these values, the
Guggenheim Foundation.
transmission from left to right is blocked and the two
particles havea high probabilityof remaining within the
left wires. A further analysis of Figs. 2 and 4 in the
APPENDIX
light of the results shown in Fig. 5, then reveals that
the highvaluesofconcurrenceintheplateausawayfrom
Φ/Φ =0(π),mod(2π)inthecaseoftheleft(right)pan- A. Reduced Density Matrix
0
els of Fig. 4, correspondto states which have a very low
probability of taking place. Instead, the resonant situa- We evaluate the post-selected state of two electrons
tion with two degenerate chiral states of the ring at the at opposite leads, ρ(2), in terms of Green functions (the
Fermienergy,leadstoahighorbitalentanglementwhich Fouriertransformof ec.(7)). The explicitexpressionof a
is clearly witnessed by a high noise amplitude S. general matrix element, up to a normalization factor, is:
6
4
δ(ε ǫ ) c† (t)c† (t)c (t)c (t) =Γ2 f (ε)f (ε)GR (ε)GR (ε) f (ε)f (ε)GR (ε)GA (ε)
− kν h kα kβ kλ kδ i α β λ,α δ,β − α δ λ,α δ,β
νY=1Xkν h
4
f (ε)f (ε)GR (ε)GA (ε) i Γ3 f (ε)f (ε)GR (ε)GR (ε)GA (ε)
− λ β δ,β λ,α − β γ λ,γ δ,β γ,α
i γX=1 h
4
f (ε)f (ε)GR (ε)GR (ε)GA (ε) + Γ4f (ε)f (ε)GR (ε)GR (ε)GA (ε)GA (ε)
α γ λ,α δ,γ γ,β η γ λ,γ δ,η η,β γ,α
i γX,η=1
4
+Γ if (ε)f (ε)GR (ε)+if (ε)f (ε)GA (ε) Γ f (ε)f (ε)GR (ε)GA (ε) δ
"− α β δ,β α δ δ,β − γ α δ,γ γ,β # kα,kλ
γ=1
X
4
+Γ if (ε)f (ε)GR (ε)+if (ε)f (ε)GA (ε) Γ f (ε)f (ε)GR (ε)GA (ε) δ
"− δ α λ,α λ δ λ,α − γ δ λ,γ γ,α # kδ,kβ
γ=1
X
δ(ε ǫ )δ(ε ǫ )f (ε)f (ε)δ δ , (12)
− − kα − kδ δ α kα,kλ kδ,kβ
Xkα Xkδ
where we considered all spectral densities equal Γ = Γ Note that GR (ε)= GR (x ,x ,ε).The retarded
α α,β λ,λ′ λ,λ′ α β
α. f (ε)isdeFermidistributionfunctionoftheleadα. GreenfunctionsareevaluatedbysolvingtheDysonequa-
α
∀ P
tion,
4
GRλ,λ′(x,x′,ε)=gλR′(x,x′,ε)δ(x−x′)δλ,λ′ + GRλ,λ′′(x,xγ,ε)ΣRγ(ε)gλR′(xγ,x′,ε), (13)
γ=1 λ′′
XX
being of reservoir α.
1 k0 e−ik(x−x′)
gλR′(x,x′,ε)= ε εk,λ′(Φ)+iη, (14)
Mk=X−k0 − B. Shot Noise Calculation
wherek istheenergycutoffand =2k +1anormal-
0 0
M
ization factor, the uncoupled retarded Green function of Here we show the shot noise expression of ec.(8) in
the ring and ΣR(ε) = w 2gR (ε) the self energy terms of retarded Green functions.15
γ kα| kα| kα
P
1
S (0)= dτ[ J (τ),J (0) 2 J (τ) J (0) ] (15)
α,β α β α β
2 h{ }i− h ih i
Z
e2 dε 4
= Γ Γ Γ Γ [(1 f (ε))f (ε)+(1 f (ε))f (ε)]GR (ε)GR∗(ε)GR (ε)GR∗(ε)
2~ 2π α β δ γ − γ δ − δ γ β,γ α,γ α,δ β,δ
Z δ,γ=1
X (cid:2) (cid:3)
2Γ Γ [(1f (ε))f (ε)+(1 f (ε))f (ε)] e GR (ε)GR (ε)
− α β − β α − α β R { α,β β,α }
4
2Γ Γ e iGR (ε) GR (ε)GR∗(ε)Γ [(1 f (ε))f (ε)+(1 f (ε))f (ε)]
− α βR { β,α α,δ β,δ δ − α δ − δ α }
δ=1
X
4
2Γ Γ e iGR (ε) GR (ε)GR∗(ε)Γ [(1 f (ε))f (ε)+(1 f (ε))f (ε)]
− α βR { α,β β,δ α,δ δ − β δ − δ β }
δ=1
X
4
+2δ Γ Γ GR (ε)2[(1 f (ε))f (ε)+(1 f (ε))f (ε)] .
α,β α δ| α,δ | − δ α − α δ
δ=1
X

7
1 R. Horodecki et al,Rev. Mod. Phys. 81, 865-942 (2009). D. Mahalu, and V. Umansky, Nature 448, 333 (2007); V.
2 R. Hanbury Brown and R. Q. Twiss, Nature 178 , 1046 Giovannetti,D.Frustaglia,F.Taddei,andR.Fazio,Phys.
(1956); C. K.Hong, Z. Y.Ou, and L. Mandel, Phys. Rev. Rev.B 74, 115315 (2006).
Lett. 59, 2044 (1987). 8 I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu, V.
3 D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 Umansky,Nature448, 333 (2007).
(1998); R. Hanson, L .P. Kouwenhoven, J. R. Petta, S. 9 P.Samuelsson, E.V.Sukhorukov,M.Bu¨ttiker, Phys.Rev.
Tarucha,andL.M.K.Vandersypen,Rev.Mod.Phys.79, Lett. 92, 026805 (2004).
1217 (2007). 10 P. Samuelsson, M. Bu¨ttiker, Phys. Rev. B 73, 041305
4 J.Wei,andV.Chandrasekhar,NaturePhys.6,494(2010). (2006).
5 C.W.J.Beenakker,C.Emary,M.Kindermann,andJ.L. 11 I. Klich and L. Levitov, Phys. Rev. Lett. 102, 100502
van Velsen, Phys.Rev. Lett.91, 147901 (2003). (2009); B. Hsu, E. Grosfeld and E. Fradkin Phys. Rev.
6 C.W.J.Beenakker,in ”Quantum Computers, Algorithms B 80, 235412 (2009).
and Chaos”, Proceedings of the International School of 12 V. A. Gopar, D. Frustaglia, Phys. Rev. B 77, 153403
PhysicsE.Fermi,Varenna,2005,(IOSPress,Amsterdam, (2008).
2006); C. W. J. Beenakker, M. Kindermann, C. M. Mar- 13 D Frustaglia and A.Cabello, Phys.Rev.B 80,201312(R)
cus, and A. Yacoby in ”Fundamental Problems of Meso- (2009).
scopic Physics”, edited by I. V. Lerner, B. L. Altshuler 14 W. K. Wooters, Phys. Rev.Lett. 80, 2245 (1998).
and Y. Gefen, NATO Science Series II, vol 154 (Kluwer, 15 H. J. W. Haug and A-P. Jauho, ”Quantum kinetics in
Dordrecht, 2004). transport and optics of semiconductors”, Springer (2008).
7 M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. En- 16 D.S.FisherandP.A.Lee, Phys.Rev.B23,6851 (1981).
sslin,M.Holland,andC.Sch¨onenberger;Science284,296 17 H. Schomerus, and J. P. Robinson, New J. Phys 9, 67
(1999);W.D.Oliver,J.Kim,R.C.Liu,andY.Yamamoto, (2007).
ibid. 284, 299 (1999); H. Kiesel, A. Renz, and F. Hassel- 18 G. F`eve, A. Mahe, J.-M. Berroir, T. Kontos, B. Placais,
bach, Nature 418, 392 (2002); Y. Ji, Y. Chung, D. Sprin- DC Glattli, A. Cavanna, B. Etienne, Y. Jin, Science 316,
zak, M. Heiblum, D. Mahalu, and H. Shtrikman, Nature 1169 (2007); J. Splettstoesser, M. Moskalets, M. Bu¨ttiker
422,415(2003);I.Neder,N.Ofek,Y.Chung,M.Heiblum, Phys. Rev.Lett. 103, 076804 (2009).