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. 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