Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Beenakker, C.W.J.

Emary, C.

Kindermann, M.

Velsen, J.L. van

Leiden University Scholary Publications

VOLUME 91, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R Sweek ending3 OCTOBER 2003Proposal for Production and Detection of Entangled Electron-Hole Pairsin a Degenerate Electron GasC W J Beenakkei, C Emaiy, M Kmdeimann, and J L van VelsenInstituut-Loientz, Umvetsiteit Leiden PO Box 9506, 2300 RA Leiden The Netheilands(Received 6 May 2003, pubhshed l Octobei 2003)We demonstiate theoieUcally lhat the shot noise pioduced by a tunnel bainei in a two-channelconductoi violates a Bell mequahty The nonlocahty is shown to oiigmate from entangled electron-holepaus cieated by tunnelmg events—without lequinng election election mteiactions The degiee ofentanglement (concunence) equals 2(7Ί7Λ)'/2(7Ί + 7Λ)~', with T\, 7Λ <3C l the tiansmission eigenval-ues A pau of edge channels in the quantum Hall effect is pioposed äs an expeiimental reahzationDOI 101103/PhysRevLett91 147901 PACS numbeis 03 67 Mn, 03 65 Ud 73 43 Qt 7350TdThe contiolled pioduction and detection of entangledpaiticles is the fiist step on the load towaids quantummfoimation piocessing [1] In optics this step was takenlong ago [2], but in the solid state it lemains an expeii-mental challenge A vai lety of methods to entangle elec-tions have been pioposed, based on quite diffeientphysical mechanisms [3] A common staitmg pomt is aspin-singlet election pan pioduced by mteiactions, suchäs the Coulomb mteiaction in a quantum dot [4-6], thepaning mteiaction in a supeiconductoi [7-10], 01 Kondoscatteimg by a magnetic impunty [11] A veiy lecentpioposal based on oibital entanglement also makes useof the supeiconducting paning mteiaction [12]It is known that photons can be entangled by means ofhneai optics usmg a beam sphttei [13-15] Theelectionicanalog would be an entanglei that is based entnely onsingle-election physics, without lequuing mteiactionsBut a dnect analogy with optics falls Election leseivonsaie m local theimal equihbiium, while in optics a beamsphttei is incapable of entangling photons fiom a theimalsouice [16] That is why pievious pioposals [11,17] to en-tangle elections by means of a beam sphttei stait fiom atwo-election Fock state, mthei than a many-electiontheimal state To contiol the extiacüon of a single paii ofelections fiom an election leseivon lequnes stiong Cou-lomb mteiaction in a tightly confined aiea, such äs a semi-conductoi quantum dot 01 caibon nanotube [3] Indeed, ithas been aigued [18] that one cannot entangle a spatiallysepaiated cunent of elections fiom a noimal (not-supeiconductmg) souice without lecouise to mteiactionsWhat we piopose heie is an altogethei diffeient,inteiaction-fiee souice öl entangled quasipaiticles in thesolid state The entanglement is notbetween election pausbut between election-hole paus in a degeneiate electiongas The entanglement and spatial sepaiatton aie leahzedpuiely by elastic scatteimg at a tunnel baniei in a two-channel conductoi We quant i fy the degiee of entangle-ment by calculating how much the cunent fluctuationsviolate a Bell mequahtyAny two-channel conductoi contammg a tunnel bai-nei could be used in pnnciple foi oui puipose, and theanalysis which follows apphes geneially The paiticulaiImplementation descnbed m Fig l uses edge channeltianspoit in the integei quantum Hall effect [19] It hasthe advantage that the individual building blocks haveaheady been leahzed expenmentally foi diffeient pui-poses If the two edge channels he in the same Landaulevel, then the entanglement is between the spin degieesFIG l Schemalic descnplion oi the method to pioduce anddetect entangled edge channels in the quantum Hall effect Thethick black lines indicate the boundaiies of a two dimensionalelection gas A stiong pcipendiculai magnetic field B ensuicsthat the lianspoit neai the Feimi level Er takes place in twocdgc channels, extended along a pan of equipotenlials (Ihmsolid and dashed Imes, with anows that give the duection ofpiopagation) A spht gate electiode (shaded leclangles at thecentei) divides the conducloi into two halves, coupled bytunnelmg thiough a nanow opening (dashed anow, scatteimgmatiix S) If a voltage V is apphed between the two halves, thcntheie is a naiiow eneigy lange eV above E/ m which the edgechannels aie piedommanlly filled m the Icft half (solid lines)and picdommantly empty m the iight half (dashed lines)Tunnelmg events mtioduce hlled states m the iight half (blackdots) and cmpty states in the left halJ (open cnclcs) Theenlanglcmenl öl these pait icle hole excitations is dctected bythe violation öl a Bell inequahty This lequnes two galc elecliodcs to locally mix the edge channels (scatlenng maincesUL UR) and two paus öl contacts l 2 to sepaiately measuie thecunenl in each tiansmilled and leflected edge channel147901-1 0031-9007/03/91(14)/147901(4)$20 00 © 2003 The Amencan Physical Society 147901-1VOLUME 91, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R Sweek endmg3 OCTOBER 2003of fieedom Altematively, if the spm degeneiacy is notlesolved by the Zeeman eneigy and the two edge channelshe in diffeient Landau levels, then the entanglement isbetween the oibital degiees of fieedom The beam splitteiisfoimedby asphtgateelectrode, asm Ref [20] In Fig lwe show the case that the beam splittei is weakly tians-mittmg and stiongly leflectmg, but it could also be theothei way aiound To analyze the Bell inequahty an extiapan of gates mixes the oibital degiees of fieedom of theoutgomg states independently of the mcoming states(Altematively, one could apply a local mhomogeneityin the magnetic field to mix the spm degiees of fieedom)Fmally, the cunent in each edge channel can be measuiedsepaiately by usmg theii spatial sepaiation, äs m Ref [21](Altematively, one could use the fenomagnetic method tomeasuie spm cunent äs desciibed m Refs [3,22])It is easiest to undeistand what happens if the beamsplittei does not mix the edge channels An election cantunnel fiom eithei Landau level mto the empty nght halfof the System, leaving behmd a hole in the filled left halfwith the same Landau level mdex This conelation en-tangles the election hole pan Let us assume, foi thesimplest example, that each edge channel tunnels withthe same piobabihty T The lesultmg state is a supei-position of the vacuum state |0) (all states filled at the leftand empty at the nght) with weight Vl — T and themaximally entangled Bell pan (|||) + |Ι|))/Λ/2 withweight VT The iole of the spm mdices |, | is played bythe Landau level mdices ι = 1,2 The fiist mdex m the ket11|) lefeis to the hole at the left and the second mdex to theelection at the nght We now geneiahze this elementaiyexample to an a ib i t ia iy scatteimg matiix, mcludmgchannel mixmg and unequal tiansmission piobabihtiesElections aie incident on the beam splittei fiom the leftin a lange eV above the Feimi eneigy EF (The statesbelow Er aie all occupied at low tempeiatuies, so they donot contnbute to tianspoit piopeities) The incident statehas the foi m| Ψ , η > = Π α ,ηΐ( ε ) α .π 2 (ε )Ι ° ) 0)0<e<tVThe feimion cieation opeiatoi α,,1((ε) excites the ;thchannel incident f iom the left at eneigy ε above theFei m i level S imi la i ly , b^ni(s) excites a channel incidentfiom the nght Each excitation is noimahzed such that itcanies uni t cunent It is convement to collect the cieationopeiatois in two vectois am,bin and to use a matnxnotation,Ό Oj\b\(2)with σ} a Pauh matnxThe mput Output lelation of the beam sphltei ist'\(amThe 4 X 4 umtaiy scatteimg matnx S has 2 X 2 sub-matiices r, r1, t, t1 that desciibe leflection and tiansmis-sion of states incident fiom the left 01 fiom the nghtSubstitution of Eq (3) mto Eq (2) gives the outgomg stateu t 2(4)The supeisciipt "T" mdicates the tianspose of a matiixTo identify the entangled election-hole excitations wetiansfoim fiom paiticle to hole opeiatois at the left of thebeam splittei couu = alotl The new vacuum state isaom ialu\.2\fy To leadmg oidei in the tiansmission matiixthe outgomg state becomes(5)|φ) = w~]'2clulyboat\Q), γ = σ ι σ tr (6)It lepiesents a supeiposition of the vacuum state and apaiticle-hole state Φ with weight w = Ti yy^The degiee of entanglement of Φ is quantified by theconcunence [23,24],' (7)which langes fiom 0 (no entanglement) to l (maximalentanglement) Substituting Eq (6) and usmg the unitai-ity of the scattei mg matiix we find aftei some algebia thatr _2V(i-r 1 )( i-r 2 )r ] r 2 ,T7 - 2T,i f T } , T 2 « l(8)t r'(3)The concunence is entnely deteimmed by the eigenval-ues T\, TI €Ξ (0, 1) of the tiansmission matnx pioductfit = l - >^ι The eigenvectois do not contnbute Thismeans, in paiticulai, that channel mixmg does not degiade the entanglement äs long äs the tiansmission eigen-values lemam unaffected Maximal entanglement isachieved if the two tiansmission eigenvalues aie equalC = l if T, = T2The paiticle-hole entanglement is a nonlocal coiiela-tion that can be detected thiough the violation of a Bellinequahty [25,26] We follow the loimulation m teims ofnieducible cui lent conelatois in the tiequency domam ofSamuelsson, Sukhoiukov, and Buttikei [12], which in thetunnehng hmit T\ T2 « l is equivalent to a moie geneialfoimulation in the time domam [18] We will demonstiateexphcitly latei on that we need the tunnehng assumptionThe quanti ty C,, = /ÜM dt8IL,(t)8IR /O) conelatesthe time-dependent cunent fluctuations <§// ( m channel i = l 2 at the left with the cunent fluctuations 8IRlin channel j = l 2 at the nght It can be measuied di-lectly in the iiequency domam äs the covaiiance of the147901 2 147901-2VOLUME 9l, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R Sweek endmg3 OCTOBER 2003low-fiequency component of the cunent fluctuations Atlow tempeiatuies (kT <äC eV) the conelatoi has the gen-eial expiession [27](9)We need the followmg lational function of conelatois"̂* α- Λ* r* r^+ <-22 ~~ «-12 ~ «-21E =C, '2l Tir1";/ 1"/(10)By mixmg the channels locally in the left and nght aimof the beam sphttei, the tiansmission and leflection ma-tnces aie tiansfoimed äs r — > ULi,t—>· URt, with unitaiy2 X 2 matnces UL, UR The conelatoi tiansfoims äsThe Bell-CHSH (Clausei-Home-Shimony-Holt) paiame-tei is [25,28]8 = E(UL, U R) + E(U'L, UR) + E(UL, U'R} ~ E(U'L, U'R)(12)The state is entangled if |£| > 2 foi some set of unitaiymatuces UL, UR, U'L, U'R By lepeating the calculation ofRel [29] we find the maximum [30]'-'max ^ l ~r1(1 ~τΊ)τ^τ2^η-τ2-r2)2 J,U l τ i 2 J l^2[\+4Τ]Τ2(Τ]+Τ2Γ2]ι/2 (13)Companson with Eq (8) confiims the expected lelation£max = 2(1 + C2)1/2 between the concunence and themaximal violation of the CHSH mequality [31] As men-tioned above, we need the tunnelmg hmit If T\ and 72aie not <SC l theie is no one-to-one lelation between £maxm Eq (13) and C in Eq (8)As a final consistency check we considei the effect ofdephasing [32] Dephasmg is modeled by intioducinglandom phase factois m each edge channel, whichamounts to the substitutionsU» 0L 0UR 0 e"(14)We aveiage E(UL, UR) ovei theiandomphases, uni fo imlym (Ο 2ττ·), and find_ 2|Tio-(15)So foi stiong dephasing theie is no violation of the Bellmequality |£| < 2 The inteimediate legime betweenweak and stiong dephasing is moie complex Theie existsa lange oi dephasing stiengths foi which E < 2 but theelection-hole state is st i l l entangled [33] All of this is äsexpected foi entanglement of a mixed state [26]In conclusion, we have demonstrated theoietically thata tunnel bamei cieates spatially sepaiated cuiients ofentangled election-hole paus m a degeneiate electiongas Because no Coulomb 01 paning mteiaction is m-volved, this is an attiactive alternative to existmg pio-posals foi the mteiaction-mediated pioduction ofentanglement in the solid state We have descnbed apossible leahzation usmg edge channel tianspoit in thequantum Hall effect Theie is a lemaikable contiast withquantum optics, wheie a beam sphttei cannot cieate en-tanglement if the souice is in local theimal equihbnumThis might well explam why the elementaty mechamsmfoi entanglement pioduction descnbed here was not no-ticed befoieWe have benefited fiom coriespondence withP Samuelsson This woik was suppoited by the DutchScience Foundation NWO/FOM and by the US AimyReseaich Office (Giant No DAAD 19-02-0086)[1] B M Teihal, M M Wolf, and A C Doheily, Phys Today56, No 4, 46 (2003)[2] A Aspect, P Giangiei, and G Rogci, Phys Rev Lett 47,460 (1981)[3] Two lecent levicws aie T Maitin, A Ciepieux, andN Chtchelkatchev, in Quantum Noise in MesoscopicPhyMcs, edited by Yu V Nazaiov, NATO Science SeuesII.Vol 97 (Kluwei, Doidiecht, 2003), p 313, J C Egues,P Rechei, D S Saiaga, V N Golovach, G Buikaid, E VSukhoiukov, and D Loss, ibid, p 241[4] G Buikaid, D Loss, and E V Sukhoiukov, Phys Rev B61, R16303 (2000), D Loss and E V Sukhoiukov, PhysRev Lett 84, 1035 (2000)[5] W D Ohvei, F Yamaguchi, and Υ Yamamoto, Phys RevLett 88, 037901 (2002)[6] D S Saiaga and D Loss, Phys Rev Lett 90, 166803(2003)[7] G B Lesovik, T Maitin, and G Blattei, Ein Phys J B24 287 (2001)[8] P Rechei, E V Sukhoiukov, and D Loss, Phys Rev B63, 165314 (2001), P 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Schonenbeiger, Science284, 296 (1999)[21] B J van Wees, E M M Willems, C J P M Kaimans,C W J Beenakkei, H van Houten, J G Wilhamson,C T Foxon, and J J Hains, Phys Rev Lett 62, 1181(1989)[22] S Kawabata, J Phys Soc Jpn 70, 1210 (2001)[23] W K Wootteis, Phys Rev Lett 80, 2245 (1998)[24] The concui i ence C quanüfies the entanglement of a two-qubit state It is meanmgful m the tunneling hmit, whenthe füll state Ψοϋ( in Eq (4) can be leduced to the twoqubil state Φ supciimposed on the vacuum We wi l l showthat m this hmit C may be dnectly measuied by a cunentcornelatoi For a ib i t ia iy liansmission, the degiee ofentanglement οί Ψ0111 can be quantificd by the entangle-ment of foimation J·' (measuied m bits pei second) Wefind J = -(eV/h)\Ti log?", (I - Γ,) + Γ21οεΓ,(1 - Γ,) +(l-Tl-T2)\og(l-Ti)(l-T2)] Although the füllstate is entangled foi aibitraiy tiansmission, we knowhow to measuie this entanglement only in the tunnelinglimit[25] J S Bell, Physics (Long Island City N Y) l, 195 (1964),J F Clausei, M A Home, A Shimony, and R A Holt,Phys Rev Lett 23, 880 (1969)[26] Violation of the Bell inequality is a necessaiy andsufficient condition toi entanglement of a puie stale,such äs Ψ011, Foi a mixed slate (such äs resulting fiomdephasmg) the condition is sufficient but not necessaiy,cf R F Wernei, Phys Rev A 40, 4277 (1989), R ABeitlmann, H Namhofei, and W Thirnng, PhysRev A 66, 032319 (2002)[27] G B Lesovik, JETP Lett 49, 592 (1989), M Buttikei,Phys Rev Lett 65, 2901 (1990)[28] Instead of seaiching foi violations of the CHSH inequal-ity \£\ s 2, one could equivalently considei the CH(Clausei-Home) inequality <?CH := 0, with £CH=(-e3V/hr]{C„(UL. UR) + C,j(U'L, UR) + C,j(UL, U'R) ~c^u'^u'g) - c„a/L i) - c,2(uL,i) - CU^.UK)-C2j(l, UR)} Substitutmg C,j(U, V) = (-e^V/h) X\(Uit^V~t),j\2 one obtains the relation £CH =j(£ - 2)Tir;ti?t belween the CH and CHSHpaiameteis[29] S Popescu and D Rohihch, Phys Lett. A 166, 293(1992)[30] The maximum (13) is attamed foi UR = X, U'R =2 '/2(1 + ι σ , ) Χ , UL = (l cos« + ισ, sma)Y, U'L =(l cosa — ;cr, sma)Y, with tan2a = C The uni ta iy matuces Χ, Υ aie choscn such lhat Yit^X^ is ical diagonal[31] N Gism, Phys Lett A 154, 201 (1991)[32] One souice of dephasmg (pointed out to us by P GSilvestiov) is the finite eneigy lange eV of the entanglededge channels Refeinng to Fig l, considei the area Abetween the two equipotentials stait ing at UL, thiough S,and ending at UR This enclosed aiea vaiies by 3A whenthe eneigy of the equipotentials vanes by eV Dephasmglesults if ΒδΑ Ζ h/e The latio δΑ/Α ^ V\ME\/\E\2depends on the giadient of the electnc field E neai theedge Foi B = 5 T, A = 10 n m2, one would need V S10~2[£Ί2/|ν£Ί to avoid dephasmg by eneigy aveiaging[33] J L van Velsen, M Kindermann, and C W J Beenakkei,cond-mal/0307198147901-4 147901-4

Proposal for production and detection of entangled electron-hole pairs in a degenerate electron gas

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Proposal for production and detection of entangled electron-hole pairs in a degenerate electron gas

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