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

Proposal for production and detection of entangled electron-hole

pairs in a degenerate electron gas

Beenakker, C.W.J.; Emary, C.; Kindermann, M.; Velsen, J.L. van

Citation

Beenakker, C. W. J., Emary, C., Kindermann, M., & Velsen, J. L. van. (2003). Proposal for

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

Retrieved from https://hdl.handle.net/1887/1285

Version:

Not Applicable (or Unknown)

License:

Leiden University Non-exclusive license

Downloaded from:

https://hdl.handle.net/1887/1285

VOLUME 91, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 3 OCTOBER 2003week ending

Proposal for Production and Detection of Entangled Electron-Hole Pairs

in a Degenerate Electron Gas

C W J Beenakkei, C Emaiy, M Kmdeimann, and J L van Velsen

Instituut-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-channel conductoi violates a Bell mequahty The nonlocahty is shown to oiigmate from entangled electron-hole paus cieated by tunnelmg events—without lequinng election election mteiactions The degiee of entanglement (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 reahzation

DOI 101103/PhysRevLett91 147901 PACS numbeis 03 67 Mn, 03 65 Ud 73 43 Qt 7350Td The contiolled pioduction and detection of entangled

paiticles is the fiist step on the load towaids quantum mfoimation piocessing [1] In optics this step was taken long 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 diffeient physical mechanisms [3] A common staitmg pomt is a spin-singlet election pan pioduced by mteiactions, such äs the Coulomb mteiaction in a quantum dot [4-6], the paning mteiaction in a supeiconductoi [7-10], 01 Kondo scatteimg by a magnetic impunty [11] A veiy lecent pioposal based on oibital entanglement also makes use of the supeiconducting paning mteiaction [12]

It is known that photons can be entangled by means of hneai optics usmg a beam sphttei [13-15] Theelectionic analog would be an entanglei that is based entnely on single-election physics, without lequuing mteiactions But a dnect analogy with optics falls Election leseivons aie m local theimal equihbiium, while in optics a beam sphttei is incapable of entangling photons fiom a theimal souice [16] That is why pievious pioposals [11,17] to en-tangle elections by means of a beam sphttei stait fiom a two-election Fock state, mthei than a many-election theimal state To contiol the extiacüon of a single paii of elections 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, it has been aigued [18] that one cannot entangle a spatially sepaiated cunent of elections fiom a noimal (not-supeiconductmg) souice without lecouise to mteiactions What we piopose heie is an altogethei diffeient, inteiaction-fiee souice öl entangled quasipaiticles in the solid state The entanglement is notbetween election paus but between election-hole paus in a degeneiate election gas The entanglement and spatial sepaiatton aie leahzed puiely by elastic scatteimg at a tunnel baniei in a two-channel conductoi We quantify the degiee of entangle-ment by calculating how much the cunent fluctuations violate a Bell mequahty

Any two-channel conductoi contammg a tunnel bai-nei could be used in pnnciple foi oui puipose, and the

analysis which follows apphes geneially The paiticulai Implementation descnbed m Fig l uses edge channel tianspoit in the integei quantum Hall effect [19] It has the advantage that the individual building blocks have aheady been leahzed expenmentally foi diffeient pui-poses If the two edge channels he in the same Landau level, then the entanglement is between the spin degiees

FIG l Schemalic descnplion oi the method to pioduce and detect entangled edge channels in the quantum Hall effect The thick black lines indicate the boundaiies of a two dimensional election gas A stiong pcipendiculai magnetic field B ensuics that the lianspoit neai the Feimi level Er takes place in t w o

cdgc channels, extended along a pan of equipotenlials (Ihm solid and dashed Imes, with anows that give the duection of piopagation) A spht gate electiode (shaded leclangles at the centei) divides the conducloi into two halves, coupled by tunnelmg thiough a nanow opening (dashed anow, scatteimg matiix S) If a voltage V is apphed between the two halves, thcn theie is a naiiow eneigy lange eV above E/ m which the edge channels 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 (black dots) and cmpty states in the left halJ (open cnclcs) The enlanglcmenl öl these paiticle hole excitations is dctected by the violation öl a Bell inequahty This lequnes two galc elec liodcs to locally m i x the edge channels (scatlenng mainces

UL UR) and two paus öl contacts l 2 to sepaiately measuie the

cunenl in each tiansmilled and leflected edge channel

VOLUME 91, NUMBER 14

P H Y S I C A L R E V I E W L E T T E R S 3 OCTOBER 2003week endmg

of fieedom Altematively, if the spm degeneiacy is not lesolved by the Zeeman eneigy and the two edge channels he in diffeient Landau levels, then the entanglement is between the oibital degiees of fieedom The beam splittei isfoimedby asphtgateelectrode, asm Ref [20] In Fig l we show the case that the beam splittei is weakly tians-mittmg and stiongly leflectmg, but it could also be the othei way aiound To analyze the Bell inequahty an extia pan of gates mixes the oibital degiees of fieedom of the outgomg states independently of the mcoming states (Altematively, one could apply a local mhomogeneity in the magnetic field to mix the spm degiees of fieedom) Fmally, the cunent in each edge channel can be measuied sepaiately by usmg theii spatial sepaiation, äs m Ref [21] (Altematively, one could use the fenomagnetic method to measuie spm cunent äs desciibed m Refs [3,22])

It is easiest to undeistand what happens if the beam splittei does not mix the edge channels An election can tunnel fiom eithei Landau level mto the empty nght half of the System, leaving behmd a hole in the filled left half with the same Landau level mdex This conelation en-tangles the election hole pan Let us assume, foi the simplest example, that each edge channel tunnels with the same piobabihty T The lesultmg state is a supei-position of the vacuum state |0) (all states filled at the left and empty at the nght) with weight Vl — T and the maximally entangled Bell pan (|||) + |Ι|))/Λ/2 with

weight VT The iole of the spm mdices |, | is played by the Landau level mdices ι = 1,2 The fiist mdex m the ket 11|) lefeis to the hole at the left and the second mdex to the election at the nght We now geneiahze this elementaiy example to an aibitiaiy scatteimg matiix, mcludmg channel mixmg and unequal tiansmission piobabihties

Elections aie incident on the beam splittei fiom the left in a lange eV above the Feimi eneigy EF (The states

below Er aie all occupied at low tempeiatuies, so they do

not contnbute to tianspoit piopeities) The incident state has the foi m

| Ψ , η > = Π α, η ΐ (ε)α. π2(ε) Ι ° ) 0 )

0<e<tV

The feimion cieation opeiatoi α,,1((ε) excites the ;th

channel incident fiom the left at eneigy ε above the Fei m i level Similaily, b^ni(s) excites a channel incident

fiom the nght Each excitation is noimahzed such that it canies unit cunent It is convement to collect the cieation opeiatois in two vectois am,bin and to use a matnx

notation,

Ό Oj\b\ (2)

with σ} a Pauh matnx

The mput Output lelation of the beam sphltei is

t'\(am

The 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 nght Substitution of Eq (3) mto Eq (2) gives the outgomg state

u t 2

(4) The supeisciipt "T" mdicates the tianspose of a matiix

To identify the entangled election-hole excitations we tiansfoim fiom paiticle to hole opeiatois at the left of the beam splittei couu = alotl The new vacuum state is aom ialu\.2\fy To leadmg oidei in the tiansmission matiix

the outgomg state becomes

(5) |φ) = w~]'2clulyboat\Q), γ = σ ι σ tr (6)

It lepiesents a supeiposition of the vacuum state and a paiticle-hole state Φ with weight w = Ti yy^

The degiee of entanglement of Φ is quantified by the concunence [23,24],

' (7) which langes fiom 0 (no entanglement) to l (maximal entanglement) Substituting Eq (6) and usmg the unitai-ity of the scattei mg matiix we find aftei some algebia that

r

_2V(i-r

1

)(i-r

2

)r

]

r

2

,

T7 - 2T,

i f T}, T2 « l

(8)

t r' (3)

The concunence is entnely deteimmed by the eigenval-ues T\, TI €Ξ (0, 1) of the tiansmission matnx pioduct

fit = l - >^ι The eigenvectois do not contnbute This

means, in paiticulai, that channel mixmg does not de giade the entanglement äs long äs the tiansmission

eigen-values lemam unaffected Maximal entanglement is achieved if the two tiansmission eigenvalues aie equal

C = l if T, = T2

The paiticle-hole entanglement is a nonlocal coiiela-tion that can be detected thiough the violacoiiela-tion of a Bell inequahty [25,26] We follow the loimulation m teims of nieducible cui lent conelatois in the tiequency domam of Samuelsson, Sukhoiukov, and Buttikei [12], which in the tunnehng hmit T\ T2 « l is equivalent to a moie geneial

foimulation in the time domam [18] We will demonstiate exphcitly latei on that we need the tunnehng assumption

The quantity C,, = /ÜM dt8IL,(t)8IR /O) conelates

the time-dependent cunent fluctuations <§// ( m chan

nel i = l 2 at the left with the cunent fluctuations 8IRl

in channel j = l 2 at the nght It can be measuied di-lectly in the iiequency domam äs the covaiiance of the

VOLUME 9l, NUMBER 14

P H Y S I C A L R E V I E W L E T T E R S 3 OCTOBER 2003week endmg low-fiequency component of the cunent fluctuations At

low 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 ~ «-21 E =

C, '2l Tir1";/1"/

(10)

By mixmg the channels locally in the left and nght aim of the beam sphttei, the tiansmission and leflection ma-tnces aie tiansfoimed äs r — > ULi,t—>· URt, with unitaiy

2 X 2 matnces UL, UR The conelatoi tiansfoims äs

The 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 unitaiy matuces UL, UR, U'L, U'R By lepeating the calculation of

Rel [29] we find the maximum [30]

'-'max ^ l ~r

1(1 ~τ

Ί

)τ^τ

2

^η

-τ

2

-r

2

)

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 the maximal violation of the CHSH mequality [31] As men-tioned above, we need the tunnelmg hmit If T\ and 72 aie not <SC l theie is no one-to-one lelation between £max

m Eq (13) and C in Eq (8)

As a final consistency check we considei the effect of dephasing [32] Dephasmg is modeled by intioducing landom phase factois m each edge channel, which amounts to the substitutions

U

» 0

L 0 U

R

0 e" (14)

We aveiage E(UL, UR) ovei theiandomphases, unifoimly m (Ο 2ττ·), and find

_

2|Tio-(15)

So foi stiong dephasing theie is no violation of the Bell mequality |£| < 2 The inteimediate legime between weak and stiong dephasing is moie complex Theie exists a lange oi dephasing stiengths foi which E < 2 but the election-hole state is still entangled [33] All of this is äs expected foi entanglement of a mixed state [26]

In conclusion, we have demonstrated theoietically that a tunnel bamei cieates spatially sepaiated cuiients of entangled election-hole paus m a degeneiate election gas Because no Coulomb 01 paning mteiaction is m-volved, this is an attiactive alternative to existmg pio-posals foi the mteiaction-mediated pioduction of entanglement in the solid state We have descnbed a possible leahzation usmg edge channel tianspoit in the quantum Hall effect Theie is a lemaikable contiast with quantum optics, wheie a beam sphttei cannot cieate en-tanglement if the souice is in local theimal equihbnum This might well explam why the elementaty mechamsm foi entanglement pioduction descnbed here was not no-ticed befoie

We have benefited fiom coriespondence with P Samuelsson This woik was suppoited by the Dutch Science Foundation NWO/FOM and by the U S Aimy Reseaich Office (Giant No DAAD 19-02-0086)

[1] B M Teihal, M M Wolf, and A C Doheily, Phys Today 56, 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, and N Chtchelkatchev, in Quantum Noise in Mesoscopic

PhyMcs, edited by Yu V Nazaiov, NATO Science Seues

II.Vol 97 (Kluwei, Doidiecht, 2003), p 313, J C Egues, P Rechei, D S Saiaga, V N Golovach, G Buikaid, E V Sukhoiukov, and D Loss, ibid, p 241

[4] G Buikaid, D Loss, and E V Sukhoiukov, Phys Rev B 61, R16303 (2000), D Loss and E V Sukhoiukov, Phys Rev Lett 84, 1035 (2000)

[5] W D Ohvei, F Yamaguchi, and Υ Yamamoto, Phys Rev Lett 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 B 24 287 (2001)

[8] P Rechei, E V Sukhoiukov, and D Loss, Phys Rev B 63, 165314 (2001), P Rechei and D Loss, Phys Rev B 65, 165327 (2002)

[9] C Bena, S Vishveshwaia L Balents, and M P A Fishei, Phys Rev Lett 89, 037901 (2002)

[10] V Bouchiat, N Chlchelkatchev, D Fembeig, G B Lesovik, T Maitin, and J Tones, Nanotechnology 14, 77 (2003)

[11] A T Costa, Ji and S Böse, Phys Rev Leu 87, 277901 (2001)

[12] P Samuelsson, E V Sukhoiukov, and M Bultikei cond-mat/0303531

[13] E Knill, R Laflamme and G J Milbum, Natuie (London) 409, 46 (2001)

[14] S Scheel and D-G Welsch Phys Rev A 64, 063811 (2001)

[15] M S Kim, W Son, V Buzek and PL Knight, Phys Rev A 65 032323 (2002)

[16] W Xiang-bm, Phys Rev A 66, 024303 (2002)

VOLUME 9l, NUMBER 14

P H Y S I C A L R E V I E W L E T T E R S 3 OCTOBER 2003week ending [17] S Böse and D Home, Phys Rev Lett 88, 050401 (2002)

[18] N M Chtchelkatchev, G Blattei, G B Lesovik, and T Maitm, Phys Rev B 66, 161320(R) (2002)

[19] C W J Beenakkei and H van Houten, Solid State Phys 44, l (1991)

[20] M Henny, S Obeiholzei, C Strunk, T Hemzel, K Ensshn, M Holland, and C Schonenbeiger, Science 284, 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, when the füll state Ψοϋ( in Eq (4) can be leduced to the two

qubil state Φ supciimposed on the vacuum We will show that m this hmit C may be dnectly measuied by a cunent cornelatoi For aibitiaiy liansmission, the degiee of entanglement οί Ψ0111 can be quantificd by the

entangle-ment of foimation J·' (measuied m bits pei second) We find J = -(eV/h)\Ti log?", (I - Γ,) + Γ21οεΓ,(1 - Γ,) +

(l-Tl-T2)\og(l-Ti)(l-T2)] Although the füll state is entangled foi aibitraiy tiansmission, we know how to measuie this entanglement only in the tunneling limit

[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 and sufficient condition toi entanglement of a puie stale, such äs Ψ011, Foi a mixed slate (such äs resulting fiom

dephasmg) the condition is sufficient but not necessaiy, cf R F Wernei, Phys Rev A 40, 4277 (1989), R A Beitlmann, H Namhofei, and W Thirnng, Phys Rev 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 £C H =

(-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

(u

L

,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 unitaiy ma tuces 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 G Silvestiov) is the finite eneigy lange eV of the entangled edge channels Refeinng to Fig l, considei the area A between the two equipotentials staiting at UL, thiough S,

and ending at UR This enclosed aiea vaiies by 3A when

the eneigy of the equipotentials vanes by eV Dephasmg lesults if ΒδΑ Ζ h/e The latio δΑ/Α ^ V\ME\/\E\2

depends on the giadient of the electnc field E neai the edge Foi B = 5 T, A = 10 n m2, one would need V S

10~2[£Ί2/|ν£Ί to avoid dephasmg by eneigy aveiaging

[33] J L van Velsen, M Kindermann, and C W J Beenakkei, cond-mal/0307198