arXiv:cond-mat/0605025v1 [cond-mat.mes-hall] 30 Apr 2006 htn.Mc ftepyia nomto tcontains on experiments it for from conductors. information recently, than nanoscale only physical slowly appreciated the been more of has vacuum progressed Much on elec- has 1920’s of photons. noise the understanding shot in mechanical tronic work quantum observations from our first date tubes, the noise Although are shot charge noise”. of electrical “shot the of as fluctu- discreteness Current known the role. to electrical diagnostic due similar the ations a in play fluctuations to expect current might one photons, material. to the textbook (due and is statistics Glauber) optics photon Roy quantum of measurements in theory Noise technique mechanical body. quantum routine black a a now for wave while are the small absent, laser merely entirely a is is For it fluctuations body: that the black a to discovered from contribution radiation was and the laser between it a difference from when the tell 1960’s, can fluctuations the in developed of power the is Such photon. needing individual actually noise. an without would the on observe — he Huygens light to with noise, of dispute measured nature his settle have corpuscular to frequen- could able been radio Newton have at If over takes cies. the contribution while (wave) frequencies, linear (particle optical con- square-root at square-root dominates the and contribution Typically, linear wave the classical coexist. clas- a particle, tributions for neither classical energy is a photon mean nor a the it Since of but particles. root waves, sical square classical the for or with energy waves scales mean by scales the fluctuations carried with energy is of linearly magnitude energy The the fluctu- particles. if by electromagnetic Einstein different that Albert are 1909) was ations (in It realized first duality. who particle-wave the with the in in present delight actually not may is noise. measurement physicist that a A information of value. of time-averaged time source in a fluctuations be What can that physics. 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These shot of current. characteristics the the to are proportional inde- linearly frequency and becomes pendent density spectral the frequencies ogtmr u fnieoehst rn h electrons the bring to has one noise of out more get To . hra os xed vralfeunisu othe to up frequencies all over extends noise Thermal hoeial,i sesett ecieeetia os i noise electrical describe to easiest is it Theoretically, tlwfeunis(yial eo 0kz h noise the kHz) 10 below (typically frequencies low At h em“htnie rw naaoybtenelec- between analogy an draws noise” “shot term The kT ∆ f V needn of independent , f hra os hnhsa lcrclpower electrical an has then noise Thermal . ye feetia noise electrical of Types sapidoe h odco,tenoise the conductor, the over applied is 0 6= S V 4 = S kT/h ,o hr-icie conductor. short-circuited a or 0, = ( f S TG kT = ) natpcleprmn n fil- one experiment typical a In . fteniei h mean-squared the is noise the of ∗ h R needn ffeuny If frequency. of independent , δI . f ( f “ht”nie.Oecan One noise). (“white” /f ) TR kT 2 i / os”bcueo the of because noise” ∆ ∆ f. f ftedissipated the of . f rudsome around I ¯ thigher At . V δI ( be- (1) f n y ) 2 would have two intrinsic sources of time-dependent cur- rent fluctuations: Noise from the thermal agitation of e e/3 electrons (thermal noise) and noise from the discreteness of the electrical charge (shot noise). In a vacuum tube, electrons are emitted by the cath- 5.6 ode randomly and independently. Such a Poisson process has the property that the mean squared fluctuation of the –28 2 number of emission events is equal to the average count. The corresponding spectral density equals S =2eI¯. The 5.5 factor of 2 appears because positive and negative frequen- cies contribute identically. 3 noise 5.4

noise power (10 A / Hz) 2 1 Measuring the unit of transferred charge V 5.3 Schottky proposed to measure the value of the elemen- 0 100 200 300 400 tary charge from the shot noise power, perhaps more ac- current (pA) curately than in the oil drop measurements which Robert Millikan had published a few years earlier. Later experi- FIG. 1: Current noise for tunneling across a Hall bar on ments showed that the accuracy is not better than a few the 1/3 plateau of the fractional quantum Hall effect. The percent, mainly because the repulsion of electrons in the slopes for e/3 charge quasiparticles and charge e electrons space around the cathode invalidates the assumption of are indicated. The data points with error bars (from the independent emission events. experiment of Saminadayar et al.5) are the measured values It may happen that the granularity of the current is at 25 mK, the open circles include a correction for finite tunnel not the elementary charge. The mean current can not probability. The inset shows schematically the setup of the ¯ experiment. Most of the current flows along the lower edge of tell the difference, but the noise can: S = 2qI if charge the Hall bar from contact 1 to contact 2 (solid red line), but is transferred in independent units of q. The ratio F = some quasiparticles tunnel to the upper edge and end up at ¯ S/2eI, which measures the unit of transferred charge, is contact 3 (dashed). The tunneling occurs predominantly at called the “Fano factor”, after Ugo Fano’s 1947 theory of a narrow constriction, created in the Hall bar by means of a the statistics of ionization. split gate electrode (shown in green). The current at contact A first example of q 6= e is the shot noise at a tunnel 3 is first spectrally filtered, then amplified, and finally the mean squared fluctuation (the noise power) is measured. junction between a normal metal and a superconductor. Charge is added to the superconductor in Cooper pairs, so one expects q = 2e and F = 2. This doubling of the Quiet electrons Poisson noise has been measured very recently.2 (Earlier experiments3 in a disordered system will be discussed later on.) Correlations reduce the noise below the value

A second example is offered by the fractional quan- SPoisson =2eI¯ (2) tum Hall effect. It is a non-trivial implication of Robert Laughlin’s theory that tunneling from one edge of a Hall expected for a Poisson process of uncorrelated current bar to the opposite edge proceeds in units of a fraction pulses of charge q = e. Coulomb repulsion is one source q = e/(2p + 1) of the elementary charge.4 The integer of correlations, but it is strongly screened in a metal and p is determined by the filling fraction p/(2p +1) of the ineffective. The dominant source of correlations is the lowest Landau level. Christian Glattli and collaborators Pauli principle, which prevents double occupancy of an of the Centre d’Etudes´ de Saclay in France and Michael electronic state and leads to Fermi statistics in thermal Reznikov and collaborators of the Weizmann Institute equilibrium. In a vacuum tube or tunnel junction the in Israel independently measured F = 1/3 in the frac- mean occupation of a state is so small that the Pauli tional quantum Hall effect5 (see figure 1). More recently, principle is inoperative (and Fermi statistics is indistin- the Weizmann group extended the noise measurements guishable from Boltzmann statistics), but this is not so to p = 2 and p = 3. The experiments at p = 2 show that in a metal. the charge inferred from the noise may be a multiple of An efficient way of accounting for the correlations e/(2p + 1) at the lowest temperatures, as if the quasipar- uses Landauer’s description of electrical conduction as a ticles tunnel in bunches. How to explain this bunching is transmission problem. According to the Landauer for- still unknown. mula, the time-averaged current I¯ equals the conduc- 3 tance quantum 2e2/h (including a factor of two for spin), with a quantized conductance. The quantization occurs times the applied voltage V , times the sum over trans- because the transmission probabilities are either close to mission probabilities Tn: 0 or close to 1. Eq. (4) predicts that the shot noise should vanish when the conductance is quantized, and N 2e2 this was indeed observed. (The experiment was reviewed I¯ = V Tn. (3) h X by Henk van Houten and Beenakker in Physics Today, n=1 July 1996, page 22.) The conductor can be viewed as a parallel circuit of A more stringent test used a single-atom junction, ob- N independent transmission channels with a channel- tained by the controlled breaking of a thin aluminum wire.8 The junction is so narrow that the entire current dependent transmission probability Tn. Formally, the † is carried by only three channels (N = 3). The trans- Tn’s are defined as the eigenvalues of the product t · t of the N × N transmission matrix t and its Hermitian con- mission probabilities T1,T2,T3 could be measured inde- jugate. In a one-dimensional conductor, which by defini- pendently from the current–voltage characteristic in the 2 superconducting state of aluminum. By inserting these tion has one channel, one would have simply T1 = |t| , with t the transmission amplitude. three numbers (the “pin code” of the junction) into eq. The number of channels N is a large number in a typ- (4), a theoretical prediction is obtained for the shot noise 2 power — which turned out to be in good agreement with ical metal wire. One has N ≃ A/λF up to a numeri- cal coefficient for a wire with cross-sectional area A and the measured value. Fermi wave length λF . Due to the small Fermi wave 7 length λF ≃ 1 A˚ of a metal, N is of order 10 for a typ- ical metal wire of width 1 µm and thickness 100 nm. In Detecting open transmission channels a semiconductor typical values of N are smaller but still ≫ 1. The analogy between an electron emitted by a cathode At zero temperature the noise is related to the trans- and a bullet shot by a gun works well for a vacuum tube mission probabilities by6 or a point contact, but seems a rather naive description of the electrical current in a disordered metal or semi- N 2e2 conductor. There is no identifiable emission event when S =2e V Tn(1 − Tn). (4) h X current flows through a metal and one might question n=1 the very existence of shot noise. Indeed, for three quar- The factor 1 − Tn describes the reduction of noise due to ters of a century after the first vacuum tube experiments the Pauli principle. Without it, one would have simply there did not exist a single measurement of shot noise in a metal. A macroscopic conductor (say, a piece of copper S = SPoisson. The shot noise formula (4) has an instructive statistical wire) shows thermal noise, but no shot noise. interpretation.7 Consider first a one-dimensional conduc- We now understand that the basic requirement on tor. Electrons in a range eV above the Fermi level enter length scale and temperature is that the length L of the the conductor at a rate eV/h. In a time τ the number wire should be short compared to the inelastic electron- of attempted transmissions is τeV/h. There are no fluc- phonon scattering length lin, which becomes longer and tuations in this number at zero temperature, since each longer as one lowers the temperature. For L>lin each occupied state contains exactly one electron (Pauli prin- segment of the wire of length lin generates independent ciple). Fluctuations in the transmitted charge Q arise voltage fluctuations, and the net result is that the shot because the transmission attempts are succesful with a noise power is reduced by a factor lin/L. Thermal fluc- probability T1 which is different from 0 or 1. The statis- tuations, in contrast, are not reduced by inelastic scat- tics of Q is binomial, just as the statistics of the number tering (which can only help the establishment of ther- of heads when tossing a coin. The mean-squared fluctu- mal equilibrium). This explains why only thermal noise ation hδQ2i of the charge for binomial statistics is given could be observed in macroscopic conductors. (As an by aside, we mention that inelastic electron-electron scatter- ing, which persists until much lower temperatures than 2 2 hδQ i = e (τeV/h)T1(1 − T1). (5) electron-phonon scattering, does not suppress shot noise, but rather enhances the noise power a little bit.9) The relation S = (2/τ)hδQ2i between the mean-squared Early experiments10 on mesoscopic semiconducting fluctuation of the current and of the transmitted charge wires observed the linear relation between noise power brings us to eq. (4) for a single channel. Since fluctuations and current that is the signature of shot noise, but could in different channels are independent, the multi-channel not accurately measure the slope. The first quantitative version is simply a sum over channels. measurement was performed in a thin-film silver wire by The quantum shot noise formula (4) has been tested Andrew Steinbach and John Martinis at the US National experimentally in a variety of systems. The groups of Institute of Standards and Technology in Boulder, collab- Reznikov and Glattli used a quantum point contact: A orating with Michel Devoret from Saclay.11 narrow constriction in a two-dimensional electron gas The data shown in figure 2 (from a more recent exper- 4

F=1 12 closed open P( T ) channels channels 10 1

2 F= = 8 3

6

F=1/3 probability distribution, 4

0 2 0 transmission eigenvalue, T 1 noise power (pA / Hz) FIG. 3: Bimodal probability distribution of the transmis- 0 sion eigenvalues, with a peak at 0 (closed channels) and a 0 10 20 30 40 peak at 1 (open channels). The functional form of the dis- − − current ( A) tribution (derived by Dorokhov) is P (T ) ∝ T 1(1 − T ) 1/2, with a mean-free-path dependent cutoff at exponentially small T . The one-third Fano factor follows directly from the ratio 2 FIG. 2: Sub-Poissonian shot noise in a disordered gold wire R T P (T )dT/ R T P (T )dT = 2/3. The cutoff affects only the (dimensions 940 nm × 100 nm). At low currents the noise sat- normalization of P (T ) and drops out of this ratio, which takes urates at the level set by the temperature of 0.3 K. [Adapted on a universal value. from M. Henny et al.11 ] Distinguishing particles from waves iment) presents a puzzle: If we calculate the slope, we find a Fano factor of 1/3 rather than 1. Surely there are So far we have encountered two diagnostic properties of no fractional charges in a normal metal conductor? shot noise: It measures the unit of transferred charge in a tunnel junction and it detects open transmission chan- A one-third Fano factor in a disordered conductor had nels in a disordered wire. A third diagnostic appears actually been predicted prior to the experiments. The in semiconductor microcavities known as quantum dots prediction was made independently by Kirill Nagaev of or electron billiards. These are small confined regions the Institute of Radio-Engineering and Electronics in in a two-dimensional electron gas, free of disorder, with Moscow and by one of the authors (Beenakker) with 12 two narrow openings through which a current is passed. Markus B¨uttiker of the University of Geneva. To un- If the shape of the confining potential is sufficiently ir- derstand the experimental finding we recall the general regular (which it typically is), the classical dynamics is shot noise formula (4), which tells us that sub-Poissonian chaotic and one can search for traces of this chaos in noise (F < 1) occurs when some channels are not weakly the quantum mechanical properties. This is the field of transmitted. These socalled “open channels” have Tn quantum chaos. close to 1 and therefore contribute less to the noise than Here is the third diagnostic: Shot noise in an electron expected for a Poisson process. billiard can distinguish deterministic scattering, charac- The appearance of open channels in a disordered con- teristic for particles, from stochastic scattering, charac- ductor is surprising. Oleg Dorokhov of the Landau Insti- teristic for waves. Particle dynamics is deterministic: A tute in Moscow first noticed the existence of open chan- given initial position and momentum fixes the entire tra- nels in 1984, but the physical implications were only un- jectory. In particular, it fixes whether the particle will be derstood some years later, notably through the work of transmitted or reflected, so the scattering is noiseless on Yoseph Imry of the Weizmann Institute. The one-third all time scales. Wave dynamics is stochastic: The quan- Fano factor follows directly from the probability distri- tum uncertainty in position and momentum introduces bution of the transmission eigenvalues, see figure 3. a probabilistic element into the dynamics, so it becomes We conclude this section by referring to the experimen- noisy on sufficiently long time scales. tal demonstrations3 of the interplay between the dou- The suppression of shot noise in a conductor with de- bling of shot noise due to superconductivity and the 1/3 terministic scattering was predicted many years ago13 reduction due to open channels, resulting in a 2/3 Fano from this qualitative argument. A better understand- factor. These experiments show that open channels are ing, and a quantitative description, of how shot noise a general and universal property of disordered systems. measures the transition from particle to wave dynamics 5

data is included in figure 4. An electron billiard (area 1 A ≈ 53 µm2) with two openings of variable width was cre- 4 ated in a two-dimensional electron gas by means of gate ∗ electrodes. The dwell time (given by τdwell = m A/¯hN, with m∗ the electron effective mass) was varied by chang- ing the number of modes N transmitted through each of the openings. The Fano factor has the value 1/4 for long dwell Fano factor Fano times, as expected for stochastic chaotic scattering. The 1/4 Fano factor for a chaotic billiard has the same ori- 0 gin as the 1/3 Fano factor for a disordered wire, ex- 0 1 2 3 4 5 6 7 plained in figure 3. (The different number results be- dwell time (ns) cause of a larger fraction of open channels in a bil- liard geometry.) The reduction of the Fano factor below 1/4 at shorter dwell times fits the exponential function FIG. 4: Dependence of the Fano factor F of an electron 1 billiard on the average time τdwell that an electron dwells F = 4 exp(−τE/τdwell) of Agam, Aleiner, and Larkin. inside. The data points with error bars are measured in a However, the accuracy and range of the experimental two-dimensional electron gas, the solid curve is the theoreti- data is not yet sufficient to distinguish this prediction 1 − cal prediction F = 4 exp( τE/τdwell) for the transition from from competing theories (notably the rational function 1 −1 stochastic to deterministic scattering (with Ehrenfest time F = 4 (1 + τE/τdwell) predicted by Sukhorukov for τE = 0.27 ns as a fit parameter). The inset shows graphically short-range impurity scattering). the sensitivity to initial conditions of the chaotic dynamics. (Adapted from ref. 14, with experimental data from ref. 15.) Entanglement detector was developed recently by Oded Agam of the Hebrew The fourth and final diagnostic property that we would University in Jerusalem, Igor Aleiner of the State Uni- like to discuss, shot noise as detector of entanglement, versity of New York in Stony Brook, and Anatoly Larkin 14 was proposed by Sukhorukov with Guido Burkard and of the University of Minnesota in Minneapolis. The key Daniel Loss from the University of Basel.16 concept is the Ehrenfest time, which is the characteristic A multi-particle state is entangled if it can not be time scale of quantum chaos. factorized into a product of single-particle states. En- In classical chaos, the trajectories are highly sensi- tanglement is the primary resource in quantum comput- tive to small changes in the initial conditions (although ing, in the sense that any speed-up relative to a classical uniquely determined by them). A change δx(0) in the ini- computer vanishes if the entanglement is lost, typically tial coordinate is amplified exponentially in time: δx(t)= αt through interaction with the environment (see the article δx(0)e . Quantum mechanics introduces an uncertainty by John Preskill, Physics Today, July 1999, page 24). in δx(0) of the order of the Fermi wave length λF . One Electron-electron interactions lead quite naturally to an can think of δx(0) as the initial size of a wave packet. entangled state, but in order to make use of the entan- The wave packet spreads over the entire billiard (of size glement in a computation one would need to be able to L) when δx(t)= L. The time spatially separate the electrons without destroying the −1 entanglement. In this respect the situation in the solid τE = α ln(L/λF ) (6) state is opposite to that in quantum optics, where the at which this happens is called the Ehrenfest time. production of entangled photons is a complex operation, The name refers to Paul Ehrenfest’s 1927 principle that while their spatial separation is easy. quantum mechanical wave packets follow classical, deter- One road towards a solid-state based quantum com- ministic, equations of motion. In quantum chaos this puter has as its building block a pair of quantum dots, correspondence principle loses its meaning (and the dy- each containing a single electron. The strong Coulomb namics becomes stochastic) on time scales greater than repulsion keeps the electrons separate, as desired. In the τE. An electron entering the billiard through one of the ground state the two spins are entangled in the singlet openings dwells inside on average for a time τdwell before state |↑i|↓i − |↓i|↑i. This state may already have been exiting again. Whether the dynamics is deterministic or realized experimentally,17 but how can one tell? Noise stochastic depends, therefore, on the ratio τdwell/τE. The has the answer. theoretical expectation for the dependence of the Fano To appreciate this we contrast “quiet electrons” with factor on this ratio is plotted in figure 4. “noisy photons”. We recall that Fermi statistics causes An experimental search for the suppression of shot the electron noise to be smaller than the Poisson value (2) noise by deterministic scattering was carried out at the expected for classical particles. For photons the noise is University of Basel by Stefan Oberholzer, Eugene Sukho- bigger than the Poisson value because of Bose statistics. rukov, and one of the authors (Sch¨onenberger).15 The What distinghuishes the two is whether the wave function 6

V metry of the spatial part of the wave function matters for entangler beam splitter the noise. Although the full many-body electron wave- function, including the spin degrees of freedom, is always antisymmetric, the spatial part is not so constrained. In e 1 particular, electrons in the spin-singlet state have a sym- metric wave function with respect to exchange of coordi- nates, and will therefore bunch together like photons. e 2 The experiment proposed by the Basel theorists is sketched in figure 5. The two building blocks are the en- tangler and the beam splitter. The beam splitter is used to perform the electronic analogue of the optical Hanbury V Brown and Twiss experiment.18 In such an experiment one measures the cross-correlation of the current fluctua- FIG. 5: Proposal for production and detection of a spin- tions in the two arms of a beam splitter. Without entan- entangled electron pair. The double quantum dot (shown glement, the correlation is positive for photons (bunch- in yellow) is defined by gate electrodes (green) on a two- ing) and negative for electrons (antibunching). The ob- dimensional electron gas. The two voltage sources at the far servation of a positive correlation for electrons is a signa- left inject one electron in each dot, resulting in an entangled ture of the entangled spin-singlet state. In a statistical spin-singlet ground state. A voltage pulse on the gates then sense, the entanglement makes the electrons behave as forces the two electrons to enter opposite arms of the ring. photons. Scattering of the electron pair by a tunnel barrier (red ar- rows) creates shot noise in each of the two outgoing leads at An alternative to the proposal shown in figure 5 is to the far right, which is measured by a pair of amplifiers (1,2). start from Cooper pairs in a superconductor, which are The observation of a positive correlation between the current also in a spin-singlet state.16 The Cooper pairs can be fluctuations at 1 and 2 is a signature of the entangled spin- extracted from the superconductor and injected into a singlet state. (Figure courtesy of L.P. Kouwenhoven and A.F. normal metal by application of a voltage over a tunnel Morpurgo, Delft University of Technology.) barrier at the metal–superconductor interface. Experimental realization of one these theoretical pro- is symmetric or antisymmetric under exchange of parti- posals would open up a new chapter in the use of noise as cle coordinates. A symmetric wave function causes the a probe of quantum mechanical properties of electrons. particles to bunch together, increasing the noise, while an Although this range of applications is still in its infancy, antisymmetric wave function has the opposite effect (“an- the field as a whole has progressed far enough to prove tibunching”). The key point here is that only the sym- Landauer right: There is a signal in the noise.

∗ Carlo Beenakker is at the Instituut-Lorentz of vik, JETP Lett. 49, 592 (1989); M. B¨uttiker, Phys. Rev. Leiden University (The Netherlands). Christian Lett. 65, 2901 (1990). Schonenberger¨ is at the Physics Department of 7 L.S. Levitov and G.B. Lesovik, JETP Lett. 58, 230 (1993). the University of Basel (Switzerland). 8 R. Cron, M. F. Goffman, D. Esteve, and C. Urbina, Phys. 1 For an overview of the entire literature on quantum shot Rev. Lett. 86, 4104 (2001). noise, we refer to: Ya.M. Blanter and M. B¨uttiker, Phys. 9 K.E. Nagaev, Phys. Rev. B 52, 4740 (1995); V.I. Kozub Rep. 336, 1 (2000). and A.M. Rudin, Phys. Rev. B 52, 7853 (1995). 2 F. Lefloch, C. Hoffmann, M. Sanquer, and D. Quirion, 10 F. Liefrink, J.I. Dijkhuis, M.J.M. de Jong, L.W. Phys. Rev. Lett. 90, 067002 (2003). Molenkamp, and H. van Houten, Phys. Rev. B 49, 14066 3 A.A. Kozhevnikov, R.J. Schoelkopf, and D.E. Prober, (1994). Phys. Rev. Lett. 84, 3398 (2000); X. Jehl, M. Sanquer, 11 A.H. Steinbach, J.M. Martinis, and M.H. Devoret, Phys. R. Calemczuk, and D. Mailly, Nature 405, 50 (2000). Rev. Lett. 76, 3806 (1996). More recent experiments in- 4 C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 72, 724 clude R.J. Schoelkopf, P.J. Burke, A.A. Kozhevnikov, (1994). D.E. Prober, and M.J. Rooks, Phys. Rev. Lett. 78, 3370 5 L. Saminadayar, D.C. Glattli, Y. Jin, and B. Etienne, (1997); M. Henny, S. Oberholzer, C. Strunk, and C. Phys. Rev. Lett. 79, 2526 (1997); R. de-Picciotto, M. Sch¨onenberger, Phys. Rev. B 59, 2871 (1999). Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. 12 C.W.J. Beenakker and M. B¨uttiker, Phys. Rev. B 46, 1889 Mahalu, Nature 389, 162 (1997); M. Reznikov, R. de- (1992); K.E. Nagaev, Phys. Lett. A 169, 103 (1992). Picciotto, T.G. Griffiths, M. Heiblum, and V. Umansky, 13 C.W.J. Beenakker and H. van Houten, Phys. Rev. B 43, Nature 399, 238 (1999); Y. Chung, M. Heiblum, and V. 12066 (1991). Umansky, preprint. 14 O. Agam, I. Aleiner, and A. Larkin, Phys. Rev. Lett. 85, 6 V.A. Khlus, Sov. Phys. JETP 66, 1243 (1987); G.B. Leso- 3153 (2000). 7

15 S. Oberholzer, E.V. Sukhorukov, and C. Sch¨onenberger, J.P. Kotthaus, Science 297, 70 (2002); W.G. van der Wiel, Nature 415, 765 (2002). S. De Franceschi, J.M. Elzerman, T. Fujisawa, S. Tarucha, 16 G. Burkard, D. Loss, and E.V. Sukhorukov, Phys. Rev. B and L.P. Kouwenhoven, Rev. Mod. Phys. 75, 1 (2003). 61, 16303 (2000). The alternative entangler using Cooper 18 M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. En- pairs is described in: M.-S. Choi, C. Bruder, and D. Loss, sslin, M. Holland, and C. Sch¨onenberger, Science 284, 296 Phys. Rev. B 62, 13569 (2000); G.B. Lesovik, T. Martin, (1999); W.D. Oliver, J. Kim, R.C. Liu, and Y. Yamamoto, and G. Blatter, Eur. Phys. J. B 24, 287 (2001). Science 284, 299 (1999). 17 A.W. Holleitner, R.H. Blick, A.K. H¨uttel, K. Eberl, and