arXiv:1312.2001v3 [cond-mat.supr-con] 28 Jan 2014 tr r ieryrltd ewl hwta h Majo- the noise that shot oper- high-frequency show antiparticle in will itself and We manifests particle relation related. rana that linearly and are form ators real a on domain energy the in by hidden probed remains nature Majorana ]—mda tts(tteFrilevel Fermi the (at states midgap [5– zero-modes — Majorana 7] [4]. for on fermions concentrated search has Majorana search experimental The of and realizations matter theoretical in condensed intense [3] inspir- returned an superconductors, has low-dimensional ing antiparticle of own context its the be might trino) odco aecag xetto au ¯ value expectation super- charge nontopological have a conductor of quasiparticles Bogoliubov dis- the a with in states signature edge tinct charge-neutral Majorana as where appear fermions [15–25], apply superconductors proposals topological Existing to superconductors. non- conventional, topological, in quasiparticles Bogoliubov of nature broken or topology, dimensionality, symmetries. irrespective of fermions, considerations Majorana of produces superconduc- itself plus by by statistics tivity forcefully fermionic argued [14], al. 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U since , [12], (or n ieo h emslte n h utaigcur- fluctuating the and splitter beam at the sources rents of voltage two side from one injected are electrons which supercurrent superconductors. any Joseph- the of absence nonlocal between the flowing the in detect existing [26], to effect way son a provide could This ed(ih le,wt hrleg hnesa h de (ar and edges motion the of at direction channels the edge magneti indicate chiral perpendicular rows with a in blue), gas (light electron 2D field a quasip is Bogoliubov Shown for ticles. interferometer Two-particle 1: FIG. emslte.Creti netda h w ns(red), ends two the at injected is for voltages Current center at the biased splitter. at beam constriction a A operators). quasiparticle usprilscnanhlt o ozr ¯ nonzero These for phase). annihilate their can in quasiparticles of only pair identical (differing two a superconductors from of originating annihilation quasiparticles the Bogoliubov detect can that correlators, fteeBgluo usprilsi eetdb correlat by detected is quasiparticles annihila the pairwise Bogoliubov and superpositi these collision coherent The of a holes. and into r electrons electrons of Andreev the repeated converts S), flection labeled (grey, electrode conducting toclae ihtepaedifference phase the with oscillates it ecluaeteanhlto probability var annihilation variance the (with calculate We charge the of fluctuations quantum ecnie h emslte emtyo i.1 in 1, Fig. of geometry splitter beam the consider We ac ± s hyaeMjrn emos meaning fermions, Majorana are They es. I cos d(ihthe (with ed currents 1 n ftesprodcigpaei the in phase superconducting the of ent eanhlto rbblt tabeam a at probability annihilation he ( eetsproiin feetosand electrons of superpositions herent t and ) φ 0R edn h Netherlands The Leiden, RA 00 eedneo h hs difference phase the on dependence P I 1 I 2 = and V l hi aoaanature Majorana their als ( ± t 1 r orltda h te ieat side other the at correlated are ) 2 1 indsigihn singlet distinguishing sign and 1+cos + (1 I 2 . V 2 pnpsigaogasuper- a along passing Upon . φ a ( var ) q/e φ , ,b c b, a, ) . P q n n that find and p eas of because eoethe denote tion (1) ing q ms ar- on e- ). c - 2 microwave frequencies ω > 0, The current correlator (2) then takes the form

∞ ∞ ∞ ∞ iωt 1 ′ ′′ P (ω)= dt e I1(0)I2(t) . (2) P (ω)= 4 G0 dE dE dE h i −∞ −∞ −∞ −∞ Z Z Z Z † ′ ′′ † b (E )τzb(E )a (E ω)τza(E) , (6) Such two-particle interferometers have been implemented × − using the quantum Hall edge channels of a two- 2 with G0 = e /h the conductance quantum. dimensional (2D) electron gas as chiral (uni-directional) Using the Majorana relation (3) we can rewrite Eq. (6) wave guides, to realize the electronic analogues of the so that only positive energies appear [38]. Only products Hanbury-Brown-Twiss (HBT) experiment [27–29] and of an equal number of creation and annihilation operators the Hong-Ou-Mandel (HOM) experiment [30, 31]. contribute, resulting in The setup we propose here differs in one essential as- ∞ ∞ ∞ pect: Before reaching the beam splitter, the electrons ′ ′′ are partially Andreev reflected at a superconducting elec- P (ω)= G0 dE dE dE 0 0 0 trode. Andreev reflection in the quantum Hall effect Z Z Z b†(E′)τ b(E′′)a†(E)τ a(E + ω) regime has been reported in InAs quantum wells [32, 33] × z z 1 † ′ † ′′ and in graphene monolayers [34–36]. In graphene, which + 4 θ (ω E)b (E )τyb (E )a(ω E)τya(E) , (7) has small spin-orbit coupling, the Andreev reflected hole − − is in the opposite spin band as the electron (spin-singlet with θ(x) the unit step function. Both terms describe an pairing). The strong spin-orbit coupling in InAs permits inelastic process accompanied by the emission of a photon spin-triplet pairing (electron and hole in the same spin at frequency ω. The difference is that the term with τz band). is a single-particle process (relaxation of a quasiparticle from energy E +ω E), while the term with τ is a two- We contrast these two cases by taking a twofold spin- 7→ y degenerate edge channel for spin-singlet pairing and one particle process (pairwise annihilation of quasiparticles at energy E and ω E). The appearance of this last term is single spin-polarized edge channel for spin-triplet pairing. − For spin-singlet pairing we therefore need a vector of four a direct consequence of the Majorana relation (3), which † annihilation operators a = (a ↑,a ↓,a ↑,a ↓) = a , transforms a (E ω)τza(E) a(ω E)τxτza(E). e e h h { τσ} − 7→ − to accomodate electrons and holes (τ = e,h) in both The quasiparticle operators cp injected towards the spin bands (σ = , ), while for spin-triplet pairing the beam splitter by voltage contact p = 1, 2 are related ↑ ↓ two operators a = (ae↑,ah↑) suffice. The creation and to their counterparts a,b behind the beam splitter by annihilation operators of these Bogoliubov quasiparticles a scattering matrix. Since the voltage contacts are in are related by particle-hole symmetry [37], local equilibrium, the expectation value of the cp opera- tors is known, and in this way one obtains an expression † for the noise correlator in terms of scattering matrix el- a(E)= τxa ( E). (3) − ements — an approach pioneered by B¨uttiker [39] and used recently to describe the electronic HBT and HOM The Pauli matrices τi and σi act, respectively on the electron-hole and spin degree of freedom. The anticom- experiments [40–42]. mutation relations thus have an unusual form, Our new ingredient is the effect of the superconductor on the injected electrons. Propagation of the edge chan- † nel along the superconductor transforms the quasiparti- ′ ′ ′ ′ aτσ(E),aτ ′σ′ (E ) = δ(E E )δττ δσσ , (4a) cle operators cp(E) Mp(E)cp(E) through a unitary { ′ } − a (E),a ′ ′ (E ) = 7→ { τσ τ σ } transfer matrix constrained by particle-hole symmetry, ′ ′ δ(E + E )δσσ′ if τ, τ is e,h or h,e ∗ (4b) Mp(E)= τxMp ( E)τx. (8) (0 otherwise. − The effect of the beam splitter is described by the unitary The nonzero anticommutator of two annihilation opera- transformation tors is the hallmark of a Majorana fermion [14]. The electrical current operator is represented by a = √RM1c1 + √1 RM2c2, − (9) √ √ ∞ b = 1 RM1c1 RM2c2. † 1 −iEt − − I(t)= ea (t)τza(t), a(t)= dE e a(E). √4π (For simplicity, we take an energy independent reflection Z−∞ (5) probability R.) The voltage contacts inject quasiparti- The Pauli matrix τz accounts for the opposite charge of cles in local equilibrium at temperature T and chemical electron and hole. (For notational convenience we set potential eV > 0 (the same at both contacts), corre- ~ = 1 and take the electron charge e> 0.) To distinguish sponding to expectation values the currents I1 and I2, we will denote the quasiparticle † ′ ′ operators at contact 2 by a and those at contact 1 by b. c (E)c ′ ′ (E ) = δ ′ δ ′ δ δ(E E )f (E), (10) h p,τσ q,τ σ i ττ σσ pq − τ 3 with electron and hole Fermi functions,

1 fe(E)= , fh(E)=1 fe( E). (11) 1+ e(E−eV )/kBT − −

In what follows we focus on the low-temperature regime kBT eV , when only electrons are injected by the voltage≪ contacts: f (E) = θ(eV E) f(E), while e − ≡ fh(E)=0 for E,V > 0. The full correlator (7) decomposes into four terms,

P (ω)= P11(ω)+ P22(ω)+ P12(ω)+ P21(ω), (12) ∞ P (ω)= G R(1 R)( 1)p+q dE f(E) pq − 0 − − Z0 Tr f(E + ω)Z (E + ω, E)Z (E, E + ω) × pq qp + 1 θ(ω E)f(ω E)Y ∗ (E,ω E)Y (ω E, E) , 2  − − pq − qp − (13)  ′ 1 † ′ Zpq(E, E )= 4 (1 + τz)Mp (E)τzMq(E )(1 + τz), (14) ′ 1 T ′ Ypq(E, E )= 4 (1 + τz)Mp (E)τyMq(E )(1 + τz). (15) FIG. 2: Noise correlator as a function of frequency (a) and as a function of superconducting phase difference (b). Panel a The partial correlators P11 and P22 can be measured sep- arately by biasing only voltage contact 1 or 2, respec- shows both the collision term Pcoll = 2P12 and the full corre- tively. The terms P and P describe the collision at lator Pfull = Pcoll + P11 + P22, while panel b shows only Pcoll 12 21 (the full correlator differs by a phase-independent offset, such the beam splitter of particles injected from contacts 1 that Pfull =0at φ12 = 0). The curves are calculated from the and 2. general result (23) for spin-singlet pairing, with parameters Transfer matrices of quantum Hall edge channels prop- α¯1 =α ¯2 = π/4. agating along a superconducting contact (so-called An- dreev edge channels) have been calculated in Ref. [43]. Their general form is constrained by unitarity and by Substitution into Eq. (13) gives the partial correlators the electron-hole symmetry relation (8). A single spin- ∞ degenerate Andreev edge channel has transfer matrix p+q Ppq (ω)= G0R(1 R)( 1) dE f(E) − − − 0 ′ Z iEtp iγpτz iγpτz 1 Mp = e e U(αp, φp,βp)e , (16) f(E + ω)(g +1)+ θ(ω E)f(ω E)(g 1) , × pq 2 − − pq − U(α, φ, β) = exp iασ (τ cos φ + τ sin φ)+ iβτ . (20) y ⊗ x y z   (17) g = cos2¯α cos2¯α cos φ sin 2¯α sin 2¯α , (21)   pq p q − pq p q φpq = φp φq 2(γp + δγp γq δγq). (22) The τz terms account for relative phase shifts of elec- − − − − trons and holes in the magnetic field, while the terms The phase φ12 represents the gauge invariant phase dif- σy τx cos φp and σy τy sin φp describe the electron- ference between the two superconductors. Substituting ⊗ ⊗ hole mixing by a spin-singlet pair potential with phase f(E)= θ(eV E), the integral over energy evaluates to φp. For a superconducting interface of width W one has − p+q α W/l and β W/l , with l = (~/eB)1/2 the Ppq(ω)= G0R(1 R)( 1) S m m − − − ≃ ≃ ~ 1 magnetic length and lS = vedge/∆ the superconducting 2Θ(eV ω)+ Θ(2eV ω)(gpq 1) , (23) coherence length (for induced gap ∆ and edge velocity × − 2 − − vedge). where we have defined the function Θ(x)= xθ(x). Sub- The presence of a τz term in the electron-hole rotation stitution into Eq. (12) then gives the full correlator matrix (17) is inconvenient. With some algebra, it can P (ω)= 1 G R(1 R)Θ(2eV ω) g +g 2g . (24) be eliminated, resulting in − 2 0 − − 11 22− 12 ′
iEtp i(γp+δγp)τz i(γ +δγp)τz In Fig. 2 we compare the collision term Pcoll = 2P12 Mp = e e U(¯αp, φp, 0)e p , (18) and the full correlator Pfull = 2P12 + P11 + P22. The sinα ¯p = (αp/ξp) sin ξp, tan2δγp = (βp/ξp) tan ξp. (19) linear voltage dependence of Pfull shown in Fig. 2a, with a singularity (discontinuous derivative) at ω = 2eV , is 2 2 1/2 We have abbreviated ξp = (αp + βp ) . Typically one known from two-terminal normal-superconducting junc- has lm . lS, which impliesα ¯ (lm/lS) sin(W/lm) and tions [44]. The collision term has an additional singu- δγ W/2l . ≃ larity at ω = eV , signaling the frequency beyond which ≃ m 4 only pairwise annihilation of Bogoliubov quasiparticles and then varying the flux through this ring. The result- contributes to the noise. ing h/2e oscillations in the noise correlator would have Fig. 2b shows the dependence on the superconducting the largest amplitude forα ¯1 =α ¯2 = π/4, but there is no phase difference of the collision term, for the case of two need for fine tuning of these parameters. For example, if identical superconductors,α ¯1 =α ¯2 α¯. In the annihila- onlyα ¯1 = π/4, the amplitude of the oscillations varies as ≡ 2 tion regime eV <ω< 2eV the general formula (23) then sin 2α2, so it remains substantial for a broad interval of simplifies to α¯2 around π/4. The main experimental bottleneck is the coupling 1 2 P12(ω)= G0R(1 R)(2eV ω)(1 + cos φ12) sin 2¯α. strength of the edge channel to the superconductor, − 2 − − (25) which is of order lm/lS (magnetic length over proximity- 2 The factor sin 2¯α = var (q/e) is the variance of the induced superconducting coherence length). The am- quasiparticle charge, cf. Eq. (1). The annihilation prob- plitude of the nonlocal Josephson oscillations depends ability is maximal for vanishing phase difference. This quadratically on this ratio, so for l 10 nm (in a 4 T m ≃ “nonlocal Josephson effect” is the superconducting ana- magnetic field) one would hope for a lS below 100 nm. logue of the nonlocal Aharonov-Bohm effect [45] and the In summary, we have proposed an experiment for Bo- solid-state counterpart of the interferometry of superfluid goliubov quasiparticles that is the condensed matter ana- Bose-Einstein condensates [26]. logue of the way in which Majorana fermions are searched The spin-singlet pairing considered so far corresponds for in particle physics [46]: By detecting their pair- to a spin-1/2 Bogoliubov quasiparticle (electron and hole wise annihilation upon collision. The Majorana fermions from opposite spin bands). The spin-up quasiparticle in a topologically trivial superconductor lack the non- then annihilates with its spin-down counterpart. This is Abelian statistics and the associated nonlocality of Ma- closest in analogy to the spin-1/2 Majorana fermion from jorana zero-modes in a topological superconductor [11], particle physics (where neutrinos of opposite helicities but a different kind of nonlocality remains: We have would annihilate). In superconductors with strong spin- found that the annihilation probability of quasiparticles orbit coupling one can also consider a spinless Majorana originating from two identical superconductors depends fermion, with electron and hole from the same spin band on their phase difference — even in the absence of any (spin-triplet pairing). It is instructive to contrast the two supercurrent coupling. Observation of the h/2e oscil- cases. lations of the annihilation probability would provide a The transfer matrix for a spin-triplet Andreev edge striking demonstration of the Majorana nature of Bo- channel has the form (18) with a different electron-hole goliubov quasiparticles. rotation matrix [43], I dedicate this paper to the memory of Markus B¨uttiker. I have benefited from discussions with A. U(α, φ, β) = exp iα(τx cos φ + τy sin φ)+ iβτz . (26) R. Akhmerov and from the support by the Foun- dation for Fundamental Research on Matter (FOM), The Pauli matrix σ is no longer present, because elec- y the Netherlands Organization for Scientific Research tron and hole are from the same spin band. To pre- (NWO/OCW), and an ERC Synergy Grant. serve the particle-hole symmetry (8) the mixing strength α should be an odd function of energy: α(E)= α( E). In particular, electron and hole are uncoupled− at− the Fermi energy (E = 0). If we consider frequencies ω = Appendix A: Response to feedback 2eV δω near the upper cutoff, this energy dependence − does not play a role (since the annihilating Bogoliubov The following two appendices are in response to feed- quasiparticles then have the same energy eV ). Ifwe again back on the manuscript that I received from Claudio Cha- take two identical superconductors we arrive at mon and Yuli Nazarov.

P = 1 G R(1 R)δω(1 cos φ ) sin2 2¯α. (27) 12 − 4 0 − − 12 1. Symmetry of the current correlator The factor of two difference with Eq. (25) is due to the absence of spin degeneracy. The annihilation probability now vanishes for φ12 = 0. We interpret this in terms of In some configurations the current correlator defined Pauli blocking, operative because two Bogoliubov quasi- as in Eq. (2), particles from the same spin band are indistinguishable ∞ for φ12 = 0. In the spin-singlet case, in contrast, they iωt P (ω)= dt e I1(0)I2(t) , (A1) remain distinguished by their opposite spin. −∞ h i To detect the nonlocal Josephson effect in an experi- Z ment, one would like to vary the superconducting phase differs from the symmetrized version difference φ12 without affecting the edge channels. This ∞ could be achieved by joining the two superconductors via 1 P (ω)= dt eiωt I (0)I (t)+ I (t)I (0) . (A2) a ring in the plane perpendicular to the 2D electron gas sym 2 h 1 2 2 1 i Z−∞ 5

In our beam splitter configuration there is no difference: absence of any supercurrent coupling. The absence of su- The two correlators are identical, because the current percurrent can be understood from Fig. 1 by noting that ′ operators I1(t) and I2(t ) commute. the chirality of the edge states prevents the transfer of To see this, we start from the definition a Cooper pair between the two superconductors. More

∞ ∞ formally, one can calculate the density of states and as- e ′ I (t)= dE dE′ eit(E−E ) certain that it is phase independent. p 4π Z−∞ Z−∞ We use the relation c†(E) (E, E′)c(E′), (A3) × Mp,z (E, E′)= S†(E) τ S(E′)= † (E′, E), (A4) Mp,α Pp α Mp,α d ρ(E)=(2πi)−1 ln Det S(E) (A10) where p projects onto contact p. The scattering matrix dE S(E) relatesP quasiparticle operators before and after the beam splitter. It is a unitary matrix, constrained by particle-hole symmetry, between the density of states and the scattering matrix, which we construct from the scattering matrix Sbeam of ∗ S(E)= τxS ( E)τx. (A5) the beam splitter and the two transfer matrices M1, M2 − of the edge states along the superconductors: The fact that = 0 if p = q implies that PpPq 6 ′ ′ ′′ p,α(E, E ) q,β(E , E ) = 0 if p = q. (A6) M1(E) 0 M M 6 S(E)= Sbeam(E) . (A11) (E, E′)τ T (E′′, E′) = 0 if p = q, (A7) 0 M2(E) Mp,α xMq,β − 6   T ( E, E′)τ (E, E′′) = 0 if p = q. (A8) Mp,α − xMq,β 6 The determinant factors into the product The Bogoliubov quasiparticle operators have the Ma- jorana anticommutation relation, cf. Eq. (4):

† ′ ′ Det S = Det Sbeam Det M1 Det M2. (A12) c(E),c (E ) = δ(E E )τ0, { ′ } − ′ (A9) The scattering matrix of the beam splitter is independent c(E),c(E ) = δ(E + E )τx. { } of φp, while Mp depends on φp according to Eq. (18): ′ ′ Substitution into the commutator I1(t)I2(t ) I2(t )I1(t) produces four terms, which all vanish in view− of Eqs. M exp iα¯ σ (τ cos φ + τ sin φ ) (A6)–(A8). p ∝ p y ⊗ x p y p −iφp  cosα ¯p ie σy sinα ¯p  = iφp . (A13) ie σy sinα ¯p cosα ¯p 2. Absence of supercurrent  

The dependence of the current correlator on the su- The φp-dependence drops out of Det Mp, so Det S(E) perconducting phases φ1, φ2 is remarkable in view of the and hence ρ(E) are φp-independent.

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