Annihilation of colliding Bogoliubov quasiparticles reveals their Majorana nature
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Annihilation of colliding Bogoliubov quasiparticles reveals their Majorana nature
C. W. J. Beenakker
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands (Dated: December 2013)The single-particle excitations of a superconductor are coherent superpositions of electrons andholes near the Fermi level, called Bogoliubov quasiparticles. They are Majorana fermions, meaningthat pairs of quasiparticles can annihilate. We calculate the annihilation probability at a beamsplitter for chiral quantum Hall edge states, obtaining a 1 ± cos φ dependence on the phase difference φ of the superconductors from which the excitations originated (with the ± sign distinguishing singletand triplet pairing). This provides for a nonlocal measurement of the superconducting phase in theabsence of any supercurrent. Condensed matter analogies of concepts from particlephysics are a source of much inspiration, and many ofthese involve superconductors or superfluids [1]. Majo-rana’s old idea [2] that a spin-1 / E = 0) bound toa defect in a superconductor with broken spin-rotationand time-reversal symmetry (a so-called topological su-perconductor [8, 9]). The name Majorana zero-mode (or Majorino [10]) is preferred over Majorana fermion , sincethey are not fermions at all but have a non-Abelian ex-change statistics [11].Majorana fermions , in the original sense of the word,do exist in superconductors, in fact they are ubiqui-tous: The time-dependent four-component Bogoliubov-De Gennes wave equation for quasiparticle excitations(so-called Bogoliubov quasiparticles) can be brought toa real form by a 4 × U [12],in direct analogy to the real Eddington-Majorana waveequation of particle physics [2, 13]. A real wave equationimplies the linear relation Ψ † ( r , t ) = U Ψ( r , t ) betweenthe particle and antiparticle field operators, which is thehallmark of a Majorana fermion. As argued forcefully byChamon et al. [14], fermionic statistics plus superconduc-tivity by itself produces Majorana fermions, irrespectiveof considerations of dimensionality, topology, or brokensymmetries.Here we propose an experiment to probe the Majorananature of Bogoliubov quasiparticles in conventional, non-topological, superconductors. Existing proposals applyto topological superconductors [15–25], where Majoranafermions appear as charge-neutral edge states with a dis-tinct signature in dc transport experiments. In contrast,the Bogoliubov quasiparticles of a nontopological super-conductor have charge expectation value ¯ q = 0, so theirMajorana nature remains hidden in the energy domainprobed by dc transport.It is in the time domain that the wave equation takeson a real form and that particle and antiparticle oper-ators are linearly related. We will show that the Majo-rana relation manifests itself in high-frequency shot noise FIG. 1: Two-particle interferometer for Bogoliubov quasipar-ticles. Shown is a 2D electron gas in a perpendicular magneticfield (light blue), with chiral edge channels at the edges (ar-rows indicate the direction of motion and a, b, c p denote thequasiparticle operators). A constriction at the center formsa beam splitter. Current is injected at the two ends (red),biased at voltages V and V . Upon passing along a super-conducting electrode (grey, labeled S), repeated Andreev re-flection converts the electrons into a coherent superpositionof electrons and holes. The collision and pairwise annihilationof these Bogoliubov quasiparticles is detected by correlatingthe ac currents I and I . correlators, that can detect the annihilation of a pair ofBogoliubov quasiparticles originating from two identicalsuperconductors (differing only in their phase). Thesequasiparticles can annihilate for nonzero ¯ q because ofquantum fluctuations of the charge (with variance var q ).We calculate the annihilation probability P and find thatit oscillates with the phase difference φ , P = (1 + cos φ ) var ( q/e ) . (1)This could provide a way to detect the nonlocal Joseph-son effect [26], existing in the absence of any supercurrentflowing between the superconductors.We consider the beam splitter geometry of Fig. 1, inwhich electrons are injected from two voltage sources atone side of the beam splitter and the fluctuating cur-rents I ( t ) and I ( t ) are correlated at the other side atmicrowave frequencies ω > P ( ω ) = Z ∞−∞ dt e iωt h I (0) I ( t ) i . (2)Such two-particle interferometers have been implementedusing the quantum Hall edge channels of a two-dimensional (2D) electron gas as chiral (uni-directional)wave guides, to realize the electronic analogues of theHanbury-Brown-Twiss (HBT) experiment [27–29] andthe Hong-Ou-Mandel (HOM) experiment [30, 31].The setup we propose here differs in one essential as-pect: Before reaching the beam splitter, the electronsare partially Andreev reflected at a superconducting elec-trode. Andreev reflection in the quantum Hall effectregime has been reported in InAs quantum wells [32, 33]and in graphene monolayers [34–36]. In graphene, whichhas small spin-orbit coupling, the Andreev reflected holeis in the opposite spin band as the electron (spin-singletpairing). The strong spin-orbit coupling in InAs permitsspin-triplet pairing (electron and hole in the same spinband).We contrast these two cases by taking a twofold spin-degenerate edge channel for spin-singlet pairing and onesingle spin-polarized edge channel for spin-triplet pairing.For spin-singlet pairing we therefore need a vector of fourannihilation operators a = ( a e ↑ , a e ↓ , a h ↑ , a h ↓ ) = { a τσ } ,to accomodate electrons and holes ( τ = e, h ) in bothspin bands ( σ = ↑ , ↓ ), while for spin-triplet pairing thetwo operators a = ( a e ↑ , a h ↑ ) suffice. The creation andannihilation operators of these Bogoliubov quasiparticlesare related by particle-hole symmetry [37], a ( E ) = τ x a † ( − E ) . (3)The Pauli matrices τ i and σ i act, respectively on theelectron-hole and spin degree of freedom. The anticom-mutation relations thus have an unusual form, { a τσ ( E ) , a † τ ′ σ ′ ( E ′ ) } = δ ( E − E ′ ) δ ττ ′ δ σσ ′ , (4a) { a τσ ( E ) , a τ ′ σ ′ ( E ′ ) } = ( δ ( E + E ′ ) δ σσ ′ if τ, τ ′ is e, h or h, e . (4b)The nonzero anticommutator of two annihilation opera-tors is the hallmark of a Majorana fermion [14].The electrical current operator is represented by I ( t ) = ea † ( t ) τ z a ( t ) , a ( t ) = 1 √ π Z ∞−∞ dE e − iEt a ( E ) . (5)The Pauli matrix τ z accounts for the opposite charge ofelectron and hole. (For notational convenience we set ~ = 1 and take the electron charge e > I and I , we will denote the quasiparticleoperators at contact 2 by a and those at contact 1 by b . The current correlator (2) then takes the form P ( ω ) = G Z ∞−∞ dE Z ∞−∞ dE ′ Z ∞−∞ dE ′′ × (cid:10) b † ( E ′ ) τ z b ( E ′′ ) a † ( E − ω ) τ z a ( E ) (cid:11) , (6)with G = e /h the conductance quantum.Using the Majorana relation (3) we can rewrite Eq. (6)so that only positive energies appear [38]. Only productsof an equal number of creation and annihilation operatorscontribute, resulting in P ( ω ) = G Z ∞ dE Z ∞ dE ′ Z ∞ dE ′′ × (cid:10) b † ( E ′ ) τ z b ( E ′′ ) a † ( E ) τ z a ( E + ω )+ θ ( ω − E ) b † ( E ′ ) τ y b † ( E ′′ ) a ( ω − E ) τ y a ( E ) (cid:11) , (7)with θ ( x ) the unit step function. Both terms describe aninelastic process accompanied by the emission of a photonat frequency ω . The difference is that the term with τ z is a single-particle process (relaxation of a quasiparticlefrom energy E + ω E ), while the term with τ y is a two-particle process (pairwise annihilation of quasiparticles atenergy E and ω − E ). The appearance of this last term isa direct consequence of the Majorana relation (3), whichtransforms a † ( E − ω ) τ z a ( E ) a ( ω − E ) τ x τ z a ( E ).The quasiparticle operators c p injected towards thebeam splitter by voltage contact p = 1 , a, b behind the beam splitter bya scattering matrix. Since the voltage contacts are inlocal equilibrium, the expectation value of the c p opera-tors is known, and in this way one obtains an expressionfor the noise correlator in terms of scattering matrix el-ements — an approach pioneered by B¨uttiker [39] andused recently to describe the electronic HBT and HOMexperiments [40–42].Our new ingredient is the effect of the superconductoron the injected electrons. Propagation of the edge chan-nel along the superconductor transforms the quasiparti-cle operators c p ( E ) M p ( E ) c p ( E ) through a unitarytransfer matrix constrained by particle-hole symmetry, M p ( E ) = τ x M ∗ p ( − E ) τ x . (8)The effect of the beam splitter is described by the unitarytransformation a = √ R M c + √ − R M c ,b = √ − R M c − √ R M c . (9)(For simplicity, we take an energy independent reflectionprobability R .) The voltage contacts inject quasiparti-cles in local equilibrium at temperature T and chemicalpotential eV > h c † p,τσ ( E ) c q,τ ′ σ ′ ( E ′ ) i = δ ττ ′ δ σσ ′ δ pq δ ( E − E ′ ) f τ ( E ) , (10)with electron and hole Fermi functions, f e ( E ) = 11 + e ( E − eV ) /k B T , f h ( E ) = 1 − f e ( − E ) . (11)In what follows we focus on the low-temperature regime k B T ≪ eV , when only electrons are injected by thevoltage contacts: f e ( E ) = θ ( eV − E ) ≡ f ( E ), while f h ( E ) = 0 for E, V > P ( ω ) = P ( ω ) + P ( ω ) + P ( ω ) + P ( ω ) , (12) P pq ( ω ) = − G R (1 − R )( − p + q Z ∞ dE f ( E ) × Tr (cid:2) f ( E + ω ) Z pq ( E + ω, E ) Z qp ( E, E + ω )+ θ ( ω − E ) f ( ω − E ) Y ∗ pq ( E, ω − E ) Y qp ( ω − E, E ) (cid:3) , (13) Z pq ( E, E ′ ) = (1 + τ z ) M † p ( E ) τ z M q ( E ′ )(1 + τ z ) , (14) Y pq ( E, E ′ ) = (1 + τ z ) M T p ( E ) τ y M q ( E ′ )(1 + τ z ) . (15)The partial correlators P and P can be measured sep-arately by biasing only voltage contact 1 or 2, respec-tively. The terms P and P describe the collision atthe beam splitter of particles injected from contacts 1and 2.Transfer matrices of quantum Hall edge channels prop-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 bythe electron-hole symmetry relation (8). A single spin-degenerate Andreev edge channel has transfer matrix M p = e iEt p e iγ p τ z U ( α p , φ p , β p ) e iγ ′ p τ z , (16) U ( α, φ, β ) = exp (cid:2) iασ y ⊗ ( τ x cos φ + τ y sin φ ) + iβτ z (cid:3) . (17)The τ z terms account for relative phase shifts of elec-trons and holes in the magnetic field, while the terms σ y ⊗ τ x cos φ p and σ y ⊗ τ y sin φ p describe the electron-hole mixing by a spin-singlet pair potential with phase φ p . For a superconducting interface of width W one has α ≃ W/l S and β ≃ W/l m , with l m = ( ~ /eB ) / themagnetic length and l S = ~ v edge / ∆ the superconductingcoherence length (for induced gap ∆ and edge velocity v edge ).The presence of a τ z term in the electron-hole rotationmatrix (17) is inconvenient. With some algebra, it canbe eliminated, resulting in M p = e iEt p e i ( γ p + δγ p ) τ z U (¯ α p , φ p , e i ( γ ′ p + δγ p ) τ z , (18)sin ¯ α p = ( α p /ξ p ) sin ξ p , tan 2 δγ p = ( β p /ξ p ) tan ξ p . (19)We have abbreviated ξ p = ( α p + β p ) / . Typically onehas l m . l S , which implies ¯ α ≃ ( l m /l S ) sin( W/l m ) and δγ ≃ W/ l m . FIG. 2: Noise correlator as a function of frequency ( a ) and asa function of superconducting phase difference ( b ). Panel a shows both the collision term P coll = 2 P and the full corre-lator P full = P coll + P + P , while panel b shows only P coll (the full correlator differs by a phase-independent offset, suchthat P full = 0 at φ = 0). The curves are calculated from thegeneral result (23) for spin-singlet pairing, with parameters¯ α = ¯ α = π/ Substitution into Eq. (13) gives the partial correlators P pq ( ω ) = − G R (1 − R )( − p + q Z ∞ dE f ( E ) × (cid:2) f ( E + ω )( g pq + 1) + θ ( ω − E ) f ( ω − E )( g pq − (cid:3) , (20) g pq = cos 2 ¯ α p cos 2 ¯ α q − cos φ pq sin 2 ¯ α p sin 2 ¯ α q , (21) φ pq = φ p − φ q − γ p + δγ p − γ q − δγ q ) . (22)The phase φ represents the gauge invariant phase dif-ference between the two superconductors. Substituting f ( E ) = θ ( eV − E ), the integral over energy evaluates to P pq ( ω ) = − G R (1 − R )( − p + q × (cid:2) eV − ω ) + Θ(2 eV − ω )( g pq − (cid:3) , (23)where we have defined the function Θ( x ) = x θ ( x ). Sub-stitution into Eq. (12) then gives the full correlator P ( ω ) = − G R (1 − R )Θ(2 eV − ω ) (cid:0) g + g − g (cid:1) . (24)In Fig. 2 we compare the collision term P coll = 2 P and the full correlator P full = 2 P + P + P . Thelinear voltage dependence of P full shown in Fig. 2 a , witha singularity (discontinuous derivative) at ω = 2 eV , isknown from two-terminal normal-superconducting junc-tions [44]. The collision term has an additional singu-larity at ω = eV , signaling the frequency beyond whichonly pairwise annihilation of Bogoliubov quasiparticlescontributes to the noise.Fig. 2 b shows the dependence on the superconductingphase difference of the collision term, for the case of twoidentical superconductors, ¯ α = ¯ α ≡ ¯ α . In the annihila-tion regime eV < ω < eV the general formula (23) thensimplifies to P ( ω ) = − G R (1 − R )(2 eV − ω )(1 + cos φ ) sin α. (25)The factor sin α = var ( q/e ) is the variance of thequasiparticle charge, cf. Eq. (1). The annihilation prob-ability is maximal for vanishing phase difference. This“nonlocal Josephson effect” is the superconducting ana-logue of the nonlocal Aharonov-Bohm effect [45] and thesolid-state counterpart of the interferometry of superfluidBose-Einstein condensates [26].The spin-singlet pairing considered so far correspondsto a spin-1 / / U ( α, φ, β ) = exp (cid:2) iα ( τ x cos φ + τ y sin φ ) + iβτ z (cid:3) . (26)The Pauli matrix σ y is no longer present, because elec-tron and hole are from the same spin band. To pre-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 theFermi energy ( E = 0). If we consider frequencies ω =2 eV − δω near the upper cutoff, this energy dependencedoes not play a role (since the annihilating Bogoliubovquasiparticles then have the same energy eV ). If we againtake two identical superconductors we arrive at P = − G R (1 − R ) δω (1 − cos φ ) sin α. (27)The factor of two difference with Eq. (25) is due to theabsence of spin degeneracy. The annihilation probabilitynow vanishes for φ = 0. We interpret this in terms ofPauli blocking, operative because two Bogoliubov quasi-particles from the same spin band are indistinguishablefor φ = 0. In the spin-singlet case, in contrast, theyremain distinguished by their opposite spin.To detect the nonlocal Josephson effect in an experi-ment, one would like to vary the superconducting phasedifference φ without affecting the edge channels. Thiscould be achieved by joining the two superconductors viaa ring in the plane perpendicular to the 2D electron gas and then varying the flux through this ring. The result-ing h/ e oscillations in the noise correlator would havethe largest amplitude for ¯ α = ¯ α = π/
4, but there is noneed for fine tuning of these parameters. For example, ifonly ¯ α = π/
4, the amplitude of the oscillations varies assin α , so it remains substantial for a broad interval of¯ α around π/ l m /l S (magnetic length over proximity-induced superconducting coherence length). The am-plitude of the nonlocal Josephson oscillations dependsquadratically on this ratio, so for l m ≃
10 nm (in a 4 Tmagnetic field) one would hope for a l S below 100 nm.In summary, we have proposed an experiment for Bo-goliubov quasiparticles that is the condensed matter ana-logue of the way in which Majorana fermions are searchedfor in particle physics [46]: By detecting their pair-wise annihilation upon collision. The Majorana fermionsin a topologically trivial superconductor lack the non-Abelian statistics and the associated nonlocality of Ma-jorana zero-modes in a topological superconductor [11],but a different kind of nonlocality remains: We havefound that the annihilation probability of quasiparticlesoriginating from two identical superconductors dependson their phase difference — even in the absence of anysupercurrent coupling. Observation of the h/ e oscil-lations of the annihilation probability would provide astriking demonstration of the Majorana nature of Bo-goliubov quasiparticles.I dedicate this paper to the memory of MarkusB¨uttiker. I have benefited from discussions with A.R. Akhmerov and from the support by the Foun-dation for Fundamental Research on Matter (FOM),the Netherlands Organization for Scientific Research(NWO/OCW), and an ERC Synergy Grant. Appendix A: Response to feedback
The following two appendices are in response to feed-back on the manuscript that I received from Claudio Cha-mon and Yuli Nazarov.
1. Symmetry of the current correlator
In some configurations the current correlator definedas in Eq. (2), P ( ω ) = Z ∞−∞ dt e iωt h I (0) I ( t ) i , (A1)differs from the symmetrized version P sym ( ω ) = 12 Z ∞−∞ dt e iωt h I (0) I ( t ) + I ( t ) I (0) i . (A2)In our beam splitter configuration there is no difference:The two correlators are identical, because the currentoperators I ( t ) and I ( t ′ ) commute.To see this, we start from the definition I p ( t ) = e π Z ∞−∞ dE Z ∞−∞ dE ′ e it ( E − E ′ ) × c † ( E ) M p,z ( E, E ′ ) c ( E ′ ) , (A3) M p,α ( E, E ′ ) = S † ( E ) P p τ α S ( E ′ ) = M † p,α ( E ′ , E ) , (A4)where P p projects onto contact p . The scattering matrix S ( E ) relates quasiparticle operators before and after thebeam splitter. It is a unitary matrix, constrained byparticle-hole symmetry, S ( E ) = τ x S ∗ ( − E ) τ x . (A5)The fact that P p P q = 0 if p = q implies that M p,α ( E, E ′ ) M q,β ( E ′ , E ′′ ) = 0 if p = q. (A6) M p,α ( E, E ′ ) τ x M T q,β ( E ′′ , − E ′ ) = 0 if p = q, (A7) M T p,α ( − E, E ′ ) τ x M q,β ( E, E ′′ ) = 0 if p = q. (A8)The Bogoliubov quasiparticle operators have the Ma-jorana anticommutation relation, cf. Eq. (4): { c ( E ) , c † ( E ′ ) } = δ ( E − E ′ ) τ , { c ( E ) , c ( E ′ ) } = δ ( E + E ′ ) τ x . (A9)Substitution into the commutator I ( t ) I ( t ′ ) − I ( t ′ ) I ( t )produces four terms, which all vanish in view of Eqs.(A6)–(A8).
2. Absence of supercurrent
The dependence of the current correlator on the su-perconducting phases φ , φ is remarkable in view of the absence of any supercurrent coupling. The absence of su-percurrent can be understood from Fig. 1 by noting thatthe chirality of the edge states prevents the transfer ofa Cooper pair between the two superconductors. Moreformally, one can calculate the density of states and as-certain that it is phase independent.We use the relation ρ ( E ) = (2 πi ) − ddE ln Det S ( E ) (A10)between the density of states and the scattering matrix,which we construct from the scattering matrix S beam ofthe beam splitter and the two transfer matrices M , M of the edge states along the superconductors: S ( E ) = S beam ( E ) (cid:18) M ( E ) 00 M ( E ) (cid:19) . (A11)The determinant factors into the productDet S = Det S beam Det M Det M . (A12)The scattering matrix of the beam splitter is independentof φ p , while M p depends on φ p according to Eq. (18): M p ∝ exp (cid:2) i ¯ α p σ y ⊗ ( τ x cos φ p + τ y sin φ p ) (cid:3) = (cid:18) cos ¯ α p ie − iφ p σ y sin ¯ α p ie iφ p σ y sin ¯ α p cos ¯ α p (cid:19) . (A13)The φ p -dependence drops out of Det M p , so Det S ( E )and hence ρ ( E ) are φ p -independent. [1] G. E. Volovik, The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003).[2] E. Majorana,
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