Probing Majorana bound states via counting statistics of a single electron transistor
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Probing Majorana bound states via counting statistics of a single electron transistor
Zeng-Zhao Li,
1, 2
Chi-Hang Lam, ∗ and J. Q. You † Laboratory for Quantum Optics and Quantum Information,Beijing Computational Science Research Center, Beijing 100094, China Department of Applied Physics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
We propose an approach for probing Majorana bound states (MBSs) in a nanowire via countingstatistics of a nearby charge detector in the form of a single-electron transistor (SET). We considerthe impacts on the counting statistics by both the local coupling between the detector and anadjacent MBS at one end of a nanowire and the nonlocal coupling to the MBS at the other end.We show that the Fano factor and the skewness of the SET current are minimized for a symmetricSET configuration in the absence of the MBSs or when coupled to a fermionic state. However, theminimum points of operation are shifted appreciably in the presence of the MBSs to asymmetricSET configurations with a higher tunnel rate at the drain than at the source. This feature persistseven when varying the nonlocal coupling and the pairing energy between the two MBSs. We expectthat these MBS-induced shifts can be measured experimentally with available technologies and canserve as important signatures of the MBSs.
PACS numbers: 73.21.-b, 85.35.Gv
I. INTRODUCTION
Majorana fermions are particles that are their ownantiparticles. In high-energy physics, neutrino beingan elementary particle was suggested as a Majoranafermion . Experiments aiming to prove this proposalare still on going. Besides the high-energy contextwhere they arose, it is believed that Majorana fermionscan also emerge as quasiparticles in condensed-mattersystems . The search for Majorana bound states(MBSs) in these systems has attracted much interest notonly due to their exotic properties (e.g., non-Abelianstatistics) but also because they are promising candi-dates for topological quantum computation . Severalphysical systems have been suggested to support MBSs,including fractional quantum Hall states , chiral p -wave superconductors/superfluids , surfaces of three-dimensional (3D) topological insulators in proximity toan s -wave superconductor , superfluids in the 3He-Bphase , and helical edge modes of 2D topologicalinsulators in proximity to both a ferromagnet and asuperconductor . More recently, it has been shown thata spin-orbit coupled semiconducting 2D thin film or a1D nanowire with Zeeman spin splitting, which isin proximity to an s -wave superconductor, can also hostMBSs.Providing experimental evidences for the realizationof MBSs is of great importance. Techniques pro-posed to detect MBSs include the analysis of the tun-neling spectroscopy , the verification of the natureof nonlocality or the observation of the periodicMajorana-Josephson current . In particular, the veryrecent observation of a zero-bias peak in the differ-ential conductance through a semiconductor nanowirein contact with a superconducting electrode indicatedthe possible existence of a midgap Majorana state .Such a zero-bias peak was also observed in subsequentexperiments . However, this zero-bias peak could be due to the Kondo resonance and also occur in the pres-ence of either disorders or a singlet-doublet quantumphase transition , corresponding to ordinary Andreevbound states rather than MBSs. Moreover, a study ofa more realistic model of a nanowire with MBSs fur-ther indicates a different origin for this observed zero-bias peak . There are several recent works devel-oped, for example, to distinguish between the Majoranaand Kondo origins of the zero-bias conductance peak,but a definite evidence for the zero-bias anomaly due toMBSs is still missing. Therefore, further investigationsare needed to convincingly reveal the existence of MBSs.We will focus on the detection of MBSs which existin pairs at the two ends of a nanowire. Most previousstudies based on a variety of setups considered a detec-tor coupled locally to an adjacent MBS at one end ofthe nanowire only , as the coupling to the otherMBS farther away is neglected. For example, a quantumdot coupled to a MBS was studied in Ref. 40. The cur-rent and the shot noise through the quantum dot werecalculated. A characteristic feature in the frequency de-pendence of the shot noise was proposed as a signature forthe MBS. The coupling of a quantum dot to two MBSs atboth ends of a nanowire has also been studied , but onlythe conductance was reported. In this work, we studyboth the local and nonlocal coupling of a single electrontransistor (SET) (consisting of a quantum dot and twoelectrodes) to two MBSs at both ends of a nanowire. Wecalculate the full counting statistics (FCS) of electrontransport through the SET. FCS yields all zero-frequencycurrent correlations at once and provides detailed insightsinto the nature of charge transfer beyond what is avail-able from conductance measurements alone . Impor-tantly, it has also become an experimentally accessibletechnique in recent years . Using the FCS, we calcu-late the current, Fano factor and skewness as functionsof a tunnel rate ratio of the SET. The calculations areperformed for various couplings of the SET island withthe MBSs. The results are also compared with those forcoupling to a fermionic state instead. We will show inthe following that in the absence of the MBSs or whencoupled to fermionic states, the Fano factor and the skew-ness are minimized for a symmetric SET. However, in thepresence of the MBSs, the minimum points shift appre-ciably to occur for an asymmetric SET with a highertunnel rate at the drain than at the source. We proposethat these MBS-induced shifts of the minimum points ofthe Fano factor and the skewness can be used as signa-tures for the identification of the MBSs. II. THE MBS-SET MODEL
The hybrid system consists of two MBSs and a SETas schematically shown in Figure 1. With a conventional s -wave superconductor and a modest magnetic field, theMBSs as electron-hole quasiparticle excitations have beensuggested to exist at the two ends of a semiconductornanowire with strong spin-orbit coupling . TheSET consists of a metallic island coupled via tunnel-ing barriers to two electrodes. The energy levels andthe tunneling barriers can be tuned by the gate volt-ages. By assuming a Zeeman splitting much larger thanthe MBS-SET coupling strength, the source-drain biasvoltage across the SET, and the tunneling rates withthe source and drain electrodes, the SET island can bemodeled by a single resonant level occupied by a spin-polarized electron.The interaction between the MBSs and the SET islandcan be derived from a second quantization Hamiltonianas (see Methods) H t = (cid:0) d † − d (cid:1) ( λγ L + µγ R ) , (1)where the coupling coefficients λ and µ are assumed tobe real and independent of k for simplicity. This Hamil-tonian involves both the local coupling λ to an adja-cent MBS at one end of the nanowire and the non-local coupling µ to the MBS at the other end of thenanowire (see Figure 1). Due to its smaller magnitude,the nonlocal coupling was neglected in most previousstudies with an exception of Ref. 21. Wenote that this nonlocal coupling can give rise to furtherdetector-position-dependent measurement results whichmay also be used for the identification of the MBSs. Thenonlocal coupling is therefore also considered here.The coupling between two separated MBSs at the twoends of the nanowire can be described by H γ = i ε M γ L γ R , (2)where ε M ∼ e − l/ζ is the coupling energy with l being thewire length and ζ the superconducting coherent length.The pair of MBSs can constitute a regular fermion withoperators f = γ L + iγ R , f † = γ L − iγ R . (3) island s-wave superconductor λ μ Drain Source g V L γ R γ SET V Figure 1: (Color online) Schematic diagram of the hybridquantum system consisting of two MBSs and a SET. TheMBSs locate at the two ends of a nanowire with large Zeemansplitting and strong spin-orbit coupling, which is in proximityto an s -wave superconductor. The SET island is coupled tothe source and drain electrodes via tunneling barriers andcapacitively biased by an external gate voltage V g . The energylevel of the SET island is tuned to be zero, i.e., in resonancewith the MBSs. Also, the SET island couples to the adjacentMBS with a coupling strength λ and the MBS at the otherend of the nanowire with a coupling strength µ . In this regular-fermion representation, the Hamilto-nian H sys = ε I d † d + H t + H γ of the hybrid MBS-SETsystem becomes H sys = ε I d † d + ε M (cid:18) f † f − (cid:19) − ( λ + iµ ) (cid:0) d − d † (cid:1) f † + ( λ − iµ ) (cid:0) d † − d (cid:1) f, (4)where ε I is the resonant-level energy of the SET islandand d † ( d ) is the corresponding creation (annihilation) op-erator. Note that this energy can be tuned by the gatevoltage V g to be zero (i.e., ε I = 0) to ensure resonanttunnelings between the SET island and the zero-energyMBSs. The basis states of the system of interest aregiven by | n d n f i , with n d and n f being 0 and 1 , i.e., a ≡ | i , b ≡ | i , c ≡ | i , d ≡ | i . To comparethe transport behaviors of the SET in the presence ofthe MBSs with those of a regular fermionic bound statein the nanowire, we also consider the following systemHamiltonian H sys = ǫ I d † d + ǫ M (cid:18) f † f − (cid:19) − ( λ + iµ ) df † + ( λ − iµ ) d † f, (5)which describes the SET when coupled to a regularfermionic state.The Hamiltonian for the source and the drain elec-trodes of the SET is described by H leads = X k ( ω sk c † sk c sk + ω dk c † dk c dk ) , (6)where c sk ( c dk ) is the annihilation operator for electronsin the source (drain) electrode. The tunneling Hamil-tonian between the SET island and the two electrodesis H T = X k [(Ω sk c † sk d + Ω dk Υ † c † dk d ) + H . c . ] , (7)where Ω sk ( dk ) is the coupling strength between the SETisland and the source (drain) electrode. The countingoperator Υ (Υ † ) decreases (increases) the number of elec-trons that have tunneled into the drain electrode in orderto keep track of the tunnelings of successive electrons.Thus, the total Hamiltonian of the system is given by H tot = H sys + H leads + H T . III. COUNTING STATISTICS
To study the FCS, it is essential to know the probabil-ity P ( n, t ) of n electrons having been transported fromthe SET island to the drain electrode during a period oftime interval t . It is related to the cumulant generatingfunction G ( χ, t ) defined by e − G ( χ,t ) = X n P ( n, t ) e inχ . (8)We will consider the time interval t much longer than thetime for an electron to tunnel through the SET island(i.e., the zero-frequency limit), so that transient prop-erties are insignificant. The derivatives of G ( χ, t ) withrespect to the counting field χ at χ = 0 yield the cumu-lant of order m as C m = − ( − i∂ χ ) m G ( χ, t ) | χ → . (9)These cumulants carry complete information of the FCSon the SET island. For instance, the average current and the shot noise can be expressed as I = eC /t and S = 2 e C /t . Thus, the Fano factor F is given by F = S/ eI = C /C , which is used to characterize thebunching and anti-bunching phenomena in the transportprocess. The third-order cumulant C gives rise to theskewness K = C /C of the distribution of transportedelectrons.On the other hand, the probability distribution func-tion of the transported electrons can be expressed as P ( n, t ) = ρ ( n ) aa ( t ) + ρ ( n ) bb ( t ) + ρ ( n ) cc ( t ) + ρ ( n ) dd ( t ) , (10)where ρ ( n ) ij ( t ) ( i, j ∈ { a, b, c, d } ) denote the reduced den-sity matrix elements of the SET island at a given number n of electrons being transported from the SET island tothe drain electrode at time t . We will calculate these re-duced density matrix elements using a master equation(see Methods) which assume a large bias voltage acrossthe SET. In fact, this large-bias case was considered inmany previous studies as it is easy to implement inexperiments. Moreover, this makes the problem simplerand more transparent because the broadening effect ofthe SET level can be neglected (see, e.g., Refs. 53 and 54).Using the discrete Fourier transform of the density ma-trix elements given by ρ ij ( χ, t ) = X n ρ ( n ) ij ( t ) e inχ , (11)we can convert the master equation into ddt ̺ = M ( χ ) ̺, (12)with M ( χ ) = A A A A A A A A , (13)where ̺ = ( ρ aa , ρ bb , ρ cc , ρ dd , ̺ , ̺ ) T with ̺ =(Re [ ρ ab ] , Im [ ρ ab ] , Re [ ρ ac ] , Im [ ρ ac ] , Re [ ρ ad ] , Im [ ρ ad ]) , and ̺ = (Re [ ρ bc ] , Im [ ρ bc ] , Re [ ρ bd ] , Im [ ρ bd ] , Re [ ρ cd ] , Im [ ρ cd ]), and A = − Γ S D e iχ − Γ S D e iχ Γ S − Γ D
00 Γ S − Γ D , A = − µ − λ µ − λ − µ λ µ λ , (14) A = − Γ S − ε M µ − λε M − Γ S λ µ − µ − λ − (Γ S + Γ D ) − ε I λ − µ ε I − (Γ S + Γ D ) , A = − µ − λ Γ D e iχ − λ µ D e iχ − µ − λ − λ µ , (15) A = µ − µλ − λ − µ µ λ − λ , A = − (Γ S + Γ D ) − ( ε M + ε I ) 0 0 ε M + ε I − (Γ S + Γ D ) 0 00 0 − (Γ S + Γ D ) ε M − ε I − ( ε M − ε I ) − (Γ S + Γ D ) , (16) A = µ λ λ − µ S µ λ S λ − µ , A = − (Γ S + Γ D ) − ε I µ λε I − (Γ S + Γ D ) − λ µ − µ λ − Γ D − ε M − λ − µ ε M − Γ D . (17)Here Γ S ( D ) is the tunneling rate of the electrons throughthe barrier between the SET island and the source (drain)electrode and it is given byΓ S ( D ) = 2 πg S ( D ) Ω sk ( dk ) , (18)where g i ( i = S , or D ) denotes the density of states at thesource or drain electrode and is assumed to be constantover the relevant energy range.The formal solution to the dynamical equation of ̺ ( χ, t ) can be readily obtained as ̺ ( χ, t ) = e M ( χ ) t ̺ ( χ, G ( χ, t ) = − ln Tr ̺ ( χ, t ). At long time t (i.e., zero-frequency limit),the cumulant generating function is simplified to G ( χ, t ) = − Λ min ( χ ) t, (19)where Λ min ( χ ) is the minimal eigenvalue of M ( χ ) andsatisfies Λ min | χ → → P n P ( n, t ) = 1. IV. SIGNATURES OF THE MBSsA. Current
Below we consider the zero-temperature case for theSET system since related experiments are usually per-formed at extremely low temperatures (see, e.g., Refs. 54and 56). Figure 2(a) shows the current flowing from theSET island to the drain electrode as a function of Γ D / Γ S for ε M = 0 and various values of λ and µ . In particular,the case of λ = µ = 0 represents the absence of the MBSs.Our calculation shows that it also equivalently representsthe case of coupling to a fermion in the nanowire. Thisis expected because a regular fermion state does not af-fect the counting statistics of a nearby SET in the zero-frequency limit (or stationary behaviors) considered. Itis clear from Figure 2(a) that for a symmetric SET inwhich the tunneling rates between the SET island andthe two electrodes are the same, i.e., Γ D = Γ S , the cur-rent does not vary with λ and µ (see also the analyticalresult below). However, when Γ D = Γ S , the currentin the presence of the MBSs deviates appreciably fromthat in the absence of the MBSs, especially in the regionΓ D > Γ S . Moreover, Figure 2(b) shows that the current also changes, albeit slightly, when varying the couplingenergy ε M of the two MBSs . From Figures 2(a) and 2(b),although coupling to the MBSs does change the currentquantitatively, a distinct qualitative feature is lacking.Thus, it is insufficient to use only the current to showthe existence of the MBSs.Much of the above numerical results can also be ob-tained from analytic expressions in some special cases.For ε M = 0, we obtain from equation (19) an analyticalresult for the current: I = e Γ D (cid:0) ξ + Γ S Γ tot (cid:1) ξ + Γ , (20)where Γ tot = Γ S + Γ D and ξ = p λ + µ . Although thissymmetric property of the two couplings λ and µ has beennoticed before , we emphasize that we apply full count-ing statistics (including the Fano factor and the skewnessas shown below) to reveal signatures of the MBSs, whichgoes beyond the conductance results reported in Ref. 21.When the MBSs are absent, i.e., ξ →
0, equation (20)recovers the well-known result I = e Γ S Γ D Γ tot . (21)Alternatively, with the MBSs coupled and Γ S = Γ D = Γ,the current is reduced to I = e Γ /
2, independent of thevalues of λ and µ .When the MBSs are absent, the current through theSET island at zero temperature can also be calculatedusing I = e Γ S Γ D Γ tot Z µ S µ D dED ( E ) , (22)where µ S ( D ) is the chemical potential of the source(drain) electrode, and D ( E ) is the density of states(DOS) of the SET island. When including the electrode-induced level broadening of the SET island, the broad-ened DOS can be described by a Lorentzian function centered around E = ε I : D ( E ) = Γ tot / π ( E − ε I ) + (Γ tot / . (23)Therefore, the current can be calculated as I = e Γ S Γ D Γ tot F ( µ S , µ D ) , (24) (a) C u rr e n t ( I / e S ) = = 0 ( Fermion case ) = S , = 0 = S , = 0.5 S = S , = S A D / S M = 0 M = 0.8 S (b) Figure 2: (Color online) Current I through the SET island tothe drain electrode versus the tunneling-rate ratio Γ D / Γ S for ε I = 0. In (a), ε M = 0 as λ and µ are varied. In (b), λ = Γ S ,and µ = 0 . S as ε M is varied. where F ( µ S , µ D ) = 1 π arctan (cid:20) tot eV Γ + 4 ( ε I − µ S ) ( ε I − µ D ) (cid:21) . (25)If the bias eV ≡ µ S − µ D applied on the SET is large sothat the SET level ε I is deeply inside the bias window,i.e., eV ∼ | ε I − µ S ( D ) | ≫ Γ tot , the factor F ( µ S , µ D ) issimply reduced to F ( µ S , µ D ) = 1 . (26)Equation (24) then recovers equation (21). B. Fano factor
It is known that the current from the SET island tothe drain electrode is related to the first-order cumulantof the generating function G ( χ, t ) by I = eC /t . Thecorresponding shot noise is related to the second-ordercumulant of G ( χ, t ) as S = 2 e C /t . Thus, the Fanofactor F = S/ eI can be written as F = C /C . InFigure 3(a), we show the Fano factor as a function ofΓ D / Γ S for ε M = 0 and various values of λ and µ . The black dotted curve in this figure represents the result notonly for the case without the MBSs but also for the iden-tical result for the fermion case similar to that in Figure1(a). It is clear that the Fano factor in the absence ofthe MBSs reaches its minimum (i.e., F min = 1 /
2) for asymmetric SET with Γ D / Γ S = 1, as indicated by pointB on the black dotted curve in Figure 3(a). This min-imum point of the Fano factor shifts appreciably in thepresence of the MBSs, e.g., F min ≈ .
49 at Γ D / Γ S ≈ . λ = Γ S and µ = 0. Interestingly, this shift is ro-bust against varying either the nonlocal coupling µ to themore distant MBS or the coupling energy ε M between thetwo MBSs [see Figures 3(a) and 3(b)].For ε M = 0, an analytical result for the Fano factorcan also be obtained as F = 8 ξ (cid:0) ξ + 2Γ S − Γ D + 5Γ S Γ D (cid:1) + Γ (cid:0) Γ S + Γ D (cid:1) (8 ξ + Γ ) . (27)Without the MBSs, i.e., ξ → , we recover the exper-imentally verified result F = (cid:0) a (cid:1) / <
1, where a = (Γ S − Γ D ) / Γ tot is the asymmetry of the SET. Inthe presence of the MBSs and when Γ S = Γ D = Γ, theFano factor follows F = (Γ + 4 ξ ) / (2Γ + 4 ξ ). It de-pends non-trivially on the couplings between the SETisland and the MBSs, which does not occur for the cur-rent (see Figure 2). In Figure 4(a), we show the depen-dence of the minimum point from equation (27) on theSET-MBS coupling. We observe that the minimum point(Γ D / Γ S ) min of the Fano factor increases with ξ/ Γ S . ThisMBS-induced shift of the minimum point of the Fano fac-tor can be used as one of the signatures of the MBSs.Such a shift does not occur when coupled to a fermionstate instead [see the black dotted curve in Figure 3(a)].We emphasize that we have considered a nanowire withboth MBSs coupled to the quantum dot . This gen-eralizes results on the Fano factor from Ref. 40 whichconsidered coupling to only one MBS, which may notbe applicable when the distances between the detector(e.g., SET) and two MBSs are comparable. In addition,instead of considering the Fano factors (or currents) atboth the source and drain electrodes as in Ref. 40, wefind it sufficient to characterize the MBSs by studyingthe statistics only at the drain electrode. This is in factmore directly related to typical experimental measure-ments. In particular, our results on the Fano factor (andalso on the skewness as demonstrated below) can reduceback to the experimentally verified ones when the MBSsare decoupled as explained above. Moreover, tuning thegate voltages to control Γ D / Γ S for identifying the MBSsin our proposal is a new alternative to the frequencytuning suggested in Ref. 40. Note that the shot noiseof a quantum dot coupled to a MBS was explored in amore recent work to distinguish the Majorana originof the zero-bias anomaly from that due to Kondo effect.However, their results of the shot-noise power spectra aswell as the tunneling conductance were obtained undera smaller bias voltage (i.e., eV ≪ Γ tot ). These are quite B (a) F a no f ac t o r ( F ) = = 0 ( Fermion case ) = S , = 0 = S , = 0.5 S = S , = S D / S M = 0 M = 0.8 S (b) Figure 3: (Color online) Fano factor F versus the tunneling-rate ratio Γ D / Γ S . The parameters in (a) are the same asthose in Figure 2(a), and the parameters in (b) are the sameas those in Figure 2(b). different from our results of Fano factor (or shot noise)and current which correspond to the case of a large biasvoltage (i.e., eV ≫ Γ tot ). In addition, we further explorethe skewness below, which goes beyond the differentialconductance (or current) and shot noise to revealthe signatures of MBSs. C. Skewness
The skewness of the distribution of transferred elec-trons is defined as K = C /C , which involves the third-order cumulant C . Figure 5(a) shows the skewness for ε M = 0 and various values of λ and µ . It is clear thatthe skewness in the absence of the MBSs takes its min-imum value (i.e., K = 1 /
4) at Γ D / Γ S = 1, as indicatedby point C on the dotted curve. This dotted curve alsorepresents the results for the fermion case due to thesame reason as that for the result of the current or Fanofactor as explained above. Also, the minimum point ofthe skewness shifts appreciably in the presence of MBSs,e.g., K min ≈ .
08 at Γ D / Γ S ≈ .
16 when λ = Γ S and µ = 0. Moreover, similar to the Fano factor, this shift ofthe minimum point is also robust against varying µ and Fano factor ( D / S ) m i n (a)(b) S Skewness
Figure 4: The minimum points (Γ D / Γ S ) min of (a) Fano factorand (b) skewness as a function of ξ = p λ + µ . ε M [see Figures 5(a) and 5(b)].If ε M = 0, the skewness can be obtained analyticallyas K = 16(2 ξ ) + 8(2 ξ ) A + 12(2 ξ ) B + 8 ξ Γ C + Γ D (8 ξ + Γ ) , (28)where A = 4Γ + 3Γ D (Γ S − D ) , B = Γ − Γ D [7Γ tot Γ S − (Γ S − Γ D ) ] , C = 4Γ − D { S Γ +Γ D [3Γ +2Γ S (5Γ S − D )] } , D = (Γ S − Γ D ) + 2Γ S Γ D (Γ − S Γ D ) . As expected, when ξ → K = D / Γ = (cid:0) a (cid:1) /
4, which was verified experimen-tally in Ref. 47. In the presence of the MBSs, theskewness takes its minimum value K min at the mini-mum point (Γ D / Γ S ) min which can be accurately iden-tified from equation (28) and is shown in Figure 4(b).Similar to the Fano factor, this MBS-induced shift of theminimum point of the skewness can be used as anothersignature of the MBSs. C (a) S k e w n e ss ( K ) = = 0 ( Fermion case ) = S , = 0 = S , = 0.5 S = S , = S D / S M = 0 M = 0.8 S (b) Figure 5: (Color online) Skewness K versus the tunneling-rate ratio Γ D / Γ S . The parameters in (a) are the same as inFigure 2(a), and the parameters in (b) are the same as inFigure 2(b). V. DISCUSSION AND CONCLUSION
Note that the coupling strengths λ and µ of the SETisland to the two MBSs at the ends of the nanowire de-pend on the position of the detector [see equation (A6)].Varying the position of the detector, one can reveal theinfluence of each MBS on the counting statistics (e.g.,the Fano factor and the skewness) of the detector.In our work, we use the Born-Markov master equationbecause it is applicable when both the couplings betweenthe system and the electrodes are weak and each elec-trode has a wide flat energy spectrum. These conditionsare valid in our system. Moreover, in studying the count-ing statistics of the SET island, we need to calculate the n -resolved reduced density matrix elements of the SET is-land [see equation (10)]. They are conveniently obtainedusing the master equation approach.In summary, we have proposed an experimentally ac-cessible approach to probe the MBSs via the countingstatistics of a charge detector in the form of a SET. Westudy the effects of both the local coupling (to an adja-cent MBS at one end of the nanowire) and the nonlocalcoupling (to a MBS at the other end of the nanowire) on the counting statistics of the SET island. We findthat in the presence of the MBSs, the minimum point ofboth the Fano factor and the skewness shifts appreciablyfrom a symmetric SET configuration to an asymmetricone. This feature persists even when varying the non-local coupling to the farther MBS or the pairing energybetween the two MBSs. These MBS-induced shifts canbe used as signatures of the MBSs. Moreover, becauseour approach is based on the FCS, it can be readily gen-eralized to higher-order cumulants to study if they canalso be used to probe the MBSs. Acknowledgements
This work is supported by the National Natural Sci-ence Foundation of China Grant Nos. 91121015 and11404019, the National Basic Research Program ofChina Grant No. 2014CB921401, the NSAF GrantNo. U1330201, Hong Kong GRF Grant No. 501213,and China Postdoctoral Science Foundation GrantNo. 2012M520146.
Appendix A: Derivation of the tunnelingHamiltonian
For the two MBSs at the ends of a 1D p -wave super-conductor, which can form at the interface between asemiconductor nanowire with strong spin-orbit couplingand an s -wave superconductor , the Majorana oper-ator can be defined as γ i = X σ Z dx (cid:2) f ∗ iσ ( x ) ψ σ ( x ) + f iσ ( x ) ψ † σ ( x ) (cid:3) , (A1)where f iσ ( x ), i = L or R , is the Majorana wave functionand ψ σ ( x ) is the superconductor electron field operatorwith spin σ (= ↑ , ↓ ).From equation (A1), it follows that the Majorana op-erator satisfies γ † i = γ i . The anticommutation relationfor the Majorana operators can be obtained as { γ i , γ j } = Z Z dxdy (cid:2) f ∗ i ↑ ( x ) f j ↑ ( y ) n ψ ↑ ( x ) , ψ †↑ ( y ) o + f i ↑ ( x ) f ∗ j ↑ ( y ) n ψ †↑ ( x ) , ψ ↑ ( y ) o + f ∗ i ↓ ( x ) f j ↓ ( y ) n ψ ↓ ( x ) , ψ †↓ ( y ) o + f i ↓ ( x ) f ∗ j ↓ ( y ) n ψ †↓ ( x ) , ψ ↓ ( y ) o (cid:3) = Z dx (cid:2) f ∗ i ↑ ( x ) f j ↑ ( x ) + f i ↑ ( x ) f ∗ j ↑ ( x )+ f ∗ i ↓ ( x ) f j ↓ ( x ) + f i ↓ ( x ) f ∗ j ↓ ( x ) (cid:3) = (cid:26) i = j, i = j, (A2)because of the anticommutation relations for thefermionic field operators n ψ α ( x ) , ψ † β ( y ) o = δ αβ δ ( x − y ) , (A3) { ψ α ( x ) , ψ β ( y ) } = 0 , (A4)and the relations of the completeness and orthogonalityof the Majorana wave functions Z dx X σ f iσ ( x ) f ∗ jσ ( x ) = δ ij . (A5)Obviously, it follows from equation (A2) that γ i = { γ i , γ i } = 1 . In the Nambu representation of the superconduc-tor electron field operator, Ψ = (cid:16) ψ ↑ , ψ ↓ , ψ †↓ , ψ †↑ (cid:17) .Projecting these field operators onto the mani-fold of Majorana states, we have Ψ ( x ) = P i γ i h f i ↑ ( x ) , f i ↓ ( x ) , f ∗ i ↓ ( x ) , f ∗ i ↑ ( x ) i . The electron tunnelingsbetween the MBSs and the SET island are then describedby the Hamiltonian H t = X σ Z dx [ t ∗ ( x ) d † σ ψ σ ( x ) + t ( x ) ψ † σ ( x ) d σ ]= X iσ (cid:0) V ∗ iσ d † σ − V iσ d σ (cid:1) γ i , (A6)where V ∗ iσ = R dxt ∗ ( x ) f iσ ( x ) , d σ is the annihilation op-erator of the electron with spin σ in the SET island, and t ( x ) is the position-dependent coupling strength betweenthe MBSs and the SET island. Note that one can alwaysfind suitable linear combinations of d †↑ and d †↓ to formspinless fermions d † coupled to the MBSs. Then, thetunneling Hamiltonian becomes H t = X i (cid:0) g ∗ i d † − g i d (cid:1) γ i , (A7)where operators d † and d are defined as d † = V ∗ i ↑ d †↑ + V ∗ i ↓ d †↓ g ∗ i , d = V i ↑ d ↑ + V i ↓ d ↓ g i . (A8)If g L = g ∗ L = λ and g R = g ∗ R = µ, we have H t = (cid:0) d † − d (cid:1) ( λγ L + µγ R ) . (A9)This is just the Hamiltonian in equation (1), which de-scribes the electron tunnelings between the MBSs andthe SET island. Note that it includes both the local cou-pling λ to the adjacent MBS at one end of the nanowireand the nonlocal coupling µ to the MBS at the otherend of the nanowire. Equation (A7) reduces to the tun-neling Hamiltonian widely used in previous studies (e.g.Refs. 40 and 41 ) by choosing µ = 0. Appendix B: Quantum dynamics of the SET
Applying the Born-Markov approximation and tracingover the degrees of freedom of the electrodes coupled tothe SET island, the master equation of the hybrid MBS-SET system in the Schr¨odinger picture can be obtainedas ˙ ρ = − i [ H sys , ρ ] + Γ S D (cid:2) d † (cid:3) ρ + Γ D D (cid:2) Υ † r d (cid:3) ρ, (B1)where ρ ( t ) is the reduced density operator of the MBS-SET system, and the superoperator D , acting on anysingle operator, is defined as D [ A ] ρ = AρA † − A † Aρ − ρA † A. From equation (B1) and the relations h n | Υ † ρ Υ | n i = ρ ( n − , h n | Υ ρ Υ † | n i = ρ ( n +1) , (B2) h n | Υ † Υ ρ | n i = ρ ( n ) , h n | ΥΥ † ρ | n i = ρ ( n ) , (B3)where n is the number of electrons that have tunneled tothe drain electrode, we obtain the equation of motion foreach n -resolved reduced density matrix element:˙ ρ ( n ) aa = iλ (cid:16) ρ ( n ) ad − ρ ( n ) da (cid:17) − µ (cid:16) ρ ( n ) ad + ρ ( n ) da (cid:17) − Γ S ρ ( n ) aa +Γ D ρ ( n − cc , ˙ ρ ( n ) bb = iλ (cid:16) ρ ( n ) bc − ρ ( n ) cb (cid:17) + µ (cid:16) ρ ( n ) bc + ρ ( n ) cb (cid:17) − Γ S ρ ( n ) bb +Γ D ρ ( n − dd , ˙ ρ ( n ) cc = − iλ (cid:16) ρ ( n ) bc − ρ ( n ) cb (cid:17) − µ (cid:16) ρ ( n ) bc + ρ ( n ) cb (cid:17) + Γ S ρ ( n ) aa − Γ D ρ ( n ) cc , ˙ ρ ( n ) dd = − iλ (cid:16) ρ ( n ) ad − ρ ( n ) da (cid:17) + µ (cid:16) ρ ( n ) ad + ρ ( n ) da (cid:17) + Γ S ρ ( n ) bb − Γ D ρ ( n ) dd , ˙ ρ ( n ) ab = iε M ρ ( n ) ab − ( iλ + µ ) ρ ( n ) db + ( iλ + µ ) ρ ( n ) ac − Γ S ρ ( n ) ab +Γ D ρ ( n − cd , ˙ ρ ( n ) ac = iε I ρ ( n ) ac − ( iλ + µ ) ρ ( n ) dc + ( iλ − µ ) ρ ( n ) ab − Γ S + Γ D ρ ( n ) ac , (B4)˙ ρ ( n ) ad = i ( ε I + ε M ) ρ ( n ) ad + ( iλ + µ ) ρ ( n ) aa − ( iλ + µ ) ρ ( n ) dd − Γ S + Γ D ρ ( n ) ad , ˙ ρ ( n ) bc = − i ( ε M − ε I ) ρ ( n ) bc − ( iλ − µ ) ρ ( n ) cc + ( iλ − µ ) ρ ( n ) bb − Γ S + Γ D ρ ( n ) bc , ˙ ρ ( n ) bd = iε I ρ ( n ) bd − ( iλ − µ ) ρ ( n ) cd + ( iλ + µ ) ρ ( n ) ba − Γ S + Γ D ρ ( n ) bd , ˙ ρ ( n ) cd = iε M ρ ( n ) cd − ( iλ + µ ) ρ ( n ) bd + ( iλ + µ ) ρ ( n ) ca + Γ S ρ ( n ) ab − Γ D ρ ( n ) cd . With the n -resolved matrix elements obtained, the re-duced density matrix elements are given by ρ ij = h i | ρ | j i = P n ρ ( n ) ij , i, j ∈ { a, b, c, d } . ∗ Electronic address: [email protected] † Electronic address: [email protected] Wilczek, F. Majorana returns.
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