Nonlocality and dynamic response of Majorana states in fermionic superfluids
NNonlocality and dynamic response of Majoranastates in fermionic superfluids
I. M. Khaymovich , , J. P. Pekola , and A. S. Melnikov , Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38,01187 Dresden, Germany Institute for Physics of Microstructures, Russian Academy of Sciences - 603950Nizhny Novgorod, GSP-105, Russia Low Temperature Laboratory, Department of Applied Physics, AaltoUniversity School of Science, P.O. Box 13500, FI-00076 Aalto, Finland Chair of Excellence of the Nanosciences Foundation, 23 rue des Martyrs, 38000Grenoble, France Lobachevsky State University of Nizhny Novgorod, 23 Prospekt Gagarina,603950, Nizhny Novgorod, RussiaE-mail: [email protected]
Abstract.
We suggest a microscopic model describing the nonlocal ac responseof a pair of Majorana states in fermionic superfluids beyond the tunnelingapproximation. The time-dependent perturbations of quasiparticle transport areshown to excite finite period beating of the wavefunction between the distantMajorana states. We propose an experimental test to measure the characteristictime scales of quasiparticle transport through the pair of Majorana states defining,thus, quantitative characteristics of nonlocality known to be a generic feature ofMajorana particles.PACS numbers: 74.78.Na, 73.40.-c, 72.90.+y, 72.10.-d,
Keywords : Majorana fermions, superconductivity, nonequilibrium dynamics
Submittedto:
New J. Phys. a r X i v : . [ c ond - m a t . s up r- c on ] D ec onlocality and dynamic response of Majorana states in fermionic superfluids
1. Introduction
Search for Majorana bound states (MBS) has recently become an active topic inthe condensed matter community [1, 2, 3]. These exotic states are known to becharacterized by the coinciding annihilation and creation operators. This is whyit is quite natural to look for such states in superconducting systems where theorder parameter ∆ is known to mix particles (electrons) and anti-particles (holes)because of the Andreev scattering processes. Standard singlet superconductivitystill does not allow the formation of this kind of excitations while the more exotictriplet state can host MBS. Among the available superfluids there exist only a fewpossible candidates for the triplet pairing such as He-3, Sr RuO and heavy fermioncompounds [4, 5]. Alternatively, the effective triplet pairing can be induced, e.g., insemiconducting nanowires [6, 7] in the presence of rather strong spin-orbit couplingand external magnetic field. Despite the clear and reliable observation of zero biaspeaks (ZBP) in the differential conductance measurements [8, 9] and on the changein the charge periodicity of conductance in Coulomb blockade regime [10] consistentwith the existence of MBS it would be extremely important to probe other attributesof these states especially keeping in mind alternative explanations of the ZBP basedon Kondo physics [11]. S wire Right gateLeft gate (cid:2) (cid:2)
R CV V RL L (cid:3) R (cid:3) X Figure 1.
Setup of a possible experiment on Majorana dynamics.
The goal of this paper is to suggest a test revealing the nonlocal dynamic responseof the MBS. This issue has recently become a subject of intensive debate in the contextof so-called quantum teleportation [12, 13, 14, 15, 16]. The Majorana partner statesare localized at the length scales of the order of the coherence length ξ and are usuallystrongly separated provided the distance L between them well exceeds this length ξ (see Fig. 1). From the standard quantum mechanics one could naively expect that thetime τ of the particle transfer between these localized states should be determined bythe inverse tunneling rate roughly proportional to the value ∆ e − L/ξ . Such scenariocan be questioned if we remind that two Majorana states form a single fermioniclevel and, thus, the injected particle should appear simultaneously in both partnerstates [12, 13, 14]. This conclusion is in obvious contradiction with the analysis of thecurrent noise correlations [15, 16]: the latter points towards the existence of a finitecharge transfer time between the MBS. Later on the teleportation phenomenon hasbeen argued to be restored due to the nonlocal coupling via the Coulomb blockage[14]. It was concluded that the key omission of the previous studies was related to thetreating of the superconducting phase as a constant, and not as a dynamic variable.According to the work [14] the recovering of the nonlocal coupling between the MBSshould occur if we consider the phase of the superconducting order parameter as aquantum variable canonically conjugate to the charge of the island.In the present manuscript we show that the previous studies of the nonlocalityin the system of the MBS suffer from another key omission, namely they do not take onlocality and dynamic response of Majorana states in fermionic superfluids τ ∼ ω − caused by their coupling ω . This result imposes restrictionson the time scales of adiabatic manipulation of the Majorana states giving a criterionof their topological protection in time-dependent phenomena. For comparison it isinteresting to mention here the work [17] where the dynamics is governed by time offlight of excitations in the normal metal wire coupled to the MBS.
2. Model
The low frequency dynamics of quasiparticles (QPs) can be described within the time-dependent generalization of the BdG equations (cf. [18]) i ∂∂t ˆ g n = (cid:18) ˆ H − µ ˆ∆ˆ∆ † µ − ˆ H ∗ (cid:19) ˆ g n . (1)Here ˆ H is the normal state Hamiltonian, µ is the chemical potential, and ˆ g n ( r , t ) =( u α,n , v α,n ). The condition of adiabaticity naturally assumes that all the characteristicfrequencies are much lower than the superconducting gap ∆, otherwise a fullnonequilibrium description of a superconductor should be applied [19]. The coefficients u α,n and v α,n are usually interpreted as electronic- and hole- like parts of the QP wavefunctions defined by the Bogolubov transformation,ˆΨ α ( r , t ) = (cid:88) n (cid:0) u α,n ( r , t )ˆ c n + v ∗ α,n ( r , t )ˆ c † n (cid:1) , (2)ˆΨ † α ( r , t ) = (cid:88) n (cid:0) u ∗ α,n ( r , t )ˆ c † n + v α,n ( r , t )ˆ c n (cid:1) . (3)Here α is the spin index and ˆ c † n , ˆ c n are the fermionic QP creation and annihilationoperators, respectively. The index n enumerates the solutions of time-dependent BdGequations for different initial conditions at t = 0 when the expressions (2) take the formof expansion over a certain full set of functions. In equilibrium the time dependenceof the wave functions reduces to the standard form u α,n ( r , t ) = ¯ u α,n ( r ) e − iE n t , v α,n ( r , t ) = ¯ v α,n ( r ) e − iE n t , where E n and (¯ u α,n ( r ) , ¯ v α,n ( r )) are the spectrum andeigenfunctions of the stationary BdG equations. Only the states with E n ≥ g n ( r , t ) may contain the contributions from all positive and negative levels of thestationary Hamiltonian.The Majorana - type states in the stationary case can appear provided we havean isolated eigenfunction satisfying the condition v ∗ α, = u α, corresponding to zeroenergy. The inverse transformation for this zero energy state can specify only the sumof the fermionic operatorsˆ c + ˆ c † (cid:88) α (cid:90) d r (cid:16) u ∗ α, ( r ) ˆΨ α ( r ) + u α, ( r ) ˆΨ † α ( r ) (cid:17) . (4)This relation does not naturally yield the full fermionic operator ˆ c = ˆ γ L + i ˆ γ R butonly its part ˆ γ L = (ˆ c + ˆ c † ) / onlocality and dynamic response of Majorana states in fermionic superfluids γ R ) of the QP operator remains undefined and in this sense the ground state ofthe superconductor with an isolated zero energy mode appears to be degenerate. Theambiguity of the operator ˆ γ R can be resolved by introducing a coupling mechanism ofthe above isolated state either to the second Majorana - type state or to a fermionicbath [16, 20]. Both these mechanisms destroy the symmetry of the isolated level v ∗ α, = u α, and shift its energy from zero. Each Majorana pair of states gives onepositive and one negative energy level. In equilibrium it is natural to keep only thepositive energy level and the corresponding hybridized wave function. Consideringthe nonequilibrium dynamics at a finite time interval t we can no more disregardthe contribution of the negative energy level to the wave function dynamics whenthe energy uncertainty δE ∼ (cid:126) /t exceeds the splitting of levels in a Majorana pair.Thus, despite of the obvious fact that both levels correspond to the only fermionthe nonequilibrium time-dependent solutions ˆ g n ( r , t ) of the BdG equations containcontributions corresponding to both levels.
3. Nonequilibrium dynamics of a pair of Majorana states
To probe the nonlocal dynamics of coupled Majorana states we suggest to studytransport through the wire hosting these MBS at its ends modulated by the changesin the coupling of the wire to the external normal metal leads (see Fig. 1). Anatural way to tune this coupling in conditions of the real experiment (see, e.g.,[21, 22, 10]) is to apply time-dependent voltages at the gate electrodes controllingthe transparencies t L,R ( t ) of the barriers between the wire and the normal lead atthe left (L) and right (R) end, respectively. Tuning these transparencies at the twoends of the wire one can easily determine the spatial correlations in the dynamicresponse of the Majorana partners as well as the scale 1 /τ of the frequency dispersion.Considering a possible experimental setup based on a semiconducting nanowire withinduced superconductivity one should take this system in a topologically nontrivialstate [6, 7] which allows to get the subgap quasiparticle states bound to the wire ends.Further derivation has been carried out by applying a general approach [23] for thesolution of the scattering problem with the quasiparticle waves incoming from the leftor right leads at a certain energy ε and propagating along the one-dimensional p -wavesuperconducting wire hosting two MBS. We focus here on the case of a weak chargingenergy of the wire which is different from the situation studied in Ref. [14]. The p -wave order parameter is chosen in the form ∆( x ) ∼ e iθ p , where θ p = 0 , π is thetrajectory orientation angle. Assuming low energies ( ε, ω (cid:28) ∆) and considering thesolution of Eq. (1) near the left end of the wire one can write it as a superposition g ( r , t ) = e − iεt + ik F s (cid:104) a + L w (1) ( s ) + b + L w (2) ( s ) (cid:105) + e − iεt − ik F s (cid:104) a − L w (1) ( − s ) + b − L w (2) ( − s ) (cid:105) (5)of two independent solutions w (1) ( s ) = e i ˆ σ z θ p / (cid:104) e − D ( s ) / (cid:18) − i (cid:19) + i ε ˜∆ sign( s ) e D ( s ) / (cid:18) i (cid:19)(cid:105) , (6) w (2) ( s ) = e i ˆ σ z θ p / e D ( s ) / (cid:18) i (cid:19) , (7) onlocality and dynamic response of Majorana states in fermionic superfluids s = ( L/
2) cos θ p + x . A similar expression can be written near the rightend of the wire by changing the subscripts L → R and the angle θ p from 0 to π , whichshifts the origin x → x − L corresponding to s ( θ p = 0) > s ( θ p = π ) < v F is the Fermi velocity in the wire, ˜∆ − = v F (cid:82) L/ e − D ( s ) ds , D ( s ) = v F (cid:12)(cid:12)(cid:82) s ∆( s (cid:48) ) ds (cid:48) (cid:12)(cid:12) ∼ | s | ξ , and Pauli matrices ˆ σ k act in the electron – hole Gor’kov –Nambu space. An appropriate matching of the wavefunctions at the wire ends withthe ones in the leads gives us the equations for the coefficients a ± k = e ± iφ k / ( A k ± a k ) / k = L ) and right ( k = R ) wire ends (see Appendix A for details ofcalculations) (Γ k − iε ) A k = F k , ( ˜∆ − iε Γ k / ˜∆) a k = F k . (8)Here for simplicity we neglect the MBS coupling ω ∼ ˜∆ e − L/ξ , Γ k = ˜∆(1 − r k ) / (1+ r k )is the rate characterizing the coupling of wire states to the k th external lead with r k = (cid:112) − | t k | being the real-valued reflection coefficient of the insulating barrier, φ k are the scattering phases. F k = ˜∆ t k / (1 + r k ) ∝ (cid:112) Γ k ˜∆ are the tunneling sourcescharacterizing the incoming QP flows. Applying the Fourier transform with respectto the energy variable ε and considering the parameter ω / ˜∆ ∼ e − L/ξ pertubativelyone can obtain the equations describing the dynamics of a model two-level system inthe time frame (cf. [26, 27]), i.e. the dynamics of the Majorana pair: (cid:18) ∂∂t + Γ L (cid:19) A L + ω A R = F L e − iεt , (9) (cid:18) ∂∂t + Γ R (cid:19) A R − ω A L = F R e − iεt . (10)In the non-stationary regime the localized states at the wire ends (being of Majorananature in the stationary regime) can be described by the wave function amplitudes A k which are in fact the quantum mechanical amplitudes describing the probability tofind the quasiparticle at the k th wire end. The amplitudes a k correspond to the off-resonant fast-decaying contributions from the states above the gap. The amplitudes A k and a k together describe in fact the low frequency dynamics of the functionˆ g n ( r , t ) including contributions from positive and negative levels of the stationaryHamiltonian. Note that in the absence of incoming QP flows, F k = 0, Eqs. (9,10) have purely real-valued coefficients corresponding to the Hermitian nature ofMajorana operators ˆ γ k . In this case the average (cid:104) Ψ † α ( r , t )Ψ α ( r , t ) (cid:105) of the electronnumber operator is conserved since its change is determined by the sum | A L | + | A R | of probabilities | A k | to find the quasiparticle at the k th wire end. This conservationfixes, in particular, the quasiparticle parity number in the wire by fixing the parameter | A L | + | A R | even for non-trivial dynamics of | A k | themselves. Note that thisstatement is independent of a strength of Coulomb interaction as the latter onlygoverns the correlations between tunneling rates. The rates Γ L,R are determinedby the local Andreev reflection processes [16] while the energy splitting of coupledMajorana states ω = ˜∆ e − D ( L/ sin ϕ is related to the probability of the quasiparticletransfer through the system. Parameters D ( L/ ∼ L/ξ and ϕ = k F L + ( φ L − φ R ) / L .The current flowing from the left and right electrodes can be calculated as [28](see also Appendix B for details of calculations) I L,R = e/π (cid:90) g L,R ( ε )( f T ( ε − eV L,R ) − f T ( ε − eV s )) dε , (11) onlocality and dynamic response of Majorana states in fermionic superfluids f T ( ε ) = ( e ε/T + 1) − is the Fermi-Dirac distribution function with the bathtemperature T , g k ( ε ) (cid:39) A k a ∗ k ] = 2 (cid:112) Γ k Re ( A k e iεt ) , (12) V k is the potential of the k th electrode, and V s is the potential of a superconductor.Generally, the definition of the potential V s in a nonstationary problem follows fromthe solution of the equations describing the particular electric circuit [29], e.g., the onein Fig. 1: I L + I R = CdV s /dt + V s /R , where C and R are the capacitance and shuntresistance of the ground connection, respectively. Considering a constant applied bias V = V L − V R and putting A L,R ∝ e − iεt we obtain a dc differential conductance peakat eV (cid:39) ω attributed to MBS [8, 9, 30, 31, 32].
4. Results
We now proceed with the analysis of the dynamic response of a pair of Majoranapartners and consider two generic examples of the time – dependent transport realizedby the modulating tunnel barrier (see Fig. 1): (i) the phase-shifted sinusoidal drivingwith Γ L ( t ) = Γ + ˜Γ cos( ωt ) and Γ R ( t ) = Γ + ˜Γ cos( ωt + φ ); (ii) pump-probe drivingby ∆ t -broadened delta-functional pulses with different amplitudes G tk applied with atime delay τ , i.e., with Γ k ( t ) = G k δ ∆ t ( t ) + G τk δ ∆ t ( t − τ ).To start with, our consideration of the dynamic response of MBS within Eqs. (9,10) through a single fermionic state formed of a superposition of two partner Majoranastates. Indeed, the levels ± ω around the zero energy can be introduced as a basisof hybridized states with the amplitudes A ± = A L ± iA R . In Eqs. (9, 10) eachof quasiparticle sources F k excites both amplitudes A ± simultaneously. Due to thecoupling to the reservoirs both amplitudes evolve then in time as separate quantitiesand, thus, cannot be described as an empty and filled state of a single level. As a result,we find beating of the wavefunction between the edge states at the frequency ω .The above arguments concerning the sources of the injected particles should be validirrespective to the strength of the Coulomb effects and for the sake of simplicity westart our consideration of time - dependent problems from the limit of large capacitance C when these effects can be neglected.Starting from the case of sinusoidal driving we consider for simplicity the acamplitude ˜Γ (cid:28) Γ as a perturbation and solve Eqs. (9, 10). For the zero-biasdifferential conductance we find1 G Γ dI L dV L (cid:12)(cid:12)(cid:12)(cid:12) V L =0 (cid:39) (cid:104) cos ωt + ω − Γ (cid:88) η = ± L η F η − ω (cid:88) η = ± η L η F ηφ − π/ (cid:105) , (13)where G Γ = ( e /π )2Γ / (Γ + ω ), L η = Γ / [( ω + ηω ) + Γ ], F ± φ = cos( ωt + φ ) +sin( ωt + φ )( ω ± ω ) / Γ . One can see that for low-frequencies ω (cid:46) ω the aboveexpression contains an essential phase φ dependence, while with increasing ω thesecontributions decay faster than the other time-dependent terms.Indeed, this statement is clearly visible in the most interesting and representativecase ω ∼ Γ in which dc results [30, 31, 32] (see also (C.1) in Appendix C)are already broadened and inconclusive. In this case to clarify the results we onlocality and dynamic response of Majorana states in fermionic superfluids ω t/2 π ( φ − π / ) / π ω t/2 π (b)(a) dI/dV,e / π Figure 2.
Color plot of differential conductance (14) versus time t and phasedifference φ for (a) ω = ω and (b) ω = 10 ω . The other parameters areΓ = ω = 10˜Γ. One can see strong phase dependence at ω ∼ ω , whichdiminishes as ω grows. rearrange the functions F ± φ = cos( ωt + φ ) + sin( ωt + φ )( ω ± ω ) / Γ = F cφ ± F sφ to F cφ = cos( ωt + φ ) + ( ω/ Γ ) sin( ωt + φ ) and F sφ = ( ω / Γ ) sin( ωt + φ ) getting1 G Γ dI L dV L (cid:12)(cid:12)(cid:12)(cid:12) V L =0 (cid:39) cos ωt + ˜ΓΓ (cid:104) ω − Γ ω { C F c − C F s }− Γ ω (cid:110) C F sφ − π/ − C F cφ − π/ (cid:111)(cid:105) , (14)with the dimensionless coefficients C k = ( πω / ) (cid:80) η = ± ( − η ) k L η accumulating the ω -dependence of the prefactors as follows C ( ω ) = ( ω + ω + Γ ) ω [( ω + ω ) + Γ ][( ω − ω ) + Γ ] , (15) C ( ω ) = 2 ωω [( ω + ω ) + Γ ][( ω − ω ) + Γ ] . (16)Now considering two limits: (a) ω ∼ ω , Γ and (b) ω (cid:29) ω , Γ illustrated inthe corresponding panels of Fig. 2, one can see that in the first limit (a) ω ∼ ω , Γ the above mentioned coefficients C , ∼ ω andthe phase dependent corrections (the last line in (14)) are of order of the main term ˜Γ2Γ cos ωt .In the second limit (b) due to the smallness of C (cid:39) ω /ω and C (cid:39) ω /ω theconductance has relatively small ω /ω phase-dependent corrections to the oscillatingterms 1 G Γ dI L dV L (cid:12)(cid:12)(cid:12)(cid:12) V L =0 (cid:39) ˜Γ2Γ (cid:104) cos ωt + ω − Γ ω F c − ω Γ ω F sφ − π/ (cid:105) . (17)In Fig. 2 the ( t, φ )-dependence of the differential conductance (14) is plotted for thefollowing parameters ω = Γ = 10˜Γ at (a) ω = ω and (b) ω = 10 ω demonstratingthe above mentioned arguments. onlocality and dynamic response of Majorana states in fermionic superfluids ω (cid:29) Γ many beating periods pass before a tunneling eventoccurs leading to the efficient transport of the charge between the localized states A k .This can be in some sense viewed as a signature of “teleportation”. If additionally∆ ω = ω − ω ∼ Γ (cid:28) ω one can neglect the contributions from η = +1 and obtain πe dI L dV L (cid:12)(cid:12)(cid:12)(cid:12) V L =0 (cid:39) ω + ˜ΓΓ ω cos ωt + ˜Γ2Γ Γ ∆ ω + Γ (cid:104) F − + 2Γ ω F − φ − π/ (cid:105) . (18)Clearly this limit describes the sharp peaks at ω in the frequency dependence of thedynamic response with the amplitude that depends on the phase shift. In the oppositelimit of broad peaks the nonlocal correlations in the dynamic response are naturallymore difficult to observe since their contributions in the dynamic response becomesmall when ω / Γ (cid:28) eV L,R = nω ± ω due to theresonant effect similar to the Shapiro phenomenon in Josephson junctions [33]. Notethat the periodic backgate voltage modulation can give another opportunity to observethe resonant features on the current – voltage curve controlling the chemical potentialof the wire as a whole. This modulation should cause the change in the energy splitting ω through its dependence on the Fermi momentum k F . Assuming k F L = ωt to belinear in time one can obtain resonances at eV L,R = nω .In the case of the pump-probe driving the differential conductance of the leftelectrode contains three contributions πe ∆ t dI L dV L = G L δ ∆ t ( t ) + G τL δ ∆ t ( t − τ ) + (cid:113) G L G τL cos( ω τ ) cos( eV L τ ) δ ∆ t ( t − τ ) . (19)Here we impose zero initial conditions on both amplitudes A ± . The terms in the firstline of Eq. (19) correspond to the local charging of the single fermionic level, whilethe term on the second line reflects correlations in the response to two pulses withthe time delay τ and shows the non-trivial dynamics of MBS at frequencies | eV L ± ω | (see Fig. 3). The first pulse excites the quantum beatings between the Majorana edgestates at the frequency ω modifying the response of the system to the second pulse. t τ τ τ LL dVdI Figure 3.
Differential conductance vs delay time τ in two-pulse pump-probesetup. The second pulse amplitude is shown by solid blue line. Taking for the estimate ∆ ∼ . ξ ∼
100 nm for Al, and L (cid:38) µ m we find ω (cid:46)
15 MHz which gives us a reasonable range of frequencies ω ∼ ω of the driveand typical time delay 1 /ω ∼ . µ s for the pump-probe setup. The conditions onbias for the observation of the beating phenomenon are less restrictive comparing tothe ones of a dc conductance peak (with the restriction of V ∼ ω ∼ . µ V [32]) onlocality and dynamic response of Majorana states in fermionic superfluids
L,R (cid:46) ω we should take the barriers with resistances R L,R ∼ . −
5. Discussion and outlook
Certainly, the above dynamic response of the MBS will be modified in Coulombblockade regime. This difference arises from the obvious fact that in the case ofCoulomb blockade the charge tunneling processes between the island and left/rightelectrodes are strongly correlated. The entry and exit of charged particles are alwayscontrolled by the overall charge of the island. However, this correlation doesn’tdestroy the beating phenomenon and cannot cause the formation of a single eigenstateresponsible for the non-local transport through Majorana states (teleportation) for theoperating frequencies above the energy splitting of Majorana partners. Let us takethe limit of high Coulomb energy and imposing, thus, the restriction on two possiblecharge states of the island and assume the operating frequencies and energy splitting ω to be small comparing to the tunneling rates Γ L,R . The latter limit allows oneto consider the charging/discharging processes as instantaneous events changing thefermion parity. On the longer time scales than the injection/ejection rates the fermionparity is fixed due to the fixed electron charge. However, the beating phenomenon asan internal dynamics of Majorana states is present due to the nonequilibrium time-dependent nature of the electron injection and further transformation of the wavefunction of the injected electron into the Andreev eigenstates both with positive andnegative energies. Therefore the current through the system is fully determined bythe interplay of two time scales, namely, the inverse beating frequency ω − and thedelay time τ between the opening of the left/right junctions. The latter is determinedeither by the operating frequency f and the phase shift φ ( τ = ( n + φ/ π ) /f withan integer n value) for the periodic driving or by the delay time τ for the pump-probe experimental setup. Certainly the above comment on the influence of Coulombblockade on the beating phenomenon is only qualitative and should be verified byfurther quantitative analysis based on the use of more elaborated methods takingaccount of the interaction effects.To conclude the solution of the above dynamic problems allows us to predict abeating effect at the frequency ω which is a hallmark of the topologically nontrivialstate of the nanowire. We show that due to the exponentially small coupling ω the MBS are strongly sensitive to any external perturbation. According to ourconsideration any driving of Majorana states with the typical operating frequency ω exceeding ω brings the system to the non-equilibrium regime imposing, thus, animportant restriction on the operating frequencies of such a device The Majorananature of these states needed for quantum calculations recovers only in the adiabaticregime ω (cid:28) ω . On the other hand, the measurement of the characteristic frequencythreshold ω separating the regimes of weak and strong perturbations of the Majoranapairs could be considered as their hallmark characterizing the nonlocality of thesepairs. Certainly the beating phenomenon similar to the one discussed in our workshould appear in other superconducting systems with subgap Andreev states. Todistinguish the beating phenomenon in topological situation from the one caused bythe presence of usual Andreev states it may be helpful to study the behavior of thebeating frequency as a function of system parameters, gate potentials and magneticfield so that to reveal the features peculiar to the topologically protected levels.The beating phenomenon may also affect non-stationary Josephson-type transport onlocality and dynamic response of Majorana states in fermionic superfluids Acknowledgments
We are pleased to thank A. A. Bespalov, Yu. G. Makhlin, C. Marcus, G. E. Volovik,and A. D. Zaikin for valuable comments and A. J. Leggett for correspondence.This work has been supported in part by Microsoft Project Q, by the NanosciencesFoundation, foundation under the aegis of the Joseph Fourier University Foundation(Grenoble, France), by Academy of Finland Projects No. 284594, 272218 (J. P. P.), bythe Russian Foundation for Basic Research and German Research Foundation (DFG)Grant No. KH 425/1-1 (I. M. K.), and by the Russian Science Foundation, Grant No.17-12-01383 (A. S. M.) and Foundation for the advancement of theoretical physics“BASIS”.
Appendix A. Derivation of Eqs. (9, 10)
In this section we present the derivation of the Eqs. (9, 10) from the main text foran exemplary system consisting of a one dimensional (1D) p -wave superconducting(S) wire of the length L connected to the left and right one-dimensional normal-metalleads. We choose the x axis along the wire, the origin to be in the middle of the wireand the order parameter in the form ∆ ∝ ˆ k x + i ˆ k y . Such system is known to hostthe subgap edge states at rather small energies ± ω ∼ ± ∆ e − L/ξ . To describe theselocalized states we start from the quasiclassical version of the Bogolubov – de Gennesequations, i.e., Andreev equations for the envelopes w = ( u, v ) of the electron and holewaves propagating along the quasiclassical trajectory k = k F (cos θ p , sin θ p ) − iv F ˆ σ z ∂∂s w + ˆ σ x ∆( s ) w = εw (A.1)where v F is the Fermi velocity in the wire, s = ( L/
2) cos θ p + x is the coordinatein the wire along the trajectory, and Pauli ˆ σ k matrices act in the electron – holeGor’kov – Nambu space, Considering the p -wave symmetry of superconducting orderparameter one can put ∆( x ) ∼ e iθ p . Note that in 1D geometry of the p -wave S wireit is natural to align the trajectory in the positive or negative direction of the x axiswhich correspond to θ p = 0 , π . The phase θ p can be removed from the gap operator∆ by the standard transformation u ( x ) → u ( x ) e iθ p / and v ( x ) → v ( x ) e − iθ p / . Appendix A.1. Low energy modes inside the wire
Considering the low energy modes with ε (cid:28) ˜∆ ∼ ∆ inside the wire one can takethe sum of two independent solutions w (1 , ( s ) of Andreev equations (A.1) found inRefs. [24, 25] w (1) ( s ) = e i ˆ σ z θ p / (cid:104) e − D ( s ) / (cid:18) − i (cid:19) + i ε ˜∆ sign( s ) e D ( s ) / (cid:18) i (cid:19)(cid:105) , (A.2) w (2) ( s ) = e i ˆ σ z θ p / e D ( s ) / (cid:18) i (cid:19) , (A.3) onlocality and dynamic response of Majorana states in fermionic superfluids ε is the energy variable, ˜∆ − = 2 (cid:82) L/ e − D ( s ) ds/v F and D ( s ) = 2 v F (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s ∆( s (cid:48) ) ds (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ∼ | s | ξ . (A.4)The full wave function near the left end of the wire being an eigenfunction of thestationary version of Bogolubov – de Gennes equations (1) in the main text can bewritten as a combination of the above envelopes with the corresponding oscillatingfactors e − iεt ± i k · r for the left and right movers with certain coefficients a ± k and b ± k g ( r , t ) = e − iεt + ik F s (cid:104) a + L w (1) ( s ) + b + L w (2) ( s ) (cid:105) + e − iεt − ik F s (cid:104) a − L w (1) ( − s ) + b − L w (2) ( − s ) (cid:105) . (A.5)A similar expression can be also written near the right end of the wire by changingthe subscripts L → R and the angle θ p from 0 to π , which shifts the origin x → x − L corresponding to s ( θ p = 0) > s ( θ p = π ) <
0. Matching the wave functions ofthe left and right movers one can find the equations for the coefficients ib ± R = a ± L e − D ( L/ ± ik F L ∓ ε ˜∆ a ± R , (A.6) − ib ± L = a ± R e − D ( L/ ∓ ik F L ∓ ε ˜∆ a ± L . (A.7)Here for simplicity we assume the following symmetry D ( x + L/ − D ( L/
2) = D ( L/ − D ( x − L/
2) originated from the assumption of a symmetric order parameter∆( L − s ) = ∆( s ). As a result, we get a smooth function describing the solution withinthe interval | x | < L/ g ( x ) = (cid:88) η = ± e − iεt + iηk F x (cid:104) a ηL e iηk F L/ e − D ( x + L/ / (cid:18) − i (cid:19) + ia ηR e − iηk F L/ e − D ( x − L/ / (cid:18) i (cid:19)(cid:105) . (A.8)At the ends of the wire we should put g ( − L/
2) = (cid:88) η = ± (cid:18) a ηL + b ηL − ia ηL + ib ηL (cid:19) e − iεt + iη , (A.9) g ( L/
2) = i (cid:88) η = ± (cid:18) a ηR + b ηR ia ηR − ib ηR (cid:19) e − iεt + iη , (A.10)where we marked the left (right) movers by the exponents e ± i . One can see that inthe vicinity of the wire ends the wave function exhibits a “jump” which occurs at thelength scale of the coherence length ξ [24, 25]. Appendix A.2. Scattering problem
As a next step we use the scattering matrix approach to get the solution of ascattering problem for an electron plane wave α L ( R ) e ± ik F x incident from the left orright normal electrode. Note that it is enough to consider only incoming electrons, butnot holes, if one integrates over the whole energy interval of the Fermi distributionto calculate the current. Moreover all the sources should be considered separatelyby putting only one of them to be non-zero at the same time and summing overall contributions in the observable to avoid any fake interference effects. Assuming onlocality and dynamic response of Majorana states in fermionic superfluids (cid:184)(cid:184)(cid:185)(cid:183)(cid:168)(cid:168)(cid:169)(cid:167) (cid:16)(cid:16) (cid:14)(cid:14) )( * LLL L baiT (cid:68) (cid:184)(cid:184)(cid:185)(cid:183)(cid:168)(cid:168)(cid:169)(cid:167) (cid:16)(cid:16) (cid:14)(cid:14) (cid:14)(cid:14) (cid:16)(cid:16) )( )(
LL LLLLL bai baRT (cid:68) (cid:184)(cid:184)(cid:185)(cid:183)(cid:168)(cid:168)(cid:169)(cid:167) (cid:16)(cid:16) (cid:14) (cid:14)(cid:14)(cid:16)(cid:16) )( )( * LLL LL baiR ba (cid:184)(cid:184)(cid:185)(cid:183)(cid:168)(cid:168)(cid:169)(cid:167) (cid:14)(cid:14)(cid:16) (cid:16)(cid:16) )(/ * LLLLLLL baTTTR (cid:68) (cid:184)(cid:184)(cid:185)(cid:183)(cid:168)(cid:168)(cid:169)(cid:167) (cid:16)(cid:16) (cid:14) (cid:16)(cid:16)(cid:14)(cid:14) )( * RRR RR baiR ba (cid:184)(cid:184)(cid:185)(cid:183)(cid:168)(cid:168)(cid:169)(cid:167) (cid:16)(cid:16) (cid:14)(cid:14) (cid:16)(cid:16) (cid:14)(cid:14) )( )(
RR RRRRR bai baRT (cid:68) (cid:184)(cid:184)(cid:185)(cid:183)(cid:168)(cid:168)(cid:169)(cid:167) (cid:16)(cid:16) (cid:16)(cid:16) )( * RRRR baiT (cid:68) (cid:184)(cid:184)(cid:185)(cid:183)(cid:168)(cid:168)(cid:169)(cid:167) (cid:14)(cid:14)(cid:16) (cid:14)(cid:14) )(/ * RRRRRRR baTTTR (cid:68) superconducting wire left normal lead right normal lead x
Figure A1.
The scheme shows the electron (upper lines in the brackets) andhole (lower lines) amplitudes in the left and right leads and in the vicinity ofthe interfaces of the superconducting wire in the scattering problem with theamplitudes α L ( α R ) of incoming electronic waves from the left (right) normallead. Right (left) arrows correspond to the factors e ( ± ) ik F x in the full wavefunction (A.8). By matching the amplitudes in the latter equation with thoseshown in this figure one can obtain the matching conditions (A.9,A.10). the absence of the electron-hole conversion in the barriers and using the electron-hole symmetry in a superconductor one can separate complex conjugate electron andhole blocks in the total scattering matrix of the k th barrier ˆ S k = (cid:16) s k s ∗ k (cid:17) . Wetake a standard representation of the unitary matrix s k = (cid:16) R k T k T k − R ∗ k T k /T ∗ k (cid:17) whichtransforms the incoming electron plane waves from the superconductor (e.g., a − L + b − L for k = L ) and from the normal reservoir ( α L ) to the outgoing ones ( a + L + b + L and u L = − α L R L T L /T ∗ L + T L ( a − L + b − L ) for k = L ) at both interfaces (see Fig. A1 forall notations). Here R L = r L e iφ L , R R = r R e − iφ R and T k are the reflection andtransmission matrix coefficients, r k = (cid:112) − | T k | and φ k are reflection amplitude andphase.The scattering matrices impose the following boundary conditions on the planewave amplitudes T L α L + R L ( a − L + b − L ) = a + L + b + L , (A.11) R ∗ L ( a + L − b + L ) = a − L − b − L , (A.12) T R α R + R R ( a + R + b + R ) = a − R + b − R , (A.13) R ∗ R ( a − R − b − R ) = a + R − b + R . (A.14)Substituting Eqs. (A.6,A.7) and introducing the notations a ± k = e ± iφ k / ( A k ± a k ) / Γ L − iε ω i ˜ ω − ω Γ R − iε i ˜ ω i ˜ ω Γ L − iε ω i ˜ ω − ω Γ R − iε A L A R a L a R = ˜∆ ρ L α L ρ R α R ρ L α L ρ R α R , (A.15)where Γ k = ˜∆ ρ k , ρ k = (1 − r k ) / (1 + r k ), ω = ˜∆ e − D ( L/ sin ϕ , ˜ ω = ˜∆ e − D ( L/ cos ϕ , ϕ = k F L + ( φ L − φ R ) /
2. The phase χ k of the transmission coefficients T k = | T k | e iχ k doesn’t affect any measurable quantity, therefore we choose it equal to χ k = φ k / ω ) dynamics is to replace theenergy ε by the time derivative i∂/∂t . In the isolated wire, ρ k →
0, the fast decayingmodes a k ∼ α k ρ k disappear as they correspond to the states of the continuous onlocality and dynamic response of Majorana states in fermionic superfluids ε = ± ω and correspond to the beating between A L and A R in the time domain (see Eqs. (A.16, A.17) below). Assuming naturally thatΓ k ω (cid:28) ˜∆ one can find that fast decaying modes a k ≈ α k ρ k − i ˜ ω ρ k A k (cid:48) ∓ ω ρ k ρ k (cid:48) α k (cid:48) corresponding to the continuous spectrum contributions give small corrections in e − L/ξ to the equations for the low-decaying ones A k . Here and further k (cid:48) = R ( L ) for k = L ( R ). Indeed, this leads to a relative renormalization of the decay rates Γ k and sources ρ k by a small values ∼ ω / ˜∆ and to the addition of the α R ( L ) sourceproportional to ˜ ω / ˜∆ ∼ e − L/ξ to the equation for A L ( R ) . All these terms correspondsto a direct tunneling of electron(s) from the lead to the opposite end of the wire.Further we neglect these contributions taking into account only a local tunneling fromthe k th leads to the k th end of the wire and considering therefore only first twoequations for the amplitudes A L,R of Majorana states in (A.15) without a k .Transforming the equations to the Schr¨odinger representation one can obtainEqs. (9, 10) from the main text (cid:18) ∂∂t + Γ L (cid:19) A L + ω A R = F L e − iεt , (A.16) (cid:18) ∂∂t + Γ R (cid:19) A R − ω A L = F R e − iεt . (A.17)with the choice of sources F k = ˜∆ ρ k α k appropriate to the replacement ε → i∂/∂t .Beyond the stationary regime one can consider the parameters ω , Γ k and ρ k to betime-dependent keeping the Eqs. (A.16, A.17) intact for the typical frequency ω of thedrive small compared to the gap ω (cid:28) ˜∆.Note that the equations of motion for Majorana amplitudes A k in the Heisenbergrepresentation (see the first two lines in Eqs. (A.15)) correspond to the scatteringmatrix through a scatterer with an internal structure described in [36] and applied forthe p -wave superconducting wire, e.g., in the Refs. [16, 27]. Appendix B. Expression for the differential conductance
In this section we consider for simplicity only the case of the non-zero left source α L , since the results for the right source can be derived using the symmetry L ↔ R .According to [28] the energy resolved contribution to the differential conductance g k of the k th interface can be written as a sum of the quasiparticle flows of the left andright moving electrons and holes with the corresponding signs g k ( ε ) = 1 − R ek + R hk = | a + k + b + k | + | a + k − b + k | − | a − k + b − k | − | a − k − b − k | . (B.1)We used here the conservation of the quasiparticle flow at the interface which resultsin the unitarity of the scattering matrix. Substituting the expressions for b ± k (A.6,A.7)and for a ± k = e ± iφ k / ( A k ± a k ) / A k and a k into the Eq. (B.1)one can obtain g k ( ε ) = 2˜∆ Re[ A k a ∗ k (cid:16) ˜∆ + ε (cid:17) − A k A ∗ k (cid:48) ε ˜ ω − iA k (cid:48) a ∗ k εω − a k (cid:48) a ∗ k ε ˜ ω − iA k a ∗ k (cid:48) εω + A k (cid:48) a ∗ k (cid:48) ( ω + ˜ ω )] . (B.2) onlocality and dynamic response of Majorana states in fermionic superfluids ω / ˜∆ , ˜ ω / ˜∆ ∼ e − L/ξ (see theprevious section) one can keep only the first term in the latter equation g k ( ε ) = 2Re[ A k a ∗ k ] + O (cid:18) A k ε ˜∆ e − L/ξ (cid:19) . (B.3)In the stationary regime one can express both A k and a k from Eqs. (A.15) A k = ˜∆ ρ k α k (Γ k (cid:48) − iε ) ∓ ωρ k (cid:48) α k (cid:48) (Γ L − iε )(Γ R − iε ) + ω (B.4) a k ≈ α k ρ k (B.5)and show that Eq. (B.3) transforms into Eq. (C.1) of the next section, due toincoherence of the left and right sources ( α L α ∗ R → L/ξ .In general to calculate the zero temperature differential conductance g k ( eV ) ofthe k th interface of the wire in the system with time-dependent parameters one shouldsolve equations (A.16, A.17) and substitute the solutions A k into the Eq. (B.3) togetherwith the expression (B.5) for a k . Appendix C. Dc differential conductance
Here we consider the dc transport for a constant applied bias V = V L − V R using theformalism of the previous section and putting A L,R ∝ e − iεt . As a result we obtain g L,R ( ε ) = 2Γ L,R (Γ R,L ω + Γ L,R ( ε + Γ R,L ))( ε − ω − Γ L,R Γ R,L ) + ε (Γ L,R + Γ
R,L ) . (C.1)It is convenient to discuss separately the limits R → ∞ and R →
0. For the first limitthe zero-temperature differential conductance of the device in the symmetric caseΓ L = Γ R and V R = − V L takes the form dI/dV = e g L ( eV / / π with I = I L = − I R and g L ( ε ) = g R ( ε ). In the opposite limit of R → dI L,R /dV
L,R = e g L,R ( eV L,R ) /π .Thus, in both limits we obtain the conductance peak near the zero bias at eV ∼ ω .It is this peak which is usually considered [8, 9] as an experimental evidence for theMajorana states in semiconducting wires with the induced superconducting order. Thenonlocal nature of the Majorana pair reveals itself in the zero bias dip which is of coursesmeared due to finite rates Γ L,R of tunneling to the fermionic baths. As a result, for theexponentially small splitting ω the dip completely disappears for Γ L ∼ Γ R ∼ ω andcan survive only in a rather exotic limit of strong asymmetry between the couplingsto the left and right reservoirs. The latter situation can be realized, in particular, forthe dip in the curve dI L /dV L for R → R = 0 [30, 31, 32]. A more realisticcase with both nonzero tunneling rates and the peak broadening due to the finitetemperature and inelastic effects can make the experimental observation of the ω scale in dc transport difficult. References [1] J. Alicea, Rep. Prog. Phys. , 076501 (2012).[2] C. W. J. Beenakker, Annu. Rev. Cond. Mat. Phys. , 113 (2013).[3] S. R. Elliott and M. Franz, Rev. Mod. Phys. , 137 (2015).[4] M. A. Silaev, G. E. Volovik, Zh. Eksp. Teor. Fiz. , 1192 (2014) [JETP , 1042 (2014)]. onlocality and dynamic response of Majorana states in fermionic superfluids [5] C. Kallin, Rep. Prog. Phys. , 042501 (2012).[6] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. , 177002 (2010).[7] T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. B , 144522 (2011).[8] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. Bakkers, L. P. Kouwenhoven, Science , 1003 (2012).[9] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Nat. Phys. , 887 (2012).[10] S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nygerd, P.Krogstrup, and C. M. Marcus Nature , 2016 (2016).[11] E. J. H. Lee, X. Jiang, R. Aguado, G. Katsaros, C. M. Lieber, and S. De Franceschi, Phys. Rev.Lett. , 186802 (2012).[12] G. W. Semenoff and P. Sodano, J. Phys. B: At. Mol. Opt. Phys. , 1479 (2007).[13] S. Tewari, C. Zhang, S. Das Sarma, C. Nayak, and D.-H. Lee, Phys. Rev. Lett. , 027001(2008).[14] L. Fu, Phys. Rev. Lett. , 056402 (2010).[15] C. J. Bolech and E. Demler, Phys. Rev. Lett. , 237002 (2007).[16] J. Nilsson, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. Lett. , 120403 (2008).[17] Dahan et al. Phys. Rev. B , 201114(R) (2017).[18] Ketterson, J. B. and Song, S. N. Superconductivity . (Cambridge Univ. Press, 1st edit.,Cambridge, 1999).[19] Kopnin, N. B.
Theory of Nonequilibrium Superconductivity . (Oxford Univ. Press, Oxford, 2001).[20] J. Li, G. Fleury, and M. B¨uttiker Phys. Rev. B , 125440 (2012).[21] W. Chang, S. M. Albrecht, T. S. Jespersen, F. Kuemmeth, P. Krogstrup, J. Nyg˚ard, and C. M.Marcus, Nat. Nanotech. , 232 (2014).[22] P. Krogstrup, N. L. B. Ziino, W. Chang, S. M. Albrecht, M. H. Madsen, E. Johnson, J. Nyg˚ard,C. M. Marcus, and T. S. Jespersen, Nat. Mat. , 400 (2015).[23] D. Averin, A. Bardas, Phys. Rev. Lett. , 1831 (1995).[24] N. B. Kopnin, A. S. Mel’nikov, and V. M. Vinokur, Phys. Rev. B , 054528 (2003).[25] N. B. Kopnin, A. S. Mel’nikov, V. I. Pozdnyakova, D. A. Ryzhov, I. A. Shereshevskii, and V.M. Vinokur, Phys. Rev. B , 024514 (2007).[26] M. G. Vavilov, V. Ambegaokar, I. L. Aleiner, Phys. Rev. B , 195313 (2001).[27] B. Tarasinski, D. Chevallier, Jimmy A. Hutasoit, B. Baxevanis, and C. W. J. Beenakker, Phys.Rev. B , 144306 (2015).[28] G. E. Blonder, M. Tinkham and T. M. Klapwijk, Phys. Rev. B , 1925 (1998).[30] R. Hutzen, A. Zazunov, B. Braunecker, A. L. Yeyati, and R. Egger, Phys. Rev. Lett. , 166403(2012).[31] R. Egger, A. Zazunov, and A. L. Yeyati, Interaction Effects on Transport in Majorana Nanowires,in Topological Insulators: Fundamentals and Perspectives, ed. by Frank Ortmann, StephanRoche and Sergio O. Valenzuela, Wiley-VCH Verlag, Weinheim, 2015.[32] P. A. Ioselevich and M. V. Feigel’man, New J. Phys. , 055011 (2013).[33] Tinkham, M. Introduction to Superconductivity . (Dover Publications, 2nd edit., 2004).[34] R. S. Deacon, J. Wiedenmann, E. Bocquillon, F. Dominguez, T. M. Klapwijk, Ph. Leubner, Ch.Br¨une, E. M. Hankiewicz, S. Tarucha, K. Ishibashi, H. Buhmann, L. W. Molenkamp, arXiv:1603.09611.[35] F. Dominguez, O. Kashuba, E. Bocquillon, J. Wiedenmann, R. S. Deacon, T. M. Klapwijk, G.Platero, L. W. Molenkamp, B. Trauzettel, E. M. Hankiewicz arXiv:1701.07389.[36] M. G. Vavilov, V. Ambegaokar, and I. L. Aleiner Phys. Rev. B63