ac Josephson Effect in Finite-Length Nanowire Junctions with Majorana Modes

Pablo San-Jose,1 Elsa Prada,2 and Ramon Aguado2

1Instituto de Estructura de la Materia (IEM-CSIC), Serrano 123, 28006 Madrid, Spain2Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), Cantoblanco, 28049 Madrid, Spain

(Received 22 February 2012; published 18 June 2012)

It has been predicted that superconducting junctions made with topological nanowires hosting

Majorana bound states (MBS) exhibit an anomalous 4�-periodic Josephson effect. Finding an experi-

mental setup with these unconventional properties poses, however, a serious challenge: for finite-length

wires, the equilibrium supercurrents are always 2� periodic as anticrossings of states with the same

fermionic parity are possible. We show, however, that the anomaly survives in the transient regime of the

ac Josephson effect. Transients are, moreover, protected against decay by quasiparticle poisoning as a

consequence of the quantum Zeno effect, which fixes the parity of Majorana qubits. The resulting long-

lived ac Josephson transients may be effectively used to detect MBS.

DOI: 10.1103/PhysRevLett.108.257001 PACS numbers: 74.50.+r, 03.65.Vf, 73.21.Hb

Recently, it has been argued that Majorana bound states(MBS) should appear in topological insulators [1] andsemiconductors with strong spin-orbit (SO) coupling[2–5] which, in proximity to s-wave superconductors,may behave as topological superconductors (TS). MBSin these TS can be understood as Bogoliubov–de Gennes(BdG) quasiparticles appearing inside the superconductinggap, exactly at zero energy (for a review, see Ref. [6]). TheTS phase is tunable, which has spurred a great deal ofexperimental activity toward detecting MBS. Tunnelingthrough such TS is expected to show a zero-bias anomalysignaling the presence of MBS [7–13]. However, suchanomaly, which has been recently observed [14], onlyproves a necessary condition. A more stringent test canbe established by measuring the Josephson effect through ajunction between two TS nanowires. Kitaev predicted [15]that such a Josephson effect has an anomalous 4� period-icity in the superconducting (SC) phase difference � ��1 ��2 between the two wires. This fractional Josephson

current IJ � sinð�2Þ [4,5,15–20] can be understood in terms

of fermion parity (FP): if FP is preserved, there is a protectedcrossing of Majorana states at � ¼ � [21] with perfectpopulation inversion, namely the system cannot remain inthe ground state as � evolves from 0 to 2� adiabatically,unlike for standard Andreev bound states (ABS) [22].However, any finite length of the two TS regions givesrise to two additional MBS at the wire ends (which areconnected to topologically trivial regions), allowing for thehybridization of two states of the same FP. This results inresidual splittings at � ¼ � which, despite being exponen-tially small, destroy the fractional effect as the systemremains in the ground state for all � [24].

The 4� periodicity can be restored in two ways. On theone hand, one may employ a fully topological circuit [25],which is free of additional MBS. In other words, byemploying an S0NS0 geometry (where S0 is a TS and Nis a normal, nontopological region), as opposed to an

SS0NS0S junction (where S is a topologically trivialsuperconductor). Unfortunately, this is presently a difficultexperimental challenge using nanowires. Alternatively, the4� Josephson effect may be recovered by biasing thejunction and thereby sweeping � fast enough through theanticrossing such that Landau-Zener (LZ) processes [26]induce nonadiabatic transitions between states of the sameFP. By the same token, however, transitions to the contin-uum of states above the gap and, hence, parity mixingprocesses will be unavoidable [27].The above hindrances pose relevant questions. Is the

Josephson effect a useful tool to detect MBS in realisticTS wires? Do finite TS length and quasiparticle poisoninginevitably destroy the fractional periodicity? We here provethat the ac currents in the TS phase contain anomalouslylong-lived 4�-periodic transients which are tunable throughboth bias and gate voltages. Thus, the Josephson effect maystill be exploited to provide unequivocal proof of the exis-tence of MBS in finite-length TS. Interestingly, the durationof the long 4�-periodic transients may be increased by thewell-known, though somewhat counterintuitive, quantumZeno effect, whereby a strong coupling to a decoheringenvironment helps to confine the dynamics of a quantumsystem into a desired sector of the Hilbert space [28].Low-energy description.—Consider a semiconducting

one-dimensional wire with chemical potential �S0 , SO cou-pling �, and Zeeman splitting B (given by B ¼ g�BB=2,where B is an in-plane magnetic field, �B is the Bohrmagneton and g is the wire g-factor). When the wire isproximity coupled to an s-wave superconductor of gap�S, aSC pairing term�S0 <�S may be induced. The wire is thenarranged into a Josephson device as illustrated in Fig. 1(a).The resulting hybrid system SS’NS’S can be driven intoa TS phase where two MBS are formed at the ends of

each S0 region, denoted by �1;2 and �3;4, when B>Bc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

S0 þ �2S0

q[4,5]. The low-energy Majorana sector appears

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for energies " < �eff , where �eff is the effective SC gap[Fig. 2(b)].

Isolated MBS �i are zero-energy superpositions of aparticle and a hole. A pair of Majoranas �1;2 may be fused

into a Dirac fermion cy ¼ ð�1 � i�2Þ=ffiffiffi2

p, such that the

operator 2cyc� 1 ¼ 2i�1�2 with eigenvalues�1 and 1 instates j0i and j1i ¼ cyj0i defines FP [29]. Thus, a spatialoverlap of two MBS hybridizes them into eigenstates ofopposite FP. In a TS wire of length LS0 , the decay distanceof the two MBS pinned at the wire ends is the effectivecoherence length �eff ¼ @vF=��eff . Their overlap will

induce a splitting �� �effe�LS0=�eff . Similarly, two MBS

at either side of a Josephson junction with phase difference� and transparency T will hybridize into even or odd

fermion states with energies �� �eff

ffiffiffiffiT

pcos�=2,

[16,17,27]. In the setup of Fig. 1, four MBS (two ‘‘inner’’�2;3 and two ‘‘outer’’ �1;4) hybridize both through the

Josephson junction (region N) and the finite length S0regions. The resulting eigenstates are empty and filled

states of two Dirac fermions dy1;2ð�Þ, constructed as two

�-dependent (orthogonal) superpositions of the two fermi-

ons cyin ¼ ð�2 þ i�3Þ=ffiffiffi2

pand cyout ¼ ð�1 þ i�4Þ=

ffiffiffi2

p,

which are themselves obtained from the fusion of the innerand outer MBS, respectively. We denote eigenenergies byEn1n2 and eigenstates as jn1n2i, where n1; n2 ¼ 0; 1 are the

occupations of fermions dy1 and dy2 . Two of them have even

total parity, j#ei � j00i, j"ei � j11i ¼ dy2dy1 j00i, and the

other two are odd, j#oi � j10i ¼ dy1 j00i, j"oi � j01i ¼dy2 j00i [33]. En1n2ð�Þ anti-cross at � ¼ � within same-

parity sectors [Fig. 1(b)] and, hence, the supercurrents are2� periodic.A 4�-periodic Josephson effect can, nevertheless, be

recovered by inducing LZ transitions with a voltage bias,such as j#ei ! j"ei [green arrow in Fig. 1(b)] [34]. Todescribe the response of a realistic biased junction, how-ever, an extension of the simplified Majorana model aboveis required. Indeed, nonadiabatic driving may also induceinelastic transitions into delocalized states above the TSgap [yellow arrow in Fig. 1(b)]. These latter transitionsinduce an effective parity-mixing rate (yellow wigglyarrow) which couples even and odd sectors (quasiparticlepoisoning). A proper description of such dynamicsinvolves a calculation of all the Andreev levels (both belowand above �eff) coupled to the continuum (above �S) ofthe junction, which we develop in what follows.Andreev levels.—The full spectrum of single-particle

eigenstates may be obtained by diagonalizing theBdG equations for the geometry in Fig. 1(a) [23]. We

obtain HBdG ¼ 12

Pnðdyndn � dnd

yn Þ"n, with eigenenergies

"nð�Þ plotted in Figs. 2(a) and 2(b) for a representativejunction. The corresponding single particle excitations,called Andreev bound states (ABS), are defined as

0 20

1

2

3

4

5

6

S'

Topologically triviala

0 20

1

2

3

4

5

6

S'

Topologically nontrivialb

FIG. 2 (color online). Andreev bound states (ABS) for aSS’NS’S junction with normal conductance G ¼ 0:72G0 (withG0 ¼ 2e2=h), an induced gap �S0 ¼ 218 �eV, and wire’s lengthof 4 �m (where lso ¼ 216 nm for a InSb wire). In the topologi-cally trivial phase (a), B ¼ 0:36 meV ¼ 0:5Bc, whereas in thenontrivial phase (b), B ¼ 1:1 meV ¼ 1:5Bc. The dashed linedenotes the wire effective gap �eff ¼ 130 �eV that separateslocalized ABS from the quasicontinuum.

FIG. 1 (color online). (a) Schematics: A nanowire of lengthL ¼ LS0 þ LN þ LS0 in contact with two s-wave superconduc-tors (with gap �S), develops a proximity-induced SC gap�S0 <�S and four Majorana modes �1;2;3;4. The transparency

of the junction can be controlled by a gate voltage Vg. (b) Energy

of the four lowest many-body states in the TS phase, as afunction of SC phase difference � across N. They correspondto even (solid) and odd (dashed) fillings of the two lowest ABS,see Fig. 2(b). Note the avoided crossing of size �� at � ¼ � andthe detachment gap �0 at � ¼ 0.

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jnð�Þi ¼ dyn ð�Þj�ð�Þi, with j�ð�Þi denoting the groundstate. In the short junction limit,LN � �eff , only two ABSlie below the effective gap�eff . Upon crossing into the TSphase, these two levels, initially localized around the Nregion in the nontopological phase [purple and orange(bottom) curves in Fig. 2(a)], reconnect into the two

dy1;2ð�Þ Majorana branches. The lowest Majorana branch

[blue (bottom) curve in Fig. 2(b), level dy1 ] has a weak �energy dependence, and is dominated by the outer MBS

(cyout fermion, "1 � �effe�LS0=�S0 ), whereas the one at

higher energy [red (second from bottom) curve, dy2 ] resultsmostly from the fusion of the inner MBS (cyin fermion,

"2 � �eff

ffiffiffiffiT

pcos�=2).

Due to the finite length LS0 > �S0 , a dense set (quasicon-tinuum) of ABS delocalized across the TS wire appearswithin a large energy window �S > " > �eff above thelow-energy sector. This quasicontinuum (which approachesa continuum as LS0 ! 1), separates the Majorana branchesfrom the true continuum of the problem at " > �S.Importantly, for finite transparency junctions, the Majoranalevels detach from the quasicontinuum, as opposed to thetopologically trivial case or a standard SNS junction [35],where the ABS touch the continuum at� ¼ 0, 2�. We haveverified that the detachment, denoted by �0 in Fig. 1(b),increases as the transmissionT of the normal region becomes

smaller, �0=�eff � 1� ffiffiffiffiT

p, in agreement with simpler

models [16,17,27].ac Josephson effect.—We now study the Josephson cur-

rent across the junction when biased with a voltage V.The bias makes the phase difference � time dependent,�ðtÞ ¼ 2 eVt=@ ¼ !Jt, where !J is the Josephson fre-quency. We calculate the Josephson current through thebiased junction in terms of a reduced density matrix in theMajorana sector. Notably, its evolution contains parity-mixing terms as a result of the Andreev levels above �eff

that have been traced out [23]. The goal is to identifysmoking-gun features in IðtÞ, in the form of fractional fre-quency components, that may unambiguously determine theexistence of MBS in the TS phase. It has been shown,however, that in the presence of finite dissipation (e.g., non-zero parity mixing), the current becomes, in the long timelimit, strictly 2� periodic, as a consequence of Floquettheorem, and that, as a result, nothing remains in the sta-tionary current to qualitatively distinguish the trivial from theTS phase [27]. The two phases, however, exhibit crucialdifferences in their spectra that give rise to very differentfeatures in their transient regimes, Iðt < tTÞ. The spectralfeatures of the TS phase (sharp � ¼ � anticrossing anddetachment of theMajorana sector from the quasicontinuumin the TS phase, see Fig. 2) allows the even and oddMajoranasubspaces in the TS phase to evolve almost coherently per-forming 4�-periodic population oscillations during a poten-tially long transient time tT � T2�, where T2� ¼ 2�=!J isthe period of the Josephson driving.We now characterize thetransient regime in detail. At bias 2 eV< ��, the junction

will remain in the initial ground state j00i, resulting in anadiabatic 2�-periodic current [blue curve (top) in Fig. 3(c)].Increasing 2 eV above �� induces LZ transitions at � ¼ �from j#ei ¼ j00i into j "ei ¼ j11i, which tend to produce aperfect population inversion as 2 eV � ��. Then, by virtueof the finite detachment �0, if V remains smaller than acertain VMAR (quantified below), the population escape intothe quasicontinuum as � crosses 2� will be negligible, andthe junction will subsequently re-invert into its ground statej#ei at� ¼ 3�. The repetition of this inversion process givesrise to a transient 4�-periodic Josephson current I � sin�=2[red curve (middle) in Fig. 3(c)], with a superimposed

FIG. 3 (color online). Top: diagrams of the different dynamicalregimes of the ac Josephson current as a function of junction biasV and dimensionless conductance G=G0, for (a) the topologi-cally trivial and (b) nontrivial phases of a 8:7-�m long InSb wiresimilar to that of Fig. 2. Typical current profiles are presentedin (c). Conventional adiabatic (blue, top) and multiple Andreevreflection (MAR) (green, bottom) regimes are both 2� periodic.A wide region of beating 4�-periodic ac Josephson current (red,middle) appears in the nonadiabatic TS phase and is absent in thetrivial case. Each regime has a distinct frequency transform (d).The inset shows the beat splitting��! of the anomalous peak at!J=2 as a function of the bias V, both for the even (solid) and theodd sectors (dashed). Panel (e) presents a typical dissipativedecay from the 4� regime into the stationary MAR regime,which develops a dc current component (peak at ! ¼ 0).

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beating envelope (due to imperfect LZ transitions at �, 3�,etc.). The parameterwindow required by this solution,�� �2 eV � 2eVMAR, is experimentally relevant, since �� andVMARmaybe independently controlled by the lengthLS0 and,e.g., the junction transparency, respectively. In contrast, inthe topologically trivial phase, only the lowest of the twoAndreev levels is detached from the quasicontinuum. Thus,the state j11iwill immediately decay into j10i upon crossing� ¼ 2�. As a consequence, the 4�-periodic current cannotdevelop. The anomalous 4�-periodic component in the tran-sient regime, therefore, is an unequivocal signature of theexistence of MBS in the junction. At high V � VMAR, thetransition into the continuum becomes significant in eachperiodT2�, for both phases. For such voltages, tT � T2�, andclose to two fermions escape into the contacts per cycle. Thisprocess yields a finite dc current [green curve (bottom) inFig. 3(c)], which is the analogue of the multiple Andreevreflection (MAR) mechanism of conventional junctions. Itshould be noted that the current for any finite bias V even-tually develops, in the stationary regime, a finite dc compo-nent, that has the same origin asMAR, i.e., the promotion ofCooper pairs into the quasicontinuum by iterated scatteringprocesses [see Fig. 3(e)].

A parametric diagram of the MAR (green), the 2�adiabatic (blue), and the 4� beating (red) regimes possibleare presented in Figs. 3(a) and 3(b). The boundaries be-tween regimes are crossovers, defined by V¼VMAR and2eV¼��. The different regimes may be clearly distin-guished using a finite time spectral analysis of the current,shown in Fig. 3(d). The adiabatic regime exhibitscomponents at ! ¼ n!J, for integer n > 0, while theMAR regime develops an additional ! ¼ 0 peak. The4�-periodic transient regime in the TS phase, on the otherhand, exhibits a distinct half-integer spectrum !¼ðnþ1

2Þ!J, together with a peak splitting due to the accom-

panying beating envelope. The splitting ��! is plotted inFig. 3(d) (inset) as a function of the applied bias, whichreveals oscillations due to the interference of successiveLZ processes at odd multiples of � ¼ �.

Transient times.—The characteristic decay time of the

transient current IðtÞ � e�t=tT is given by tT � @2�0

ð2 eVÞ2 �,which scales with bias V and the characteristic relaxationrate �0 that models the escape of quasiparticles from thequasicontinuum levels into the reservoirs [23]. The dimen-sionless function �ðG=G0Þ that depends solely on thejunction’s normal conductance is numerically computedin Fig. 4(a). It is maximum in the tunneling regime, and itdecays to zero as transparency goes to one. This means thattT decreases when the detachment �0 ! 0, since then theescape probability into the quasicontinuum in each cyclegoes to one. The characteristic escape voltage VMAR isdetermined from the condition tT ¼ T2�, 2 eVMAR ¼@�0�=2�. One can see that, while a sufficiently slowdriving V will suppress quasiparticle poisoning and willthus increase the duration of the transient regime, a

complementary, and potentially preferable, route is to en-gineer the environment’s �0 in order to increase the lifetimeof the Majorana qubits without slowing operation. Indeed,by increasing the escape rates, parity mixing will be sup-pressed, increasing tT . This is known as the quantum Zenoeffect, whereby a quantum system coupled to a bath maydevelop longer coherence times when increasing the speedat which information is lost into the bath. In a very fastenvironment, the transitions into the quasicontinuum at� ¼2�n will be suppressed by this same Zeno mechanism. Thisis demonstrated in the simulation of Fig. 4(b), which showshow tT is tripled as a result of increasing the �0. Usingrealistic parameters for InSb nanowires, we estimate thattypical tT reach into the �s range at �V bias voltages.Owing to these long transients, the spectrum of micro-

wave radiation should show clear features of the fractionalfrequencies [23]. Such measurement can be performedwith an on-chip detector by using, for example, the photonassisted tunneling current of quasiparticles across asuperconductor-insulator-superconductor junction capaci-tively coupled to the TS one. Importantly, it has beenalready demonstrated [36] that such technique allows adirect detection of fractional Josephson frequencies whichpaves the way for MBS on-chip detection [23].We are grateful to Y.V. Nazarov, S. Frolov, L.

Kouwenhoven, and S. Kohler for useful discussions. Weacknowledge the support from the CSIC JAE-Doc programand the Spanish Ministry of Science and Innovationthrough Grants No. FIS2008-00124 and FIS2009-08744.

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0 0.5 1

1

2

3

4

5

G G0

a

tT

tT

t

It 0 1 2

0

50

100

0 S'

t TT

2

b

FIG. 4 (color online). Transient time scales as tT � @2�0

ð2 eVÞ2 �,with dimensionless function �ðG=G0Þ plotted in (a). The tran-sient 4� regime [red shaded region, (b)] becomes longer as therelaxation rates to the quasicontinuum �0 increase (QuantumZeno effect). The two main panels in (b) show the ac Josephsoncurrent vs. time (2 eV ¼ 0:03�S0 ) and two values of �0 high-lighted in the inset, which shows the transient time tT / �0.

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[23] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.108.257001 for adiscussion about analogies with the Josephson effectthrough a single resonant level, details of the calculationof Andreev levels, a description of the nonequilibriumformalism, the Lindblad description of quasiparticle poi-soning, and a discussion of various aspects of experimen-tal detection.

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[29] These parity states constitute an Ising qubit, z ¼�2i�1�2. One can represent a full spin with three MBS[30,31] and thus a logical qubit may be constructed [32]:z ¼ �2i�1�2; y ¼ �2i�3�1; x ¼ �2i�2�3.

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