Fusion of Majorana Bound States with Mini-Gate Control in Two-Dimensional Systems
Tong Zhou, Matthieu C. Dartiailh, Kasra Sardashti, Jong E. Han, Alex Matos-Abiague, Javad Shabani, Igor Zutic
FFusion of Majorana Bound States with Mini-Gate Control in Two-Dimensional Systems
Tong Zhou, ∗ Matthieu C. Dartiailh, Kasra Sardashti, Jong E. Han, Alex Matos-Abiague, Javad Shabani, and Igor ˇZuti´c † Department of Physics, University at Bu ff alo, State University of New York, Bu ff alo, New York 14260, USA Center for Quantum Phenomena, Department of Physics, New York University, New York 10003, USA Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA (Dated: January 26, 2021)A hallmark of topological superconductivity is the non-Abelian statistics of Majorana bound states (MBS),its chargeless zero-energy emergent quasiparticles. The resulting fractionalization of a single electron, storednonlocally as a two spatially separated MBS, provides a powerful platform for implementing fault-toleranttopological quantum computing. However, despite intensive e ff orts, experimental support for MBS remainsindirect and does not probe their non-Abelian statistics. Here we propose how to overcome this obstacle in mini-gate controlled planar Josephson junctions (JJ) and demonstrate non-Abelian statistics through MBS fusion,detected by charge sensing using a quantum point contact. The feasibility of preparing, manipulating, andfusing MBS in two-dimensional (2D) systems is supported in our experiments which demonstrate the control ofsuperconducting properties with five mini gates in InAs / Al-based JJs. While we focus on this well-establishedplatform, where the topological superconductivity was already experimentally detected, our proposal to identifyelusive non-Abelian statistics motivates also further MBS studies in other gate-controlled 2D systems.
I. INTRODUCTION
Superconducting proximity e ff ects, known for ninetyyears [1, 2], provide a platform to realize exotic topolog-ical superconductivity even within common materials andhost Majorana bound states (MBS) [3–7]. These MBS arepredicted to emerge as chargeless and zero-energy modesat the boundary between topological superconductors and atopologically-trivial region [8]. The non-Abelian statistics ofMBS as Ising anyons and the possibility of using them tononlocally store quantum information make them particularlysuitable for topological quantum computing [9–11].The studies of MBS predominantly focus on proximity-induced superconductivity in one-dimensional (1D) sys-tems [12–16], detected through spectral signatures, such as thezero-bias conductance peak [17]. These systems have enabledimportant materials advances in high-quality transparent in-terfaces and pronounced proximity e ff ects [18–20]. However,1D systems typically require highly-tuned parameters to sup-port topological superconductivity [4, 5] and pose inherentlimitations to probe non-Abelian statistics through exchange(braiding) or fusion of MBS [9, 10].To overcome these limitations, 2D proximitized materialsare sought [21–33]. Here we reveal how mini-gate controlin planar Josephson junctions (JJs), based on common nor-mal (N) and superconducting (S) regions, provides a versa-tile platform to realize multiple MBS and implement their fu-sion. Building on the experimentally tested epitaxial growthof semiconductor / superconductor planar JJs [34–37], we pro-pose two geometries, depicted in Fig. 1, to fuse MBS.Demonstrating non-Abelian statistics for MBS would bea major milestone for topological quantum computing andbridge a gap between the debated observation of MBS [38]and advances in topological quantum algorithms which arelargely detached from their materials implementation [39].Before discussing a close connection between the two keysignatures of non-Abelian anyons, revealed by braiding and FIG. 1. (a) Straight and (b) V-shaped junction (SJ, VJ) formed by su-perconducting, S , , , regions (blue), partially covering a 2D electrongas (yellow). The electron density in the uncovered part is locallytuned using mini gates, V , , , , (left to right). With an in-plane mag-netic field B x and superconducting phases, ϕ , , , controlled by theexternal fluxes Φ , Φ , and Φ , the chemical potential of the normalregion, µ N , is locally changed to support topological (orange) andtrivial (green) regions, by imposing the mini-gate voltage V + and V − .MBS γ , , , (stars) form at the ends of the topological regions. fusion of MBS, a useful guidance for the operation of theschemes from Fig. 1 comes from the trends in 1D systems.A well-known condition for the transition to 1D topologicalsuperconductivity in proximitized nanowires [4, 5], E Z = ( ∆ + µ ) / , (1)where E Z = g µ B B / g is the g -factor, Ban applied magnetic field, ∆ the proximity-induced supercon-ducting gap, and µ the chemical potential, is already suggest-ing a direct role of mini-gate control in Fig. 1. A uniform gatevoltage, through changes in µ , determines the MBS creationat the end of a nanowire if | µ | < µ c = ( E Z − ∆ ) / or MBSdestruction if | µ | > µ c . | µ | = µ c implies the closure of thebulk excitation gap and the removal of topological protection.A spatially-dependent µ locally controlled with mini gates al-ters the topological condition from Eq. (1) thereby modifying a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n FIG. 2. (a) Topological superconductor (blue) hosting MBS, γ ,..., . They behave as non-Abelian anyons and lead to the four-fold degeneracyin topological ground states, separated by the energy gap, ∆ , from the trivial excited states. (b) and (c) Di ff erent fusion outcomes: trivial fusionof γ and γ , 100% probability to access vacuum, I (Cooper pair condensate), and nontrivial fusion of γ and γ , equal probabilities to access I or an unpaired fermion, ψ . Red dashed lines: paired MBS. In each case bringing closer MBS leads to the level splitting from the initialzero-energy modes. Filling the lower level, corresponding to I with even parity, means the absence of a given particle, while filling the upperlevel refers to ψ with odd parity. We assume initially even parity of the system. The net change in the charge characterizes nontrivial fusion. the location of boundaries between topological and trivial re-gions which host MBS. Moving these boundaries would inturn move MBS and even force their pairs to coalesce or fuse.While MBS mini-gate control scenario in 1D nanowires hasbeen proposed a decade ago [40], the envisioned experimentalrealization would be di ffi cult, not just due to fine-tuned param-eters to satisfy Eq. (1), but due to the common nanowire ge-ometries which are surrounded by superconductors and thusthrough screening make the attempted gating ine ff ective. Incontrast, Fig. 1 geometry of planar JJs leaves the portion of aproximitized 2D electron gas (2DEG) uncovered by supercon-ductors (blue regions), allowing for an e ffi cient modificationof µ through mini gates, located in that uncovered part.Our implementation builds on recent experiments in planarJJs [34, 35, 41], which reveal that topological superconductiv-ity exists over a large parameter space and is particularly ro-bust when the phase di ff erence, φ , between two superconduct-ing regions is close to π . Such φ can be directly measured [35],and imposed externally [34, 41]. The observed stability oftopological superconductivity for di ff erent in-plane B -fieldorientations, with respect to the normal region / superconductor(N / S) interface [35], supports the feasibility of non-collinearjunctions from Fig. 1(b). In addition to considered prox-imitized 2DEG platforms, recent advances in selective areagrowth and stencil litography for high-transparency JJs basedon topological insulators [42], are encouraging for other ma-terials implementations of our proposal. We propose to detectdi ff erent fusion outcomes through charge measurements, wellestablished in semiconductor nanostrucutres [43, 44].Following this Introduction, in Sec. II we summarize somemotivation to pursue the fusion of MBS and why it would beimportant. Our theoretical framework is discussed in Sec. IIIprovides important considerations that distinguish 2D systemsfrom better studied 1D counterparts. The main results for theMBS fusion in straight and V-shaped junctions are discussedin Secs. IV and V using the parameters already experimentallymeasured in epitaxial InAs / Al planar Josephson junctions. We also address a possible readout scheme for the performed fu-sion using a charge detection. In Sec. VI we provide Conclu-sions and Outlook for the future work.
II. WHY FUSION?
The presence of defects or quasiparticles in topological su-perconductors, such as vortices, skyrmions, domain walls,or boundaries between topological and trivial regions, canbind localized Majorana zero-energy modes which behave asnon-Abelian Ising anyons [10, 11]. By using two Majoranafermion operators to express an ordinary (complex) fermion, f = ( γ + i γ ) /
2, the corresponding parity operator becomes P , = − n = − f † f = − i γ γ , (2)where ˆ n is the usual number operator. P , has eigenvalues ± n =
0) and odd ( n =
1) fermion parity [45].Majorana operators, which satisfy { γ i , γ j } = δ i , j , γ i = H , γ i ] = i , j = , ..., N and H is theHamiltonian of the system, ensure the existence of degener-ate ground states, expected from non-Abelian anyons. Thesetopologically-protected degenerate states, in which quantuminformation can be stored, are separated by the energy ∆ from the excited states, as depicted in Fig. 2(a) for the case2 N =
4. The ground states, nonlocally storing ordinary N fermions, can be labelled by the eigenvalues of parity opera-tors, − i γ γ , ..., − i γ N − γ N , and have the 2 N degeneracy.For an ordinary fermion, f , composed of non-overlappingMajoranas, the ground state is twofold degenerate since bothfermion parities correspond to zero energy. However, bring-ing the two Majoranas closer removes this degeneracy, as de-picted in Fig. 2(b). The resulting multiple fusion outcomescan be labeled by the corresponding parity and they reflect theunderlying non-Abelian statistics [10, 46]. The fusion rules γ × γ = I + ψ, (3)summarize that the fusion of the two MBS behaves either asvacuum (Cooper pair condensate, the parity is unchanged), I ,or a fermion (Bogoliubov quasiparticle, an unpaired fermion,the parity is changed), ψ , resulting in an extra charge. Thetwo situations are shown in Figs. 2(b) and (c). For the triv-ial fusion, when MBS with a defined parity within the samepair coalesce, the outcome is deterministic, it leads to the un-changed parity (shown to be even) with no extra charge. Forthe nontrivial fusion, when MBS from di ff erent pair coalesce,both parities are equally likely and a probabilistic measure-ment could reveal an extra charge related to the negative par-ity. Additional rules can be simply understood. ψ × ψ = I implies that two fermions brought together behave as if thereis no particle, while ψ × γ = γ signals that the fusion of ψ and γ is indistinguishable from a single γ [46].The MBS fusion provides projective measurements ofthe ground-states degeneracy inherent to non-Abelian statis-tics with topological protection from local perturbations.While these non-Abelian signatures are complementary tothose from the unitary transformation of Majorana operatorsthrough braiding, experimentally implementing fusion is sim-pler. There are even schemes for topological quantum com-puting implemented through fusion where the braiding is notrequired [47–49]. A further motivation for our study of fu-sion is given in a recent review [38]. Before Majorana qubitscould be used in topological quantum computing, it notes:“Researchers first must establish unambiguously that in thelab they can make fully nonlocal Majoranas with the requisitetopological protection, demonstrate ground-state degeneracy,and test for simple measurements, such as fusion rules.” III. THEORETICAL FRAMEWORK
In characterizing topological superconductivity using pla-nar JJs, it is important to understand the evolution of the phasedi ff erence, φ = ϕ − ϕ , between the two superconducting re-gion, S and S . While in 1D systems an increase of an in-plane field B ≡ B x leads to the closing of the trivial and thenreopening of the topological gap [4, 5], in planar JJs a similarincrease in B x (recall Fig. 1) leads to the first-order phase tran-sition between trivial and topological regime, accompanied bythe corresponding phase jump from φ = φ = π , or slightlyadjusted values, depending on the strength of the spin-orbitcoupling (SOC) [35, 36]. With the increase in B x the systemself-adjusts to the new ground state which, using the SQUIDgeometry [35], is detected through direct phase measurementsidentifying the corresponding jump in φ .Unlike in 1D systems, the parameter range for topologicalsuperconductivity in planar JJs is much larger than suggestedfrom Eq. (1). The range of suitable values of the chemical po-tentials can be further extended by externally setting φ ≈ π ,for example, by a flux control Φ [34, 41] as shown in Fig. 1.Some guidance of what is expected for the onset of topologi- cal superconductivity follows from the expression [24] E Z = E T / , (4)where E T = ( π/ v F / W N is the Thouless energy in the bal-listic regime [50], with v F as the Fermi velocity and W N thewidth of the normal region (recall Fig. 1). Similar to topo-logical superconductivity in nanowires, for a larger B x , when E Z ≥ E T /
2, the system is expect to be in the topologicalregime. Experiments on InAs-based planar JJs show that theobserved topological superconductivity never reaches the re-quired E Z ≥ E T / φ measurements in InAs / AlJJs [35], reveal the phase jump at B x ≈ . E Z (cid:28) E T .Instead of relying on simplified topological conditions,these findings necessitate developing a theoretical frameworkwhich, building on the explanation of the observed topologi-cal superconductivity in InAs / Al JJs, can guide demonstratingthe MBS fusion. To this end, we describe planar JJs using theBogoliubov-de Gennes (BdG) Hamiltonian, H = (cid:34) p m ∗ − µ S + V ( x , y ) + α (cid:126) (cid:16) p y σ x − p x σ y (cid:17)(cid:35) τ z − g µ B B · σ + ∆ ( x , y ) τ + + ∆ ∗ ( x , y ) τ − , (5)and numerically solve the corresponding eigenvalue problemon a discretized lattice as implemented in Kwant [51]. Here p is the momentum, m ∗ is the e ff ective electron mass, µ S isthe chemical potential in the considered S i , α is the RashbaSOC strength, unless explicitly specified, B ≡ B x . We use σ i ( τ i ) as the Pauli (Nambu) matrices in the spin (particle-hole) space and τ ± = ( τ x ± τ y ) / ∆ ( x , y ) is the proximity-induced superconducting pair potential, for the 2DEG belowthe superconducting regions, which can be expressed, usingthe BCS relation for the B-field suppression, as ∆ ( x , y ) = ∆ (cid:113) − (B / B c ) e i ϕ i , (6)where ∆ is the superconducting gap at B =
0, B c is the criticalmagnetic field, and ϕ i is the corresponding superconductingphase. The function V ( x , y ) ≡ µ N ( x , y ) − µ S describes the localchanges of µ N ( x , y ) in the N region due to the application ofthe mini gate voltages, V , ..., V , as shown in Fig. 1.In all the calculations, we choose the parameters consis-tent with our fabricated junctions (SJ and VJ) that also matchexperimental observation of robust proximity-induced super-conductivity and topological states in epitaxial InAs / Al-basedJJs [35], m ∗ = . m , where m is the electron mass, and g =
10 for InAs, ∆ = α =
10 meVnm, B c = µ S = . x is not aligned along the N / Sinterface and instead forms a misalignment angle θ , as shownin Fig. 3(a). Topological superconductivity surviving such FIG. 3. (a) Schematic of a tilted junction with a misalignment angle θ from from the applied B x . MBS (stars) reside at the opposite ends ofthe N region (yellow). (b) Energy spectra for an SJ with B x = . ff erence, φ = ϕ − ϕ . (c) Same as (b) butfor a tilted junction with θ = . π . (d) Energy spectra for a tilted junction, φ = π and θ = . π as a function of B x . (e) Energy spectra for atilted junction, φ = π and B x = . θ . (f) Probability density, ρ P , for the lowest (red) energy states with φ = π , θ = . π , andB x = . x P , Q = L / y P , Q = ( W ± W N ) /
2. The geometricparameters are L = µ m, W = . µ m and W N = . µ m. Other parameters are specified in the main text. a misalignment supports the feasibility of the proposed VJsfrom Fig. 1(b). The influence of a misalignment can be seenby comparing the φ -dependent spectrum for an SJ [Fig. 3(b)]and a tilted junction [Fig. 3(c)], as depicted in Fig. 3(a).While the topological gap and the range of φ support-ing topological superconductivity become smaller comparedto that in the SJ, the MBS still survive in a large range of φ ∈ (0.56 π , 1.35 π ) and localize at the end of the N regions[Fig. 3(f)]. Similar to that in an SJ, when φ = π , the MBS canappear in the tilted junction under a very small B x ∼ . π phase di ff erencestill favors the superconducting state with a sizable topologi-cal gap, even with θ (cid:44)
0, as predicted in Ref. [52]. Thus, un-less specified, in the following sections we fix φ = π betweenthe adjacent S regions. To get the range of θ which could sup-port MBS in a tilted junction, we plot the θ -dependent energyspectra in Fig. 3(e). The MBS zero-energy modes oscillateand eventually disappear as θ increases, in agreement with theexperimental observations [35]. However, the MBS still sur-vive when θ ≤ . π , guiding the VJ design. IV. MBS FUSION IN A STRAIGHT JUNCTION (SJ)
The proposed setup for an SJ with mini gates from Fig. 1(a)implies that a proximity-induced superconductivity, in the2DEG part uncovered by the S region, can be manipulatedby B x , µ N and φ , the phase di ff erence between the S and S ,imposed by the magnetic flux, Φ . With fixed B x and Φ , thetopological condition can then be directly controlled by thegate voltage through the changes in µ N [25]. We assume thatgate voltage V + and V − support topological and trivial states, respectively. As discussed in Sec. III, a simple topologicalcondition for planar JJs from Eq. (4) is not su ffi cient to ac-curately describe our system and we need to explicitly calcu-late the relevant V − and V + . With mini gates, as depicted inFig. 1(a), we expect to electrostatically create multiple topo-logical ( + ) and trivial ( − ) regions along the N channel by im-posing the corresponding voltage V + and V − in the mini gates FIG. 4. (a), (b) Scanning electron microscope (SEM) image of anInAs / Al SJ with 5 mini gates covering the normal region. Mini-gatescan be controlled independently and those overlaid in red indicate theregions in which the applied bias current, I B can flow. The 2DEG isdepleted under the other gates. (c) and (d) Di ff erential resistance ofthe device as a function of the I B and out-of-plane B ⊥ , correspondingto the gate configuration presented in (a) and (b), respectively. FIG. 5. (a) Schematic (top) and energy spectra (bottom) for a planar π -SJ at B x = . V = µ N − µ S ),tuned by the top gate (not shown), which covers the whole N region. The black dashed line indicate the critical gate voltage between thetrivial region (green) and topological region (orange). (b)-(d) Schematic (top) of a planar π -SJ with MBS (stars) for + + + + + , + + − − − , and + + − + + mini-gate configurations and the corresponding energy spectra (bottom), where V − = − V + = − and + states, respectively. Red and black lines: evolution of finite-energy states into MBS inside the topological gap. The geometric parametersare L = µ m, W S = . µ m, and W N = . µ m. The other parameters are specified in the main text. ( V , ..., V ). Multiple MBS residing at the ends of topologicalregions can then be moved and fused.An experimental feasibility of the proposed mini-gate con-trolled MBS fusion builds on the demonstrated topological su-perconductivity in epitaxial InAs / Al planar JJs [35, 53]. Thisis further corroborated by using the same platform to demon-strate that mini gates can modulate the superconducting statein our fabricated SJ, shown with scanning electron microscope(SEM) images in Figs. 4(a) and (b). With five gold mini gatescovering the N region, µ N for the each region under the minigates can be independently tuned by the bias current, I B .With the three inner gates depleted, the current can onlyflow through the two outermost regions (marked in red) as de-picted in Fig. 4(a). In this configuration, the device behaves asa SQUID [54], as seen from the map of the measured di ff er-ential resistance as a function of I B and out-of-plane magneticfield in Fig. 4(c) which indicates interference between the cur-rent going though the two open channels.In contrast, when the three middle gates allow current toflow, and the outer most gates are used to deplete the 2DEGin Fig. 4(b), the measured di ff erential resistance in Fig. 4(d)shows a Fraunhofer pattern, typical of a single JJ [54]. Asexpected, its periodicity is close to the one of the SQUID con-figuration which contains the same region.Distinct features between Figs. 4(c) and 4(d) show that lo-cally µ N can indeed be strongly changed by the mini gates,providing a clear advantage over an attempt of gate control innanowire systems [40, 55], where the screening by supercon-ductors diminishes changing µ N . The feasibility of such gate-controlled superconducting response also provides a strongsupport for our proposal of manipulating MBS with mini-gatecontrol, when the topological superconductivity is achievedwith an in-plane B x and a phase bias, φ .Based on our fabricated device in Fig. 4, to obtain the rele-vant voltages V + ( V − ) for topological (trivial) state, we choosegeometrical parameters depicted in Fig. 1(a) as L = µ m, W S = . µ m, W N = . µ m, with each mini gate 1 µ m long.To identify such V + and V − , we calculate the gate-voltage de-pendent energy spectrum with B x = . φ = π , as shownin Fig. 5(a). The evolution of the lowest-energy states intozero-energy modes reveals that the MBS states emerge whenthe voltage exceeds the critical value V c = − . V + ∈ ( − . ρ P , and the vanishing chargedensity, ρ C [56, 57], while V − < V c gives trivial states [56].We choose V + = V − = − V + and V − gives us a chance to create and manipulate multiple MBSbased on di ff erent mini-gate configurations.It is instructive to examine the topological robustness of the + + + + + configuration, where all the mini gates are set at V + ,which is similar to a single topological SJ without mini gates.The whole N region is expected to be topological with MBSat its ends [Fig. 5(b)]. The calculated B x -dependent energyspectrum shows that MBS indeed exist in a very large rangeof B x , and a small B x ∼ . + + + + + into + + − − − , the MBS at theright end can be moved to the left part [Fig. 5(c)], while break-ing the topological region into two separate ones, by changing + + + + + into + + − + + , creates two MBS pairs [Fig. 5(d)].These SJ configurations are revisited in Sec. V, where we willsee that the expected control of MBS is further corroboratedby the calculated ρ P .We note that in the + + + + + configuration for a larger B x ( > . ρ P and ρ C [56]. Such an increase in B x reduces V c and makes themiddle region ( V = − x > . x from 0 . > . ++ − ++ becomes +++++ . As a result, the two FIG. 6. Probing non-Abelian statistics through MBS fusion with the corresponding probability and charge densities, ρ P and ρ C , respectively.The red dashed lines link the same MBS pair, the yellow dashed line indicate the N region covered by the mini gates. (a) Initial trivial statewith − − − − − mini gates. (b) A : changing − − − − − into + + − − − , MBS pair ( γ , γ ) is created. (c) A : changing + + − − − into + + − + + ,a second MBS pair ( γ , γ ) is created. (d) A : changing + + − + + into + + + + + , the MBS ( γ , γ ) are nontrivially fused at the center,accessing both vacuum, I , and an unpaired fermion, ψ , with 50% probability. For I , the system has no extra charge, supported by the ρ C in (i)for the ground state after the fusion [marked as G in Fig. 7]. For ψ , the system has an extra charge, shown in the sum of ρ C for the ground andfirst excited state [H in Fig. 7] after the fusion. (f) B : changing − − − − − into + + + + + , the MBS ( γ , γ ) are created. (g) B : changing + + + + + into + + − + + , a second MBS pair ( γ , γ ) is created. (h) B : changing + + − + + into + + + + + , the MBS ( γ , γ ) are triviallyfused, corresponding to I state with 100% probability. A or B : changing + + + + + to − − − − − , the MBS ( γ , γ ) are fused and the systemreturns to the initial mini-gate configuration. MBS fusion can be repeated following such operations. The (minimum, maximum) values in (e)and (i) are (-3.5, 2.9) and (-0.00009, 0.00004), respectively. e is the charge of the electron. inner MBS located in the middle mini gate are fused. While itis tempting to consider B x -controlled MBS fusion, this is notfeasible given that experimental changes of an applied mag-netic field are too slow and exceed the MBS lifetime [59–61].In contrast, gate-controlled changes of µ N at fixed B x are muchfaster and o ff er a natural choice for the MBS fusion.Following this previous analysis, we propose a scenariofor probing non-Abelian statistics based on fusion rules us-ing mini-gate control as shown in the Fig. 6. The system isinitially prepared in a trivial state (no MBS) with − − − − − configuration. Subsequently, we can follow paths A and B toprobe nontrivial and trivial fusion rules. For path A, in A wefirst generate one MBS pair ( γ , γ ) by changing V and V from V − to V + , and in A the second MBS pair ( γ , γ ) byacting on V and V . These two MBS pairs build two complexfermions f = ( γ + i γ ) / f = ( γ + i γ ) /
2, whichcan be described by the occupation numbers n and n , re-spectively [recall Eq. (2)]. The system therefore evolves intoa state with well-defined values n and n .Without loss of generality, we assume that the twofermion states are unoccupied, giving an initial state | n , n (cid:105) = | , (cid:105) . In A the change of V from V − to V + nontrivially fuses ( γ , γ ), which accesses both the I and ψ fusion channels with equal probability. To bet-ter understand such nontrivial fusion, we reexpress theground state in the basis of f = ( γ + i γ ) / f = ( γ + i γ ) /
2, i.e. | , (cid:105) = / √ | , (cid:105) − i | , (cid:105) ),where f f | , (cid:105) =
0, while | , (cid:105) = f † f † | , (cid:105) .Fusing ( γ , γ ) induces a finite energy to f , lifting the degen-eracy between | , (cid:105) and | , (cid:105) . As a result, measuringsuch state then collapses the wave function with 50% proba-bility onto either the ground state, I , or excited state with anextra quasiparticle, ψ . In A fusing the remaining ( γ , γ ),by changing + + + + + into − − − − − , drives the system tothe initial mini-gate configuration. To verify the non-Abelianstatistics, we examine a trivial fusion scheme B -B . Unlikein the nontrivial fusion, ( γ , γ ) is first created and then ( γ , γ ) by changing − − − − − to + + + + + and then to + + − + + .Therefore, fusing ( γ , γ ) can only access the I channel witha trivial fusion because ( γ , γ ) belong to the same pair.To simplify the description of MBS fusion it is helpful thatconsidered scheme from Fig. 6 is adiabatic, which requiresthat the topological gap remains open during the entire fusionprocess. We show the corresponding evolution of the calcu-lated low-energy spectra during the fusion in Fig. 7. For anyvalue of the continuously changing mini gates, the MBS are FIG. 7. Intermediate states for the MBS fusion scheme in an SJ. The calculated energy spectra evolution for the operations in (a) nontrivial(A - A ) and (b) trivial (B - B ) fusion, shown as a function of the relevant mini-gate voltage. Red and black lines: evolution of finite-energystates into MBS inside the topological gap. E and F indicate the two MBS pairs (degenerate ground states) in the + + − + + configuration(before fusion), while G and H indicate the ground and first excited state at V = − . + + + + + configuration (after fusion). Theparameters are taken from Fig. 5. protected by the topological gap between the ground and firstexcited states which, from our results in Fig. 7, has the mini-mum value, ∆ min ≈ µ eV, which could be enhanced by usingSn or Nb with a higher bulk ∆ than in Al. An animation of theentire nontrivial fusion process is provided in Ref. [56].Through uncertainty relations this ∆ min imposes a lowerbound for the switching time, τ , during the mini-gate oper-ation, which can be estimated as t ∆ ∼ (cid:126) / ∆ min ∼ .
04 ns. Theupper bound for the gate-switching time can be inferred fromthe quasiparticle poisoning time, t poisoning . From the previousmeasurements in InAs / Al systems, t poisoning was reported to bebetween 1 µ s and 10 ms [59, 62]. Together, these requirementsfor the adiabatic fusion, 0 . n s < τ < µ s are readily real-ized with the existing gate controlled employed in JJ-basedqubits which are reaching GHz operation [63].The feasibility of this adiabatic evolution and distinct out-comes between the nontrivial and trivial MBS fusion are im-portant prerequisites for using the fusion rules as an experi-mental verification of the non-Abelian statistics. A guidancefor how the fusion rules could be measured comes from theprior proposals in nanowires, suggesting using Josephson cur-rent, fermion-parity, or cavity detection [10, 13, 40, 64–66].In finite junctions with overlapping MBS we expect de-viations from idealized fusion rules since there are also de-viations from the properties of idealized Majorana operatorsnoted in Sec. II. For example, [ H , γ i ] (cid:44)
0, instead the commu-tator should be exponentially small in the ratio between theMBS coherence length and MBS spacing [11]. Nevertheless,the idealized outcome for an infinite system of initially non-overlapping MBS still holds in our finite-size calculations.This implies that the trivial fusion deterministically gives I ,preserving the charge of the system, while in the nontrivialfusion there is 50% probability for creating an extra charged quasiparticle ψ . Comparing the calculations in Figs. 6(e) and(i) reveals a four orders of magnitude di ff erence in the cor-responding ρ C [56]. Therefore, charge sensing is promisingfor fusion detection. Repeating operations A -A from Fig. 6should give charge fluctuations. In contrast, the fluctuationsshould be absent when repeating operations B -B . V. MBS FUSION IN A V-SHAPED JUNCTION (VJ)
The previous SJ geometry provides a plausible path to MBSfusion and distinguishing the resulting outcomes. However,the corresponding charge fluctuations emerge in the interiorof the central part of the N region, which is challenging toaccess experimentally due to the screening of superconductorsand the presence of the top mini gates. To overcome thesedi ffi culties, we propose a V-shaped geometry for the N-regionwhere its apex is exposed to the edge, as shown in Fig. 1(b)and further described in Fig. 8.With two external fluxes, Φ , Φ , and mini-gate control, FIG. 8. (a) Schematic of an experimental setup incorporating a quan-tum point contact (QPC) to detect the results of fusion operations ina V-shaped junction (VJ) with mini-gates control. (b) SEM image ofa device incorporating a VJ with mini-gates and a QPC.
FIG. 9. (a) Schematic (top) and energy spectra (bottom) for a VJ with superconducting phases ( π , 0, π ) set by Φ = Φ = Φ , Φ is themagnetic flux quantum at B x = . V , tuned by the top gate (not shown), which covers the whole N region.The black dashed line indicate the critical gate voltage between the trivial (green) and topological region (orange). (b)-(d) Schematic (top)of a VJ with MBS (stars) for the + + + + + , + + − − − , and + + − + + mini-gate configurations supported by the probability density, ρ P (middle), at B x = . L = . µ m, W = . µ m, W N = . µ m, and θ = . π . Each mini gate has the same lengthof ∼ . µ m. Other parameters are from Fig. 5. the MBS can be fused at the apex in a similar way to thatin the SJ. An advantage in the VJ is that its apex provides aplace to detect the additional charge induced by MBS fusionusing quantum point contact (QPC) measurements, success-fully used in semiconducting nanostructures [43, 44] and alsoproposed for detection of topological superconductivity in 1Dsystems [66]. An experimental realization of the VJ with 5mini gates, fabricated using standard electron-beam lithogra-phy and InAs / Al JJs, is shown in Fig. 8(b). The correspondingSEM image also shows a gold electrode attached at the apexof the VJ to pinch o ff the QPC.A key di ff erence from the SJ is that for the VJ, B x andthe N / S interfaces are nor longer aligned. To support MBSin VJs, the topological superconductivity should survive tosuch a misalignment, characterized by the angle θ , shownin Fig. 9(a). The role of the misalignment was already ex-plored in Sec. III. Our calculations reveal that topological su-perconductivity is supported for θ ≤ . π [Fig. 3(e)]. Fora larger θ , the topological states become eventually fully sup-pressed [Fig. 3(e)], consistent with the trends measured in pla-nar JJs [35]. Based on the misalignment angle in the geometryof the fabricated VJ from Fig. 8(b), we fix θ = π in the fol-lowing calculations.The VJ geometry resembles a half of an X-junction [58],where various MBS can be created at the ends of the N regionsby phase control. Similar as discussed for an SJ, a phase dif-ference of π between the two adjacent S regions supports topo-logical superconductivity at a lower B x . Therefore, as shownin Fig. 9(a), we fix the phases ( ϕ , ϕ , ϕ ) of S , S , and S as ( π , 0, π ) with external fluxes Φ = Φ = Φ , where Φ isthe magnetic flux quantum, forming a π -VJ. A similar phase control with two external fluxes has been realized experimen-tally [67]. Such a π -VJ is expected to exhibit topological su-perconductivity in the whole N region with MBS localized atits two ends. This can be seen in Fig. 9(b) when the gate volt-age gives rise to topological states, analogous to the long-edgeMBS in the X-junction [58].To identify the V + and V − in the π -VJ, we calculate the V -dependent energy spectrum at B x = . V c = -3meV in the VJ, where V smaller (larger) than V c yields trivial(topological) states, further verified by the calculated ρ P and ρ C [56]. The chosen V + = V − = − +++++ configuration,the MBS are located at the ends of the N region, supportedby the calculated zero-energy modes and ρ P in Fig. 9(b). Bychanging + + + + + into + + − − − , the MBS can be moved tothe left side [Fig. 9(c)], while changing ++ −−− into ++ − ++ creates another MBS pair on the the right side [Fig. 9(d)]. Thezero-energy bands have small oscillations in the + + − − − and + + − + + configurations because of the limited length ofthe topological regions. These oscillations are suppressed byreducing the MBS overlap with an increased system size [56].The MBS trivial and nontrivial fusion in a VJ, shown inRef. [56], can be implemented following similar operations asfor the SJ described in Fig. 6. Starting from the initial −−−−− configuration, we again consider a sequence where it is firstchanged to + + − − − then to + + − + + and finally to + + + + + . γ and γ from two di ff erent MBS pairs can then be fused atthe apex of the VJ as shown in Fig. 8(a), corresponding to anontrivial fusion. For an adiabatic fusion in a VJ, the required FIG. 10. (a) Intermediate states for the nontrivial MBS fusion in a VJ. The evolution of the energy spectra for the operations A - A as afunction of the relevant mini-gate voltage. Red and black lines: evolution of finite-energy states into MBS inside the topological gap. E and Findicate the two MBS pairs (degenerate ground states) in the + + − + + configuration (before fusion). G and H indicate the ground and firstexcited state in the + + + + + configuration (after fusion). (b) Sum of the probability densities, ρ P , for the ground states, E and F, before fusion.(e)-(g) The same as (b)-(d), but shown for charge densities, ρ C . The dashed line mark the N regions covered by the mini gates. The (minimum,maximum) values in (e), (f) and (g) are (-0.52, 0.45), (-0.02, 0.03) and (-3.0, 1.9), respectively. The parameters are taken from Fig. 9. switching time of the mini-gate control could be estimatedanalogously as for the SJ. We obtain 0 .
036 ns < τ < µ sby calculating the MBS evolution in the energy spectra duringthe whole fusion process, shown in Fig. 10(a). During the adi-abatic evolution, the MBS remain protected by the topologicalgap.The connection between di ff erent fusion outcomes and thenon-Abelian statistics, introduced in Sec. II and examinedfor an SJ in Sec. IV, also follows directly for the consideredVJ in Fig. 9. The two MBS pairs, marked as E and F inFig. 10(a), are localized at the ends of the topological minigates [Fig. 10(b)] in the + + − + + configuration. They arechargeless before the fusion, supported by the calculated ρ C in Fig. 10(e). For a nontrivial fusion of chargeless MBS thereis a 50% probability of attaining the ground state, G , local-ized at the ends of the N regions [Fig. 10(c)] with vanishing ρ C [Fig. 10(f)], accessing the I fusion channel. The otheroutcome, to attain with 50% probability the excited state, H , bound at the VJ apex [Fig. 10(d)], is accompanied with ρ C [Fig. 10(g)] more than 1000 times larger than that of theground state at the VJ apex, accessing the ψ fusion channel.The resulting charge di ff erence for the fusion outcomes withdi ff erent fermion parity can be distinguished by the QPC at-tached next to the VJ apex [Fig. 8], since the QPC current,I QPC , is very sensitive to the charge di ff erence [43, 44, 66]. In contrast, for a trivial fusion of γ and γ , from the samepair, which is realized by changing − − − − − to + + + + + and then to + + − + + and, finally, to + + + + + , the resultingoutcome I is achieved with 100% probability. Therefore, theprobabilistic presence or absence of an extra charge is a signa-ture of di ff erent fusion outcomes. In a VJ this can be probedby repeating the fusion protocols which lead to charge fluc-tuations that are readily detected by the I QPC , to identify thenon-Abelian statistics expected from Ising anyons.
VI. CONCLUSIONS AND OUTLOOK
The proposed platform for the MBS fusion builds on re-cent experimental advances in InAs / Al-based planar JJs [34–36], where the topological superconductivity and the keyelements discussed here have been already experimentallydemonstrated. The phase-bias through flux control was re-alized in the studies of the topological superconductivity [34].Using a SQUID geometry, on the same sample, has revealedseveral independent signatures of topological superconductiv-ity. A π -jump in the phase di ff erence was measured with in-creasing in-plane magnetic field, accompanied by a minimumof the critical current, indicating a closing and reopening ofthe superconducting gap, shown to be strongly anisotropic0with in-plane magnetic field [35].Together with the mini-gate control of the superconductingresponse realized here, these prior results support the feasi-bility of the MBS fusion, which is based on our calculationsusing experimentally measured parameters in InAs / Al JJs. Tofacilitate the charge sensing as the detection method for di ff er-ent fusion outcomes, expected from the non-Abelian statisticsof Ising anyons, we have focused on the V-shaped planar JJs.While using the V-shaped junction requires some care inits design, such that magnitude of the misalignment angle be-tween the N / S interface and the applied in-plane magneticfield is not too large, there are also important advantages ofemploying similar non-collinear structures which can morecompletely manipulate MBS in 2D platforms and overcomethe geometrical constraints of 1D systems. Within the samedevice footprint it is possible to pattern non-collinear struc-tures where MBS are further separated and their hybridizationis reduced to better attain the limit of chargeless zero-energystates. These 2D opportunities allow using zigzag structuresfor an improved robustness of MBS [68] or creating multipleMBS [58]. Experimental progress in fabricating supercon-ducting structures with topological insulators [42, 67] furtherexpands materials candidates to implement non-collinear JJsas 2D platforms for MBS.Even though we have emphasized the MBS fusion as an ex-perimentally simpler path to demonstrate non-Abelian statis-tics than by realizing MBS braiding, our proof-of-concept fora mini-gate control in planar JJs is likely to stimulate manyother e ff orts in manipulating MBS. A number of previousproposal from 1D systems, including proximitized nanowires,could simply be implemented using planar JJs and their en-hanced parameter space supporting topological superconduc-tivity. For example, the proposal for Majorana qubits [55]could benefit from planar JJs, where placing the mini gatesin the uncovered 2DEG region can avoid constraints in theiroperation from the superconducting screening expected innanowire-based structures.In the present work we have considered using the externalflux control which can be conventionally realized through out-of-plane applied magnetic field. However, future e ff orts mayalso take advantage of tunable magnetic textures as a methodto implement a highly-localized flux control. Such texturescould be implemented with an array of magnetic elements ormagnetic multilayers [22, 23, 29, 32, 69–75] as well as byusing magnetic skyrmions [76–80]. The presence of magnetictextures also extends the control of the spin-orbit coupling, be-yond the usual classification into Rashba or Dresslhaus con-tribution [27], as such textures generate synthetic spin-orbitcoupling [22, 23, 81–84] and allow supporting MBS even insystems with inherently small spin-orbit coupling [64, 71, 85]. ACKNOWLEGDEMENTS
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