Abstract
A hallmark of topological superconductivity is the nonAbelian statistics of Majorana bound states (MBS), its chargeless zeroenergy emergent quasiparticles. The resulting fractionalization of a single electron, stored nonlocally as a two spatiallyseparated MBS, provides a powerful platform for implementing faulttolerant topological quantum computing. However, despite intensive efforts, experimental support for MBS remains indirect and does not probe their nonAbelian statistics. Here we propose how to overcome this obstacle in minigate controlled planar Josephson junctions (JJs) and demonstrate nonAbelian statistics through MBS fusion, detected by charge sensing using a quantum point contact, based on dynamical simulations. The feasibility of preparing, manipulating, and fusing MBS in twodimensional (2D) systems is supported in our experiments which demonstrate the gate control of topological transition and superconducting properties with five mini gates in InAs/Albased JJs. While we focus on this wellestablished platform, where the topological superconductivity was already experimentally detected, our proposal to identify elusive nonAbelian statistics motivates also further MBS studies in other gatecontrolled 2D systems.
Similar content being viewed by others
Introduction
Proximity effects can transform common materials to acquire exotic properties^{1}. A striking example is a topological superconductivity hosting Majorana bound states (MBS)^{2,3,4,5,6}. Their nonAbelian statistics support a peculiar state of matter, where quantum information stored nonlocally is preserved under local perturbation and disorder, particularly suitable for faulttolerant quantum computing^{7,8,9}. Detecting MBS is mainly focused on onedimensional (1D) systems^{10,11,12,13} through spectral features, such as the zerobias conductance peak (ZBCP)^{14}. However, even stable quantized ZBCP may not correspond to MBS^{15,16,17}. While it is critical to identify MBS signatures that directly probe nonAbelian statistics, 1D systems require finetuned parameters for topological superconductivity^{4,5} and limit probing nonAbelian statistics through MBS exchange (braiding) or fusion^{7,8}.
Defects and quasiparticles in topological superconductors, or boundaries between topological and trivial regions, can bind localized Majorana zeroenergy modes which behave as nonAbelian anyons^{8,9}. These zeroenergy topologicallyprotected degenerate states, in which quantum information can be stored, are separated by the energy Δ from the excited states, as depicted in Fig. 1a. The ground states, nonlocally storing ordinary fermions, can be labeled by the fermionparity (even or odd), reflecting 0 or 1 fermion occupancy. For an ordinary fermion, f, composed of nonoverlapping Majoranas, the ground state is twofold degenerate since both fermion parities correspond to zero energy. However, bringing the two Majoranas closer removes this degeneracy, as depicted in Fig. 1b, c. The resulting multiple fusion outcomes^{8,18}
reflect the underlying nonAbelian statistics and summarize that the fusion of the two MBS behaves either as vacuum, I, or an unpaired fermion ψ, resulting in an extra charge. For the trivial fusion in Fig. 1b, when MBS with a defined parity within the same pair coalesce, the outcome is deterministic, it leads to the unchanged parity (shown to be even) with no extra charge. For the nontrivial fusion in Fig. 1c, both parities are equally likely, a probabilistic measurement would yield an extra charge. While a pioneering proposal for MBS fusion in 1D nanowires envisions gatecontrol realization of Fig. 1b, c^{19}, it has important obstacles. (i) Common nanowire geometries are surrounded by superconductors, the screening makes attempted gating ineffective. (ii) Topological superconductivity requires finetuned parameters^{4,5}. (iii) 1D geometry complicates detecting an extra charge from fusion. (iv) Without an accurate preparation of the initial state, the distinction between trivial and nontrivial fusion outcomes is unclear.
Here we overcome these limitations by recognizing the opportunities in 2D proximitized materials^{20,21,22,23,24,25,26,27,28,29,30}. We reveal how minigate control in planar Josephson junctions (JJs) with 2D electron gas (2DEG) provides a versatile platform to realize MBS fusion. Our 2D InAs/Al JJs have proximitized 2DEG only partially covered by superconductors. Minigates placed in the uncovered part strongly change the proximitized 2DEG. Unlike finetuned parameters for 1D nanowires, recent experiments^{31,32,33} reveal that in planar JJs topological superconductivity exists over a large parameter space, and is particularly robust when the phase difference, ϕ, between two superconducting regions is close to π.
By proposing a Vshaped geometry, our JJ has its apex exposed edges where the locations of the bound states, formed through fusion, simplifies the charge detection in the adjacent quantum dot (QD) using quantum point contact (QPC)^{34,35,36}. To distinguish the fusion outcomes in the charge detection, we reveal the importance of an accurate preparation of the initial state. We theoretically demonstrate the fundamental aspect of nonAbelian fusion that we can transform an MBS pair into an unpaired fermion while using experimental parameters for topological superconductivity from our JJs^{31}. The feasibility of these findings is corroborated experimentally through the gate control of topological transition and superconducting properties and dynamical simulations of the MBS fusion.
Demonstrating fusion would be a major milestone for topological quantum computing and bridge the gap between the still controversial MBS observation^{37} and topological quantum algorithms, largely detached from their materials implementation^{38}. While the nonAbelian signatures from MBS fusion are complementary to those obtained from braiding, experimentally the fusion is simpler. There are even schemes in topological quantum computing implemented through fusion without braiding^{39,40,41}.
Results
Setup and model
Building on our fabrications and experimental minigate control, we propose two geometries to fuse MBS, the straight and Vshaped planar Josephson junctions (SJ, VJ). Figure 2a shows the SJ setup, formed by two epitaxial superconducting layers covering a 2DEG with mini gates. A 1D normal region (N), defined between the superconducting leads S_{1,2} with phases φ_{1,2}, can be tuned into the topological regime by the magnetic field B_{x}, the 2DEG chemical potential μ_{N} and the phase difference ϕ = φ_{1} − φ_{2} between S_{1,2}, imposed by the magnetic flux Φ. For ϕ ≈ π, the topological superconductivity exists over a large parameter space and is particularly robust^{23,24}. With ϕ = π, for a certain B_{x}, the topological condition can then be directly controlled by the gate voltage through the changes in μ_{N}^{24}. We assume that gate voltage V_{+} and V_{−} support topological and trivial states, respectively. With mini gates, as depicted in Fig. 2a, we expect to electrostatically create multiple topological (+) and trivial (−) regions along the N channel by imposing the corresponding voltage V_{+} and V_{−} in the five mini gates. Multiple MBS residing at the ends of topological regions can then be moved and fused. The setup of VJ is shown in Fig. 2b. It is similar to the SJ but has a Vshaped channel with an exposed apex defined by the three superconducting leads S_{1,2,3}. The corresponding phases φ_{1,2,3} can be tuned by the magnetic flux Φ_{1,2}. An advantage of the VJ is that its apex provides a place to detect the fusion outcome using QPC charge sensing.
Considering the topological condition for realistic planar JJs is complicated and strongly dependent on the system parameters^{31,32,33}, we need to explicitly calculate the relevant V_{+} and V_{−}. To this end, we simulate our fabricated planar JJs using the Bogoliubov–de Gennes (BdG) Hamiltonian,
where p is the momentum, m^{*} is the effective electron mass, μ_{S} is the chemical potential in the considered S_{i}, α is the Rashba SOC strength, unless explicitly specified, B ≡ B_{x}. We use σ_{i} (τ_{i}) as the Pauli (Nambu) matrices in the spin (particlehole) space and τ_{±} = (τ_{x} ± iτ_{y})/2. Δ(x, y) is the proximityinduced superconducting pair potential, for the 2DEG below the superconducting regions, which can be expressed, using the BCS relation for the Bfield suppression, as
where Δ_{0} is the superconducting gap at B = 0, B_{c} is the critical magnetic field, and φ_{i} is the corresponding S_{i} phase. The function V(x, y) ≡ μ_{N}(x, y) − μ_{S} describes the local changes of μ_{N}(x, y) in the N region due to the application of the minigate voltages, V_{1}, . . . , V_{5}, as shown in Fig. 2.
In all the calculations, we choose the parameters consistent with our fabricated junctions (SJ and VJ) that also match experimental observation of robust proximityinduced superconductivity and topological states in epitaxial InAs/Albased JJs^{31}, m^{*} = 0.03m_{0}, where m_{0} is the electron mass, and g = 10 for InAs, Δ_{0} = 0.23 meV, α = 10 meV nm, B_{c} = 1.6 T, and μ_{S} = 0.5 meV. By switching V_{+} and V_{−} through minigate control, we expect to generate, manipulate, and fuse MBS electrostatically. We will first demonstrate how this is realized in an SJ and then extend it to a VJ to show how the QPC charge sensing can distinguish the trivial and nontrivial fusion.
MBS fusion in an SJ
Experimental feasibility of the proposed minigate controlled MBS fusion builds on the demonstrated topological superconductivity in epitaxial InAs/Al planar JJs^{31,42}. This is further corroborated by using the same platform to demonstrate that mini gates can modulate the superconducting state in our fabricated SJ, shown with scanning electron microscope (SEM) images in Fig. 3a, b. With five gold mini gates covering the N region, μ_{N} for the each region under the mini gates can be independently tuned by the bias current, I_{B}.
With the three inner gates depleted, the current can only flow through the two outermost regions (marked in red) as depicted in Fig. 3a. In this configuration, the device behaves as a SQUID^{43}, as seen from the map of the measured differential resistance as a function of I_{B} and outofplane magnetic field in Fig. 3c which indicates interference between the current going through the two open channels. In contrast, when the three middle gates allow current to flow, and the outermost gates are used to deplete the 2DEG in Fig. 3b, the differential resistance in Fig. 3d shows a Fraunhofer pattern, typical of a single JJ^{43}. As expected, its periodicity is close to the one of the SQUID configuration which contains the same region.
Distinct features in Fig. 3c, d show that locally μ_{N} is strongly changed by the mini gates, providing a clear advantage over an attempt of gate control in nanowire systems^{19,44}, where the screening by superconductors diminishes changing μ_{N}. Such gatecontrolled superconducting response strongly supports our proposal of manipulating MBS with mini gates, when the topological superconductivity is achieved with B_{x} and a phase bias, ϕ. This demonstration of the minigate control, first established in our work, was later extended to experiments with even a larger number of mini gates^{45}.
Based on our fabricated device in Fig. 3, to obtain the relevant voltages V_{+} (V_{−}) for the topological (trivial) state, we do simulations based on the geometrical parameters depicted in Fig. 2a as L = 5 μm, W_{S} = 0.3 μm, W_{N} = 0.1 μm, with each mini gate 1 μm long. The calculated gatevoltagedependent energy spectrum with B_{x} = 0.6 T and ϕ = π, is shown in Fig. 4a. The evolution of the lowestenergy states into zeroenergy modes reveals that the MBS states emerge when the voltage exceeds the critical value V_{c} = −0.7 meV. This gives V_{+} ∈ (−0.7 meV, 1 meV), confirmed by the spatiallylocalized probability density, ρ_{P}, and the vanishing charge density, ρ_{C}, while V_{−} < V_{c} gives trivial states as shown in Supplementary Fig. 1. Such gatecontrolled topological transition has been confirmed by the gap closing and reopening in our experiments as shown in Supplementary Fig. 2. We choose V_{+} = 0 meV and V_{−} = −1 meV for the following simulations of minigate control. This identification of V_{+} and V_{−} gives us a chance to create and manipulate multiple MBS based on different minigate 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 MBS at its ends (Fig. 4b). The calculated B_{x}dependent energy spectrum shows that MBS indeed exists in a very large range of B_{x}, and a small B_{x} ~0.1 T already supports MBS, in agreement with the previous works^{23,24,46}. With minigate control changing + + + + + into + + − − −, the MBS at the right end can be moved to the left part (Fig. 4c), while breaking the topological region into two separate ones, by changing + + + + + into + + − + +, creates two MBS pairs (Fig. 4d). These SJ configurations are revisited in Fig. 5, where we will see that the expected control of MBS is further corroborated by the calculated ρ_{P}.
Following the above analysis, we propose a scenario for probing nonAbelian statistics based on fusion rules using minigate control as shown in Fig. 5. The system is initially prepared in a trivial state (no MBS) with − − − − − configuration. Subsequently, we can follow paths A and B to probe nontrivial and trivial fusion rules. For path A, in A_{1} we first generate one MBS pair (γ_{1}, γ_{2}) by changing V_{1} and V_{2} from V_{−} to V_{+}, and in A_{2} the second MBS pair (γ_{3}, γ_{4}) by changing V_{4} and V_{5}. These two MBS pairs build two complex fermions \({f}_{12}=\left({\gamma }_{1}+{{{{{\rm{i}}}}}}{\gamma }_{2}\right)/2\) and \({f}_{34}=\left({\gamma }_{3}+{{{{{\rm{i}}}}}}{\gamma }_{4}\right)/2\), which can be described by the occupation numbers n_{12} and n_{34}.
Without loss of generality of demonstrating the fusion rules, we assume that the two fermion states are unoccupied, giving an initial state \(\left{n}_{12},{n}_{34}\right\rangle =\left{0}_{12},{0}_{34}\right\rangle \). To facilitate experimentally probing the fusion rules, it is important to keep the same initial states in both trivial and nontrivial fusion. We will discuss later how to prepare the initial states by adding a new operation of initialization before the MBS manipulations. In A_{3} the change of V_{3} from V_{−} to V_{+} nontrivially fuses (γ_{2}, γ_{3}), which accesses both the I and ψ fusion channels with equal probability. To better understand such nontrivial fusion, we reexpress the ground state in the basis of \({f}_{14}=\left({\gamma }_{1}+{{{{{\rm{i}}}}}}{\gamma }_{4}\right)/2\) and \({f}_{23}=\left({\gamma }_{2}+{{{{{\rm{i}}}}}}{\gamma }_{3}\right)/2\), i.e., \(\left{0}_{12},{0}_{34}\right\rangle =1/\sqrt{2}\left(\left{0}_{14},{0}_{23}\right\rangle {{{{{\rm{i}}}}}}\left{1}_{14},{1}_{23}\right\rangle \right)\), where \({f}_{14}{f}_{23}\left{0}_{14},{0}_{23}\right\rangle = 0\), while \(\left{1}_{14},{1}_{23}\right\rangle ={f}_{14}^{{{\dagger}} }{f}_{23}^{{{\dagger}} }\left{0}_{14},{0}_{23}\right\rangle \). Fusing (γ_{2}, γ_{3}) induces finite energy to f_{23}, lifting the degeneracy between \(\left{0}_{14},{0}_{23}\right\rangle \) and \(\left{1}_{14},{1}_{23}\right\rangle \). As a result, measuring such a state then collapses the wavefunction with 50% probability onto either the ground state, I, or excited state with an extra quasiparticle, ψ. In A_{4} fusing the remaining (γ_{1}, γ_{4}), by changing + + + + + into − − − − −, drives the system to the initial minigate configuration. To verify the nonAbelian statistics, we examine a trivial fusion scheme B_{1}B_{4}. Unlike in the nontrivial fusion, first (γ_{1}, γ_{2}) and then (γ_{3}, γ_{4}) are created by changing − − − − − to + + + + + and then to + + − + +. Therefore, fusing (γ_{3}, γ_{4}) can only access the I channel with a trivial fusion because (γ_{3}, γ_{4}) belong to the same pair.
To simplify the description of MBS fusion it is helpful that the considered scheme from Fig. 5 is adiabatic, which requires that the topological gap remains open during the entire fusion. We show the corresponding evolution of the calculated lowenergy spectra during the fusion in Supplementary Fig. 3. For any value of the continuously changing mini gates, the MBS are protected by the topological gap between the ground and first excited states which has the minimum value, \({{{\Delta }}}_{\min }\approx 6\,\mu \)eV. The \({{{\Delta }}}_{\min }\) could be enhanced by increasing the Rashba SOC or using Sn or Nb with a higher bulk Δ than in Al^{47,48}. An animation for the evolution of the energy spectrum and wavefunction probability during the nontrivial fusion process is provided in Supplementary Movie 1.
Through uncertainty relations, this \({{{\Delta }}}_{\min }\) imposes a lower bound for the switching time, τ, during the minigate operation, which can be estimated as \({\tau }_{0} \sim \hslash /{{{\Delta }}}_{\min }\). In a realistic system, since the fusion involves multiple finitesize MBS pairs, their energies are not exactly zero and are characterized by their splitting, Δ_{S}. Therefore, the switching time should be sufficiently short to ensure the nonadiabatic transition between these nearlydegenerate MBS levels, giving an upper bound τ < τ_{S} = ℏ/Δ_{S}. The upper bound is also constrained by the quasiparticle poisoning time, τ_{P}. From the previous measurements in InAs/Al systems, τ_{P} was reported to be between 1 μs and 10 ms^{49,50}. Together, \({\tau }_{0} \, < \, \tau \, < \, \min ({\tau }_{S},{\tau }_{P})\) is required for adiabatic fusion. In our SJ, this constraint implies 0.1 ns < τ < 13 ns, which is readily realized with the existing gatecontrolled employed in JJbased qubits which are reaching GHz operation^{51}. The feasibility of this adiabatic evolution and distinct outcomes between the nontrivial and trivial MBS fusion are important prerequisites for using the fusion rules as an experimental verification of the nonAbelian statistics. A guidance for how the fusion rules could be measured comes from the prior proposals in nanowires, suggesting using Josephson current, fermionparity, or cavity detection^{8,11,19,52,53,54}.
As shown in Fig. 1, the trivial fusion deterministically gives rise to the fusion channel I, preserving the charge of the system, while in the nontrivial fusion there is a 50% probability for creating an extra charged quasiparticle ψ, which opens ways for charge detection. We expect the dynamical process of the charge creation is associated with a Cooper pair which is then quickly absorbed into the spatiallyseparated condensate for which the BCS formalism is adequate. Such an extra charge residing at a bound state [Supplementary Fig. 3] shows a huge local charge density difference compared to that in the I fusion channel, which is verified by the four orders of magnitude difference in the corresponding ρ_{C} as shown in Fig. 5e, i. When the initial states are fixed, repeating operations A_{1}–A_{4} from Fig. 5 is expected to give rise to charge fluctuations. In contrast, the fluctuations should be absent when repeating operations B_{1}–B_{4} in the nontrivial fusion process. Detecting such charge fluctuations can be a direct evidence for the MBS nontrivial fusion and nonAbelian statistics.
MBS fusion in a VJ
The previous SJ geometry provides a plausible path to MBS fusion and distinguishing the resulting outcomes. However, the corresponding charge fluctuations emerge in the interior of the central part of the N region, which is challenging to access experimentally due to the screening of superconductors and the presence of the top mini gates. Furthermore, it is unclear how to prepare the initial states, which is important to distinguish different experimental outcomes between the trivial and nontrivial fusion.
To overcome these difficulties, we propose a Vshaped geometry for the Nregion where its apex is exposed to the edge, as shown in Fig. 2b. To control preparing initial states, we also add halflength mini gates (“L”, and “R”), with voltages V_{L} and V_{R} at the ends of VJ, as shown in Fig. 6a. In their topological regime, L, and R, behave as effective quantum islands, supported by our calculations [Supplementary Note 1]. With two external fluxes, Φ_{1}, Φ_{2}, and minigate control, the MBS can be fused at the apex in a similar way to that in the SJ. An advantage in the VJ is that its apex provides a place to detect the additional charge induced by MBS fusion with QPC charge sensing, successfully used in semiconducting nanostructures^{34,35,36} and also proposed for detection of topological superconductivity in 1D systems^{54}. An experimental realization of the VJ with five mini gates, fabricated using standard electronbeam lithography and InAs/Al JJs, is shown in Supplementary Fig. 5.
A key difference from the SJ is that for the VJ, B_{x} and the N/S interfaces are no longer aligned. To support MBS in VJs, the topological superconductivity should survive such a misalignment, characterized by the angle θ in Fig. 6a. As shown in Supplementary Fig. 4, our calculations reveal that topological superconductivity is supported for θ ≤ 0.15π. For a larger θ, the topological states become eventually fully suppressed, consistent with the trends measured in planar JJs^{31}. Based on the misalignment angle in the geometry of the fabricated VJ from Supplementary Fig. 5, we fix θ = 0.1π in the following calculations.
The VJ geometry resembles half of an Xjunction^{46}, where various MBS can be created at the ends of the N regions by phase control. Similar as discussed for an SJ, a phase difference of π between the two adjacent S regions supports topological superconductivity at a lower B_{x}. Therefore, as shown in Fig. 6c, we fix the phases (φ_{1}, φ_{2}, φ_{3}) of S_{1}, S_{2}, and S_{3} as (π, 0, π) with external fluxes Φ_{1} = Φ_{2} = 0.5Φ_{0}, where Φ_{0} is the magnetic flux quantum, forming a πVJ. A similar phase control with two external fluxes has been realized experimentally^{55}. Such a πVJ is expected to exhibit topological superconductivity in the whole N region with MBS localized at its two ends. This can be seen in Fig. 6d when the gate voltage gives rise to topological states, analogous to the longedge MBS in the Xjunction^{46}.
To identify the V_{+} and V_{−} in the πVJ, we calculate the Vdependent energy spectrum at B_{x} = 0.7 T (see Fig. 6c). The evolution of the lowestenergy states shows the critical V_{c} = −3 meV in the VJ, where V smaller (larger) than V_{c} yields trivial (topological) states, further verified by the calculated ρ_{P} and ρ_{C} in Supplementary Fig. 6. The chosen V_{+} = 0 meV and V_{−} = −5 meV are used to manipulate the MBS with various minigate configurations. Similar to the SJ, for the + + + + + configuration, the MBS are located at the ends of the N region, supported by the calculated zeroenergy modes (Fig. 6d) and ρ_{P} (Fig. 7c). By changing + + + + + into + + − − −, the MBS can be moved to the left side (Fig. 6e), while changing + + − − − into + + − + + creates another MBS pair on the right side (Fig. 6f). The zeroenergy bands have small oscillations in the + + − − − and + + − + + configurations because of the limited length of the topological regions. These oscillations are suppressed by reducing the MBS overlap with an increased system size as in Supplementary Fig. 7.
Similar to the fusion protocol in Fig. 5, the MBS trivial and nontrivial fusion can be implemented in a VJ as shown in Supplementary Note 2. The spectrum evolution during the nontrivial (trivial) fusion with A_{1}–A_{4} (B_{1}–B_{4}) operations is shown in Fig. 7a (Supplementary Fig. 11). We can see the two quantum island states adiabatically evolve into two MBS pairs after the operations A_{1−2}/B_{1−2}. Such two MBS pairs, marked as E and F in Fig. 7a, are localized at the ends of the topological mini gates (Fig. 7b) in the + + − + + configuration. They are chargeless before the fusion, supported by the calculated ρ_{C} in Fig. 7e. The operation A_{3} fuses the MBS (γ_{2} and γ_{3}) from different pairs and gives a nontrivial fusion. There is a 50% probability of attaining the ground state, G, localized at the ends of the N regions (Fig. 7c) with vanishing ρ_{C} (Fig. 7f), accessing the I fusion channel. The other outcome, to attain with 50% probability the excited state, H, bound at the VJ apex (Fig. 7d), is accompanied with ρ_{C} (Fig. 7g) more than 1000 times larger than that of the ground state at the VJ apex, accessing the ψ fusion channel. In contrast, operation B_{3} trivially fuses the MBS (γ_{3} and γ_{4}) from the same pair. The resulting outcome I is achieved with 100% probability. Therefore, the probabilistic presence or absence of an extra charge at the VJ apex is a signature of different fusion outcomes.
To experimentally realize the fusions, the switching time should be tuned to enable that the MBS are adiabatically evolved during the whole fusion process in a VJ. The required switching time of the minigate control could be estimated analogously as for the SJ. We obtain 0.07 ns < τ < 7 ns by calculating the spectrum evolution during the whole fusion process (Fig. 7a), which is independently confirmed from our dynamical simulations shown in Supplementary Note 3.
The presence (absence) of the charge fluctuations when repeating nontrivial (trivial) is usually viewed as evidence for the MBS fusion rules. However, if each time the initial state and its occupation change randomly, the trivial fusion may also give charge fluctuations as a false signature of the fusion rules. This issue has been overlooked in previous fusion proposals^{8,19}, which neglect the effect of the initial occupations. To overcome this problem, as shown in Supplementary Note 1, we propose an initialization A_{0} (B_{0}), further supported by our simulations, to empty the initial occupations of the quantum island and get the \(\left00\right\rangle \) state. Such initialization precedes A_{1}–A_{4} (B_{1}–B_{4}) operations to realize the initial \(\left00\right\rangle \) state in every fusion cycle, which corresponds to the operations A_{1}–A_{3} (B_{1}–B_{3}) for the nontrivial (trivial) fusion. Observing the presence (absence) of the charge fluctuations at the VJ apex for repeated nontrivial (trivial) fusion can then be a conclusive evidence for the fusion rules.
Readout of the fusion outcome
To detect the charge fluctuations from the fusion, we couple a QD to the VJ apex. The QD is created by confining gates^{34}, as shown in Fig. 6a, and its energy levels can be shifted by the gate voltage V_{QD}. When the energy of the lowest unoccupied state in the QD is aligned between the energy levels of the G and H states (Fig. 7a), the fusioninduced charge can transfer into the QD, giving a QD charge change, Q_{QD}. For trivial fusion, Q_{QD} is 0; while for nontrivial fusion, Q_{QD} is 0 or 1e with the same probability, giving an average value of 0.5e. Such a different fusion outcome is supported by the calculated Q_{QD}, shown in Fig. 6b, by using dynamical simulations based on timedependent BdG equation^{56,57}. More discussion and calculation details are given in Supplementary Note 3. The Q_{QD} can be detected by the attached QPC^{34,35,36}, because the QPC current, I_{QPC}, is very sensitive to the charge change^{35,36,54}. Such a charge sensing technique has been widely used to accurately detect the charge in QDs^{34}. After the charge sensing detection, the fusioninduced charge does not stay in the VJ. We then reset the mini gates to the initial configuration (A_{4}/B_{4}) and do the initialization to make sure that each fusion cycle has the same initial state.
By repeating the operations of A_{0}–A_{4} (B_{0}–B_{4}), we can repeat the MBS fusion with the same occupation state \(\left00\right\rangle \) as shown in the fusion protocols (Supplementary Note 2). Every time the fusion occurs (the system goes into + + + + + after A_{3}/B_{3}), we use QPC charge sensing to detect Q_{QD}. The detected current, I_{QPC}, through the QPC is denoted by I_{TF} (I_{NF}) for the trivial (nontrivial) fusion. While the expected I_{TF} remains the same, the I_{NF} fluctuates during the fusion cycles. To suppress the possible trivial background charge fluctuation, we can focus on the difference, Δ_{IF}, between the I_{TF} and I_{NF}. Measuring such a fluctuating Δ_{IF} is a direct conclusive evidence for the nonAbelian statistics of MBS.
Discussion
While using the Vshaped junction requires some care in its design, such that the magnitude of the misalignment angle between the N/S interface and the applied inplane magnetic field is not too large^{31,46}, there are also important advantages of employing similar noncollinear structures to more completely manipulate MBS in 2D platforms and overcome the geometrical constraints of 1D systems. Within the same device footprint, it is possible to pattern noncollinear structures where MBS are further separated and their hybridization is reduced to better attain the limit of chargeless zeroenergy states. These 2D opportunities allow using zigzag structures for improved robustness of MBS^{58} or creating multiple MBS^{46}. Progress in fabricating superconducting structures with topological insulators^{55,59} expands materials candidates to implement noncollinear JJs as platforms for MBS.
In the present work, we have considered using the external flux control which can be conventionally realized through an outofplane applied magnetic field. We have theoretically demonstrated the fundamental aspect of nonAbelian fusion that we can transform an MBS pair into an unpaired fermion while using experimental parameters for topological superconductivity^{31}. Our experiments on minigate controlled superconducting properties in JJ and dynamical simulations of the MBS fusion are reassuring for the feasibility of these findings. However, future efforts may also take advantage of tunable magnetic textures as a method to implement a highlylocalized flux control. Such textures could be implemented with an array of magnetic elements or magnetic multilayers^{21,22,27,60,61,62,63,64}, as well as by using magnetic skyrmions^{65,66,67,68}. The presence of magnetic textures also extends the control of the spinorbit coupling (SOC), beyond the usual classification into Rashba or Dresselhaus contribution^{25}, as such textures generate synthetic SOC^{21,22,69}, and allow supporting MBS even in systems with inherently small SOC^{52,62}.
Methods
Simulations
The calculated results are obtained by numerically solving the BdG Hamiltonian from Eq. (2), using the Kwant package^{70}. The dynamical simulations are performed by solving the timedependent BdG equations^{56,57}, as given in Supplementary Note 3.
Fabrications
The JJ structure is grown on a semiinsulating InP (100) substrate, followed by a graded buffer layer. The quantum well consists of a 4 nm layer of InAs grown on a 6 nm layer of In_{0.81}Ga_{0.25}As. The InAs layer is capped by a 10 nm In_{0.81}Ga_{0.25}As layer to produce an optimal interface while maintaining high 2DEG mobility, followed by in situ growth of epitaxial Al (111). JJs are fabricated on the same wafer exhibit a highlytransparent interface between the superconducting layer and the 2DEG. The fabrication process consists of three steps of electronbeam (ebeam) lithography using PMMA resist. After the first lithography, the deep semiconductor mesas are etched using first Transene type D to etch the Al and then an IIIV wet etch [C_{6}H_{8}O_{7}(1M):H_{3}PO_{4}(85% in mass):H_{2}O_{2}(30% in mass):H_{2}O = 18.3:0.43:1:73.3]. The second lithography is used to define the JJ gap which is etched using Transene type D. A layer of 90 nm of SiO_{x} was then deposited using ebeam evaporation and finally, the gates were patterned using ebeam lithography followed by ebeam evaporation of 5 nm of Ti followed by 45 nm of Au.
Measurements
The device has been measured in an Oxford Triton dilution refrigerator fitted with a 631.5 T vector magnet which has a base temperature of 7 mK. All transport measurements are performed using standard dc and lockin techniques at low frequencies and excitation current I_{ac} = 10 nA.
Data availability
The data that support the findings of this study are available within the paper and its Supplementary Information. Additional data are available from the corresponding authors upon reasonable request.
Code availability
The computation code information for getting the theoretical results is available from the corresponding authors upon reasonable request.
References
Žutić, I., MatosAbiague, A., Scharf, B., Dery, H. & Belashchenko, K. Proximitized materials. Mater. Today 22, 85–107 (2019).
Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys.Usp. 44, 131 (2001).
Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).
Lutchyn, R. M., Sau, J. D. & Das Sarma, S. Majorana fermions and a topological phase transition in semiconductorsuperconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010).
Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010).
Klinovaja, J., Stano, P. & Loss, D. Transition from fractional to Majorana fermions in Rashba nanowires. Phys. Rev. Lett. 109, 236801 (2012).
Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. NonAbelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).
Aasen, D. et al. Milestones toward Majoranabased quantum computing. Phys. Rev. X 6, 031016 (2016).
Das Sarma, S., Freedman, M. & Nayak, C. Majorana zero modes and topological quantum computation. NPJ Quantum Inf. 1, 150001 (2015).
Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductorsemiconductor nanowire devices. Science 336, 1003 (2012).
Rokhinson, L. P., Liu, X. & Furdyna, J. K. The fractional a.c. Josephson effect in a semiconductorsuperconductor nanowire as a signature of Majorana particles. Nat. Phys. 8, 795–799 (2012).
Deng, M. T. et al. Anomalous zerobias conductance peak in a NbInSb nanowireNb hybrid device. Nano Lett. 12, 6414–6419 (2012).
NadjPerge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014).
Sengupta, K., Žutić, I., Kwon, H.J., Yakovenko, V. M. & Das Sarma, S. Midgap edge states and pairing symmetry of quasionedimensional organic superconductors. Phys. Rev. B 63, 144531 (2001).
Zhang, H. et al. Large zerobias peaks in InSbAl hybrid semiconductorsuperconductor nanowire devices. Preprint at arXiv:2101.11456 (2021).
Yu, P. et al. NonMajorana states yield nearly quantized conductance in proximatized nanowires. Nat. Phys. 17, 482–488 (2021).
Pan, H., Cole, W. S., Sau, J. D. & Das Sarma, S. Generic quantized zerobias conductance peaks in superconductorsemiconductor hybrid structures. Phys. Rev. B 101, 024506 (2020).
Lahtinen, V. T. & Pachos, J. K. A short introduction to topological quantum computation. SciPost Phys. 3, 021 (2017).
Alicea, J., Oreg, Y., Refael, G., von Oppen, F. & Fisher, M. P. A. NonAbelian statistics and topological quantum information processing in 1D wire networks. Nat. Phys. 7, 412–417 (2011).
Shabani, J. et al. Twodimensional epitaxial superconductorsemiconductor heterostructures: a platform for topological superconducting networks. Phys. Rev. B 93, 155402 (2016).
Fatin, G. L., MatosAbiague, A., Scharf, B. & Žutić, I. Wireless Majorana bound states: from magnetic tunability to braiding. Phys. Rev. Lett. 117, 077002 (2016).
MatosAbiague, A. et al. Tunable magnetic textures: from Majorana bound states to braiding. Solid State Commun. 262, 1–6 (2017).
Pientka, F. et al. Topological superconductivity in a planar Josephson junction. Phys. Rev. X 7, 021032 (2017).
Hell, M., Leijnse, M. & Flensberg, K. Twodimensional platform for networks of Majorana bound states. Phys. Rev. Lett. 118, 107701 (2017).
Scharf, B., Pientka, F., Ren, H., Yacoby, A. & Hankiewicz, E. M. Tuning topological superconductivity in phasecontrolled Josephson junctions with Rashba and Dresselhaus spinorbit coupling. Phys. Rev. B 99, 214503 (2019).
Liu, J., Wu, Y., Sun, Q.F. & Xie, X. C. Fluxinduced topological superconductor in planar Josephson junction. Phys. Rev. B 100, 235131 (2019).
Zhou, T., Mohanta, N., Han, J. E., MatosAbiague, A. & Žutić, I. Tunable magnetic textures in spin valves: from spintronics to Majorana bound states. Phys. Rev. B 99, 134505 (2019).
Setiawan, F., Wu, C.T. & Levin, K. Full proximity treatment of topological superconductors in Josephsonjunction architectures. Phys. Rev. B 99, 174511 (2019).
Hegde, S. S. et al. A topological Josephson junction platform for creating, manipulating, and braiding Majorana bound states. Ann. Phys. 423, 168326 (2020).
Alidoust, M., Shen, C. & Žutić, I. Cubic spinorbit coupling and anomalous Josephson effect in 2D planar junctions. Phys. Rev. B 103, L060503 (2021).
Dartiailh, M. C. et al. Phase signature of topological transition in Josephson junctions. Phys. Rev. Lett. 126, 036802 (2021).
Fornieri, A. et al. Evidence of topological superconductivity in planar Josephson junctions. Nature 569, 89–92 (2019).
Ren, H. et al. Topological superconductivity in a phasecontrolled Josephson junction. Nature 569, 93–98 (2019).
Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in fewelectron quantum dots. Rev. Mod. Phys. 79, 1217–1265 (2007).
Barthel, C., Reilly, D. J., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Rapid singleshot measurement of a singlettriplet qubit. Phys. Rev. Lett. 103, 160503 (2009).
Reilly, D. J., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Fast singlecharge sensing with a rf quantum point contact. Appl. Phys. Lett. 91, 162101 (2007).
Aguado, R. & Kouwenhoven, L. P. Majorana qubits for topological quantum computing. Phys. Today 73, 6, 44–50 (2020).
Brown, B. J., Laubscher, K., Kesselring, M. S. & Wootton, J. R. Poking holes and cutting corners to cchieve Clifford gates with the surface code. Phys. Rev. X 7, 021029 (2017).
Bonderson, P., Freedman, M. & Nayak, C. Measurementonly topological quantum computation via anyonic interferometry. Ann. Phys. 324, 787–826 (2009).
Litinski, D. & von Oppen, F. Braiding by Majorana tracking and longrange CNOT gates with color codes. Phys. Rev. B 96, 205413 (2017).
Beenakker, C. W. J. Search for nonAbelian Majorana braiding statistics in superconductors. SciPost Phys. Lect. Notes 15, 1–29 (2020).
Barati, F. et al. Tuning supercurrent in Josephson fieldeffect transistors using hBN dielectric. Nano Lett. 21, 1915–1920 (2021).
Tinkham, M. Introduction to Superconductivity (McGrawHill, 1996). .
Bauer, B., Karzig, T., Mishmash, R. V., Antipov, A. E. & Alicea, J. Dynamics of Majoranabased qubits operated with an array of tunable gates. SciPost Phys. 5, 004 (2018).
Elfeky, B. H. et al. Local control of supercurrent density in epitaxial planar Josephson junctions. Nano Lett. 21, 8274–8280 (2021).
Zhou, T. et al. Phase control of Majorana bound states in a topological X junction. Phys. Rev. Lett. 124, 137001 (2020).
Sau, J. D., Tewari, S., Lutchyn, R. M., Stanescu, T. D. & Das Sarma, S. NonAbelian quantum order in spinorbitcoupled semiconductors: search for topological Majorana particles in solidstate systems. Phys. Rev. B 82, 214509 (2010).
Pakizer, J. D., Scharf, B. & MatosAbiague, A. Crystalline anisotropic topological superconductivity in planar Josephson junctions. Phys. Rev. Res. 3, 013198 (2021).
Albrecht, S. M. et al. Transport signatures of quasiparticle poisoning in a Majorana island. Phys. Rev. Lett. 118, 137701 (2017).
Higginbotham, A. P. et al. Parity lifetime of bound states in a proximitized semiconductor nanowire. Nat. Phys. 11, 1017–1021 (2015).
Krantz, P. et al. A quantum engineer’s guide to superconducting qubits. Appl. Phys. Rev. 6, 021318 (2019).
Desjardins, M. M. et al. Synthetic spinorbit interaction for Majorana devices. Nat. Mater. 18, 1060–1064 (2019).
BenShach, G. et al. Detecting Majorana modes in onedimensional wires by charge sensing. Phys. Rev. B 91, 045403 (2015).
Wimmer, M., Akhmerov, A. R., Dahlhaus, J. P. & Beenakker, C. W. J. Quantum point contact as a probe of a topological superconductor. New. J. Phys. 13, 053016 (2011).
Yang, G. et al. Protected gap closing in Josephson trijunctions constructed on Bi_{2}Te_{3}. Phys. Rev. B 100, 180501(R) (2019).
Amorim, C. S., Ebihara, K., Yamakage, A., Tanaka, Y. & Sato, M. Majorana braiding dynamics in nanowires. Phys. Rev. B 91, 174305 (2015).
Sanno, T., Miyazaki, S., Mizushima, T. & Fujimoto, S. Ab initio simulation of nonAbelian braiding statistics in topological superconductors. Phys. Rev. B 103, 054504 (2021).
Laeven, T., Nijholt, B., Wimmer, M. & Akhmerov, A. R. Enhanced proximity effect in zigzagshaped Majorana Josephson junctions. Phys. Rev. Lett. 125, 086802 (2020).
Schüffelgen, P. et al. Selective area growth and stencil lithography for in situ fabricated quantum devices. Nat. Nanotechnol. 14, 825–831 (2019).
Mohanta, N. et al. Electrical control of Majorana bound states using magnetic stripes. Phys. Rev. Appl. 12, 034048 (2019).
PalacioMorales, A. et al. Atomicscale interface engineering of Majorana edge modes in a 2D magnetsuperconductor hybrid system. Sci. Adv. 5, eaav6600 (2019).
Turcotte, S., Boutin, S., Camirand Lemyre, J., Garate, I. & PioroLadriére, M. Optimized micromagnet geometries for Majorana zero modes in low gfactor materials. Phys. Rev. B 102, 125425 (2020).
Ronetti, F., Plekhanov, K., Loss, D. & Klinovaja, J. Magnetically confined bound states in Rashba systems. Phys. Rev. Res. 2, 022052(R) (2020).
Wei, P., Manna, S., Eich, M., Lee, P. & Moodera, J. Superconductivity in the surface state of noble metal gold and its Fermi level tuning by EuS dielectric. Phy. Rev. Lett. 122, 247002 (2019).
Yang, G., Stano, P., Klinovaja, J. & Loss, D. Majorana bound states in magnetic skyrmions. Phys. Rev. B 93, 224505 (2016).
Güngördü, U., Sandhoefner, S. & Kovalev, A. A. Stabilization and control of Majorana bound states with elongated skyrmions. Phys. Rev. B 97, 115136 (2018).
Garnier, M., Mesaros, A. & Simon, P. Topological superconductivity with deformable magnetic skyrmions. Commun. Phys. 2, 126 (2019).
Mascot, E., Bedow, J., Graham, M., Rachel, S. & Morr, D. K. Topological superconductivity in skyrmion lattices. npj Quantum Mater. 6, 1–6 (2021).
Kjaergaard, M., Wölms, K. & Flensberg, K. Majorana fermions in superconducting nanowires without spinorbit coupling. Phys. Rev. B 85, 020503(R) (2012).
Groth, C. W., Wimmer, M., Akhmerov, A. R. & Waintal, X. Kwant: a software package for quantum transport. New J. Phys. 16, 063065 (2014).
Acknowledgements
We thank Jie Liu for the helpful discussion. This work is supported by US ONR Grant No. N000141712793 (I.Ž., J.H., and A.M.A.), DARPA Grant No. DP18AP900007, and the University at Buffalo Center for Computational Research.
Author information
Authors and Affiliations
Contributions
T.Z. and I.Ž. conceived the study. T.Z. performed the calculations and analysis with J.H., A.M.A., and I.Ž providing input. M.C.D., K.S., and J.S. fabricated the samples and performed the experimental measurements and analysis. T.Z. and I.Ž. wrote the paper. All authors were involved in the discussion and editing of the paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks the anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhou, T., Dartiailh, M.C., Sardashti, K. et al. Fusion of Majorana bound states with minigate control in twodimensional systems. Nat Commun 13, 1738 (2022). https://doi.org/10.1038/s41467022294636
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467022294636
This article is cited by

Vacancyengineered nodalline semimetals
Scientific Reports (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.