Fusion of Majorana bound states with mini-gate control in two-dimensional systems

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, stored nonlocally as a two spatially-separated MBS, provides a powerful platform for implementing fault-tolerant topological quantum computing. However, despite intensive efforts, experimental support for MBS remains indirect and does not probe their non-Abelian statistics. Here we propose how to overcome this obstacle in mini-gate controlled planar Josephson junctions (JJs) and demonstrate non-Abelian 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 two-dimensional (2D) systems is supported in our experiments which demonstrate the gate control of topological transition and superconducting properties with five mini gates in InAs/Al-based JJs. While we focus on this well-established platform, where the topological superconductivity was already experimentally detected, our proposal to identify elusive non-Abelian statistics motivates also further MBS studies in other gate-controlled 2D systems.


Supplementary Fig. 3: Spectrum evolution during the MBS fusion in an SJ
shows the spectrum evolution for the MBS fusion scheme in an SJ. The adiabatic evolution in the fusion scheme is verified by the calculated energy spectra evolution for the operations in nontrivial (A1-A4) and trivial (B1-B4) fusion. 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 of ~6 eV. The different signatures between the trivial and nontrivial fusions are verified by the calculated ρP and ρC of the fused MBS. Supplementary Fig. 3. Calculated energy spectrum evolution for the operations in a nontrivial (A1-A4) and b trivial (B1-B4) fusion in an SJ, shown as a function of the relevant mini-gate voltage. Red and black lines: evolution of finiteenergy states 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 V3 = -0.6 meV in the + + + + + configuration (after fusion). c Sum of the probability densities, ρP, for E and F. d-e ρP for G and sum of the ρP for G and H. f-h The same as c-e, but shown for charge densities, ρC. The dashed line mark the N regions covered by the mini gates. The (minimum, maximum) values in f, g and h are (-0.07, 0.09), (-0.00009, 0.00004) and (-3.5, 2.9), respectively. The parameters are taken from Fig. 4 in the main text. Supplementary Fig. 4: MBS robustness against the deviation of the magnetic field from the junction interface. Supplementary Fig. 4 shows the robustness of the MBS against the misalignment angle, θ, between the N/S interface and the applied in-plane magnetic field. Our calculations reveal that topological superconductivity is supported for θ ≤ 0.15π. Supplementary Fig. 4. a Schematic of a tilted junction with a misalignment angle θ from the applied Bx. MBS (stars) reside at the opposite ends of the N region (yellow). b Energy spectra for an SJ with Bx = 0.4 T as a function of the phase difference,  = 1 − 2. c Same as b but for a tilted junction with θ = 0.1π. d Energy spectra for a tilted junction,  = π and θ = 0.1π as a function of Bx. e Energy spectra for a tilted junction,  = π and Bx = 0.4 T as a function of θ. f Probability density, ρP, for the lowest (red) energy states with  = π, θ = 0.1π and Bx = 0. Supplementary Fig. 7 shows that the oscillations of the zero-energy modes can be reduced when the system size is increased. Correspondingly, ρC of the zero-energy modes decreases drastically.

Supplementary Note 1: Preparation of the initial states through initializations
For a two-fermions system (f12 and f34) discussed in the manuscript, there are four typical occupation states, depending on whether they are occupied (|1) or unoccupied (|0), i.e. |00, |11, |10, and |01. Based on the fusion rules given in Fig. 1 in the main text, irrespective of the initial state, the trivial fusion cannot change the fermion occupations, while the nontrivial fusion can always induce a superposition of the occupied and unoccupied states, having 50% probability to change the fermion occupations. As a result, we should observe charge fluctuations when repeating the nontrivial fusion, while no charge fluctuations when repeating the trivial fusion. Supplementary Fig. 8. a Schematic of the VJ having five mini gates with voltages V1-V5 and two additional halflength mini gates, with voltages VL and VR. We set VL,R = V+ = 0 (orange) and V1-5 = V-= -5 meV (green) to engineer the two short mini-gates into the effective quantum islands. b Density of states for a with VL = V+ but VR = V-. c Probability density, ρP, of the EIsland marked by the triangle in b, supporting the formation of the quantum island. Schematic d and the spectrum evolution e when the topological region is extended by changing V1 and V2 from V-to V+. f ρP of the lowest positive energy marked by the star in e, supporting the MBS formation. The parameters are taken from Fig. 6 in the main text.
However, if each time the initial state changes randomly, the trivial fusion may also give charge fluctuations (due to trivial change of the initial occupations), bringing possible false signature of the fusion rules. To overcome this problem, it is important to keep the same initial occupation in every fusion cycle. To prepare the initial occupations of f12 and f34, we first need to generate two quantum islands to store the f12 and f34. Such islands should form at the edge of the system to be accessible and flexibly controlled. Considering that our MBS are initially generated at the two VJ ends, we propose to create two quantum islands by adding half-length mini gates (L, and R), with voltages VL and VR at the ends of VJ, as shown in Supplementary  Fig. 8a. Considering the length of the L (R) is small, when it is in the topological region, it can behave as an effective quantum island with the lowest energy level, EIsland, within the superconducting gap. Taking the quantum island L as an example, we calculate its energy spectrum and find the EIsland ~ 8V as shown in Supplementary Fig. 8b. Our calculated probability density (Supplementary Fig. 8c) confirms the effective quantum island indeed forms at L. Such an island can store the fermion f12. When the EIsland is occupied (unoccupied), the f12 is in |1 (|0) state. By changing the V1 and V2 from V-to V+ (Supplementary Fig. 8d), the topological region can be extended, and the f12 in quantum island adiabatically evolves into MBS (f12 = 1 + i2), supported by the calculated spectrum evolution (Supplementary Fig. 8e). Since it is an adiabatic evolution, the f12 does not change its occupation until the nontrivial MBS fusion happens. Similarly, the fermion f34 can be stored at the quantum island R, and its occupation (|1 or |0) can be well defined depending on whether its EIsland is occupied or not. Combing the occupations of f12 and f34, we have four possible typical initial states (|00, |11, |10, and |01). Supplementary Fig. 9. Schematic for the A0/B0 operation of "initialization". a The initial occupation of the fermion f12 = 1 + i2 in the quantum island. If the f12 is unoccupied (|0), it is our desired state and the initialization does not do anything. Thus, here we only show the case when f12 is occupied (|1), where the lowest positive energy level of the L quantum island (EIsland ~ 8eV in Supplementary Fig. 8b) must be occupied. We couple the initializing QD, QDI (blue circle) with energy EQDI > EIsland, to the quantum island. b By shifting EQDI < EIsland, the charge of f12 in the quantum island can transfer into the QDI, leaving a |0 state for f12. Similarly, the |0 state for f34 can be prepared by coupling another QD to the quantum island.
In principle, the initial state should be |00 because it is the ground state of the system. However, if for any reason the initial state is different (for example |11, |10, or |01 state), we can still empty it through additional QD by performing initialization procedure. Such initialization (A0/B0) shown in Supplementary  Fig. 9 is added before the MBS generation and manipulation (A1-4/B1-4) in our fusion protocol ( Supplementary Fig. 10), which can empty the initial occupations to give an initial state |00. We couple an additional initializing QD, QDI ( Supplementary Fig. 9a), as a reservoir to receive the occupied fermion in the L quantum island and drive f12 into the |0 state. An advantage of using the QDI is that we can flexibly control its energy and coupling with the quantum island by gating [5]. Now, let us see how the QDI helps to empty the quantum island. For the |0 state, it is our desired state and the initialization operation does nothing. For the |1 state, by initially making the highest unoccupied energy level of the QDI, EQDI > EIsland ( Supplementary Fig. 9a), and then making the EQDI < EIsland ( Supplementary Fig. 9b), the charge of f12 in the L island can transfer into the QDI, leaving a |0 state for f12. Similarly, we can drive f34 into the |0 state. After such an initialization, f12 and f34 are in the desired |00 state. Then we decouple the QDI with the VJ to make sure the QDI does not perturb the MBS manipulations in the VJ.

Supplementary Note 2: Protocol and spectrum evolution for the MBS fusion in a VJ
With the initialization, the initial VJ state is prepared as |00. Then we can perform A1-A4 (B1-B4) operations to implement the MBS nontrivial (trivial) fusion as shown in Supplementary Fig. 10. The spectrum evolution of A1-A4 and B1-B4 is shown in Supplementary Fig. 11. We can see two quantum island states can adiabatically evolve into two MBS pairs, where their pair configurations depend on the A1-2/B1-2 operations. The operation A3 fuses the MBS from different pairs and gives a nontrivial fusion, accessing both vacuum, I, and an unpaired fermion, ψ, with 50% probability, while the B3 trivially fuses the MBS from the same pairs, corresponding to I with 100% probability. After A3/B3, we use the QPC to detect the QD charge number, QQD. For the trivial fusion, the QQD is 0; while for the nontrivial fusion, the QQD is 0 or 1e with the same probability, giving an expectation value of 0.5e. Such a different fusion outcome is supported by our dynamical simulations in Supplementary Fig. 12. After the QPC charge sensing detection, we perform A4/B4 operations to reset the system. By repeating such A0-A4 (B0-B4) operations, we can repeat the MBS fusion with the same |00 occupation, where every time the fusion occurs (the system goes to + + + + + after A3/B3), we use the QPC to detect the QQD. The detected current, IQPC, through the QPC is denoted by ITF (INF) for the trivial (nontrivial) fusion. While the expected ITF remains the same, the INF fluctuates during the fusion cycles. To suppress the possible trivial background charge fluctuation, we can focus on the difference, IF, between ITF and INF. Measuring such a fluctuating IF is a direct evidence for the non-Abelian statistics of MBS.