Realization of a minimal Kitaev chain in coupled quantum dots

Majorana bound states constitute one of the simplest examples of emergent non-Abelian excitations in condensed matter physics. A toy model proposed by Kitaev shows that such states can arise at the ends of a spinless $p$-wave superconducting chain. Practical proposals for its realization require coupling neighboring quantum dots in a chain via both electron tunneling and crossed Andreev reflection. While both processes have been observed in semiconducting nanowires and carbon nanotubes, crossed-Andreev interaction was neither easily tunable nor strong enough to induce coherent hybridization of dot states. Here we demonstrate the simultaneous presence of all necessary ingredients for an artificial Kitaev chain: two spin-polarized quantum dots in an InSb nanowire strongly coupled by both elastic co-tunneling and crossed Andreev reflection. We fine-tune this system to a sweet spot where a pair of Poor Man's Majorana states is predicted to appear. At this sweet spot, the transport characteristics satisfy the theoretical predictions for such a system, including pairwise correlation, zero charge and stability against local perturbations. While the simple system presented here can be scaled to simulate a full Kitaev chain with an emergent topological order, it can also be used imminently to explore relevant physics related to non-Abelian anyons.


Realization of a minimal Kitaev chain in coupled quantum dots
Engineering Majorana bound states in condensed matter systems is an intensively pursued goal, both for their exotic non-Abelian exchange statistics and for potential applications in building topologically protected qubits [1,9,10].The most investigated experimental approach looks for Majorana states at the boundaries of topological superconducting materials, made of hybrid semiconducting-superconducting heterostructures [11][12][13][14][15].However, the widely-relied-upon signature of Majorana states, zero-bias conductance peaks, is by itself unable to distinguish topological Majorana states from other trivial zero-energy states induced by disorder and smooth gate potentials [16][17][18][19][20][21].Both problems disrupting the formation or detection of a topological phase originate from a lack of control over the microscopic details of the electron potential landscape in these heterostructure devices.
In this work, we realize a minimal Kitaev chain [1] using two quantum dots (QDs) coupled via a short superconducting-semiconducting hybrid [2].
By controlling the electrostatic potential on each of these three elements, we overcome the challenge imposed by random disorder potentials.At a fine-tuned sweet spot where Majorana states are predicted to appear, we observe end-toend correlated conductance that signals emergent Majorana properties such as zero charge and robustness against local perturbations.We note that these Majorana states in a minimal Kitaev chain are not topologically protected and have been dubbed "Poor Man's Majorana" (PMM) states [3].
The elementary building block of the Kitaev chain is a pair of spinless electronic sites coupled simultaneously by two mechanisms: elastic co-tunneling (ECT) and crossed Andreev reflection (CAR).Both processes are depicted in Fig. 1a.ECT involves a single electron hopping between two sites with an amplitude t.CAR refers to two electrons from both sites tunneling back and forth into a common superconductor with an amplitude ∆ (not to be confused f.Charge stability diagram of the coupled-QD system, in the cases of t > ∆ (i), t = ∆ (ii) and t < ∆ (iii).Blue marks regions in the (µ LD , µ RD ) plane where the ground state is even and orange where the ground state is odd.g.False-colored scanning electron microscopy image of the device, prior to the fabrication of the normal leads.InSb nanowire is colored green.QDs are defined by bottom finger gates (in red) and their locations are circled.The gates controlling the two QD chemical potentials are labeled by their voltages, V LD and V RD .The central thin Al/Pt film, in blue, is grounded.The proximitized nanowire underneath is gated by V PG .Two Cr/Au contacts are marked by yellow boxes.The scale bar is 300 nm.h.Right-side zero-bias local conductance G RR in the (V LD , V RD ) plane when the system is tuned to t > ∆ (1) and t < ∆ (2).The arrows mark the spin polarization of the QD levels.The DC bias voltages are kept at zero, V L = V R = 0, and an AC excitation of 6 µV RMS is applied on the right side.
with the superconducting gap size), forming and splitting Cooper-pairs [4].
To create the two-site Kitaev chain, we utilize two spin-polarized QDs where only one orbital level in each dot is available for transport.In the absence of tunneling between the QDs, the system is characterized by a well-defined charge state on each QD: |n LD n RD ⟩, where n LD , n RD ∈ {0, 1} are occupations of the left and right QD levels.The charge on each QD depends only on its electrochemical potential µ LD or µ RD , schematically shown in Fig. 1b.
In the presence of inter-dot coupling, the eigenstates of the combined system become superpositions of the charge states.ECT couples |10⟩ and |01⟩, resulting in two eigenstates of the form α |10⟩ − β |01⟩ (Fig. 1c), both with odd combined charge parity.These two bonding and anti-bonding states differ in energy by 2t when both QDs are at their charge degeneracy, i.e., µ LD = µ RD = 0. Analogously, CAR couples the two even states |00⟩ and |11⟩ to produce bonding and anti-bonding eigenstates of the form u |00⟩ − v |11⟩, preserving the even parity of the original states.These states differ in energy by 2∆ when µ LD = µ RD = 0 (Fig. 1d).If the amplitude of ECT is stronger than CAR (t > ∆), the odd bonding state has lower energy than the even bonding state near the joint charge degeneracy µ LD = µ RD = 0 (see Methods for details).The system thus features an odd ground state in a wider range of QD potentials, leading to a charge stability diagram shown in Fig. 1f(i) [22].
The opposite case of CAR dominating over ECT, i.e., t < ∆, leads to a charge stability diagram shown in Fig. 1f(iii), where the even ground state is more prominent.Fine-tuning the system such that t = ∆ equalizes the two avoided crossings, inducing an even-odd degenerate ground state at µ LD = µ RD = 0 (Fig. 1f(ii)).This degeneracy gives rise to two spatially separated PMMs, each localized at one QD [3].
Measurements were conducted in a dilution refrigerator in the presence of a magnetic field B = 200 mT applied approximately along the nanowire axis.
The combination of Zeeman splitting E Z and orbital level spacing allows singleelectron QD transitions to be spin-polarized.Two neighbouring Coulomb resonances correspond to opposite spin orientations, enabling the QD spins to be either parallel (↑↑ and ↓↓) or anti-parallel (↑↓ and ↓↑).We report on two devices, A in the main text and B in Extended Data (Fig. ED7 and Fig. ED8).
A scanning electron microscope image of Device A is shown in Fig. 1g.
Transport measurements are used to characterize the charge stability diagram of the system.In Fig. 1h(1), we show G RR as a function of QD voltages V LD , V RD when both QDs are set to spin-down (↓↓).The measured charge stability diagram shows avoided crossing which indicates the dominance of ECT.
In Fig. 1h(2), we change the spin configuration to ↓↑.The charge stability diagram now develops the avoided crossing of the opposite orientation, indicating the dominance of CAR for QDs with anti-parallel spins.This is, to our Realization of a minimal Kitaev chain in coupled quantum dots knowledge, the first verification of the prediction that spatially separated QDs can coherently hybridize via CAR coupling to a superconductor [23].Thus, we have introduced all the necessary ingredients for a two-site Kitaev chain.
Tuning the relative strength of CAR and ECT Majorana states in long Kitaev chains are present under a wide range of parameters due to topological protection [1].Strikingly, even a chain consisting of only two sites can host a pair of PMMs despite a lack of topological protection, if the fine-tuned sweet spot t = ∆ and µ LD = µ RD = 0 can be achieved [3].
This, however, is made challenging by the above-mentioned requirement to have both QDs spin-polarized.If spin is conserved, ECT can only take place between QDs with ↓↓ or ↑↑ spins, while CAR is only allowed for ↑↓ and ↓↑.
Rashba spin-orbit coupling in InSb nanowires solves this dilemma [2,24,25], allowing finite ECT even in anti-parallel spin configurations and CAR between QDs with equal spins.
A further challenge is to make the two coupling strengths equal for a given spin combination.Refs.[24][25][26] show that both CAR and ECT in our device are virtual transitions through intermediate Andreev bound states residing in the short InSb segment underneath the superconducting film.Thus, varying V PG changes the energy and wavefunction of said Andreev bound states and thereby t, ∆.We search for the V PG range over which ∆ changes differently than t and look for a crossover in the type of charge stability diagrams.The crossover from the t > ∆ regime to the t < ∆ regime can be seen across multiple QD resonances (Fig. ED9).
To show that gate-tuning of the t/∆ ratio is indeed continuous, we repeat charge stability diagram measurements (Fig. ED3) and bias spectroscopy at more V PG values.As before, each bias sweep is conducted while keeping both QDs at charge degeneracy.Fig. 2g shows the resulting composite plot of G RR (i) and G LR (ii) vs bias voltage and V PG .The X-shaped conductance feature indicates a continuous evolution of the excitation energy, with a linear zero-energy crossing agreeing with predictions in Ref. [3].Following analysis described in Methods, we extract the peak spacing and average nonlocal conductance in Fig. 2h in order to visualize the continuous crossover from t > ∆ to t < ∆.
Poor Man's Majorana sweet spot Next, we study the excitation spectrum in the vicinity of the t = ∆ sweet spot.The predicted zero-temperature experimental signature of the PMMs is a pair of quantized zero-bias conductance peak on both sides of the devices.
These zero-bias peaks are persistent even when one of the QD levels deviates from charge degeneracy [3].We focus on the ↑↑ spin configuration since it exhibits higher t, ∆ values when they are equal (see Fig. ED4).Fig. 3a shows the charge stability diagram measured via I R under fixed No level repulsion is visible, indicating t ≈ ∆.Panel b(i) shows the excitation spectrum when both dots are at charge degeneracy.The spectra on both sides show zero-bias peaks accompanied by two side peaks.The values of t, ∆ can be read directly from the position of the side peaks, which correspond to the anti-bonding excited states at energy 2t = 2∆ ≈ 25 µeV.The height of the observed zero-bias peaks is 0.3 to 0.4 × 2e 2 /h, likely owing to a combination of tunnel broadening and finite electron temperature (Fig. ED2).Fig. 3b(ii) shows the spectrum when the right QD is moved away from charge degeneracy while µ LD is kept at 0. The zero-bias peaks persist on both sides of the device, as expected for a PMM state.In contrast, tuning both dots away from charge degeneracy, shown in Fig. 3b(iii), splits the zero-bias peaks.
In Fig. 3c,d Finally, when varying the chemical potential of both dots simultaneously (panel e), we see that the zero-bias peaks split away from zero energy.This splitting is not linear, in contrast to the case when ∆ ̸ = t (see Fig. ED5).
The profile of the peak splitting is consistent with the predicted quadratic protection of PMMs against chemical potential fluctuations [3].This quadratic protection is expected to develop into topological protection in a long-enough Kitaev chain [2].
To facilitate comparison with data, we develop a transport model (see Methods) and plot in Fig. 4a-c c).These conditions are an idealization of those in Fig. 3 (a more realistic simulation of the experimental conditions is presented in Fig. ED6).The numerical simulations capture the main features appearing in the experiments discussed above.
Particle-hole symmetry ensures that zero-energy excitations in this system always come in pairs.These excitations can extend over both QDs or be confined to one of them.In Fig. 4d  subband occupation [31].The QD-S-QD platform discussed here opens up a new frontier to the study of Majorana physics.In the long term, this approach can generate topologically protected Majorana states in longer chains [2].A shorter term approach is to use PMMs as an immediate playground to study fundamental non-Abelian statistics, e.g., by fusing neighboring PMMs in a device with two such copies.

Device fabrication
The nanowire hybrid devices presented in this work were fabricated on prepatterned substrates, using the shadow-wall lithography technique described in Refs.[32,33].Nanowires were deposited onto the substrates using an optical micro-manipulator setup.8 nm of Al was grown at a mix of 15 the surface treatment of the nanowires, the growth conditions of the superconductor, the thickness calibration of the Pt coating and the ex-situ fabrication of the ohmic contacts can be found in Ref. [34].Devices A and B also slightly differ in the length of the hybrid segment: 180 nm for A and 150 nm for B.

Transport measurement and data processing
We have fabricated and measured six devices with similar geometry.Two of them showed strong hybridization of the QD states by means of CAR and ECT.
We report on the detailed measurements of Device A in the main text and show qualitatively similar measurements from Device B in Fig. ED7 and Fig. ED8.
All measurements on Device A were done in a dilution refrigerator with base temperature 7 mK at the cold plate and electron temperature of 40∼50 mK at the sample, measured in a similiar setup using an NIS metallic tunnel junction.and with amplitudes between 2 and 6 µV RMS.In this manner, we measure the DC currents I L , I R and the conductance matrix G in response to applied voltages V L , V R on the left and right N leads, respectively.The conductance matrix is corrected for voltage divider effects (see Ref. [35] for details) taking into account the series resistance of sources and meters and in each fridge line (1.85 kΩ for Device A and 2.5 kΩ for Device B), except for the right panel of Fig. 1h and Fig. 2d.There, the left half of the conductance matrix was not measured and correction is not possible.We verify that the series resistance is much smaller than device resistance and the voltage divider effect is never more than ∼ 10% of the signal.

Characterization of QDs and the hybrid segment
To form the QDs described in the main text, we pinch off the finger gates next to the three ohmic leads, forming two tunnel barriers in each N-S junction.
V LD and V RD applied on the middle finger gates on each side accumulate electrons in the QDs.We refer to the associated data repository for the raw gate voltage values used in each measurement.See Fig. ED1a-f for results of the dot characterizations.
Characterization of the spectrum in the hybrid segment is done using conventional tunnel spectroscopy.In each uncovered InSb segment, we open up the two finger gates next to the N lead and only lower the gate next to the hybrid to define a tunnel barrier.The results of the tunnel spectroscopy are shown in Fig. ED1g,h and the raw gate voltages are available in the data repository.

Determination of QD spin polarization
Control of the spin orientation of QD levels is done via selecting from the even vs odd charge degeneracy points following the method detailed in Ref. [36].At the charge transition between occupancy 2n and 2n+1 (n being an integer), the electron added to or removed from the QD is polarized to spin-down (↓, lower in energy).The next level available for occupation, at the transition between 2n + 1 and 2n + 2 electrons, has the opposite polarization of spin-up (↑, higher in energy).To ensure the spin polarization is complete, the experiment was

Controlling ECT and CAR via electric gating
Ref. [24] describes a theory of mediating CAR and ECT transitions between QDs via virtual hopping through an intermediate Andreev bound state.
Ref. [26] experimentally verifies the applicability of this theory to our device.
To summarize the findings here, we consider two QDs both tunnel-coupled to a central Andreev bound state in the hybrid segment of the device.The QDs have excitation energies lower than that of the Andreev bound state and thus transition between them is second-order.The wavefunction of an Andreev bound state consists of a superposition of an electron part, u, and a hole part, v.Both theory and experiment conclude that the values of t and ∆ depend strongly and differently on u, v. Specifically, CAR involves converting an incoming elec-|uv| 2 .ECT, however, occurs over two parallel channels (electron-to-electron and hole-to-hole) and its coupling strength depends on u, v independently as As the composition of u, v is a function of the chemical potential of the middle Andreev bound state, the CAR to ECT ratio is strongly tunable by V PG .We thus look for a range of V PG where Andreev bound states reside in the hybrid segment, making sure that the energies of these states are high enough so as not to hybridize with the QDs directly (Fig. ED1).Next, we sweep V PG to find the crossover point between t and ∆ as described in the main text.
Additional details on the measurement of the coupled

QD spectrum
The measurement of the local and nonlocal conductance shown in Fig. 2g was conducted in a series of steps.First, the value of V PG was set, and a charge The continuous transition from t > ∆ to t < ∆ is visible in Fig. 2g via both local and nonlocal conductance.G RR shows that level repulsion splits the zero-energy resonance peaks both when t > ∆ (lower values of V PG ) and when t < ∆ (higher values of V PG ).The zero-bias peak is restored in the vicinity of t = ∆, in agreement with theoretical predictions [3].The crossover is also apparent in the sign of G LR , which changes from negative (t > ∆) to positive (t < ∆).
To better visualize the transition between the ECT-and CAR-dominated regimes, we extract V PP , the separation between the conductance peaks under positive and negative bias voltages, and plot them as a function of V PG in Fig. 2h.When tuning V PG , the peak spacing decreases until the two peaks merge at V PG ≈ 210 mV.Further increase of V PG leads to increasing V PP .In addition, to observe the change in sign of the nonlocal conductance, we follow ⟨G LR ⟩, the value of G LR averaged over the bias voltage V R between −100 and 100 µV at a given V PG .We see that ⟨G LR ⟩ turns from negative to positive at V PG ≈ 210 mV, in correspondence to a change in the dominant coupling mechanism.Model of the phase diagrams in Fig. 1f To calculate the ground state phase diagram in Fig. 1f, we write the Hamiltonian in the many-body picture, with the four basis states being |00⟩ , |11⟩ , |10⟩ , |01⟩: in block-diagonalized form.The two 2×2 matrices yield the energy eigenvalues separately for the even and odd subspaces: The ground state phase transition occurs at the boundary E o,− = E e,− .This is equivalent to Transport model in Fig. 3 and Fig. 4 We describe in this section the model Hamiltonian of the minimal Kitaev chain and the method we use for calculating the differential conductance matrices when the Kitaev chain is tunnel-coupled to two external N leads.
The effective Bogoliubov-de-Gennes Hamiltonian of the double-QD system is where is the level energy in dot-L/R relative to the superconducting Fermi surface, t and ∆ are the ECT and CAR amplitudes.Here we assume t and ∆ to be real without loss of generality [3].The presence of both t and ∆ in this Hamiltonian implies breaking spin conservation during QD-QD tunneling via either spin-orbit coupling (as done in the present experiment) or non-collinear magnetization between the two QDs (as proposed in [3]).Without one of them, equal-spin QDs cannot recombine into a Cooper pair, leading to vanishing ∆, while opposite-spin QDs cannot support finite t.The exact values of t and ∆ depend on the spin-orbit coupling strength and we refer to Ref. [24] for a detailed discussion.
To calculate the differential conductance for the double-QD system, we use the S-matrix method [37].In the wide-band limit, the S matrix is where Γ α being the tunnel coupling strength between dot-α and lead-α.The zero-temperature differential conductance is given by where α, β = L/R.Finite-temperature effect is included by a convolution between the zero-temperature conductance and the derivative of Fermi-Dirac distribution, i.e., The theoretical model presented above uses five input parameters to calculate the conductance matrix under given µ LD , µ RD , V L , V R .The input parameters are: t, ∆, Γ L , Γ R , T .To choose the parameters in Fig. 3b(i), we fix the temperature to the measured value T = 45 mK and make the simplification t = ∆, Γ ≡ Γ L = Γ R .This results in only two free parameters t, Γ, which we manually choose and compare with data.While oversimplified, this approach allows us to obtain a reasonable match between theory and data taken at µ LD = µ RD = 0 without the risk of overfitting.To obtain the other numerical curves shown in Fig. 3, we keep the same choice of t, Γ and vary µ LD , µ RD , V L , V R along various paths in the parameter space.Similarly, to model the data shown in Fig. ED5, we keep T = 45 mK and Γ the same as in Fig. 3.The free parameters to be chosen are thus t and ∆.The theory panels are obtained with the same t, ∆, and only µ LD , µ RD , V L , V R are varied in accordance with the experimental conditions.
Finally, we comment on the physical meaning of the theory predictions in   Fig. ED10 Theoretical effect of tunnel broadening on the charge stability diagrams.In some charge stability diagrams where level-repulsion is weak, e.g., Fig. 2a and Fig. ED4, some residual conductance is visible even when µ LD = µ RD = 0.This creates the visual feature of the two conductance curves appearing to "touch" each other at the center.In the main text, we argued this is due to level broadening.Here, we plot the numerically simulated charge stability diagrams at zero temperature under various dot-lead tunnel coupling strengths.We use coupling strengths t = 20 µV, ∆ = 10 µV as an example.From panel a to c, increasing the tunnel coupling and thereby level broadening reproduces this observed feature.When the level broadening is comparable to the excitation energy, |t − ∆|, finite conductance can take place at zero bias.This feature is absent in, e.g., Fig. 2c, where |t − ∆| is greater than the level broadening.

Fig. 1
Fig. 1 Coupling quantum dots through elastic co-tunneling (ECT) and crossed Andreev reflection (CAR).a. Illustration of the basic ingredients of a Kitaev chain: two QDs simultaneously coupled via ECT with amplitude t and via CAR with amplitude ∆ through the superconductor in between.b.Energy diagram of a minimal Kitaev chain.Two QDs with gate-controlled chemical potentials are coupled via both ECT and CAR.The two ohmic leads enable transport measurements from both sides.c.Energy diagram showing that coupling the |01⟩ and |10⟩ states via ECT leads to a bonding state (|10⟩ − |01⟩)/ √ 2 and antibonding state (|10⟩ + |01⟩)/ √ 2. d.Same showing how CAR couples |00⟩ and |11⟩ to form the bonding state (|00⟩−|11⟩)/ √ 2 and anti-bonding state (|00⟩+|11⟩)/ √ 2 .e. Illustration of the N-QD-S-QD-N device and the measurement circuit.Dashed potentials indicate QDs defined in the nanowire by finger gates.f.Charge stability diagram of the coupled-QD system, in the cases of t > ∆ (i), t = ∆ (ii) and t < ∆ (iii).Blue marks regions in the (µ LD , µ RD ) plane where the ground state is even and orange where the ground state is odd.g.False-colored scanning electron microscopy image of the device, prior to the fabrication of the normal leads.InSb nanowire is colored green.QDs are defined by bottom finger gates (in red) and their locations are circled.The gates controlling the two QD chemical potentials are labeled by their voltages, V LD and V RD .The central thin Al/Pt film, in blue, is grounded.The proximitized nanowire underneath is gated by V PG .Two Cr/Au contacts are marked by yellow boxes.The scale bar is 300 nm.h.Right-side zero-bias local conductance G RR in the (V LD , V RD ) plane when the system is tuned to t > ∆ (1) and t < ∆ (2).The arrows mark the spin polarization of the QD levels.The DC bias voltages are kept at zero, V L = V R = 0, and an AC excitation of 6 µV RMS is applied on the right side.

Fig. 1e illustrates
Fig. 1e illustrates our coupled QD system and the electronic measurement circuit.An InSb nanowire is contacted on two sides by two Cr/Au normal leads (N).A 200 nm-wide superconducting lead (S) made of a thin Al/Pt film covering the nanowire is grounded and proximitizes the central semiconducting segment.The chemical potential of the proximitized semiconductor can be

Fig. 2
Fig. 2 Tuning the relative strength of CAR and ECT for the ↓↑ spin configuration.a-c.Conductance matrices measured with V PG = (198, 210, 218) mV, respectively.d-f.G LR and G RR as functions of V R when V LD , V RD are set to the center of each charge stability diagram in panels a to c, indicated by the black dots in the corresponding panels above them.g.Local (G RR ) and nonlocal (G LR ) conductance as a function of V R and V PG while keeping µ LD ≈ µ RD ≈ 0, showing the continuous crossover from t > ∆ to t < ∆. h.Green dots: peak-to-peak distance (V PP ) between the positive-and negative-bias segments of G RR , showing the closing and re-opening of QD avoided crossings.Purple dots: average G LR (⟨G LR ⟩) as a function of V PG , showing a change in the sign of the nonlocal conductance.

Fig. 2a -
Fig. 2a-c shows the resulting charge stability diagrams for the ↓↑ spin configuration at different values of V PG .The conductance matrix G(V L = 0, V R = 0) at V PG = 198 mV is shown in Fig. 2a.The local conductance on both sides, G LL and G RR , exhibit level repulsion indicative of t > ∆.We emphasize that ECT can become stronger than CAR even though the spins of the two QD transitions are anti-parallel due to the electric gating mentioned above.The dominance of ECT over CAR can also be seen in the negative sign of the nonlocal conductance, G LR and G RL .During ECT, an electron enters the system through one dot and exits through the other, resulting in negative nonlocal conductance.CAR, in contrast, causes two electrons to enter or leave both dots simultaneously, producing positive nonlocal conductance [27].The residual finite conductance in the center of the charge stability diagram can be

Fig. 2c
Fig.2cshows G at V PG = 218 mV (the G RR component is also used for Fig.1h(2)).Here, all the elements of G exhibit CAR-type avoided crossings.The spectrum shown in panel f, obtained at the joint charge degeneracy point (black dots in panels c(ii, iv)), similarly has two conductance peaks surrounding zero energy.The measured nonlocal conductance is positive as predicted for CAR.The existence of both t > ∆ and t < ∆ regimes, together with continuous gate tunability, allows us to approach the t ≈ ∆ sweet spot.This is shown in panel b, taken with V PG = 210 mV.Here, G RR and G LL exhibit no avoided crossing while G LR and G RL fluctuate around zero, confirming that CAR and ECT are in balance.Accordingly, the spectrum in panel e confirms the even and odd ground states are degenerate and transport can occur at zero excitation energy via the appearance of a zero-bias conductance peak.

Fig. 3
Fig. 3 Conductance spectroscopy at the t = ∆ sweet spot for the ↑↑ spin configuration.a.I R vs V LD , V RD under V L = 0, V R = 10 µV.The spectra in panel b are taken at values of V LD , V RD marked by corresponding symbols.The gate vs bias sweeps are taken along the dashed, dotted, dash-dot lines in panels c,d,e respectively.Data are taken with fixed V PG = 215.1 mV.b.Spectra taken under the values of V LD , V RD marked in panel a.The dashed lines are theoretical curves calculated with t = ∆ = 12 µeV, Γ L = Γ R = 4 µeV, T = 45 mK and at QD energies converted from V LD , V RD using measured lever arms (see Methods for details).c, d.G as a function of the applied bias and V RD (c) or V LD (d), taken along the paths indicated by the dashed blue line and the dotted green line in panel a, respectively.e. G as a function of the applied bias and along the diagonal indicated by the dashed-dotted pink line in panel a.This diagonal represents 500 µV of change in V LD and 250 µV of change in V RD .
, we show the evolution of the spectrum when varying V RD and V LD , respectively.The vertical feature appearing in both G LL and G RR shows correlated zero-bias peaks in both QDs, which persist when one QD potential departs from zero.This crucial observation demonstrates the robustness of PMMs against local perturbations.The excited states disperse in agreement with the theoretical predictions[3].Nonlocal conductance, on the other hand, reflects the local charge character of a bound state on the side where current is measured[28][29][30]. Near-zero values of G LR in panel c and G RL in panel d are consistent with the prediction that the PMM mode on the unperturbed side remains an equal superposition of an electron and a hole and therefore chargeless.

Fig. 4
Fig. 4 Calculated conductance and Majorana localization.a. Numerically calculated G as a function of energy ω and µ RD at the t = ∆ sweet spot.b.Numerically calculated G as a function of ω and µ LD at the t = ∆ sweet spot.c.Numerically calculated G as a function of ω and µ RD , µ LD along the diagonal corresponding to µ LD = µ RD at t = ∆.All of the numerical curves use the same value of t, ∆, Γ L , Γ R as those in Fig.3.d.Illutrations of the localization of two zero-energy solutions for the following set of parameters: t = ∆, µ LD = µ RD = 0 (sub-panel i), t = ∆, µ RD = 0, µ LD > 0 (sub-panel ii), t < ∆, µ LD = −µ RD = √ ∆ 2 − t 2 (sub-panel iii).
the calculated conductance matrices as functions of excitation energy, ω, vs µ RD (panel a), µ LD (panel b), and µ ≡ µ LD = µ RD (panel Fig.3b(i) and shows that the sweet-spot zero-energy solutions are two PMMs, each localized on a different QD.The second scenario in Fig.4d(ii), illustrating Fig.3b(ii), is varying µ LD while keeping µ RD = 0.This causes some of the wavefunction localized on the perturbed left side, γ 1 , to leak into the right QD.Since the right-side γ 2 excitation has no weight on the left, it does not respond to this perturbation and remains fully localized on the right QD.As the theory confirms[3], it stays a zero-energy PMM state.Since Majorana excitations always come in pairs, the excitation on the left QD must also remain at zero energy.This provides an intuitive understanding of the remarkable stability of the zero-energy modes at the sweet spot in Fig.3c,dwhen moving one of the QDs' chemical potentials away from zero.Finally, zero-energy solutions can be found away from the sweet spot, t ̸ = ∆, as illustrated in Fig.4d(iii).These zero-energy states are only found when both QDs are off-resonance and none of them are localized Majorana states, extending over both QDs and exhibiting no gate stability.Measurements under these conditions are shown in Fig.ED5, where zero-energy states can be found in a variety of gate settings (panels a, c therein).
Device A were conducted in the presence of a magnetic field of 200 mT approximately oriented along the nanowire axis with a 3 • offset.Device B was measured similarly in another dilution refrigerator under B = 100 mT along the nanowire with 4 • offset.

Fig. 1e shows
Fig. 1e shows a schematic depiction of the electrical setup used to measure the devices.The middle segment of the InSb nanowire is covered by a thin Al shell, kept grounded throughout the experiment.On each side of the hybrid segment, we connect the normal leads to a current-to-voltage converter.The amplifiers on the left and right sides of the device are each biased through a Fig. ED1 for determination of the spin configuration).In the experiment data, a change in the QD spin orientation is visible as a change in the range of V LD or V RD .
stability diagram was measured as a function of V LD and V RD .Representative examples of such diagrams are shown in Fig. ED3.Second, each charge stability diagram was inspected and the joint charge degeneracy point (µ LD = µ RD = 0) was selected manually (V 0 LD , V 0 RD ).Lastly, the values of V LD and V RD were set to those of the joint degeneracy point and the local and nonlocal conductance were measured as a function of V R .

Fig.
Fig. 3c-e presents measurements where the conductance was measured against applied biases along some paths within the charge stability diagram (panel a).Prior to each of these measurements, a charge stability diagram was measured and inspected, based on which the relevant path in the (V LD , V RD ) plane was chosen.Following each bias spectroscopy measurement, another charge stability diagram was measured and compared to the one taken before to check for potential gate instability.In case of noticeable gate drifts between the two, the measurement was discarded and the process was repeated.The values of µ LD and µ RD required for theoretical curves appearing in panel b were calculated by µ i = α i (V i − V 0 i ) where i = LD, RD and α i is the lever arm of the corresponding QD.The discrepancy between the spectra measured with G LL and G RR likely results from gate instability, since they were not measured simultaneously.Finite remaining G LR in panel c and G RL in panel d most likely result from small deviations of µ LD , µ RD from zero during these measurements.

Fig.Fig.
Fig.4a-c.Tuning µ RD leads to symmetric G LL and asymmetric G RR , as well Fig. ED7 Reproduction of the main results with Device B. a-c.Conductance matrices measured at V PG = (976, 979.6, 990) mV, respectively.d.Conductance matrix as a function of V L , V R and V PG while keeping µ LD ≈ µ RD ≈ 0. This device shows two continuous crossovers from t > ∆ to t < ∆ and again to t > ∆.
• and 45 • angles with respect to the substrate.Subsequently, Device A was coated with 2 Å of Pt grown at 30 • .No Pt was deposited for Device B. Finally, all devices were capped with 20 nm of evaporated AlO x .Details of the substrate fabrication,