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# Realization of a minimal Kitaev chain in coupled quantum dots

## Abstract

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 chain1. Practical proposals for its realization2,3 require coupling neighbouring quantum dots (QDs) in a chain through both electron tunnelling and crossed Andreev reflection4. Although both processes have been observed in semiconducting nanowires and carbon nanotubes5,6,7,8, 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 QDs in an InSb nanowire strongly coupled by both elastic co-tunnelling (ECT) and crossed Andreev reflection (CAR). 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. Although 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.

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## Data availability

Raw data presented in this work, the data processing/plotting code and code used for the theory calculations are available at https://doi.org/10.5281/zenodo.6594169.

## References

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## Acknowledgements

This work has been supported by the Dutch Organisation for Scientific Research (NWO), a subsidy for Top Consortia for Knowledge and Innovation (TKl toeslag), Microsoft Corporation Station Q and support from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 828948, project AndQC. We thank G. de Lange, S. Goswami, D. Xu, D. Loss and J. Klinovaja for helpful discussions.

## Author information

Authors

### Contributions

G.W., G.P.M., N.v.L., A.B., J.-Y.W., D.v.D. and F.Z. fabricated the devices. G.W., T.D., S.L.D.t.H., A.B. and X.L. performed the electrical measurements. T.D. and G.W. designed the experiment with inputs from F.K.M. and analysed the data. G.W., T.D. and L.P.K. prepared the manuscript with input from all authors. T.D. and L.P.K. supervised the project. C.-X.L. developed the theoretical model and performed numerical simulations with input from G.W., T.D. and M.W. S.G., G.B. and E.P.A.M.B. performed InSb nanowire growth.

### Corresponding authors

Correspondence to Tom Dvir or Leo P. Kouwenhoven.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks Sumanta Tewari, Hongqi Xu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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## Extended data figures and tables

### Extended Data Fig. 1 Characterization of the QDs.

a, Coulomb blockade diamonds of the left QD when the right QD is off-resonance. IL is measured as a function of VL and VLD. The data are overlaid with a constant interaction model38 with 1.8-meV charging energy and gate lever arm of 0.32. b, A high-resolution scan of a with a symmetric logarithmic colour scale to show the presence of a small amount of Andreev current at sub-gap energies. This is because of the left QD being weakly proximitized by local Andreev coupling to Al. c, Field dependence of the Coulomb resonances. IL is measured as a function of VLD and B with a constant VL = 600 μV. The resonances of opposite spin polarization evolve in opposite directions with a g-factor of about 35, translating to Zeeman energy of 400 μeV at B = 200 mT. df, Characterization of the right QD, as described in the captions of panels ac. Overlaid model in d has charging energy 2.3 meV and gate lever arm of 0.33. No sub-gap transport is detectable in e. B dispersion in f corresponds to g = 40. g,h, Bias spectroscopy results of the proximitized InSb segment under the thin Al/Pt film. IL and IR are measured as a function of VL and VPG. GLL and GRL are obtained by taking the numerical derivative of IL and IR along the bias direction after applying a Savitzky–Golay filter of window length 15 and order 1. The sub-gap spectrum shows discrete, gate-dispersing Andreev bound states. The presence of nonlocal conductance correlated with the sub-gap states shows that these Andreev bound states extend throughout the entire hybrid segment, coupling to both left and right N leads30. Parts of this dataset are also presented in ref. 34. (Reproduced under the terms of the CC-BY Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Copyright 2022, The Authors, published by Wiley-VCH.).

### Extended Data Fig. 2 Theoretical temperature dependence of the height of Majorana zero-bias conductance peaks.

The height of the Majorana zero-bias peaks is only quantized to 2e2/h at zero temperature. At finite electron temperature T, the peak height is generally lower, with the exact value depending on T and tunnel broadening ΓL and ΓR owing to coupling between QDs and N leads. The local zero-bias conductance GLL at the sweet spot (t = Δ and μLD = μRD = 0) is calculated and shown in this plot as a function of T, using the parameters presented in Fig. 3: t = Δ = 12 μeV. Three curves are calculated assuming three different values of tunnel coupling Γ = ΓL = ΓR. The orange curve assumes a Γ value that matches the experimentally observed peak width (both of the zero-bias peaks and of generic QD resonant peaks at other conductance features), showing that conductance approaching quantization would only be realized at electron temperatures <5 mK, unattainable in our dilution refrigerator. The blue curve, calculated with lower Γ = 2 μeV, shows even lower conductance. Increasing Γ would not lead to conductance quantization either, as the zero-bias peaks would merge with the conductance peaks arising from the excited states (pink curve). The green dot marks the experimentally measured electron temperature and peak height (averaged between the values obtained on the left and right leads).

### Extended Data Fig. 3 Evolution of the charge-stability diagram for the ↓↑ spin configuration.

Each panel shows IL (nonlocal) and IR (local) as functions of VLD and VRD measured under fixed biases VL = 0 and VR = 10 μV. VPG is tuned from 196.5 mV, showing signatures of the t > Δ regime in both local and nonlocal currents, to 220 mV, featuring the opposite t < Δ regime.

### Extended Data Fig. 4 Evolution of the charge-stability diagram for the ↑↑ spin configuration.

Each panel shows IL (nonlocal) and IR (local) as functions of VLD and VRD measured under fixed biases VL = 0 and  VR = 10 μV. VPG is tuned from 210 mV, showing signatures of the t > Δ regime in both local and nonlocal currents, to 219 mV, featuring the opposite t < Δ regime.

### Extended Data Fig. 5 Conductance spectroscopy when t < Δ.

a, IR versus μLD and μRD with VR = 10 μV. The evolution of the spectrum with the chemical potential is taken along the dashed, dashed-dotted and dotted lines in panels b, c and d, respectively. Data taken at the spin configuration with fixed VPG = 218 mV. b, Local conductance spectroscopy taken at gate setpoints marked by corresponding symbols in panel a. c, Conductance matrix as a function of bias and VLD, taken along the dashed blue line in panel a, that is, varying the detuning between the QDs δ = (μLD − μRD)/2 while keeping the average chemical potential $$\bar{\mu }=({\mu }_{{\rm{LD}}}+{\mu }_{{\rm{RD}}})/2$$ close to 0. d, Conductance matrix as a function of bias and VLD, taken along the dotted green line in panel a, keeping the detuning between the QDs around 0. e, Conductance matrix as a function of bias and VLD, taken along the dashed-dotted pink line in panel a, keeping roughly constant non-zero detuning between the QDs. fh, Numerically calculated G as a function of energy ω and μLD and μRD along the paths shown in panel a. All of the numerical curves assume the same parameters as those in Fig. 3, except with Δ = 23 μeV and t = 6 μeV.

### Extended Data Fig. 6 Calculated conductance matrices at the t = Δ sweet spot.

a, Numerically calculated G as a function of energy ω and μLD and μRD along the path shown in Fig. 3c. The presence of finite GLR and asymmetric GRL result from a slight deviation from the μLD = 0 condition, which is depicted in Fig. 4a. These features appear in the experimental data shown in Fig. 3c. b, Numerically calculated G as a function of energy ω and μLD and μRD along the path shown in Fig. 3d. The presence of finite GRL and asymmetric GLR result from a slight deviation from the μRD = 0 condition, which is depicted in Fig. 4b. These features appear in the experimental data shown in Fig. 3d. c, Numerically calculated G as a function of energy ω and μLD and μRD along the path shown in Fig. 3e. Because the path does not obey μLD = μRD, the calculated spectral lines do not follow parallel trajectories, in slight disagreement with the experimental data. The conversion from VLD and VRD to μLD and μRD is carried out as explained in Methods with the measured lever arms of both QDs.

### Extended Data Fig. 7 Reproduction of the main results with device B.

ac, Conductance matrices measured at VPG = (976, 979.6 and 990 mV, respectively. d, Conductance matrix as a function of VL and VR and VPG while keeping μLD ≈ μRD ≈ 0. This device shows two continuous crossovers from t > Δ to t < Δ and again to t > Δ.

### Extended Data Fig. 8 Device B spectrum versus gates.

a, Charge-stability diagram measured through GRR of another t = Δ sweet spot of device B, at VPG = 993 mV. Dashed lines mark the gate voltage paths along which the corresponding panels are taken. bd, Conductance matrices when varying VRD (b), VLD (c) and the two gates simultaneously (d), similar to Fig. 3. The sticking zero-bias conductance peak feature when only one QD potential is varied around the sweet spot is clearly reproduced in GRR of panel b. The quadratic peak splitting profile when both QD potentials are varied by the same amount is also reproduced the panel d. The left N contact of this device was broken and a distant lead belonging to another device on the same nanowire was used instead. This and gate jumps in VRD complicate interpretation of other panels.

### Extended Data Fig. 9 CAR-induced and ECT-induced interactions across several QD resonances.

a,b, Local (IL) and nonlocal (IR) currents as a function of VLD and VRD measured with VPG = 200 mV and fixed VL. All resonances show an ECT-dominated structure and a negative correlation between the local and the nonlocal currents. c,d, Local (IL) and nonlocal (IR) currents as a function of VLD and VRD measured with VPG = 218 mV and fixed VL. Some resonances show the structure associated with the t = Δ sweet spot, showing both positive and negative correlations between the local and nonlocal currents. e,f, Local (IL) and nonlocal (IR) currents as a function of VLD and VRD measured with VPG = 200 mV and fixed VL. All orbitals show a CAR-dominated structure and a positive correlation between the local and the nonlocal currents. All measurements were conducted with VL = 10 μV, VR = 0 and B = 100 mT.

### Extended Data Fig. 10 Theoretical effect of tunnel broadening on the charge-stability diagrams.

In some charge-stability diagrams in which level repulsion is weak, such as Fig. 2a and Extended Data Fig. 4, some residual conductance is visible even when μLD = μRD = 0. This creates the visual feature of the two conductance curves appearing to ‘touch’ each other at the centre. In the main text, we argued that this is owing 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 and Δ = 10 μV as an example. From panels ac, increasing the tunnel coupling and thereby level broadening reproduces this observed feature. When the level broadening is comparable with the excitation energy, |t − Δ|, finite conductance can take place at zero bias. This feature is absent in, for example, Fig. 2c, in which |t − Δ| is greater than the level broadening.

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Dvir, T., Wang, G., van Loo, N. et al. Realization of a minimal Kitaev chain in coupled quantum dots. Nature 614, 445–450 (2023). https://doi.org/10.1038/s41586-022-05585-1

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