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arXiv:2504.13702 · Nature Communications

Probing Majorana localization of a phase-controlled three-site Kitaev chain with an additional quantum dot

Alberto Bordin, Florian J. Bennebroek Evertsz’, Bart Roovers, Juan Daniel Torres Luna, Wietze D. Huisman, Francesco Zatelli, Grzegorz P. Mazur, Sebastiaan L. D. ten Haaf, Ghada Badawy, Erik P. A. M. Bakkers, Chun-Xiao Liu, Rubén Seoane Souto, Nick van Loo, Leo P. Kouwenhoven

Published Feb 3, 2026• 7 citations
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Abstract

Few-site implementations of the Kitaev chain offer a minimal platform to study the emergence and stability of Majorana bound states. Here, we realize two- and three-site chains in semiconducting quantum dots coupled via superconductors, and tune them to the sweet spot where zero-energy Majorana modes appear at the chain ends. We demonstrate control of the superconducting phase through both magnetic field and sweet-spot selection, and fully characterize the excitation spectrum under local and global perturbations. All spectral features are identified using the ideal Kitaev chain model. To assess Majorana localization, we couple the system to an additional quantum dot. The absence of energy splitting at the sweet spot is compatible with high-quality Majorana modes, despite the modest chain size.

DOI
10.1038/s41467-026-68897-0

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Few-site implementations of the Kitaev chain offer a minimal platform to study the emergence and stability of Majorana bound states. Here, we realize two- and three-site chains in semiconducting quantum dots coupled via superconductors, and tune them to the sweet spot where zero-energy Majorana modes appear at the chain ends. We demonstrate control of the superconducting phase through both magnetic field and sweet-spot selection, and fully characterize the excitation spectrum under local and global perturbations. All spectral features are identified using the ideal Kitaev chain model. To assess Majorana localization, we couple the system to an additional quantum dot. The absence of energy splitting at the sweet spot is compatible with high-quality Majorana modes, despite the modest chain size.
Artificial Kitaev chains based on arrays of quantum dots are promising platforms for realizing Majorana Bound States (MBSs). In a two-site Kitaev chain, it is possible to find these non-Abelian zero-energy excitations at certain points in parameter space (sweet spots). These states, commonly referred to as Poor man's Majorana bound states (PMMs), are challenging to find and stabilize experimentally. In this work, we investigate the evolution of the sweet spots as we increase the number of sites of the Kitaev chain. To this end, we use the Bogoliubov-de Gennes representation to study the excitations of the system, and the scattering matrix and Green functions formalisms to calculate the zero-bias conductance. Our results show that the sweet spots evolve into a region that grows bigger and becomes gradually more protected as the number of sites NNN increases. Due to the protection of the MBSs, we refer to this region as a topological island. We obtain similar results by considering a realistic spinful model with finite magnetic fields in a chain of normal-superconducting quantum dots. For long chains, N≥20N \geq 20N≥20, we show the emergence of strictly zero-energy plateaus robust against disorder. Finally, we demonstrate that the topological islands can be observed by performing conductance measurements via a quantum dot side-coupled to the Kitaev chain. Our work shows that the fine-tuning required to create and detect PMMs in a 2-site Kitaev chain is significantly relaxed as the length of the chain increases and details how PMMs evolve into MBSs. Our results are consistent with experimental reports for 2 and 3-site chains.
Majorana zero modes provide a route to fault-tolerant qubits by encoding information non-locally in fermion parity. Their sensitivity to noise is expected to decrease exponentially with increasing separation between the Majoranas, a suppression known as topological protection. Kitaev chains engineered in quantum dot-superconductor arrays provide a tunable platform in which separated Majorana zero modes can emerge at the ends of the chain, even in two-site chains. These minimal-chain modes are known as poor man's Majoranas and retain characteristic Majorana properties, including near-zero energy and equal electron-hole character, but have only limited protection. A key outstanding challenge is to move beyond identifying such modes in electrical transport measurements and achieve coherent qubit control in the time domain. Here, we demonstrate a Majorana parity qubit by realizing coherent coupling between two-site Kitaev chains. Since total fermion parity is conserved, the system separates into global even and odd parity manifolds. We observe coherent parity oscillations in both manifolds with equal oscillation frequencies at the Majorana sweet spot, as predicted for isolated Majorana zero modes. We further show that the oscillation frequency and coherence depend systematically on inter-chain coupling and quantum-dot detunings, in close agreement with our model for short, partially protected chains. Our results establish the first coherent control of a Majorana qubit, encoded in the fermion parity of Majorana zero modes in minimal Kitaev chains.
Artificial Kitaev chains (AKCs), formed of quantum dot--superconductor linear arrays, provide a promising platform for hosting Majorana bound states (MBSs) and implementing topological quantum computing. The main challenges along this research direction include the tuning of AKCs for hosting MBSs and the readout of the parity of the chains. In this work, we present a step-by-step procedure for tuning a three-site AKC to its sweet spots based on the spectra of a transmon circuit which is integrated with the chain for the purpose of reading out the parity of the chain. The signatures of the transmon's plasma modes in each step, in particular those related to the appearance of MBSs in the chain, will be given. We find that the sweet spots in a three-site AKC can be classified into three types based on the relative strengths of the elastic cotunneling (ECT) and crossed Andreev reflection (CAR): ECT-dominated sweet spots, genuine sweet spots, and CAR-dominated sweet spots. We show that the ECT-dominated and CAR-dominated sweet spots can be more conveniently accessed and utilized in transmon-based measurements.
Motivated by the recent experimental realization of a minimal Kitaev chain in quantum dot systems, we present our theoretical findings on the dynamics and fusion of MZMs at or near the “sweet spot” th = Δ (where the fermionic hopping th and superconducting coupling Δ are equal). We investigated the dynamics and fusion of MZMs using time-dependent real-space local density-of-states methods. The movement of Majoranas and the detection of fusion channels are crucial for topological quantum computations. Additionally, we discuss our recent discovery of exotic “multi-site” MZMs in Y-shaped Kitaev wires, which is important for the potential braiding of Majoranas in Y-junctions formed from arrays of quantum dots. Finally, we present results on ”non-trivial” fusion using canonical Kitaev wires at the sweet spot.
Poor man's Majorana bound states (MBS) arise in minimal Kitaev chains when the parameters are fine-tuned to a sweet spot. We consider an interacting two-site Kitaev chain coupled to a single-mode cavity and show that the sweet spot condition can be controlled with the cavity frequency and the hopping between sites. Furthermore, we demonstrate that photon-mediated effective interactions can be used to screen intrinsic interactions, improving the original quality of the MBS. We describe experimental signatures in the cavity transmission to detect their presence and quality. Our work proposes a new way to tune poor man's MBS in a quantum dot array coupled to a cavity.
The Fermi-Hubbard model describes a large variety of condensed matter systems with spinful fermions and strong interactions. On the other hand, the Kitaev chain model deals with noninteracting spinless fermions and produces robust ground-state degeneracies that give rise to Majorana bound states. In this Letter, we connect these two domains by experimentally studying small arrays of quantum dots (QDs) coupled via superconductors. Without a magnetic field, this system constitutes a Fermi-Hubbard model with intersite superconducting correlations. Using two coupled QDs, we find sweet spots in parameter space where we observe zero-bias conductance peaks that are robust against on-site perturbations of the QDs, similar to the behavior expected of a two-site Kitaev chain. This observation is consistent with the presence of Majorana Kramers pairs or Z_{3} parafermions in this minimal system. Extending to three sites, we find that the spinful system scales very differently compared to the spinless Kitaev chain. When the same sweet-spot conditions are satisfied, the ground state degeneracy of the full three-site system is lifted. The degeneracy can be restored by tuning the superconducting phase difference between the hybrid segments, but the robustness against on-site detunings is lost. Our findings are a first step in studying ground-state degeneracies in strongly interacting Fermi-Hubbard systems with superconducting correlations.