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Physical review. B./Physical review. B

Procedure to tune a three-site artificial Kitaev chain to host Majorana bound states based on transmon measurements

Xiaozhou Yang, Zhaozheng Lyu, Xiang Wang, Enna Zhuo, Yunxiao Zhang, Duolin Wang, Yukun Shi, Yuyang Huang, Bing Li, Xiaohui Song, Peiling Li, Bingbing Tong, Ziwei Dou, Jie Shen, Guangtong Liu, Fanming Qu, Li Lü

Published Sep 18, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1103/2r8x-9d9m

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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.
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 would include the tuning up 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 up 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, 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 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.
Artificial Kitaev chains have emerged as a promising platform for realizing topological quantum computing. Once the chains are formed and the Majorana zero modes are braided/fused, reading out the parity of the chains is essential for further verifying the non-Abelian property of the Majorana zero modes. Here we demonstrate the feasibility of using a superconducting transmon qubit, which incorporates an end of a four-site quantum dot-superconductor chain based on a Ge/Si nanowire, to directly detect the singlet/doublet state, and thus the parity of the entire chain. We also demonstrate that for multiple-dot chains there are two types of 0-π transitions between different charging states: the parity-flip 0-π transition and the parity-preserved 0-π transition. Furthermore, we show that the inter-dot coupling, hence the strengths of cross Andreev reflection and elastic cotunneling of electrons, can be adjusted by local electrostatic gating in chains fabricated on Ge/Si core-shell nanowires. Our exploration would be helpful for the ultimate realization of topological quantum computing based on artificial Kitaev chains.
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.
Minimal Kitaev chains composed of two semiconducting quantum dots coupled via a superconductor have emerged as a promising platform to realize and study Majorana bound states (MBSs), which appear for fine-tuned configurations. We propose a hybrid qubit based on a Josephson junction between two such double quantum dots (DQDs) embedded in a superconducting qubit geometry. The qubit makes use of the $4\ensuremath{\pi}$-Josephson effect in the Kitaev junction to create a subspace based on the even/odd fermionic parities of the two DQD arrays hosting MBSs. Deep in the transmon regime, we demonstrate that by performing circuit QED spectroscopy on such a hybrid Kitaev-transmon ``Kitmon'' qubit, one could observe distinct MBS features in perfect agreement with precise analytical predictions in terms of DQD parameters only. This agreement allows us to extract the Majorana polarization in the junction from the microwave response.
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.