Majorana sweet spots in 3-site Kitaev chains

TL;DR

This work analyzes a 3-site Kitaev-chain analogue realized with an array of five quantum dots connected via two proximitized dots, revealing three distinct Majorana sweet spots: an effective 2-site regime with a central barrier, a genuine 3-site regime with well-localized end MBSs, and a delocalized regime with MBSs overlapping toward the chain center. Using both a Kitaev-model reduction and a microscopic quantum-dot–superconductor model, the authors map out ground-state degeneracy, Majorana polarization, and excitation gaps, and characterize transport and microwave absorption fingerprints. Phase control via the superconducting phase difference $\\phi$ provides additional diagnostics to distinguish the three spots, while detuning individual dots reveals how MBS localization and robustness respond to parameter variations. The results offer practical guidance for identifying robust Majorana modes in multi-site chains and suggest routes for optimization, including potential machine-learning-assisted parameter tuning and microwave- spectroscopy-based readout.

Abstract

Minimal Kitaev chains, composed of two quantum dots (QDs) connected via a superconductor, have emerged as an attractive platform to realize Majorana bound states (MBSs). These excitations exist when the ground state is degenerate. The additional requirement of isolating the MBS wavefunctions further restricts the parameter space to discrete sweet spots. While scaling up to Kitaev chains with more than two sites has the potential to improve the stability of the MBSs, longer chains offer more features to optimize, including the MBS localization length and the excitation gap. In this work, we theoretically investigate 3-site Kitaev chains and show that there are three different types of sweet spots, obtained by maximizing distinct MBS properties: genuine 3-site sweet spots with well-localized MBSs at the ends, effective 2-site sweet spots, where the middle site acts as a barrier, and sweet spots with delocalized MBSs that overlap in the middle of the chain. These three cases feature different degrees of robustness against perturbations, with the genuine 3-site being the most stable. We analyze the energy spectrum, transport, and microwave absorption associated with these three cases, showing how to distinguish them.

Figures (8)

• Figure 1: Sketch of the 3-site artificial Kitaev chain, where 5 QDs couple via tunneling. The even QDs couple to superconductors which gives rise to pairing potentials $\Delta_i$ ($i=2,4$). The system attaches to metallic leads to perform spectroscopy. The superconductors form a SQUID loop that allows for phase control and is capacitively coupled to a microwave resonator (top left).
• Figure 2: Conductances and energies for the Kitaev model. (a-d): true 2-site Kitaev chain. (e-h): effective 2-site sweet spot in a 3-site chain, where the middle site is detuned by $2\delta$ and acts as a barrier. (i-l): 3-site sweet spot, $\epsilon_{1, 2, 3} = 0$. (m)-(p): 3-site sweet spot with the outer sites detuned by $1.5\delta$. In all cases, $\tau = \delta$. In the first to fourth columns we show the wavefunctions, local conductance, nonlocal conductance, and $\delta E_0$, respectively. For the conductance calculations, we set $T = \delta/100$ and $\Gamma_L = \Gamma_R = 0.1 \delta$.
• Figure 3: (a) Energy difference between the even and odd ground states, $\delta E_0= E_0^{odd} - E_0^{even}$, (b) MP, and (c) excitation gap, as functions QD levels. (d) Local conductance at the sweet spots, indicated by the crosses in panels (a-c) as a function of the voltage bias.
• Figure 4: Illustrative sweet spots for the microscopic model. The first to third rows show the results for the green, purple, and cyan sweet spots shown in Fig. \ref{['Fig3']}, which are ordered with increasing values of the excitation gap. (a, e, i) present the wavefunctions of the MBSs. (b, f, j) and (c, g, k) show the local and nonlocal conductances, respectively. (d, h, l) show the dependence of $\delta E_0$ with respect to variations of multiple QD levels $\epsilon_i$.
• Figure 5: Phase dependence of the 3-site Kitaev chain. (a), (c), and (e) diagrams showing the couplings between the MBSs as a function of $\phi$ for the three typoe of sweet spots. (b), (d), and (f) local conductance as a function of $\phi$ and bias voltage $V_L$, revealing the energy spectrum. In the first to last rows, we show the effective 2-site sweet spot, the genuine 3-site sweet spot, and the delocalized sweet spot, respectively.