Robust poor man's Majorana zero modes using Yu-Shiba-Rusinov states

Quantum-dot-based Kitaev chains provide a minimal platform to engineer Majorana bound states. Even a two-site chain can host fine-tuned Majorana zero modes known as poor man’s Majoranas (PMMs), which, while not topologically protected, are robust to local perturbations and exhibit quadratic protection against global chemical potential fluctuations. Realizing PMMs by coupling spin-polarized quantum dots (QDs) via crossed Andreev reflection (CAR) and elastic cotunneling (ECT) is challenging because the QD–superconductor coupling must be balanced: too weak yields negligible inter-dot coupling; too strong spoils well-defined charge states. Prior demonstrations achieved the basic functionality but with small excitation gaps (comparable to electron temperature), pronounced charge-noise sensitivity, and limited device yield for strong coherent coupling. This work targets enhanced robustness and control by replacing non-proximitized QDs with spin-polarized Yu–Shiba–Rusinov (YSR) states formed in proximitized QDs. The research question is whether YSR-based sites can deterministically tune CAR and ECT to a PMM sweet spot while substantially increasing the excitation gap and reducing sensitivity to charge fluctuations, thereby improving feasibility for longer chains, parity qubits, and non-Abelian operations.

The study builds on proposals and demonstrations of PMMs in double QDs coupled through superconductors, where CAR and ECT implement the pairing and hopping terms of the Kitaev model. Previous experimental work in InSb/Al nanowires showed tunable CAR/ECT via an intermediate Andreev bound state (ABS), enabling PMMs but with small energy gaps and charge-noise issues. Theoretical analyses have highlighted that tuning the ABS electrochemical potential controls the electron–hole balance, enhancing CAR near charge neutrality while suppressing ECT, thereby guaranteeing a PMM sweet spot for a single mediating ABS. Related work on YSR physics in QDs has established strong proximitization with large charging energies distinct from ABSs, and prior theory suggested that strongly coupled YSR states can form effective Kitaev chains with enhanced gaps. Complementary, a parallel study on a 2D InAsSb/Al platform demonstrates the broader applicability of ABS-mediated coupling for minimal Kitaev chains.

Device: A hybrid InSb nanowire partially covered by an 8 nm Al shell (capped with 20 nm AlOx) sits on patterned Ti/Pd bottom gates with ALD dielectrics (10 nm AlOx, 10 nm HfO2). The central Al-covered segment (hybrid) is tuned by plunger gate V_H. On each side, a QD is defined with three gates; QD plunge gates are V_LP and V_RP; couplings to the hybrid are set by tunnel gates V_LT and V_RT. Each QD connects to a normal metal lead via another gate-defined tunnel barrier; the superconducting Al is grounded. Measurement: Three-terminal transport at base temperature 30 mK, magnetic field B = 150 mT along the nanowire. Independent DC biases V_L and V_R applied to left/right normal leads; currents I_L, I_R measured. Local conductances G_LL = dI_L/dV_L and G_RR = dI_R/dV_R, and non-local G_LR = dI_L/dV_R, G_RL = dI_R/dV_L measured simultaneously using lock-in (AC excitations 5 μV RMS; 10 μV in specified supplementary datasets). Off-chip multiplexed RF resonators enable fast reflectometry for tuning and spectroscopy. Line/series resistance corrections are applied by calibrating the superconducting gap constancy, yielding an additional 3.65 kΩ series resistance (beyond source/meter resistances) used to correct datasets where full conductance matrices are taken. Modeling: The hybrid segment is treated as a single ABS in the atomic limit with induced gap Δ (set by coupling to bulk superconductor), negligible charging energy (screened by grounded Al), and smaller Zeeman splitting E_ZH than in QDs. QDs have charging energy U, Zeeman splitting E_Z, and chemical potentials μ_L, μ_R. Spin-conserving (ω) and spin-flip (ω_SO) tunneling couple QDs to the ABS, producing proximitized YSR states when ω, ω_SO are comparable to the level broadening Γ. Identification of YSR vs ABS is based on large charging energy and characteristic spectroscopy (eye-shaped spectra near ABS charge neutrality, avoided crossings reflecting spin-conserving/flipping processes). Tuning CAR and ECT: By sweeping V_H to adjust ABS electron–hole balance, effective YSR–YSR couplings t (ECT) and Δ (CAR) are tuned. Charge-stability diagrams (CSDs) of the two proximitized QDs reveal regimes t > Δ (anti-diagonal avoided crossings with negative non-local conductance), t ≈ Δ (crossings at PMM sweet spots), and Δ > t (diagonal avoided crossings). Extraction of t and Δ: At the center of an avoided crossing (μ_1 = μ_2 = 0), subgap spectra show symmetric peaks at energies t − Δ and t + Δ. Fitting with two pairs of symmetric Gaussians around zero bias provides t and Δ as functions of V_H. Lever arm estimation of YSR excitations uses gate-dispersion slopes, showing substantial reduction relative to above-gap QD lever arms. Stability tests: PMM spectra are recorded at the sweet spot (t ≈ Δ) while detuning one QD (local perturbation) or simultaneously detuning both QDs along the antidiagonal (global perturbation) to quantify quadratic protection and charge-dispersion reduction. The linear splitting of zero-energy states versus V_H quantifies sensitivity to ABS gate fluctuations (i.e., deviations from t = Δ).

• Deterministic formation of a robust two-site Kitaev chain using spin-polarized YSR states in proximitized QDs, coupled via a single ABS.
• Large excitation gap: E_gap ≈ 76 μeV between PMM zero modes and first excited states, about threefold larger than prior QD-based demonstrations and well above the electron temperature (~30 mK), enabling faster adiabatic operations.
• Strong reduction of charge dispersion: YSR lever arm at ABS charge neutrality reduced to ~0.05e (vs ~0.4e above gap and ~0.2e for detuned subgap), indicating much weaker sensitivity to gate-induced charge fluctuations; overall, PMM robustness against QD charge noise improved by about two orders of magnitude compared to QD-based chains.
• Tunable CAR and ECT: By sweeping V_H, continuous control from t ≫ Δ (ECT-dominated) to Δ ≫ t (CAR-dominated) with intermediate PMM sweet spots (t ≈ Δ) at V_H ≈ 333 mV and ≈ 345 mV; characteristic CSD patterns and non-local conductance sign confirm regimes.
• Spectral extraction of couplings: Subgap peak positions at t − Δ and t + Δ used to quantify t(V_H) and Δ(V_H); around V_H ≈ 338 mV, Δ peaks and t dips due to constructive/destructive interference of electron/hole components of the ABS.
• Sensitivity to ABS detuning: Linear splitting slopes of the zero-energy PMM versus V_H are ~7 μeV/mV and ~3 μeV/mV at two sweet spots, indicating no protection against deviations from t = Δ in a two-site chain.
• Stability to local detuning: Zero-bias peaks persist when detuning one QD while the other remains on resonance, consistent with PMM behavior.
• Non-local signatures: Alternating sign of non-local conductance at the sweet spot; negative non-local conductance in ECT-dominated regime.
• Platform independence: Results demonstrated in InSb/Al nanowires and consistent with a parallel 2D InAsSb/Al platform study, indicating broader applicability.

Using YSR states as the sites of a minimal Kitaev chain addresses key limitations of QD-based PMMs by enhancing the excitation gap and suppressing charge dispersion. The strong QD–superconductor hybridization forming YSR states maintains localized charge configurations while enabling significant CAR and ECT mediated via a single ABS. Tuning the ABS electrochemical potential provides systematic control of t and Δ, ensuring access to PMM sweet spots without device-specific fine-tuning. The observed large gap (≈76 μeV) and two-orders-of-magnitude improvement in robustness to QD charge noise directly support the goal of more stable PMM operation, advancing prospects for parity qubits, fusion, and braiding experiments. Nonetheless, as a two-site system, the PMM remains unprotected against deviations from t = Δ and susceptible to tunnel-coupling noise, as evidenced by linear splitting with V_H. The demonstrated control and improved energy scales suggest that extending to longer chains will further mitigate coupling noise and approach the coherence expected in continuous nanowires, while the established tuning protocols are transferable across material platforms.

The work demonstrates robust poor man’s Majorana zero modes using YSR states in a two-site Kitaev chain with deterministic tuning via an intervening ABS. Key achievements include a substantially increased excitation gap (~76 μeV), systematic and continuous control of CAR and ECT to reach multiple PMM sweet spots, and a strong reduction in charge dispersion leading to enhanced stability. These advances make minimal chains more viable for prototypical Majorana qubits and non-Abelian experiments. Looking ahead, increasing the number of sites (3–5) is expected to mitigate tunnel-coupling noise and achieve dephasing times comparable to continuous-nanowire implementations. The demonstrated QD–ABS and YSR–YSR hybridization control also benefits Andreev spin qubits, long-distance spin-qubit coupling via superconductors, and analog quantum simulations of correlated electron systems with superconductivity.

• Two-site PMMs lack topological protection: zero-energy modes split linearly with deviations from the t = Δ condition (observed slopes ~7 and ~3 μeV/mV vs V_H), indicating sensitivity to ABS detuning and tunnel-coupling noise.
• Present devices remain susceptible to fluctuations in tunnel couplings (ECT/CAR), which are not intrinsically protected in minimal chains.
• Series/line resistances required post-measurement correction to stabilize the extracted superconducting gap; while corrected, this adds uncertainty to absolute conductance scaling and non-local conductance magnitude.
• Residual proximity in the right QD when ABS is detuned suggests some device-specific coupling pathways that may complicate isolation of single-ABS mediation in certain gate ranges.