A two-site Kitaev chain in a two-dimensional electron gas

a–b, Coulomb diamond measurements of the left and right QDs, in the regime used for measurements presented in Fig. 4. Charging energies are estimated to be 1.1 meV in each QD. The lever arm of the left and right dot is estimated to be α ≈ 0.11. c–d, Zoomed in views of figures (a) and (b) in a smaller energy range. Due to a large tunnel coupling between the QDs and the hybrid section, YSR-states form in the sub-gap spectrum of the QDs. Tunneling spectroscopy around the charge degeneracy points in (a-b) reveal clear sup-gap features within the Coulomb diamonds.As V_QDL and V_QDR are tuned, the sub-gap features form an eye-shape feature enclosing the doublet charge occupation. This behavior is typical for YSR-states with large charging energies²⁸. e, Crossed Andreev reflection and elastic co-tunneling require the presence of extended ABSs. Local G_l and non-local conductance G_nl of the hybrid region are measured via tunnelling spectroscopy and their identical energy dependence as a function of V_ABS highlights that ABSs extend across the entire hybrid section. Comparable behavior was observed in a wide V_ABS range from 0 to − 1 V. The measurement presented in Fig. 1 is taken at the V_ABS with the eye-shaped crossing. f, ABS spectroscopy as a function external magnetic field at V_ABS = −623 mV. The effect of splitting of the doublet state can be observed at low fields. A g-factor of 5.5 is extracted by linear fitting of the lowest sub-gap states (dashed line) in Extended Data Fig. 1d.

a, Scanning electron micro-graph of device B, used to obtain the measurements presented in Fig. 2. Scale bar shown is 100 nm. b–c, Coulomb diamond measurements of the left and right QDs. Charging energies are extracted to be 1.4 meV. The lever arm α_N is extracted to be about 0.11 for each QD. This lever arm is used for the extraction in Fig. 2d. d–e, To validate the direction of B_SO and to show the connection between interactions in strongly coupled QDs and underlying ECT and CAR processes, we first measure ECT and CAR currents in the weakly coupled dots, as detailed in²⁹. With B⊥B_SO, measurements of CAR (d) and ECT (e) show the typical blockades for same-spin and opposite-spin charge configurations respectively. In f–g, with B∥B_SO, the spin non-conserving ECT and CAR processes are observed to be revived. h, Measuring CAR and ECT rates as a function of magnetic field angle θ shows the currents for the spin non-conserving processes are indeed smoothly controlled and become suppressed when θ = 0. This supports the interpretation that B∥B_SO when B is perpendicular to the 1-D channel. i–j, Next, the QDs are operated with higher tunnelling rates between the QDs and the SC, to enable strong couplings. Similar to Fig. 1e-f, CSDs are obtained in the strongly interacting regime, taken with the verified B∥B_SO and B⊥B_SO respectively.

a, CSDs measured at various magnetic field angles θ between 0° and 90°, used to extract the data shown in Fig. 2d. b, Example of the extraction process. For each obtained CSD, V_QDL and V_QDR are converted to energies μ_L and μ_R using lever arms obtained in Extended Data Fig. 2. Next, the conductance is extracted along a μ_L = − μ_R or μ_L = μ_R line. c, Two Gaussian peaks are fitted to extract the separation between the two avoided crossings, from which the quantity \(\sqrt{| {\varGamma }_{{\rm{o}}}-{\varGamma }_{{\rm{e}}}| }\) is obtained (plotted in Fig. 2d).

To complement the data in Fig. 3, Extended Data Fig. 5 and Extended Data Fig. 10, we measure the sub-gap states in QDL and QDR (see Extended Data Fig. 1). Using this, we obtain the lever arms of V_QDL and V_QDR on the YSR-state energies (denoted α_YSR) (see Methods). a–b, Sub-gap spectroscopy of QDL and QDR at B_z = 0 mT. From the slopes of the states upon crossing V_L, V_R = 0, we estimate α_YSR ≈ 0.045. Applying an external magnetic field lowers the energy of ABSs in the hybrid region, as a result of Zeeman splitting. This in turn will affect the YSR-spectrum of the QDs, due to increased hybridization between the QDs and the ABS. c–d, Measuring sub-gap spectroscopy of QDL and QDR at B_z = 225 mT for the same settings as in (a–b) shows indeed the effective lever arm here decreases to α_YSR ≈ 0.028.

A comparison between numerically calculated conductance and measured conductance in support of Fig. 3, measured at B = 225 mT. Presented results show the evolution of G_l and G_nl for four different cases: (a–b) detuning V_QDL, (c–d) detuning V_QDR, (e–f) detuning both V_QDL and V_QDR simultaneously along a diagonal path and (g–h) detuning both anti-diagonally. For each case, we find the behavior of both G_l and G_nl is well described by the numerical results.

a, Sets of CSDs obtained while varying V_ABS in the range presented in Fig. 3. The range of V_QDL and V_QDR is constant for each measurement. The slight drift of the avoided crossing upon varying V_ABS is owed to cross-capacitance between V_ABS and the potential of the QDs.

In CSDs presented in Fig. 3, a clear sign change is observed when changing from the Γ_O > Γ_E regime to the Γ_O < Γ_E regime. This can be understood by considering the possible transport cycles that underlie the measured non-local conductance. a, When Γ_O > Γ_E, G_nl is observed to be negative in the measured CSDs (see Fig. 3c). c, When Γ_O < Γ_E, the same measurements yield a positive G_nl (see Fig. 3a). Horizontal and vertical dashed lines indicate μ_R = 0 and μ_L = 0 respectively. The state of the uncoupled system is labelled in each quadrant. b,d, In such CSD measurements, zero-bias transport can take place when the odd and even ground states are degenerate. For non-local transport to occur, the system can accept a hole/electron from one lead, and relax non-locally to its original state by either (b) donating a hole/electron to the opposite lead, giving rise to negative G_nl, or (d) accept a hole/electron from the opposite lead, giving rise to positive G_nl. The preferred path is dictated by the quadrant in μ_L, μ_R space where the odd-even degeneracy occurs. e, When μ_L, μ_R > 0 or μ_L, μ_R < 0, the former path is expected to dominate and the resulting G_nl will be negative. f, When μ_L > 0 and μ_R < 0 or vice versa, the latter path is expected to dominate and resulting G_nl will be positive.

Numerical calculations supporting the results presented in Fig. 4. Through the procedure detailed in Methods, Majorana sweet spots are obtained and analysed for fields between 0 mT and 300 mT. a, Field evolution of G_RR line-traces at each sweet spot, showing the excitations above the ZBPs gradually increasing in energy and then saturating at ± 30 μeV. b, Field evolution of G_RR line-traces when QDR detuned by 3Δ_ind, showing the excited states increase linearly in energy. From calculations in (a) and (b), E_gap and \({E}_{\det }\) are obtained, given by the energy between the lowest even-parity state and second-lowest odd-parity state. c, Extraction of E_gap (solid) and the Majorana polarization (dashed), for different values of the tunneling parameter t. In each case t_so = 0.4t. Larger tunnel coupling results in larger hybridization between ABSs, in turn lowering the MP at a specific magnetic field. d, Extraction of \({E}_{\det }\) for various values of detuning μ_R. In each case the slope at low fields corresponds to 2E_z (dashed grey line). The larger the detuning of μ_R, the longer this holds. The dashed black line shows the energy of the ABS E_ABS. At large detuning \({E}_{\det }\) will increase linearly with 2E_z, until becoming of comparable E_ABS becomes the lowest energy scale.

Obtained ‘sweet spots’ at magnetic fields between 0 mT and 300 mT. a–i, CSDs and tunnelling spectroscopy are measured at each sweet spot, where V_QDR is detuned. From these measurements \({E}_{\det }\) and E_gap are extracted, as described in the main text and in Methods.

Reproduction of the main results from Fig. 4, using the orbitals shown in Fig. 3. Data was obtained at 6 different field values B between 0 and 250 mT. At each field V_ABS is adjusted to tune to the sweet spot. a, Extraction of \({E}_{\det }\) and E_gap, similar to the analysis presented in Fig. 4l. From a linear for of \({E}_{\det }\) a g-factor of 5.7 is estimated. b, Numerically obtained \({E}_{\det }\) and E_gap, using parameters tuned to compare to (a). At 250 mT, an estimate of M ≈ 0.9 is obtained. Extrapolation for comparison to Fig. 4l yields M ≈ 0.92 at 300 mT. c,d, Waterfall plots highlighting the line-traces used to extract the data in (a). e–j, Raw datasets of CSDs and tunnelling spectroscopy measurements, from which (a-d) is extracted. Datasets at 150 mT and 225 mT datasets are repeated from Fig. 3 and Extended Data Fig. 5 respectively.