Using Andreev bound states and spin to remove domain walls in a Kitaev chain

TL;DR

This work demonstrates flux-free control of phase differences in a multi-site Kitaev chain realized with quantum-dot–superconductor hybrids by tuning either the outer QD spins or the ABS energy. The authors combine detailed transport and RF-reflectometry measurements on a three-site device with a theoretical model that maps spinful setups to an effective spinless chain, revealing that spin flips and ABS energy shifts impart phase changes between neighboring couplings that can remove domain walls, thereby preserving a bulk gap. Crucially, they observe that the phase evolution can be smooth and even exceed a strict π jump, which they explain with a spatial variation in the spin-orbit field and related parameters. These findings support the feasibility of scaling Kitaev-chain-based qubits without external flux control while highlighting the need to account for microscopic spin-orbit inhomogeneities when designing longer chains for robust Majorana-based operations.

Abstract

Quantum dot-superconductor hybrids have been established as a suitable platform for realizing Kitaev chains hosting Majorana bound states. Implementing these structures in a qubit architecture is expected to result in coherence times that scale exponentially with the lengths of the chains. To scale to longer systems, the phase differences between all superconducting segments in the chain need to be controlled. While this control has been demonstrated by using an external magnetic flux, ideally it can be achieved with control over intrinsic system parameters. In this work, we investigate whether the relevant phase differences can be tuned through the spin degree of freedom in each QD, or the chemical potential of the discrete bound states in the hybrid sections. We confirm that both these tuning knobs allow for controlling the phase difference in the couplings between neighbouring QDs, bypassing the requirement to tune an external flux. However, we find that the amplitude of the phase shifts can deviate from a discrete $π$-shift. We introduce a spatial variation in the spin-orbit field as a possible mechanism to explain the observed behaviour and comment on the consequences for experimentally creating long Kitaev chains.

Figures (14)

• Figure 1: Proposed mechanisms for controlling phase in the Kitaev chain and device layout of a three-site chain.a) Schematic of ECT and CAR sub-gap processes in a two-site Kitaev chain setup, controlled by an ABS with energy $E_{\mathrm{ABS}}$. b) ECT ($t$) and CAR ($\Delta$) amplitudes as a function of $\mu_{\mathrm{ABS}}$ for two different spin configurations. c) Simulated conductance spectra on the middle quantum dot of a three-site Kitaev chain, as a function of the phase $\phi$. Two different spin configurations are compared, demonstrating the removal of a domain wall through flipping the rightmost spin. d) Three-site Kitaev chain schematic showing the two pairs of couplings $t_1,\Delta_1$ and $t_2,\Delta_2$. Phase modulation is enabled by threading a flux $\Phi$ through the superconducting loop connecting the two hybrid sections. e) SEM of the measured device. An out-of plane magnetic field $B_z$ is used to modulate the flux, with a period of 28µT. An in-plane magnetic field $B_x$ is applied parallel to the channel, to spin-polarize the quantum dots.
• Figure 2: Spin-induced phase shifts.a) Bias spectroscopy measurement for the middle quantum dot as a function of the out-of-plane magnetic field $B_{\mathrm{z}}$ for an $\uparrow\uparrow\uparrow$ spin configuration. b) A higher resolution $G_{\mathrm{MM}}$ linetrace along $V_{\mathrm{M}}$ = 0, expressed in units of the flux quantum $\Phi_{\mathrm{0}}$. We identify a flux where the excitation gap closes. c) Repeated measurements of a), for different spin-configurations. d) Comparison of $G_{\mathrm{MM}}$ linetraces along $V_{\mathrm{M}}$=0, for the same spin-configurations as in c). Applied magnetic field along $B_{\mathrm{x}}$ is 150mT. The behaviour is reproducibly observed, shown in \ref{['fig:supp2']}.
• Figure 3: ABS-induced phase shifts.a) Hybridization between the right ABS and two spin levels of the right quantum dot QDR. Three sweet spots ($\times,\circ,\triangledown$) are identified from charge stability diagrams of the right and middle quantum dot (\ref{['fig:supp4']}). b) Flux dependence of the zero-bias conductance $G_{\mathrm{MM}}$ on the middle dot, measured at the three sweet spots indicated in a). c) Continuous measurement of the extracted flux-shift $\widetilde{\Phi}$ as a function of the right ABS energy. The extracted shift $\widetilde{\Phi}$ is defined with respect to the first conductance peak in the zero-bias trace as indicated in b) (see \ref{['fig:supp3']} for further details about the extraction). The spin-down configuration is only shown for values of $V_{\mathrm{ABS}}^{\mathrm{(2)}}$$<$ 16mV, as the extraction was unreliable for the rest of the range. Raw datasets are shown in \ref{['fig:supp6']}, \ref{['fig:supp7']}. Measurements were performed at an in-plane field $B_{\mathrm{x}}$ of 200mT. The flux dependence of the energy spectrum at all values of the ABS energy is shown in \ref{['fig:supp5']}.
• Figure 4: Theoretical analysis of $\widetilde{\phi}$ as a function of $\mathbf{\mu_{\mathrm{ABS}}}$.a) To explain the experimental result, we propose a spatial variation in the spin-orbit hopping between neighbouring sites as possible source (inset left). We implement this here equivalently through an offset angle $\theta_{\mathrm{A}}$ between the polarization of the QDs and the ABS (inset right). The more general case is discussed in Methods. The analytically obtained phase-shift $\widetilde{\phi}$ (\ref{['eq:phasedependence']}) is plotted, for various $\theta_{\mathrm{A}}$. For each curve, the two $\mu_{\mathrm{ABS}}$ values are indicated that correspond to a sweet-spot numerically. A fitting of the experimental data to the theoretical result is shown in \ref{['fig:supp5']}. b) The smoothening of the phase dependence on $\mu_{\mathrm{ABS}}$ results in a decrease in the phase difference $\delta\widetilde{\phi}$ between two sweet-spots, for two values of $E^{\mathrm{ABS}}_Z$ in units of the induced superconducting pairing in the ABS. c) The smoothening parameter $\Gamma$ as a function of $\theta_{\mathrm{A}}$, for two values of $E^{\mathrm{ABS}}_Z$.
• Figure S1: Device and characterizationa) Scanning electron microscopy (SEM) images of the measured device, showing the full flux loop (top) and the part of the loop that connects to the device region (bottom). Light areas indicate remaining Aluminium, while the three darker strips are the Ti/Pd device leads b) Zoomed in false-coloured SEM of the active region after all three gate depositions. Off-chip lumped element resonators are connected to all three normal leads, allowing for fast reflectometry measurements. c) Zeeman splitting of all three QDs used for the measurements of \ref{['fig:Fig2']}, showing the energy splitting of the two spin states. d) Charge stability diagrams of the left and middle QD, showing both CAR (left) and ECT (right) dominated coupling, indicating the existence of sweet spots for both spin states on the left QD. e) Charge stability diagrams of the middle and right QD, showing similar behaviour.