Srijit Goswami

Quantum dot (QD)-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 [ten Haaf , ; Bordin , ], ideally, it can be achieved with control over intrinsic system parameters [Liu , ]. 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 neighboring 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 behavior and comment on the consequences for experimentally creating long Kitaev chains.

The Fermi-Hubbard model describes a large variety of condensed matter systems with spinful fermions and strong interactions. On the other hand, the Kitaev chain model deals with noninteracting spinless fermions and produces robust ground-state degeneracies that give rise to Majorana bound states. In this Letter, we connect these two domains by experimentally studying small arrays of quantum dots (QDs) coupled via superconductors. Without a magnetic field, this system constitutes a Fermi-Hubbard model with intersite superconducting correlations. Using two coupled QDs, we find sweet spots in parameter space where we observe zero-bias conductance peaks that are robust against on-site perturbations of the QDs, similar to the behavior expected of a two-site Kitaev chain. This observation is consistent with the presence of Majorana Kramers pairs or Z 3 parafermions in this minimal system. Extending to three sites, we find that the spinful system scales very differently compared to the spinless Kitaev chain. When the same sweet-spot conditions are satisfied, the ground state degeneracy of the full three-site system is lifted. The degeneracy can be restored by tuning the superconducting phase difference between the hybrid segments, but the robustness against on-site detunings is lost. Our findings are a first step in studying ground-state degeneracies in strongly interacting Fermi-Hubbard systems with superconducting correlations.

Contemporary quantum devices are reaching new limits in size and complexity, allowing for the experimental exploration of emergent quantum modes. However, this increased complexity introduces significant challenges in device tuning and control. Here, we demonstrate autonomous tuning of emergent Poor Man’s Majorana zero modes in a minimal realization of a Kitaev chain. We achieve this task using cross-platform transfer learning. First, we train a tuning model on a theory model. Next, we retrain it using a Kitaev chain realization in a two-dimensional electron gas. Finally, we apply this model to tune a Kitaev chain realized in quantum dots coupled through a semiconductor-superconductor section in a one-dimensional nanowire. Utilizing a convolutional neural network, we predict the tunneling and Cooper pair splitting rates from differential conductance measurements, employing these predictions to adjust the electrochemical potential to a Poor Man’s Majorana sweet spot. The algorithm successfully converges to an immediate vicinity of a sweet spot (within ± 1.5 m V in 67.6% of attempts and within ± 4.5 m V in 80.9% of cases), typically finding a sweet spot in 45 min or less. This advancement is a stepping stone toward autonomous tuning of emergent modes in interacting systems, and toward foundational tuning machine learning models that can be deployed across a range of experimental platforms.

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.

The Fermi-Hubbard model and its rich phase diagram naturally emerges as a description for a wide range of electronic systems. Recent advances in semiconductor-superconductor hybrid quantum dot arrays have allowed to realize degenerate quantum systems in a controllable way, e.g., allowing to observe robust zero-bias peaks in Kitaev chains, indicative for Majorana bound states. In this work, we connect these two domains. Noting the strong on-site Coulomb repulsion within quantum dots, we study small arrays of spinful hybrid quantum dots implemented in a two-dimensional electron gas. This system constitutes a Fermi-Hubbard model with inter-site superconducting correlations. For two electronic sites, we find robust zero-bias peaks indicative of a strongly degenerate spectrum hosting emergent Majorana Kramers pairs or $\mathbb{Z}_3$-parafermions. Extending to three sites, we find that these spinful systems scale very differently compared to spinless Kitaev chains. When the sweet-spot conditions are satisfied pairwise, we find that the ground state degeneracy of the full three-site system is lifted. This degeneracy can be restored by tuning the superconducting phase difference between the hybrid segments. However, these states are not robust to quantum dot detuning. Our observations are a first step towards studying degeneracies in strongly interacting Fermi-Hubbard systems with superconducting correlations.

Hybrid quantum dot (QD)-superconductor system can be used to realize Majorana zero modes in artificial Kitaev chains. These chains provide a promising platform for the realization of Majorana qubits. Radio-frequency (RF) gate reflectometry is a fast, non-invasive, and sensitive technique that can be used to read out such qubits. In this work, we use gate reflectometry to probe two QDs coupled via a semiconductor-superconductor hybrid segment. We demonstrate that gate sensing can resolve charge stability diagrams and clearly distinguish between elastic cotunneling and crossed-Andreev reflection, the two key processes that allow one to form a Kitaev chain. Furthermore, we show that this information is accessible, even when the system is completely decoupled from the from the normal leads. In this closed regime, we show that the observed quantum capacitance signal is indicative of parity switching between the even and odd ground states. Our measurements in both open and closed regimes confirm that gate reflectometry captures the essential features of interdot coupling and parity dynamics.

In semiconducting-superconducting hybrid devices, Andreev bound states (ABSs) can mediate the coupling between quantum dots, allowing for the realization of artificial Kitaev chains. In order to engineer Majorana bound states (MBSs) in these systems, one must control the energy of the ABSs. In this Letter, we show how extended ABSs in a flux-tunable Josephson junction can be used to control the coupling between distant quantum dots separated by ≃1 μm. In particular, we demonstrate that the combination of electrostatic control and phase control over the ABSs increases the parameter space in which MBSs are observed. Finally, by employing an additional spectroscopic probe in the hybrid region between the quantum dots, we gain information about the spatial distribution of the Majorana wave function in a two-site Kitaev chain.

A chain of quantum dots (QDs) in semiconductor–superconductor hybrid systems can form an artificial Kitaev chain hosting Majorana bound states (MBSs)^1,2,3. These zero-energy states are expected to be localized on the edges of the chain⁴, at the outermost QDs. The remaining QDs, comprising the bulk, are predicted to host an excitation gap that protects the MBSs at the edges from local on-site perturbations. Here we demonstrate this connection between the bulk and edges in a minimal system, by engineering a three-site Kitaev chain in a two-dimensional electron gas. Through direct tunnelling spectroscopy on each site, we show that the appearance of stable zero-bias conductance peaks at the outer QDs is correlated with the presence of an excitation gap in the middle QD. Furthermore, we show that this gap can be controlled by applying a superconducting phase difference between the two hybrid segments and that the MBSs are robust only when the excitation gap is present. We find a close agreement between experiments and the original Kitaev model, thus confirming key predictions for MBSs in a three-site chain.

Majorana bound states are expected to appear in one-dimensional semiconductor-superconductor hybrid systems, provided they are homogeneous enough to host a global topological phase. In order to experimentally investigate the uniformity of the system, we study the spatial dependence of the local density of states in multiprobe devices where several local tunneling probes are positioned along a gate-defined wire in a two-dimensional electron gas. Spectroscopy at each probe reveals a hard induced gap and an absence of subgap states at zero magnetic field. However, subgap states emerging at a finite magnetic field are not always correlated between different probes. Moreover, we find that the extracted critical field and effective g-factor vary significantly across the length of the wire. Upon studying several such devices, we do however find examples of striking correlations in the local density of states measured at different tunnel probes. We discuss possible sources of variation across devices.