Andreev bound states arise in low-dimensional confined systems, coupled to superconductors. They show great similarity to the highly sought-after Majorana bound states, yet they lack the desirable non-Abelian statistics or topological protection. Except for these, Andreev bound states possess a multitude of unique and interesting properties. This thesis explores several of these properties, using semiconductor quantum dots as a measurement tool. In addition, quantum dots are hybridized with Andreev bound states to form novel systems, including Andreev molecules, and Kitaev chains.

In the theory section, we introduce several basic concepts, followed by a discussion of the properties of Andreev bound states and similar Yu-Shiba-Rusinov states. We then extend the concept of Andreev bound states to Kitaev chains in various implementations. In the first chapter, we use a quantum dot as both a spin and energy filter to probe an Andreev bound state. We observe pure spin states despite the strong spin-orbit interaction in the host semiconductor. Utilizing a three-terminal measurement setup, we can change the spin-relaxation process of the Andreev bound state by changing tunnel barrier strengths. Next, we configure a quantum dot as a charge sensor to study Andreev bound states. We observe smooth changes in ground state charge due to hybridization of the even-occupation states. We additionally detect abrupt loading of electrons during the singlet-doublet transition, which agrees with a change of ground state parity. Having used quantum dots as a measurement tool, we then hybridize them with Andreev bound states to form an Andreev molecule. We demonstrate readout of the ground state parity of the combined system using the charge sensor. We argue that parity-to-charge conversion in semiconductorsuperconductor systems is a viable scheme for reading out Kitaev chains and associated qubits.

We proceed by strongly coupling two quantum dots to a single Andreev bound state. This coupling mediates tunneling and Cooper pair splitting processes between the quantum dots, effectively constituting a Kitaev chain. For each Andreev bound state, we can find two gate voltages at which the rates of these processes are equal and non-zero. Spectroscopic measurements reveal localized Majorana zero modes on the quantum dots that are robust against local electrostatic changes.

Engineering Kitaev chain-based qubits requires consistently finding Majorana zero modes. In the final experimental chapter of this thesis, we present an algorithm that tunes gate voltages until Majorana zero modes emerge in Kitaev chains. We employ a neural network to estimate the relative Cooper pair splitting and tunneling rates from spectroscopic measurements. These estimates are then input into a gradient descent algorithm until the rates are balanced, and Majorana zero modes emerge. We present statistics on the algorithm’s performance and conclude that it is a vital tool in elevating Kitaev chains from the realm of fundamental study to quantum information.

We then propose a series of future experiments, based on our current findings. Notably, we explore the possibility of storing quantum information in the spin degree of freedom of a superconductor using Yu-Shiba-Rusinov states.
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We propose to implement a Kitaev chain based on an array of alternating normal and superconductor hybrid quantum dots embedded in semiconductors. In particular, the orbitals in the dot and the Andreev bound states in the hybrid are now on an equal footing, and both emerge as low-energy degrees of freedom in the Kitaev chain, with the couplings being induced by direct tunneling. Due to the electron and hole components in the Andreev bound state, this coupling is simultaneously of the normal and Andreev types, with their ratio being tunable by varying one or several of the experimentally accessible physical parameters, e.g., strength and direction of the Zeeman field, as well as changing the proximity effect on the normal quantum dots. As such, it becomes feasible to realize a two-site Kitaev chain in a simple setup with only one normal quantum dot and one hybrid segment. Interestingly, when scaling up the system to a three-site Kitaev chain, next-nearest-neighbor couplings emerge as a result of high-order tunneling, lifting the Majorana zero energy at the sweet spot. This energy splitting is mitigated in a longer chain, approaching topological protection. Our proposal has two immediate advantages: obtaining a larger energy gap from direct tunneling, and creating a Kitaev chain using a reduced number of quantum dots and hybrid segments.

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The proximity effect of superconductivity on confined states in semiconductors gives rise to various bound states such as Andreev bound states, Andreev molecules, and Majorana zero modes. While such bound states do not conserve charge, their fermion parity is a good quantum number. One way to measure parity is to convert it to charge first, which is then sensed. In this work, we sense the charge of Andreev bound states and Andreev molecules in an InSb-Al hybrid nanowire using an integrated quantum dot operated as a charge sensor. We show how charge sensing measurements can resolve the even and odd states of an Andreev molecule, without affecting the parity. Such an approach can be further used for parity measurements of Majorana zero modes in Kitaev chains based on quantum dots.

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The formation of a topological superconducting phase in a quantum-dot-based Kitaev chain requires nearest neighbor crossed Andreev reflection and elastic cotunneling. Here, we report on a hybrid InSb nanowire in a three-site Kitaev chain geometry - the smallest system with well-defined bulk and edge - where two superconductor-semiconductor hybrids separate three quantum dots. We demonstrate pairwise crossed Andreev reflection and elastic cotunneling between both pairs of neighboring dots and show sequential tunneling processes involving all three quantum dots. These results are the next step toward the realization of topological superconductivity in long Kitaev chain devices with many coupled quantum dots.

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Kitaev chains in quantum dot-superconductor arrays are a promising platform for the realization of topological superconductivity. As recently demonstrated, even a two-site chain can host Majorana zero modes known as “poor man’s Majorana”. Harnessing the potential of these states for quantum information processing, however, requires increasing their robustness to external perturbations. Here, we form a two-site Kitaev chain using Yu-Shiba-Rusinov states in proximitized quantum dots. By deterministically tuning the hybridization between the quantum dots and the superconductor, we observe poor man’s Majorana states with a gap larger than 70 μeV. The sensitivity to charge fluctuations is also greatly reduced compared to Kitaev chains made with non-proximitized dots. The systematic control and improved energy scales of poor man’s Majorana states realized with Yu-Shiba-Rusinov states will benefit the realization of longer Kitaev chains, parity qubits, and the demonstration of non-Abelian physics.@en

Majorana bound states constitute one of the simplest examples of emergent non-Abelian excitations in condensed matter physics. A toy model proposed by Kitaev shows that such states can arise at the ends of a spinless p-wave superconducting chain¹. Practical proposals for its realization^2,3 require coupling neighbouring quantum dots (QDs) in a chain through both electron tunnelling and crossed Andreev reflection⁴. Although both processes have been observed in semiconducting nanowires and carbon nanotubes^5–8, crossed-Andreev interaction was neither easily tunable nor strong enough to induce coherent hybridization of dot states. Here we demonstrate the simultaneous presence of all necessary ingredients for an artificial Kitaev chain: two spin-polarized QDs in an InSb nanowire strongly coupled by both elastic co-tunnelling (ECT) and crossed Andreev reflection (CAR). We fine-tune this system to a sweet spot where a pair of poor man’s Majorana states is predicted to appear. At this sweet spot, the transport characteristics satisfy the theoretical predictions for such a system, including pairwise correlation, zero charge and stability against local perturbations. Although the simple system presented here can be scaled to simulate a full Kitaev chain with an emergent topological order, it can also be used imminently to explore relevant physics related to non-Abelian anyons.

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A short superconducting segment can couple attached quantum dots via elastic cotunneling (ECT) and crossed Andreev reflection (CAR). Such coupled quantum dots can host Majorana bound states provided that the ratio between CAR and ECT can be controlled. Metallic superconductors have so far been shown to mediate such tunneling phenomena, albeit with limited tunability. Here, we show that Andreev bound states formed in semiconductor-superconductor heterostructures can mediate CAR and ECT over mesoscopic length scales. Andreev bound states possess both an electron and a hole component, giving rise to an intricate interference phenomenon that allows us to tune the ratio between CAR and ECT deterministically. We further show that the combination of intrinsic spin-orbit coupling in InSb nanowires and an applied magnetic field provides another efficient knob to tune the ratio between ECT and CAR and optimize the amount of coupling between neighboring quantum dots.

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The proximity effect in semiconductor-superconductor nanowires is expected to generate an induced gap in the semiconductor. The magnitude of this induced gap, together with the semiconductor properties like spin-orbit coupling and g-factor, depends on the coupling between the materials. It is predicted that this coupling can be adjusted through the use of electric fields. We study this phenomenon in InSb/Al/Pt hybrids using nonlocal spectroscopy. We show that these hybrids can be tuned such that the semiconductor and superconductor are strongly coupled. In this case, the induced gap is similar to the superconducting gap in the Al/Pt shell and closes only at high magnetic fields. In contrast, the coupling can be suppressed which leads to a strong reduction of the induced gap and critical magnetic field. At the crossover between the strong-coupling and weak-coupling regimes, we observe the closing and reopening of the induced gap in the bulk of a nanowire. Contrary to expectations, it is not accompanied by the formation of zero-bias peaks in the local conductance spectra. As a result, this cannot be attributed conclusively to the anticipated topological phase transition and we discuss possible alternative explanations.

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Determining Hamiltonian parameters from noisy experimental measurements is a key task for the control of experimental quantum systems. An interesting experimental platform where precise knowledge of device parameters is useful is the quantum-dot-based Kitaev chain. In these systems, the fine tuning of Hamiltonian parameters is crucial in order to reach the desired regime with stable midgap modes. In this work, we demonstrate an adversarial machine-learning algorithm to determine the parameters of a quantum-dot-based Kitaev chain. We train a convolutional conditional generative adversarial neural network (CCGAN) with simulated differential conductance data and use the model to predict the parameters at which Majorana bound states are predicted to appear. In particular, the CCGAN model facilitates a rapid, numerically efficient exploration of the phase diagram describing the transition between elastic co-tunneling and crossed Andreev reflection regimes. We verify the theoretical predictions of the model by applying it to experimentally measured conductance obtained from a minimal Kitaev chain consisting of two spin-polarized quantum dots coupled by a superconductor-semiconductor hybrid. Our model accurately predicts, with an average success probability of 97%, whether the measurement was taken in the elastic co-tunneling or crossed Andreev reflection-dominated regime. Our work constitutes a stepping stone towards fast, reliable parameter prediction for tuning quantum dot systems into distinct Hamiltonian regimes. Ultimately, our results yield a strategy to support Kitaev-chain tuning that is scalable to longer chains.

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Semiconductor nanowires coupled to superconductors can host Andreev bound states with distinct spin and parity, including a spin-zero state with an even number of electrons and a spin-1/2 state with odd-parity. Considering the difference in spin of the even and odd states, spin-filtered measurements can reveal the underlying ground state. To directly measure the spin of single-electron excitations, we probe an Andreev bound state using a spin-polarized quantum dot that acts as a bipolar spin filter, in combination with a non-polarized tunnel junction in a three-terminal circuit. We observe a spin-polarized excitation spectrum of the Andreev bound state, which can be fully spin-polarized, despite strong spin-orbit interaction in the InSb nanowires. Decoupling the hybrid from the normal lead causes a current blockade, by trapping the Andreev bound state in an excited state. Spin-polarized spectroscopy of hybrid nanowire devices, as demonstrated here, is proposed as an experimental tool to support the observation of topological superconductivity.

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We present a report on hybrid InSb-Pb nanowires that combine high spin-orbit coupling with a high critical field and a large superconducting gap. Material characterization indicates the Pb layer of high crystal quality on the nanowire side facets. Hard induced superconducting gaps and gate-tunable supercurrent are observed in the hybrid nanowires. These results showcase the promising potential of this material combination for a diverse range of applications in hybrid quantum transport devices.

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We study a Cooper pair transistor realized by two Josephson weak links that enclose a superconducting island in an InSb-Al hybrid nanowire. When the nanowire is subject to a magnetic field, isolated subgap levels arise in the superconducting island and, because of the Coulomb blockade, mediate a supercurrent by coherent cotunneling of Cooper pairs. We show that the supercurrent resulting from such cotunneling events exhibits, for low to moderate magnetic fields, a phase offset that discriminates even and odd charge ground states on the superconducting island. Notably, this phase offset persists when a subgap state approaches zero energy and, based on theoretical considerations, permits parity measurements of subgap states by supercurrent interferometry. Such supercurrent parity measurements could, in a series of experiments, provide an alternative approach for manipulating and protecting quantum information stored in the isolated subgap levels of superconducting islands.

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We measure the charge periodicity of Coulomb blockade conductance oscillations of a hybrid InSb-Al island as a function of gate voltage and parallel magnetic field. The periodicity changes from to at a gate-dependent value of the magnetic field, , decreasing from a high to a low limit upon increasing the gate voltage. In the gate voltage region between the two limits, which our numerical simulations indicate to be the most promising for locating Majorana zero modes, we observe correlated oscillations of peak spacings and heights. For positive gate voltages, the transition with low is due to the presence of nontopological states whose energy quickly disperses below the charging energy due to the orbital effect of the magnetic field. Our measurements highlight the importance of a careful exploration of the entire available phase space of a proximitized nanowire as a prerequisite to define future topological qubits.

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