Majorana zero modes are quasiparticle excitations in condensed matter systems that have been proposed as building blocks of fault-tolerant quantum computers [1]. They are expected to exhibit non-Abelian particle statistics, in contrast to the usual statistics of fermions and bosons, enabling quantum operations to be performed by braiding isolated modes around one another [1, 2]. Quantum braiding operations are topologically protected insofar as these modes are pinned near zero energy, and the pinning is predicted to be exponential as the modes become spatially separated [3, 4]. Following theoretical proposals [5, 6], several experiments have identified signatures of Majorana modes in proximitized nanowires [7][8][9][10][11] and atomic chains [12], with small modesplitting potentially explained by hybridization of Majoranas [13][14][15]. Here, we use Coulombblockade spectroscopy in an InAs nanowire segment with epitaxial aluminum, which forms a proximity-induced superconducting Coulomb island (a Majorana island) that is isolated from normal-metal leads by tunnel barriers, to measure the splitting of near-zero-energy Majorana modes. We observe exponential suppression of energy splitting with increasing wire length. For short devices of a few hundred nanometers, subgap state energies oscillate as the magnetic field is varied, as is expected for hybridized Majorana modes. Splitting decreases by a factor of about ten for each half a micrometer of increased wire length. For devices longer than about one micrometer, transport in strong magnetic fields occurs through a zero-energy state that is energetically isolated from a continuum, yielding uniformly spaced Coulomb-blockade conductance peaks, consistent with teleportation via Majorana modes [16, 17]. Our results help to explain the trivial-to-topological transition in finite systems and to quantify the scaling of topological protection with end-mode separation.The set of structures we investigate consist of InAs nanowires grown by molecular beam epitaxy in the [0001] wurtzite direction with an epitaxial Al shell on two facets of the hexagonal cross section [18]. The Al shell was removed except in a small segment of length L and isolated from normal metal (Ti/Au) leads by electrostatic gatecontrolled barriers (Fig. 1a). Charging energy, E C , of the device ranges from greater than to less than the superconducting gap of Al (∼ 0.2 meV). The thin Al shell (8 − 10 nm thickness on the two facets) gives a large critical field, B c , before superconductivity is destroyed: for fields along the wire axis, B c,|| ∼ 1 T; out of the plane of the substrate but roughly in the plane of the two Alcovered facets, B c,⊥ ∼ 700 mT (Fig. 1b). The very high achieved critical fields make these wires a suitable platform for investigating topological superconductivity [18].Five devices over a range of Al shell lengths L ∼ 0.3 − 1.5 µm were measured (see Methods for device layouts). Charge occupation and tunnel coupling to the leads were tuned via electrostatic gates. Differential conductan...

Electrons in atoms possess both spin and orbital degrees of freedom. In non-relativistic quantum mechanics, these are independent, resulting in large degeneracies in atomic spectra. However, relativistic effects couple the spin and orbital motion leading to the wellknown fine structure in their spectra. The electronic states in defect-free carbon nanotubes (NTs) are widely believed to be four-fold degenerate 1-10 , due to independent spin and orbital symmetries, and to also possess electron-hole symmetry 11 . Here we report measurements demonstrating that in clean NTs the spin and orbital motion of electrons are coupled, thereby breaking all of these symmetries. This spin-orbit coupling is directly observed as a splitting of the four-fold degeneracy of a single electron in ultra-clean quantum dots. The coupling favours parallel alignment of the orbital and spin magnetic moments for electrons and anti-parallel alignment for holes. Our measurements are consistent with recent theories 12,13 that predict the existence of spin-orbit coupling in curved graphene and describe it as a spin-dependent topological phase in NTs. Our findings have important implications for spin-based applications in carbon-based systems, entailing new design principles for the realization of qubits in NTs and providing a mechanism for all-electrical control of spins 14 in NTs.Carbon-based systems are promising candidates for spin based applications such as spinqubits [14][15][16][17][18][19] and spintronics 20-23 as they are believed to have exceptionally long spin coherence times due to weak spin-orbit interactions and the absence of nuclear spin in the 12 C atom. Carbon NTs may play a particularly interesting role in this context because in addition to spin they offer a unique two-fold orbital degree of freedom that can also be used for quantum manipulation. The latter arises from the two equivalent dispersion cones (K and K') in graphene, which lead to doubly-degenerate electronic orbits that encircle the nanotube circumference in a clockwise (CW) and counter-clockwise (CCW) fashion 24 (Fig 1a). Together, the two-fold spin degeneracy and two-fold orbital degeneracy are generally assumed to yield a four-fold-degenerate electronic energy spectrum in clean NTs. Understanding the fundamental symmetries of this spectrum is at the heart of successful manipulation of these quantum degrees of freedom.A powerful way to probe the symmetries is by confining the carriers to a quantum dot (QD) and applying a magnetic field parallel to the tube axis, || B 4,5,8,10,24,25 . The confinement creates bound states and the field interrogates their nature by coupling independently to their spin and orbital moments. In the absence of spin-orbit coupling, such a measurement should yield for a defect-free NT the energy spectrum shown in figure 1b. At 0 || = B the NT spectrum should be four-fold degenerate. With increasing || B the spectrum splits into pairs of CCW and CW states

Many present and future applications of superconductivity would benefit from electrostatic control of carrier density and tunnelling rates, the hallmark of semiconductor devices. One particularly exciting application is the realization of topological superconductivity as a basis for quantum information processing. Proposals in this direction based on the proximity effect in semiconductor nanowires are appealing because the key ingredients are currently in hand. However, previous instances of proximitized semiconductors show significant tunnelling conductance below the superconducting gap, suggesting a continuum of subgap states--a situation that nullifies topological protection. Here, we report a hard superconducting gap induced by the proximity effect in a semiconductor, using epitaxial InAs-Al semiconductor-superconductor nanowires. The hard gap, together with favourable material properties and gate-tunability, makes this new hybrid system attractive for a number of applications, as well as fundamental studies of mesoscopic superconductivity.

Carbon nanotubes are a versatile material in which many aspects of condensed matter physics come together. Recent discoveries have uncovered new phenomena that completely change our understanding of transport in these devices, especially the role of the spin and valley degrees of freedom. This review describes the modern understanding of transport through nanotube devices. Unlike in conventional semiconductors, electrons in nanotubes have two angular momentum quantum numbers, arising from spin and valley freedom. The interplay between the two is the focus of this review. The energy levels associated with each degree of freedom, and the spin-orbit coupling between them, are explained, together with their consequences for transport measurements through nanotube quantum dots. In double quantum dots, the combination of quantum numbers modifies the selection rules of Pauli blockade. This can be exploited to read out spin and valley qubits and to measure the decay of these states through coupling to nuclear spins and phonons. A second unique property of carbon nanotubes is that the combination of valley freedom and electron-electron interactions in one dimension strongly modifies their transport behavior. Interaction between electrons inside and outside a quantum dot is manifested in SU(4) Kondo behavior and level renormalization. Interaction within a dot leads to Wigner molecules and more complex correlated states. This review takes an experimental perspective informed by recent advances in theory. As well as the well-understood overall picture, open questions for the field are also clearly stated. These advances position nanotubes as a leading system for the study of spin and valley physics in one dimension where electronic disorder and hyperfine interaction can both be reduced to a low level.

We introduce a hybrid qubit based on a semiconductor nanowire with an epitaxially grown superconductor layer. Josephson energy of the transmonlike device ("gatemon") is controlled by an electrostatic gate that depletes carriers in a semiconducting weak link region. Strong coupling to an on-chip microwave cavity and coherent qubit control via gate voltage pulses is demonstrated, yielding reasonably long relaxation times (~0.8 μs) and dephasing times (~1 μs), exceeding gate operation times by 2 orders of magnitude, in these first-generation devices. Because qubit control relies on voltages rather than fluxes, dissipation in resistive control lines is reduced, screening reduces cross talk, and the absence of flux control allows operation in a magnetic field, relevant for topological quantum information.

We demonstrate a substantial improvement in the spin-exchange gate using symmetric control instead of conventional detuning in GaAs spin qubits, up to a factor of six increase in the quality factor of the gate. For symmetric operation, nanosecond voltage pulses are applied to the barrier that controls the interdot potential between quantum dots, modulating the exchange interaction while maintaining symmetry between the dots. Excellent agreement is found with a model that separately includes electrical and nuclear noise sources for both detuning and symmetric gating schemes. Unlike exchange control via detuning, the decoherence of symmetric exchange rotations is dominated by rotation-axis fluctuations due to nuclear field noise rather than direct exchange noise.

Quasiparticle excitations can compromise the performance of superconducting devices, causing high frequency dissipation, decoherence in Josephson qubits [1][2][3][4][5][6], and braiding errors in proposed Majorana-based topological quantum computers [7][8][9]. Quasiparticle dynamics have been studied in detail in metallic superconductors [10][11][12][13][14] but remain relatively unexplored in semiconductor-superconductor structures, which are now being intensely pursued in the context of topological superconductivity. To this end, we introduce a new physical system comprised of a gate-confined semiconductor nanowire with an epitaxially grown superconductor layer, yielding an isolated, proximitized nanowire segment. We identify Andreev-like bound states in the semiconductor via bias spectroscopy, determine the characteristic temperatures and magnetic fields for quasiparticle excitations, and extract a parity lifetime (poisoning time) of the bound state in the semiconductor exceeding 10 ms.Semiconductor-superconductor hybrids have been investigated for many years [15][16][17][18][19], but recently have received renewed interest in the context of topological superconductivity, motivated by the realization that combining spin-orbit interaction, Zeeman splitting and proximity coupling to a conventional s-wave superconductor provides the necessary ingredients to create Majorana modes at the ends of a one-dimensional (1D) wire. Such modes are expected to show nonabelian statistics, allowing, in principle, topological encoding of quantum information [20][21][22] among other interesting effects [23,24].Transport experiments on semiconductor nanowires proximitized by a grounded superconductor have recently revealed characteristic features of Majorana modes [25][26][27][28]. Semiconductor quantum dots with superconducting leads have also been explored experimentally [29][30][31][32], and have been proposed as a basis for Majorana chains [33][34][35]. Here, we expand the geometries investigated in this context by creating an isolated semiconductorsupercondutor hybrid quantum dot (HQD) connected to normal leads. The device forms the basis of an isolated * These authors contributed equally to this work.Majorana system with protected total parity, where both the semiconductor nanowire and the metallic superconductor are mesoscopic [36,37].The measured device consists of an InAs nanowire with epitaxial superconducting Al on two facets of the hexagonal wire, with Au ohmic contacts (Figs. 1a,b). Four devices showing similar behavior have been measured. The InAs nanowire was grown without stacking faults using molecular beam epitaxy with Al deposited in situ to ensure high-quality proximity effect [38,39]. Differential conductance, g, was measured in a dilution refrigerator with base electron temperature T ∼ 50 mK using standard ac lock-in techniques. Local side gates, patterned with electron beam lithography, and a global back gate were adjusted to form an Al-InAs HQD in the Coulomb blockade regime, with gate-controlled weak tun...

). Electron-nuclear interaction in 13C nanotube double quantum dots. Nature Physics, 5(5), 321-326. https://doi.org