Correction to: Nature Nanotechnologyhttps://doi.org/10.1038/s41565-017-0032-8, published online 15 January 2018. The Letter reports Majorana signatures in hybrid InSb semiconductor nanowire–NbTiN superconductor devices. The devices exhibit a conductance plateau near the conductance quantum 2e²/h at bias voltages above the superconducting gap (normal conductance), accompanied by an enhanced Andreev conductance at bias voltages below the superconducting gap (subgap conductance). We have attributed these experimental observations to ballistic transport as supported by a theoretical analysis¹, finding mean free paths on the order of or larger than the effective wire segment (the segment covered by the superconducting electrode). Here, we correct errors discovered on reanalysis of the original data², following concerns raised by readers. Due to the age of the paper, it cannot be corrected directly in the original publication, thus the updates are provided via this amendment. We provide additional discussion on the claim of ballistic transport so as to avoid misinterpretations. External peer review of the reanalysis concluded that the claims in the Letter remain. An extended public repository including data obtained from nanowire devices that were not included in the publication can be found in ref. ². We note the lack of a series of flat and precisely quantized conductance plateaus (a staircase), a clear ballistic transport characteristic (see the two newly included Supplementary Figs. 1 and 7 showing larger voltage ranges of Fig. 1 and the original Supplementary Fig. 5). Our earlier studies on ballistic transport in nanowire devices^3,4 indicate that vapour–liquid–solid nanowires do not have the proper geometry for observing a conductance staircase without the application of a magnetic field perpendicular to the wire axis, which requires ideal (Landauer) reservoirs interfacing the ballistic region, absorbing charge carriers with near-unit probability. Similar to our earlier studies, ohmic contacts in the present nanowire devices do not satisfy the conditions of Landauer reservoirs. However, the transport in the effective wire segment can nevertheless be ballistic whose characteristic is a plateau feature near 2e²/h in normal conductance together with an enhanced Andreev conductance. Importantly, precise quantization is not realistic, prevented by the two-terminal device geometry, inevitably decreasing the conductance. In summary, a plateau feature with an enhanced Andreev conductance together with our theoretical analyses indicate that a large fraction of transport is ballistic over distances of the order of our device length. We add a discussion to the main text of the Letter as follows: “… followed by a dip in conductance due to channel mixing²⁰ [ref. ¹ below]. We do not observe higher plateaus (Supplementary Figs. 1 and 7), which we attribute to the contacts not satisfying the conditions of Landauer reservoirs, resulting in residual scattering more effective at larger conductance. This is in line with our earlier studies^25,39 [refs. ^4,5 below] which indicated that vapour–liquid–solid nanowires do not have the proper geometry for observing a conductance staircase without the application of a perpendicular magnetic field. From the absence of quantum dots, the observed induced gap …”. The following text should also have been included in the abstract: “… exhibiting clear ballistic transport properties manifested by a conductance plateau with an Andreev enhancement, albeit lacking a quantized conductance staircase hindered by the device geometry.” The conductance values reported in the publication are ~8% lower (near 2e²/h) than the actual value (corrected Fig. 1). This deviation is due to a drop in the gain of the current-to-voltage amplifier at an ac excitation frequency of 67 Hz⁵. As a result, there is a slight change in the Andreev conductance enhancement factor and the superconducting contact transparency extracted from the enhancement (a comparison between the values quoted in the publication and the corrected ones is given below in B). The general conclusions do not rely on the exact value of the conductance as precise quantization is not expected due to the two-terminal device geometry. The subtracted series resistance of 3 kΩ in the original Fig. 1 was an overestimation (see corrected Fig. 1 in the Supplementary Data file). The subtraction of 3 kΩ was not mentioned in the original publication. A comparison of the original and corrected Fig. 1 is presented in a Supplementary Data file accompanying this correction. For all the figures in the original publication except Fig. 1, we either subtracted a contact resistance value of 0.5 kΩ, which is an underestimation¹, or no resistance at all. We note that in tunneling measurements the overall resistance is significantly higher than the normal metal contact resistance whose contribution can therefore be neglected. Figure 1, however, was used to estimate the superconducting contact transparency and Andreev enhancement in the high conductance regime, requiring a realistic exclusion of the contact resistance. Following our previous paper⁴, which found normal metal contact resistance values between 1.5–3.25 kΩ per contact and was based on fitting the measured conductance using theory (single mode interfacing a superconductor), which provided reasonable agreement after excluding 3 kΩ, we subtracted 3 kΩ to exclude the resistance of the normal metal contact. During our reanalysis, we have discovered that the minimum resistance of this device at the largest applied gate voltages is 2.9 kΩ, a value providing an upper bound on the contact resistance. Here, 2.9 kΩ would be the contact resistance under the assumption that the nanowire itself has zero resistance at largest gate voltages. The contact resistance can be estimated with an alternative method by subtracting a series resistance to match the observed conductance plateau at bias voltages above the superconducting gap to the expected quantized value, a procedure not done in the original publication. By taking the conductance averaged at positive and negative |V| ~ 1.7 mV (around the largest bias voltages available for this analysis) we find that the quantized value is reached for a contact resistance of 0.77 kΩ. (Considering only the positive bias and separately only the negative bias results in a range of 0–2.13 kΩ for the contact resistance.) In our corrected estimate of the contact resistance, we have applied the calibration procedure⁵ that corrects for ac circuit effects, uses calibrated values for the series resistance of the setup where Fig. 1 was measured and directly corrects the error listed in A above. Upon reanalysis we estimate the following contact resistance values, enhancement factors and transparencies: (Table presented.) Contact resistance Enhancement factor Transparency Lower bound 0 kΩ 1.26 0.88 Conservative estimation¹ (used in corrected Fig. 1) 0.5 kΩ 1.32 0.90 Current best estimate 0.77 kΩ 1.36 0.90 Original estimate in paper 3 kΩ >1.5 >0.93 The corrected superconducting contact transparency value of 0.9 does not affect the claim of high transparency. The claim of ballistic transport does not rest on the exact value of the conductance plateau and hence is also unaffected. The original Methods section omits the indication of subtracted series resistances which account for the normal metal contact resistance in each figure. The following is included here for the corrected Methods: The original Methods section omits the indication of subtracted series resistances which account for the normal metal contact resistance in each figure. The following is included here for the corrected Methods: “Contact resistance treatment. A fixed-value series resistance of 0.5 kΩ has been subtracted in Figs. 1 and 4, Supplementary Figs. 1, 2b,c and 4–9 to account for the contact resistance of the normal metal lead. This value is smaller than the lowest contact resistance we have obtained for InSb nanowire devices²⁵ (ref. ⁴ below), which makes the interface transparency estimated from Fig. 1 a lower bound. For the remaining figures, no series resistance has been subtracted to account for the normal metal contact resistance.” In the original Supplementary Fig. 5 (now Supplementary Fig. 6), a charge jump was corrected by removal of 12 line traces (corresponding to +0.15 V to +0.04 V in gate voltage in the measured data) and offset of the gate voltage axis by 0.12 V after the charge jump (–1 V to +0.03 V) to maintain continuity of the axis. This processing was not mentioned in the original publication. The corrected Supplementary Fig. 6 excludes this processing and represents the data as measured. In the original Supplementary Fig. 5 (now Supplementary Fig. 6), a charge jump was corrected by removal of 12 line traces (corresponding to +0.15 V to +0.04 V in gate voltage in the measured data) and offset of the gate voltage axis by 0.12 V after the charge jump (–1 V to +0.03 V) to maintain continuity of the axis. This processing was not mentioned in the original publication. The corrected Supplementary Fig. 6 excludes this processing and represents the data as measured. A comparison of the original and corrected Fig. SI5 (now Fig. SI6) is presented in a Supplementary Data file accompanying this correction. Original Supplementary Fig. 1f (now Supplementary Fig. 2f): The offset mentioned in the caption is erroneously given as 0.006 × 2e²/h but is 0.01 × 2e²/h. Original Supplementary Fig. 4a,b (now Supplementary Fig. 5a,b) were indicated to present data from Fig. 2a (or original Supplementary Fig. 1a). This is incorrect. The data used are from the original Supplementary Fig. 1b (now Supplementary Fig. 2b) which has the same measurement settings as in Fig. 2a except the barrier gate is –1.5 V (the barrier gate is –1.4 V in Fig. 2a or original Supplementary Fig. 1a). In the original panels c–e of Supplementary Fig. 7 (now Supplementary Fig. 9c–e) the bias polarity is mistakenly inverted.

Electrostatic charging affects the many-body spectrum of Andreev states, yet its influence on their microwave properties has not been elucidated. We developed a circuit quantum electrodynamics probe that, in addition to transition spectroscopy, measures the microwave susceptibility of different states of a semiconductor nanowire weak link with a single dominant (spin-degenerate) Andreev level. We found that the microwave susceptibility does not exhibit a particle-hole symmetry, which we qualitatively explain as an influence of Coulomb interaction. Moreover, our state-selective measurement reveals a large, ?-phase shifted contribution to the response common to all many-body states which can be interpreted as arising from a phase-dependent continuum in the superconducting density of states.

Quantum phenomena are typically observable at length and time scales smaller than those of our everyday experience, often involving individual particles or excitations. The past few decades have seen a revolution in the ability to structure matter at the nanoscale, and experiments at the single particle level have become commonplace. This has opened wide new avenues for exploring and harnessing quantum mechanical effects in condensed matter. These quantum phenomena, in turn, have the potential to revolutionize the way we communicate, compute and probe the nanoscale world. Here, we review developments in key areas of quantum research in light of the nanotechnologies that enable them, with a view to what the future holds. Materials and devices with nanoscale features are used for quantum metrology and sensing, as building blocks for quantum computing, and as sources and detectors for quantum communication. They enable explorations of quantum behaviour and unconventional states in nano- and opto-mechanical systems, low-dimensional systems, molecular devices, nano-plasmonics, quantum electrodynamics, scanning tunnelling microscopy, and more. This rapidly expanding intersection of nanotechnology and quantum science/technology is mutually beneficial to both fields, laying claim to some of the most exciting scientific leaps of the last decade, with more on the horizon.

Two promising architectures for solid-state quantum information processing are based on electron spins electrostatically confined in semiconductor quantum dots and the collective electrodynamic modes of superconducting circuits. Superconducting electrodynamic qubits involve macroscopic numbers of electrons and offer the advantage of larger coupling, whereas semiconductor spin qubits involve individual electrons trapped in microscopic volumes but are more difficult to link. We combined beneficial aspects of both platforms in the Andreev spin qubit: the spin degree of freedom of an electronic quasiparticle trapped in the supercurrent-carrying Andreev levels of a Josephson semiconductor nanowire. We performed coherent spin manipulation by combining single-shot circuit–quantum-electrodynamics readout and spin-flipping Raman transitions and found a spin-flip time T_S = 17 microseconds and a spin coherence time T_2E = 52 nanoseconds. These results herald a regime of supercurrent-mediated coherent spin-photon coupling at the single-quantum level.

The charge localization of single electrons on mesoscopic metallic islands leads to a suppression of the electrical current, known as the Coulomb blockade. When this correction is small, it enables primary electron thermometry, as it was first demonstrated by Pekola et al. (Phys Rev Lett 73:2903, 1994). However, in the low temperature limit, random charge offsets influence the conductance and limit the universal behavior of a single metallic island. In this work, we numerically investigate the conductance of a junction array and demonstrate the extension of the primary regime for large arrays, even when the variations in the device parameters are taken into account. We find that our simulations agree well with measured conductance traces in the submillikelvin electron temperature regime.

In this Letter the following original sentence has been amended for clarity: “As the Kramers theorem does not hold in the presence of a non-zero weak-link phase bias φ, the splitting of the spin states requires an additional ingredient.”; it has been changed to: “Although the presence of a non-zero weak-link phase bias φ breaks time-reversal symmetry so that the Kramers theorem does not hold, this alone is insufficient to lift the spin degeneracy—an additional effect is required.” The online versions of the Letter have been amended.

Fragile quantum effects such as single electron charging in quantum dots or macroscopic coherent tunneling in superconducting junctions are the basis of modern quantum technologies. These phenomena can only be observed in devices where the characteristic spacing between energy levels exceeds the thermal energy, kBT, demanding effective refrigeration techniques for nanoscale electronic devices. Commercially available dilution refrigerators have enabled typical electron temperatures in the 10 to 100 mK regime, however indirect cooling of nanodevices becomes inefficient due to stray radiofrequency heating and weak thermal coupling of electrons to the device substrate. Here, we report on passing the millikelvin barrier for a nanoelectronic device. Using a combination of on-chip and off-chip nuclear refrigeration, we reach an ultimate electron temperature of Te = 421 ± 35 μK and a hold time exceeding 85 h below 700 μK measured by a self-calibrated Coulomb-blockade thermometer.

Serial double quantum dots created in semiconductor nanostructures provide a versatile platform for investigating two-electron spin quantum states, which can be tuned by electrostatic gating and an external magnetic field. In this Rapid Communication, we directly measure the supercurrent reversal between adjacent charge states of an InAs nanowire double quantum dot with superconducting leads, in good agreement with theoretical models. In the even charge parity sector, we observe a supercurrent blockade with increasing magnetic field, corresponding to the spin singlet to triplet transition. Our results demonstrate a direct spin to supercurrent conversion, the superconducting equivalent of the Pauli spin blockade. This effect can be exploited in hybrid quantum architectures coupling the quantum states of spin systems and superconducting circuits.

Readout and control of electrostatically confined electrons in semiconductors are key primitives of quantum information processing with solid-state spin qubits^1,2. In superconductor–semiconductor heterostructures, localized electronic modes known as Andreev levels result from confinement that is provided by the pair potential^3,4. Unlike electronic modes confined exclusively via electrostatic effects, Andreev levels carry supercurrent. Therefore, they naturally integrate with the techniques of circuit quantum electrodynamics (cQED) that have been developed in the field of superconducting qubits and used to detect pairs of quasiparticles that are trapped in Andreev levels^5–8. Here, we demonstrate single-shot cQED readout of the spin of an individual quasiparticle trapped in the Andreev levels of a semiconductor nanowire Josephson element. Owing to a spin-orbit interaction in the nanowire, this ‘superconducting spin’ directly determines the flow of supercurrent through the element. We harnessed this spin-dependent supercurrent to achieve both a zero-field spin splitting and a long-range interaction between the quasiparticle and a superconducting microwave resonator^9–13. Measurement of the resultant spin-dependent resonator frequency yielded quantum non-demolition spin readout with 92% fidelity in 1.9 μs, which enabled us to monitor the quasiparticle spin in real time. These results pave the way for superconducting spin qubits that operate at zero magnetic field and for time-domain measurements of Majorana zero modes^{9,10,12,14,15}.

Inhomogeneous superconductors can host electronic excitations, known as Andreev bound states (ABSs), below the superconducting energy gap. With the advent of topological superconductivity, a new kind of zero-energy ABS with exotic qualities, known as a Majorana bound state (MBS), has been discovered. A special property of MBS wavefunctions is their non-locality, which, together with non-Abelian braiding, is the key to their promise in topological quantum computation. We focus on hybrid superconductor–semiconductor nanowires as a flexible and promising experimental platform to realize one-dimensional topological superconductivity and MBSs. We review the main properties of ABSs and MBSs, state-of-the-art techniques for their detection and theoretical progress beyond minimal models, including different types of robust zero modes that may emerge without a band-topological transition.

The Cooper-pair transistor (CPT), a small superconducting island enclosed between two Josephson weak links, is the atomic building block of various superconducting quantum circuits. Utilizing gate-tunable semiconductor channels as weak links, the energy scale associated with the Josephson tunneling can be changed with respect to the charging energy of the island, tuning the extent of its charge fluctuations. Here, we directly demonstrate this control by mapping the energy level structure of a CPT made of an indium arsenide nanowire with a superconducting aluminum shell. We extract the device parameters based on the exhaustive modeling of the quantum dynamics of the phase-biased nanowire CPT and directly measure the even-odd parity occupation ratio as a function of the device temperature, relevant for superconducting and prospective topological qubits.

Quantum computation by non-Abelian Majorana zero modes (MZMs) offers an approach to achieve fault tolerance by encoding quantum information in the non-local charge parity states of semiconductor nanowire networks in the topological superconductor regime. Thus far, experimental studies of MZMs chiefly relied on single electron tunneling measurements, which lead to the decoherence of the quantum information stored in the MZM. As a next step towards topological quantum computation, charge parity conserving experiments based on the Josephson effect are required, which can also help exclude suggested non-topological origins of the zero bias conductance anomaly. Here we report the direct measurement of the Josephson radiation frequency in indium arsenide nanowires with epitaxial aluminium shells. We observe the 4π-periodic Josephson effect above a magnetic field of ≈200 mT, consistent with the estimated and measured topological phase transition of similar devices.

The frontiers of quantum electronics have been linked to the discovery of new refrigeration methods since the discovery of superconductivity at a temperature of around 4 K, enabled by the liquefaction of helium. Since then, advances in cryogenics have led to discoveries such as the quantum Hall effect and new technologies such as superconducting and semiconductor quantum bits. Presently, nanoelectronic devices typically reach electron temperatures of around 10-100 mK by use of commercially available dilution refrigerators. However, cooling electrons via the encompassing lattice vibrations, or phonons, becomes inefficient at low temperatures. Further progress toward lower temperatures requires new cooling methods for electrons on the nanoscale, such as direct cooling with nuclear spins, which themselves can be brought to microkelvin temperatures by adiabatic demagnetization. Here we introduce indium as a nuclear refrigerant for nanoelectronics and demonstrate that solely on-chip cooling of electrons is possible down to 3.2±0.1mK, limited by the heat leak via the electrical connections of the device.

The modern understanding of the Josephson effect in mesosopic devices derives from the physics of Andreev bound states, fermionic modes that are localized in a superconducting weak link. Recently, Josephson junctions constructed using semiconducting nanowires have led to the realization of superconducting qubits with gate-tunable Josephson energies. We have used a microwave circuit QED architecture to detect Andreev bound states in such a gate-tunable junction based on an aluminum-proximitized indium arsenide nanowire. We demonstrate coherent manipulation of these bound states, and track the bound-state fermion parity in real time. Individual parity-switching events due to nonequilibrium quasiparticles are observed with a characteristic timescale Tparity=160±10 μs. The Tparity of a topological nanowire junction sets a lower bound on the bandwidth required for control of Majorana bound states.

The number of electrons in small metallic or semiconducting islands is quantised. When tunnelling is enabled via opaque barriers this number can change by an integer. In superconductors the addition is in units of two electron charges (2e), reflecting that the Cooper pair condensate must have an even parity. This ground state (GS) is foundational for all superconducting qubit devices. Here, we study a hybrid superconducting-semiconducting island and find three typical GS evolutions in a parallel magnetic field: a robust 2e-periodic even-parity GS, a transition to a 2e-periodic odd-parity GS, and a transition from a 2e- to a 1e-periodic GS. The 2e-periodic odd-parity GS persistent in gate-voltage occurs when a spin-resolved subgap state crosses zero energy. For our 1e-periodic GSs we explicitly show the origin being a single zero-energy state gapped from the continuum, i.e., compatible with an Andreev bound states stabilized at zero energy or the presence of Majorana zero modes.

Majorana modes are zero-energy excitations of a topological superconductor that exhibit non-Abelian statistics^1–3. Following proposals for their detection in a semiconductor nanowire coupled to an s-wave superconductor^4,5, several tunnelling experiments reported characteristic Majorana signatures^6–11. Reducing disorder has been a prime challenge for these experiments because disorder can mimic the zero-energy signatures of Majoranas^12–16, and renders the topological properties inaccessible^17–20. Here, we show characteristic Majorana signatures in InSb nanowire devices exhibiting clear ballistic transport properties. Application of a magnetic field and spatial control of carrier density using local gates generates a zero bias peak that is rigid over a large region in the parameter space of chemical potential, Zeeman energy and tunnel barrier potential. The reduction of disorder allows us to resolve separate regions in the parameter space with and without a zero bias peak, indicating topologically distinct phases. These observations are consistent with the Majorana theory in a ballistic system²¹, and exclude the known alternative explanations that invoke disorder^12–16 or a nonuniform chemical potential^22,23.

Parity control of superconducting islands hosting Majorana zero modes (MZMs) is required to operate topological qubits made from proximitized semiconductor nanowires. We test this control by studying parity effects in hybrid InAs-Al single-Cooper-pair transistors (SCPTs) to evaluate the feasibility of this material system. In particular, we investigate the gate-charge modulation of the supercurrent and observe a consistent 2e-periodic pattern indicating a general lack of low-energy subgap states in these nanowires at zero magnetic field. In a parallel magnetic field, an even-odd pattern develops with a gate-charge spacing that oscillates as a function of field demonstrating that the modulation pattern is sensitive to the presence of a single bound state. In addition, we find that the parity lifetime of the SCPT decreases exponentially with magnetic field as the bound state approaches zero energy. Our work shows that aluminum is the preferred superconductor for future topological qubit experiments and highlights the important role that quasiparticle traps and superconducting gap engineering would play in these qubits. Moreover, we demonstrate a means by which bound states can be detected in devices with superconducting leads.

We measured the Josephson radiation emitted by an InSb semiconductor nanowire junction utilizing photon-assisted quasiparticle tunneling in an ac-coupled superconducting tunnel junction. We quantify the action of the local microwave environment by evaluating the frequency dependence of the inelastic Cooper-pair tunneling of the nanowire junction and find the zero-frequency impedance Z(0)=492Ω with a cutoff frequency of f0=33.1GHz. We extract a circuit coupling efficiency of η≈0.1 and a detector quantum efficiency approaching unity in the high-frequency limit. In addition to the Josephson radiation, we identify a shot noise contribution with a Fano factor F≈1, consistently with the presence of single electron states in the nanowire channel.

Junctions created by coupling two superconductors via a semiconductor nanowire in the presence of high magnetic fields are the basis for the potential detection, fusion, and braiding of Majorana bound states. We study NbTiN/InSb nanowire/NbTiN Josephson junctions and find that the dependence of the critical current on the magnetic field exhibits gate-tunable nodes. This is in contrast with a well-known Fraunhofer effect, under which critical current nodes form a regular pattern with a period fixed by the junction area. Based on a realistic numerical model we conclude that the Zeeman effect induced by the magnetic field and the spin-orbit interaction in the nanowire are insufficient to explain the observed evolution of the Josephson effect. We find the interference between the few occupied one-dimensional modes in the nanowire to be the dominant mechanism responsible for the critical current behavior. We also report a strong suppression of critical currents at finite magnetic fields that should be taken into account when designing circuits based on Majorana bound states.

The superconducting proximity effect in semiconductor nanowires has recently enabled the study of new superconducting architectures, such as gate-tunable superconducting qubits and multiterminal Josephson junctions. As opposed to their metallic counterparts, the electron density in semiconductor nanosystems is tunable by external electrostatic gates, providing a highly scalable and in situ variation of the device properties. In addition, semiconductors with large g-factor and spin-orbit coupling have been shown to give rise to exotic phenomena in superconductivity, such as † 0 Josephson junctions and the emergence of Majorana bound states. Here, we report microwave spectroscopy measurements that directly reveal the presence of Andreev bound states (ABS) in ballistic semiconductor channels. We show that the measured ABS spectra are the result of transport channels with gate-tunable, high transmission probabilities up to 0.9, which is required for gate-tunable Andreev qubits and beneficial for braiding schemes of Majorana states. For the first time, we detect excitations of a spin-split pair of ABS and observe symmetry-broken ABS, a direct consequence of the spin-orbit coupling in the semiconductor.