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arXiv:2003.02852 · Physical review. B./Physical review. B

Superconducting islands with topological Josephson junctions based on semiconductor nanowires

Jesús Ávila, Elsa Prada, Pablo San-José, Ramón Aguado

Published Sep 25, 2020• 19 citations
arXivHTMLPDF

Abstract

We theoretically study superconducting islands based on semiconductor-nanowire Josephson junctions and take into account the presence of subgap quasiparticle excitations in the spectrum of the junction. Our method extends the standard model Hamiltonian for a superconducting charge qubit and replaces the Josephson potential by the Bogoliubov--de Gennes Hamiltonian of the nanowire junction, projected onto the relevant low-energy subgap subspace. This allows us to fully incorporate the coherent dynamics of subgap levels in the junction. The combined effect of spin-orbit coupling and Zeeman energy in the nanowires forming the junction triggers a topological transition, where the subgap levels evolve from finite-energy Andreev bound states into near-zero-energy Majorana bound states. The interplay between the microscopic energy scales governing the nanowire junction (the Josephson energy, the Majorana coupling, and the Majorana energy splitting), with the charging energy of the superconducting island, gives rise to a great variety of physical regimes. Based on this interplay of different energy scales, we fully characterize the microwave response of the junction, from the Cooper pair box to the transmon regimes, and show how the presence of Majoranas can be detected through distinct spectroscopic features. In split-junction geometries, the plasma mode couples to the phase-dispersing subgap levels resulting from Majorana hybridization via a Jaynes--Cummings-like interaction. As a consequence of this interaction, higher order plasma excitations in the junction inherit Majorana properties, including the $4\ensuremath{\pi}$ effect.

DOI
10.1103/physrevb.102.094518

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We theoretically study superconducting islands based on semiconductor-nanowire Josephson junctions and take into account the presence of subgap quasiparticle excitations in the spectrum of the junction. Our method extends the standard model Hamiltonian for a superconducting charge qubit and replaces the Josephson potential by the Bogoliubov--de Gennes Hamiltonian of the nanowire junction, projected onto the relevant low-energy subgap subspace. This allows us to fully incorporate the coherent dynamics of subgap levels in the junction. The combined effect of spin-orbit coupling and Zeeman energy in the nanowires forming the junction triggers a topological transition, where the subgap levels evolve from finite-energy Andreev bound states into near-zero-energy Majorana bound states. The interplay between the microscopic energy scales governing the nanowire junction (the Josephson energy, the Majorana coupling, and the Majorana energy splitting), with the charging energy of the superconducting island, gives rise to a great variety of physical regimes. Based on this interplay of different energy scales, we fully characterize the microwave response of the junction, from the Cooper pair box to the transmon regimes, and show how the presence of Majoranas can be detected through distinct spectroscopic features. In split-junction geometries, the plasma mode couples to the phase-dispersing subgap levels resulting from Majorana hybridization via a Jaynes--Cummings-like interaction. As a consequence of this interaction, higher order plasma excitations in the junction inherit Majorana properties, including the 4\ensuremathπ4\ensuremath{\pi}4\ensuremathπ effect.
We study nanowire-based Josephson junctions shunted by a capacitor and take into account the presence of low-energy quasiparticle excitations. These are treated by extending conventional models used to describe superconducting qubits to include the coherent coupling between fermionic quasiparticles, in particular the Majorana zero modes that emerge in topological superconductors, and the plasma mode of the junction. Using accurate, unbiased matrix-product state techniques, we compute the energy spectrum and response function of the system across the topological phase transition. Furthermore, we develop a perturbative approach, valid in the harmonic limit with small charging energy, illustrating how the presence of low-energy quasiparticles affects the spectrum and response of the junction. Our results are of direct interest to on-going experimental investigations of nanowire-based superconducting qubits.
We show that the microwave (MW) spectra in semiconductor-nanowire-based transmon qubits provide a strong signature of the presence of Majorana bound states in the junction.This occurs as an external magnetic field tunes the wire into the topological regime and the energy splitting of the emergent Majorana modes oscillates around zero energy owing to their wave function spatial overlap in finite-length wires.In particular, we discuss how these Majorana oscillations, and the concomitant fermion parity switches in the ground state of the junction, result in distinct spectroscopic features-in the form of an intermittent visibility of absorption lines-that strongly deviate from standard transmon behavior.In contrast, nonoscillating zero modes, such as topologically trivial Andreev bound states resulting from sufficiently smooth potentials, exhibit an overall standard transmon response.These differences in the MW response could help determine whether the junction contains topological Majoranas or not.
A practical quantum computer demands a physical quantum bit (qubit) that is scalable, has a well-defined initial state, and brings together coherence with reliable control and read-out. A compelling idea is to put together a locally protected Majorana qubit with a superconducting qubit. The superconducting qubit is used to read out the Majorana states while the Majorana qubit may act as a quantum memory. This thesis presents a series of studies on quantum transport in semiconducting nanowires coupled to superconductors to investigate various platforms for realizing Majorana. These studies are followed by microwave measurements on superconducting circuits compatible with semiconducting-superconducting heterostructures. We first evaluate the potential of Ge/Si core/shell nanowires by achieving induced superconductivity as well as estimating spin-orbit coupling. Next we explore the transport mediated by Andreev bound states formed in InSb nanowire quantum. A subgap negative differential conductance is investigated together with the coalescing Andreev resonances at zero bias relevant for the correct interpretation of Majorana experiments done on the same structures. We conclude our studies of semiconducting nanowires by exploring tunnel junctions in Sn-InSb nanowires that are prepared by in-situ shadowing using nearby nanowires and flakes. Tin shells are found to induce a hard superconducting gap persisting up to high magnetic fields. We observe the two-electron charging effect from a small superconducting island of Sn-InSb. This effect is attributed to charge parity stability, which makes this nanowire system an intriguing candidate for superconducting and topological quantum circuits. These Sn-InSb junctions are then used as the nonlinear element in a transmon qubit design where we observe a dispersive coupling between this nanowire Josephson junction and a superconducting resonator. We also present our progress towards building a magnetic field resilient superconducting circuit that allows integration with semiconducting and topological structures. In the last chapter, InSb semiconducting nanowires are utilized as shadow masks prior to superconductor deposition on an InAs quantum well. We study Josephson current properties in Josephson junctions made from these nanowire shadows. Our results point to highly transparent junctions that can be developed further for hybrid superconductor-semiconductor qubit systems.
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 T_{parity}=160±10 μs. The T_{parity} of a topological nanowire junction sets a lower bound on the bandwidth required for control of Majorana bound states.
Quantum computers can solve some problems exponentially faster than classical computers. Unfortunately, the computational power of quantum computers is currently limited by the number of working qubits. It is difficult to scale up these systems, because qubits are easily affected by noise in their environment. This noise leads to decoherence: loss of the qubit’s encoded information. A possible solution to diminish decoherence is using Majorana box qubits, as these qubits are predicted to be insensitive to local noise. However, this promising type of qubit does not exist yet. With the research described in this thesis, we aim to contribute to the development of Majorana box qubits (MBQs). In these qubits, Majorana zero modes, the basic elements of MBQs, are contained within a superconducting island to suppress Majorana parity fluctuations caused by quasiparticle poisoning. To enable parity readout of the MBQ, these modes are coupled to quantum dots within a nanowire network. To help realize MBQs, we need a better understanding of quasiparticles in superconducting islands, parity-readout techniques, and ways to fabricate nanowire networks. These three aspects are the focus of the experiments presented in this thesis. To study superconducting islands and readout techniques, we used InAs semiconductor nanowires with an epitaxially grown Al shell. Majorana signatures have already been observed in such nanowires. We addressed quasiparticle dynamics in superconducting islands by measuring the gate-charge modulation of the switching current. We found a consistent 2e-periodic modulation at zero magnetic field, and an exponential decrease of parity lifetime with increasing magnetic field. We explored MBQ readout, using a quantum dot level as a proxy for a Majorana zero mode, and measured its charge hybridization with another dot using gate-based readout. We showed that we can rapidly discriminate between two settings with different tunnel couplings, demonstrating the potential of gate-based readout to measure MBQs. And, using gate-based readout, we could study charge-transfer processes occurring in hybrid structures of superconducting islands coupled to quantum dots. Finally, to find a good material platform for nanowire networks, we characterized two two-dimensional systems. We realized quantum point contacts in InSb, which we used to measure the ggg-factor anisotropy, and effective electron mass in this system. And, we studied the spin-orbit interaction in InAs/GaSb by extracting the difference in density between electrons with different spin orientations. This thesis finishes with a proposal for a series experiments to realize MBQs. These experiments make use of superconducting islands and the reflectometry setup we developed for gate-based readout.
A topological superfluid phase characterized by an emergent chiral-p-wave pair potential is expected to form in a two-dimensional Fermi superfluid subject to s-wave pairing, spin-orbit coupling and a large-enough Zeeman splitting. Andreev bound states appear at phase boundaries, including Majorana zero modes whose existence is assured by the bulk-boundary correspondence principle. Here we study the physical properties of these subgap-energy bound states at step-like interfaces using the spin-resolved Bogoliubov-de Gennes mean-field formalism and assuming small spin-orbit coupling. Extending a recently developed spin-projection technique based on Feshbach partitioning [SciPost Phys. 5, 016 (2018)] combined with the Andreev approximation allows us to obtain remarkably simple analytical expressions for the bound-state energies as well as the majority and minority spin components of their wave functions. Besides the vacuum boundary, where a majority-spin Majorana excitation is encountered, we also consider the boundary between the topological and a nontopological superfluid phase that can appear in a coexistence scenario due to the first-order topological phase transition predicted for this system. At this superfluid-superfluid interface, we find a localized chiral Majorana mode hosted by the minority-spin sector. Our theory further predicts majority-spin subgap-energy bound states similar to those found at a Josephson junction between same-chirality p-wave superfluids. Their presence affects the Majorana mode due to a coupling of minority and majority spin sectors only in the small energy range where their spectra overlap. Our results may inform experimental efforts aimed at realizing and characterizing unconventional Majorana quasiparticles.