arXiv subjects

C. W. J. Beenakker

Publications and source records attributed to C. W. J. Beenakker.

At least 55 records · Page 3Linked to original sources

arXiv2021

Deconfinement of Majorana vortex modes produces a superconducting Landau level

A spatially oscillating pair potential $Δ(r)=Δ_0 e^{2i K\cdot r}$ with momentum $K>Δ_0/\hbar v$ drives a deconfinement transition of the Majorana bound states in the vortex cores of a Fu-Kane heterostructure (a 3D topological insulator with Fermi velocity $v$, on a superconducting substrate with gap $Δ_0$, in a perpendicular magnetic field). In the deconfined phase at zero chemical potential the Majorana fermions form a dispersionless Landau level, protected by chiral symmetry against broadening due to vortex scattering. The coherent superposition of electrons and holes in the Majorana Landau level is detectable as a local density of states oscillation with wave vector $\sqrt{K^2-(Δ_0/\hbar v)^2}$. The striped pattern also provides a means to measure the chirality of the Majorana fermions.

cond-mat.mes-hall↗

arXiv2021

Chirality inversion of Majorana edge modes in a Fu-Kane heterostructure

Fu and Kane have discovered that a topological insulator with induced s-wave superconductivity (gap $Δ_0$, Fermi velocity $v_{\rm F}$, Fermi energy $μ$) supports chiral Majorana modes propagating on the surface along the edge with a magnetic insulator. We show that the direction of motion of the Majorana fermions can be inverted by the counterflow of supercurrent, when the Cooper pair momentum along the boundary exceeds $Δ_0^2/μv_{\rm F}$. The chirality inversion is signaled by a doubling of the thermal conductance of a channel parallel to the supercurrent. Moreover, the inverted edge can transport a nonzero electrical current, carried by a Dirac mode that appears when the Majorana mode switches chirality. The chirality inversion is a unique signature of Majorana fermions in a spinful topological superconductor: it does not exist for spinless chiral p-wave pairing.

cond-mat.mes-hall↗

arXiv2021

Voltage staircase in a current-biased quantum-dot Josephson junction

We calculate the current-voltage (I-V) characteristic of a Josephson junction containing a resonant level in the weakly coupled regime (resonance width small compared to the superconducting gap). The phase $ϕ$ across the junction becomes time dependent in response to a DC current bias. Rabi oscillations in the Andreev levels produce a staircase I-V characteristic. The number of voltage steps counts the number of Rabi oscillations per $2π$ increment of $ϕ$, providing a way to probe the coherence of the qubit in the absence of any external AC driving. The phenomenology is the same as the "Majorana-induced DC Shapiro steps in topological Josephson junctions" of Phys. Rev. B 102, 140501(R) (2020) -- but now for a non-topological Andreev qubit.

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arXiv2020

Shot noise distinguishes Majorana fermions from vortices injected in the edge mode of a chiral p-wave superconductor

The chiral edge modes of a topological superconductor support two types of excitations: fermionic quasiparticles known as Majorana fermions and $π$-phase domain walls known as edge vortices. Edge vortices are injected pairwise into counter-propagating edge modes by a flux bias or voltage bias applied to a Josephson junction. An unpaired edge mode carries zero electrical current on average, but there are time-dependent current fluctuations. We calculate the shot noise power produced by a sequence of edge vortices and find that it increases logarithmically with their spacing - even if the spacing is much larger than the core size so the vortices do not overlap. This nonlocality produces an anomalous V log V increase of the shot noise in a voltage-biased geometry, which serves as a distinguishing feature in comparison with the linear-in-V Majorana fermion shot noise.

cond-mat.supr-con↗

arXiv2020

Magnetic breakdown spectrum of a Kramers-Weyl semimetal

We calculate the Landau levels of a Kramers-Weyl semimetal thin slab in a perpendicular magnetic field $B$. The coupling of Fermi arcs on opposite surfaces broadens the Landau levels with a band width that oscillates periodically in $1/B$. We interpret the spectrum in terms of a one-dimensional superlattice induced by magnetic breakdown at Weyl points. The band width oscillations may be observed as $1/B$-periodic magnetoconductance oscillations, at weaker fields and higher temperatures than the Shubnikov-de Haas oscillations due to Landau level quantization. No such spectrum appears in a generic Weyl semimetal, the Kramers degeneracy at time-reversally invariant momenta is essential.

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arXiv2020

Half-integer charge injection by a Josephson junction without excess noise

A Josephson junction in a topological superconductor can inject a charge e/2 into a normal-metal contact, carried by chiral Majorana edge modes. Here we address the question whether this half-integer charge is a sharp observable, without quantum fluctuations. Because the Majorana modes are gapless, they support charge fluctuations in equilibrium at zero temperature. But we find that the excess noise introduced out of equilibrium by the e/2 charge transfer vanishes. We discuss a strategy to reduce the equilibrium fluctuations, by means of a heavy-tailed time-dependent detection efficiency, to achieve a nearly noiseless half-integer charge transfer.

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arXiv2020

Search for non-Abelian Majorana braiding statistics in superconductors

This is a tutorial review of methods to braid the world lines of non-Abelian anyons (Majorana zero-modes) in topological superconductors. That "Holy Grail" of topological quantum information processing has not yet been reached in the laboratory, but there now exists a variety of platforms in which one can search for the Majorana braiding statistics. After an introduction to the basic concepts of braiding we discuss how one might be able to braid immobile Majorana zero-modes, bound to the end points of a nanowire, by performing the exchange in parameter space, rather than in real space. We explain how Coulomb interaction can be used to both control and read out the braiding operation, even though Majorana zero-modes are charge neutral. We ask whether the fusion rule might provide for an easier pathway towards the demonstration of non-Abelian statistics. In the final part we discuss an approach to braiding in real space, rather than parameter space, using vortices injected into a chiral Majorana edge mode as "flying qubits". Contents: I. Introduction II. Basic Concepts (The magic of braiding; Non-Abelian statistics; Fusion rules; Clifford gates; Topological protection) III. Braiding of Majorana zero-modes in nanowires (The three-point turn; Non-Abelian Berry phase; Coulomb-assisted braiding; Anyon teleportation) IV. Read-out of Majorana qubits (Majorana interferometry; Inductive coupling to a flux qubit; Microwave coupling to a transmon qubit; Capacitive coupling to a quantum dot; Random Access Majorana Memory) V. Fusion of Majorana zero-modes in nanowires (Linear junction or tri-junction; If we can fuse, do we need to braid?) VI. How to braid Majorana edge modes (Chiral edge modes in a superconductor; Edge vortex injection; Construction of the vortex operator; Edge vortex braiding)

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arXiv2020

Universal chiral magnetic effect in the vortex lattice of a Weyl superconductor

It was shown recently that Weyl fermions in a superconducting vortex lattice can condense into Landau levels. Here we study the chiral magnetic effect in the lowest Landau level: The appearance of an equilibrium current $I$ along the lines of magnetic flux $Φ$, due to an imbalance between Weyl fermions of opposite chirality. A universal contribution $dI/dΦ=(e/h)^2μ$ (at equilibrium chemical potential $μ$ relative to the Weyl point) appears when quasiparticles of one of the two chiralities are confined in vortex cores. The confined states are charge-neutral Majorana fermions.

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arXiv2020

Localization landscape for Dirac fermions

In the theory of Anderson localization, a landscape function predicts where wave functions localize in a disordered medium, without requiring the solution of an eigenvalue problem. It is known how to construct the localization landscape for the scalar wave equation in a random potential, or equivalently for the Schrödinger equation of spinless electrons. Here we generalize the concept to the Dirac equation, which includes the effects of spin-orbit coupling and allows to study quantum localization in graphene or in topological insulators and superconductors. The landscape function $u(r)$ is defined on a lattice as a solution of the differential equation $\overline{H}u(r)=1$, where $\overline{H}$ is the Ostrowsky comparison matrix of the Dirac Hamiltonian. Random Hamiltonians with the same (positive definite) comparison matrix have localized states at the same positions, defining an equivalence class for Anderson localization. This provides for a mapping between the Hermitian and non-Hermitian Anderson model.

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arXiv2019

Dynamical signatures of ground-state degeneracy to discriminate against Andreev levels in a Majorana fusion experiment

Detection of the fusion rule of Majorana zero-modes is a near-term milestone on the road to topological quantum computation. An obstacle is that the non-deterministic fusion outcome of topological zero-modes can be mimicked by the merging of non-topological Andreev levels. To distinguish these two scenarios, we search for dynamical signatures of the ground-state degeneracy that is the defining property of non-Abelian anyons. By adiabatically traversing parameter space along two different pathways one can identify ground-state degeneracies from the breakdown of adiabaticity. We show that the approach can discriminate against accidental degeneracies of Andreev levels.

cond-mat.mes-hall↗

arXiv2019

Time-resolved electrical detection of chiral edge vortex braiding

A $2π$ phase shift across a Josephson junction in a topological superconductor injects vortices into the chiral edge modes at opposite ends of the junction. When two vortices are fused they transfer charge into a metal contact. We calculate the time dependent current profile for the fusion process, which consists of $\pm e/2$ charge pulses that flip sign if the world lines of the vortices are braided prior to the fusion. This is an electrical signature of the non-Abelian exchange of Majorana zero-modes.

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arXiv2019

Effect of charge renormalization on electric and thermo-electric transport along the vortex lattice of a Weyl superconductor

Building on the discovery that a Weyl superconductor in a magnetic field supports chiral Landau level motion along the vortex lines, we investigate its transport properties out of equilibrium. We show that the vortex lattice carries an electric current $I=\tfrac{1}{2}(Q_{\rm eff}^2/h)(Φ/Φ_0) V$ between two normal metal contacts at voltage difference $V$, with $Φ$ the magnetic flux through the system, $Φ_0$ the superconducting flux quantum, and $Q_{\rm eff}<e$ the renormalized charge of the Weyl fermions in the superconducting Landau level. Because the charge renormalization is energy dependent, a nonzero thermo-electric coefficient appears even in the absence of energy-dependent scattering processes.

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arXiv2019

Pfaffian formula for fermion parity fluctuations in a superconductor and application to Majorana fusion detection

Kitaev's Pfaffian formula equates the ground-state fermion parity of a closed system to the sign of the Pfaffian of the Hamiltonian in the Majorana basis. Using Klich's theory of full counting statistics for paired fermions we generalize the Pfaffian formula to account for quantum fluctuations in the fermion parity of an open subsystem. A statistical description in the framework of random-matrix theory is used to answer the question when a vanishing fermion parity in a superconductor fusion experiment becomes a distinctive signature of an isolated Majorana zero-mode.

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arXiv2018

Electrical detection of the Majorana fusion rule for chiral edge vortices in a topological superconductor

Majorana zero-modes bound to vortices in a topological superconductor have a non-Abelian exchange statistics expressed by a non-deterministic fusion rule: When two vortices merge they may or they may not produce an unpaired fermion with equal probability. Building on a recent proposal to inject edge vortices in a chiral mode by means of a Josephson junction, we show how the fusion rule manifests itself in an electrical measurement. A $2π$ phase shift at a pair of Josephson junctions creates a topological qubit in a state of even-even fermion parity, which is transformed by the chiral motion of the edge vortices into an equal-weight superposition of even-even and odd-odd fermion parity. Fusion of the edge vortices at a second pair of Josephson junctions results in a correlated charge transfer of zero or one electron per cycle, such that the current at each junction exhibits shot noise, but the difference of the currents is nearly noiseless.

cond-mat.supr-con↗

arXiv2018

Deterministic creation and braiding of chiral edge vortices

Majorana zero-modes in a superconductor are midgap states localized in the core of a vortex or bound to the end of a nanowire. They are anyons with non-Abelian braiding statistics, but when they are immobile one cannot demonstrate this by exchanging them in real space and indirect methods are needed. As a real-space alternative, we propose to use the chiral motion along the boundary of the superconductor to braid a mobile vortex in the edge channel with an immobile vortex in the bulk. The measurement scheme is fully electrical and deterministic: edge vortices ($π$-phase domain walls) are created on demand by a voltage pulse at a Josephson junction and the braiding with a Majorana zero-mode in the bulk is detected by the charge produced upon their fusion at a second Josephson junction.

cond-mat.supr-con↗

arXiv2018

Neural network decoder for topological color codes with circuit level noise

A quantum computer needs the assistance of a classical algorithm to detect and identify errors that affect encoded quantum information. At this interface of classical and quantum computing the technique of machine learning has appeared as a way to tailor such an algorithm to the specific error processes of an experiment --- without the need for a priori knowledge of the error model. Here, we apply this technique to topological color codes. We demonstrate that a recurrent neural network with long short-term memory cells can be trained to reduce the error rate $ε_{\rm L}$ of the encoded logical qubit to values much below the error rate $ε_{\rm phys}$ of the physical qubits --- fitting the expected power law scaling $ε_{\rm L} \propto ε_{\rm phys}^{(d+1)/2}$, with $d$ the code distance. The neural network incorporates the information from "flag qubits" to avoid reduction in the effective code distance caused by the circuit. As a test, we apply the neural network decoder to a density-matrix based simulation of a superconducting quantum computer, demonstrating that the logical qubit has a longer life-time than the constituting physical qubits with near-term experimental parameters.

quant-ph↗

arXiv2018

Low-high voltage duality in tunneling spectroscopy of the Sachdev-Ye-Kitaev model

The Sachdev-Ye-Kitaev (SYK) model describes a strongly correlated metal with all-to-all random interactions (average strength $J$) between $N$ fermions (complex Dirac fermions or real Majorana fermions). In the large-$N$ limit a conformal symmetry emerges that renders the model exactly soluble. Here we study how the non-Fermi liquid behavior of the closed system in equilibrium manifests itself in an open system out of equilibrium. We calculate the current-voltage characteristic of a quantum dot, described by the complex-valued SYK model, coupled to a voltage source via a single-channel metallic lead (coupling strength $Γ$). A one-parameter scaling law appears in the large-$N$ conformal regime, where the differential conductance $G=dI/dV$ depends on the applied voltage only through the dimensionless combination $ξ=eVJ/Γ^2$. Low and high voltages are related by the duality $G(ξ)=G(π/ξ)$. This provides for an unambiguous signature of the conformal symmetry in tunneling spectroscopy.

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arXiv2018

Topologically protected Landau level in the vortex lattice of a Weyl superconductor

The question whether the mixed phase of a gapless superconductor can support a Landau level is a celebrated problem in the context of \textit{d}-wave superconductivity, with a negative answer: The scattering of the subgap excitations (massless Dirac fermions) by the vortex lattice obscures the Landau level quantization. Here we show that the same question has a positive answer for a Weyl superconductor: The chirality of the Weyl fermions protects the zeroth Landau level by means of a topological index theorem. As a result, the heat conductance parallel to the magnetic field has the universal value $G=\tfrac{1}{2}g_0 Φ/Φ_0$, with $Φ$ the magnetic flux through the system, $Φ_0$ the superconducting flux quantum, and $g_0$ the thermal conductance quantum.

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