## Publications by Karsten Flensberg

• 2021
• Readout of parafermionic states by transport measurements - Abstract
• Recent experiments have demonstrated the possibility of inducing superconducting pairing into counterpropagating fractional quantum Hall edge modes. This paves the way for the realization of localized parafermionic modes, nonabelian anyons which share fractional charges in a nonlocal way. We show that, for a pair of isolated parafermions, this nonlocal degree of freedom can be read by conductance measurements across standard metallic electrodes. We investigate first the transport through a grounded superconductor hosting two interacting parafermions. In the low-energy limit, the conductance reveals their shared fractional charge. We then examine the two-terminal electron conductance of a blockaded fractional topological superconductor. Depending on the induced pairing, parafermion overlap, and charging energy, such a device displays different patterns as a function of its induced charge. The zero-bias conductance allows us to distinguish a strongly-blockaded quantum dot regime, an intermediate pairing regime with a sixfold pattern of peaks repeating with periodicity $2e$, and a deep topological regime characterized by an $e/3$ periodicity.
• 2109.02300v1 [pdf] Ida E. Nielsen, Karsten Flensberg, Reinhold Egger, Michele Burrello [pdf]

• Demonstrating Majorana nonabelian exchange using fast adiabatic charge-transfer - Abstract
• Demonstration of Majorana nonabelian exchange properties is a major challenge in the field of topological superconductivity. In this work, we propose a minimal device and protocol for testing nonabelian exchange properties using charge-transfer operations between a quantum dot and two Majorana bound states combined with reading the parity state using a second dot. We use an adiabatic perturbation theory to find fast adiabatic paths to perform operations and to account for non-adiabatic errors. We find the ideal parameter sweep and a region in parameter space which reduces the charge-transfer operation time 1-2 orders of magnitude with respect to constant velocity driving. Using realistic parameters, we estimate that the lower bound for the time scale can be reduced to $\sim10$ ns. Deviations from the ideal parameters lead to the accumulation of an undesired dynamical phase, affecting the outcome of the proposed protocol. We furthermore suggest to reduce the influence from the dynamical phase using an angular echo.
• 2107.11833v2 [pdf] Svend Krøjer, Rubén Seoane Souto, Karsten Flensberg [pdf]

• Multilevel effects in quantum dot based parity-to-charge conversion of Majorana box qubits - Abstract
• Quantum-dot based parity-to-charge conversion is a promising method for reading out quantum information encoded nonlocally into pairs of Majorana zero modes. To obtain a sizable parity-to-charge visibility, it is crucial to tune the relative phase of the tunnel couplings between the dot and the Majorana modes appropriately. However, in the presence of multiple quasi-degenerate dot orbitals, it is in general not experimentally feasible to tune all couplings individually. This paper shows that such configurations could make it difficult to avoid a destructive multi-orbital interference effect that substantially reduces the read-out visibility. We analyze this effect using a Lindblad quantum master equation. This exposes how the experimentally relevant system parameters enhance or suppress the visibility when strong charging energy, measurement dissipation and, most importantly, multi-orbital interference is accounted for. In particular, we find that an intermediate-time readout could mitigate some of the interference-related visibility reductions affecting the stationary limit.
•  Jens Schulenborg, Michele Burrello, Martin Leijnse, Karsten Flensberg Journal reference: Phys. Rev. B 103, 245407 (2021) [pdf] DOI: 10.1103/PhysRevB.103.245407

• Spin-polarized bound states in semiconductor-superconductor-ferromagnetic insulator islands - Abstract
• We report Coulomb blockade transport studies of InAs nanowires grown with epitaxial superconducting Al and ferromagnetic insulator EuS on overlapping facets. By comparing experimental results to a theoretical model, we associate cotunneling features in even-odd bias spectra with spin-polarized Andreev levels, indicating that spin splitting exceeding the induced superconducting gap at zero applied magnetic field. Energies of the polarized subgap states can be tuned on either side of zero by electrostatic gates.
• 2104.01463v1 [pdf] S. Vaitiekėnas, R. Seoane Souto, Y. Liu, P. Krogstrup, K. Flensberg, M. Leijnse, C. M. Marcus [pdf]

• Three-phase Majorana zero modes at tiny magnetic fields - Abstract
• Proposals for realizing Majorana fermions in condensed matter systems typically rely on magnetic fields, which degrade the proximitizing superconductor and plague the Majoranas' detection. We propose an alternative scheme to realize Majoranas based only on phase-biased superconductors. The phases (at least three of them) can be biased by a tiny magnetic field threading macroscopic superconducting loops, focusing and enhancing the effect of the magnetic field onto the junction, or by supercurrents. We show how a combination of the superconducting phase winding and the spin-orbit phase induced in closed loops (Aharonov-Casher effect) facilitates a topological superconducting state with Majorana end states. We demontrate this scheme by an analytically tractable model as well as simulations of realistic setups comprising only conventional materials.
•  Omri Lesser, Karsten Flensberg, Felix von Oppen, Yuval Oreg Journal reference: Phys. Rev. B 103, 121116 (2021) [pdf] DOI: 10.1103/PhysRevB.103.L121116

• Topological superconductivity in semiconductor–superconductor–magnetic-insulator heterostructures - Abstract
• Hybrid superconductor-semiconductor heterostructures are promising platforms for realizing topological superconductors and exploring Majorana bound states physics. Motivated by recent experimental progress, we theoretically study how magnetic insulators offer an alternative to the use of external magnetic fields for reaching the topological regime. We consider different setups, where: (1) the magnetic insulator induces an exchange field in the superconductor, which leads to a splitting in the semiconductor by proximity effect, and (2) the magnetic insulator acts as a spin-filter tunnel barrier between the superconductor and the semiconductor. We show that the spin splitting in the superconductor alone cannot induce a topological transition in the semiconductor. To overcome this limitation, we propose to use a spin-filter barrier that enhances the magnetic exchange and provides a mechanism for a topological phase transition. Moreover, the spin-dependent tunneling introduces a strong dependence on the band alignment, which can be crucial in quantum-confined systems. This mechanism opens up a route towards networks of topological wires with fewer constraints on device geometry compared to previous devices that require external magnetic fields.
•  A. Maiani, R. Seoane Souto, M. Leijnse, K. Flensberg Journal reference: Phys. Rev. B 103, 104508 (2021) [pdf] DOI: 10.1103/PhysRevB.103.104508

• Engineered platforms for topological superconductivity and Majorana zero modes - Abstract
• Among the major approaches that are being pursued for realizing quantum bits, the Majorana-based platform has been the most recent to be launched. It attempts to realize qubits which store quantum information in a topologically-protected manner. The quantum information is protected by its nonlocal storage in localized and well-separated Majorana zero modes, and manipulated by exploiting their nonabelian quantum exchange properties. Realizing these topological qubits is experimentally challenging, requiring superconductivity, helical electrons (created by spin-orbit coupling) and breaking of time reversal symmetry to all cooperate in an uncomfortable alliance. Over the past decade, several candidate material systems for realizing Majorana-based topological qubits have been explored, and there is accumulating, though still debated, evidence that zero modes are indeed being realized. This paper reviews the basic physical principles on which these approaches are based, the material systems that are being developed, and the current state of the field. We highlight both the progress made and the challenges that still need to be overcome.
• 2103.05548v1 [pdf] Karsten Flensberg, Felix von Oppen, Ady Stern [pdf]

• 2020
• Long-distance coherence of Majorana wires - Abstract
• Theoretically, a pair of Majorana bound states in a topological superconductor forms a single fermionic level even at large separations, implying that the parity information is stored nonlocally. The nonlocality leads to a long-distance coherence for electrons tunneling through a Coulomb blockaded Majorana wire [Fu, Phys. Rev. Lett. 104, 056402 (2010)], an effect that can be observed, e.g., in an interferometer. Here, we examine theoretically the coherent electron transfer, taking into account that tunneling implies the long-distance transfer of charge, which is carried by one-dimensional plasmons. We show that the charge dynamics does not affect the coherence of the electron tunneling process in a topological superconductor consisting of a semiconductor wire proximitized by a single bulk superconductor. The coherence may be strongly suppressed, however, if the topological superconductivity derives from a semiconductor wire proximitized by a granular superconductor.
•  Zheng Shi, Piet W. Brouwer, Karsten Flensberg, Leonid I. Glazman, Felix von Oppen Journal reference: Phys. Rev. B 101, 241414 (2020) [pdf] DOI: 10.1103/PhysRevB.101.241414

• Parity-to-charge conversion in Majorana qubit readout - Abstract
• We study the time-dependent effect of Markovian readout processes on Majorana qubits whose parity degrees of freedom are converted into the charge of a tunnel-coupled quantum dot. By applying a recently established effective Lindbladian approximation [1-3], we obtain a completely positive and trace preserving Lindblad master equation for the combined dot-qubit dynamics, describing relaxation and decoherence processes beyond the rotating-wave approximation. This approach is applicable to a wide range of weakly coupled environments representing experimentally relevant readout devices. We study in detail the case of thermal decay in the presence of a generic Ohmic bosonic bath, in particular for potential fluctuations in an electromagnetic circuit. In addition, we consider the nonequilibrium measurement environment for a parity readout using a quantum point contact capacitively coupled to the dot charge.
•  Morten I. K. Munk, Jens Schulenborg, Reinhold Egger, Karsten Flensberg Journal reference: Phys. Rev. Research 2, 033254 (2020) [pdf] DOI: 10.1103/PhysRevResearch.2.033254

• Absence of supercurrent sign reversal in a topological junction with a quantum dot - Abstract
• Experimental techniques to verify Majoranas are of current interest. A prominent test is the effect of Majoranas on the Josephson current between two wires linked via a normal junction. Here, we study the case of a quantum dot connecting the two superconductors and the sign of the supercurrent in the trivial and topological regimes under grand-canonical equilibrium conditions, explicitly allowing for parity changes due to, e.g., quasi-particle poisoning. We find that the well-known supercurrent reversal for odd occupancy of the quantum dot (pi-junction) in the trivial case does not occur in the presence of Majoranas in the wires. However, we also find this to be a mere consequence of Majoranas being zero energy states. Therefore, the lack of supercurrent sign reversal can also be caused by trivial bound states, and is thus not a discriminating signature of Majoranas.
•  J. Schulenborg, K. Flensberg Journal reference: Phys. Rev. B 101, 014512 (2020) [pdf] DOI: 10.1103/PhysRevB.101.014512

• Weak Measurement Protocols for Majorana Bound State Identification - Abstract
• We propose a continuous weak measurement protocol testing the nonlocality of Majorana bound states through current shot noise correlations. The experimental setup contains a topological superconductor island with three normal-conducting leads weakly coupled to different Majorana states. Putting one lead at finite voltage and measuring the shot noise correlations between the other two (grounded) leads, devices with true Majorana states are distinguished from those without by strong current correlations. The presence of true Majoranas manifests itself in unusually high noise levels or the near absence of noise, depending on the chosen device configuration. Monitoring the noise statistics amounts to a weak continuous measurement of the Majorana qubit and yields information similar to that of a full braiding protocol, but at much lower experimental effort. Our theory can be adapted to different platforms and should allow for clear identification of Majorana states.
•  Jan Manousakis, Carolin Wille, Alexander Altland, Reinhold Egger, Karsten Flensberg, Fabian Hassler Journal reference: Phys. Rev. Lett. 124, 096801 (2020) [pdf] DOI: 10.1103/PhysRevLett.124.096801

• Timescales for charge transfer based operations on Majorana systems - Abstract
• In this article we analyze the efficiency of operations based on transferring charge from a quantum dot (QD) to two coupled topological superconductors, which can be used for performing nonabelian operations on Majorana bound states (MBSs). We develop a method which allows us to describe the full time-evolution of the system as the QD energy is manipulated. Using a full counting statistics analysis, we set bounds to the operation time scales. The lower bound depends on the superconducting phase difference due to a partial decoupling of the different MBSs parity sectors, while the upper bound is set by the tunneling of quasiparticles to the MBSs. Using realistic parameters, we find the existence of a regime where the operation can be carried out with a fidelity close to unity. Finally, we propose the use of a two operations protocol to quantify the effect of the dephasing and accumulated dynamical phases, demonstrating their absence for certain superconducting phase differences.
•  R. Seoane Souto, K. Flensberg, M. Leijnse Journal reference: Phys. Rev. B 101, 081407 (2020) [pdf] DOI: 10.1103/PhysRevB.101.081407

• Conductance-Matrix Symmetries of a Three-Terminal Hybrid Device - Abstract
• We present conductance-matrix measurements of a three-terminal superconductor-semiconductor hybrid device consisting of two normal leads and one superconducting lead. Using a symmetry decomposition of the conductance, we find that the antisymmetric components of pairs of local and nonlocal conductances match at energies below the superconducting gap, consistent with expectations based on a non-interacting scattering matrix approach. Further, the local charge character of Andreev bound states is extracted from the symmetry-decomposed conductance data and is found to be similar at both ends of the device and tunable with gate voltage. Finally, we measure the conductance matrix as a function of magnetic field and identify correlated splittings in low-energy features, demonstrating how conductance-matrix measurements can complement traditional tunneling-probe measurements in the search for Majorana zero modes.
•  G. C. Ménard, G. L. R. Anselmetti, E. A. Martinez, D. Puglia, F. K. Malinowski, J. S. Lee, S. Choi, M. Pendharkar, C. J. Palmstrøm, K. Flensberg, C. M. Marcus, L. Casparis, A. P. Higginbotham Journal reference: Phys. Rev. Lett. 124, 036802 (2020) [pdf] DOI: 10.1103/PhysRevLett.124.036802

• Nonlocal Conductance Spectroscopy of Andreev Bound States: Symmetry Relations and BCS Charges - Abstract
• Two-terminal conductance spectroscopy of superconducting devices is a common tool for probing Andreev and Majorana bound states. Here, we study theoretically a three-terminal setup, with two normal leads coupled to a grounded superconducting terminal. Using a single-electron scattering matrix, we derive the subgap conductance matrix for the normal leads and discuss its symmetries. In particular, we show that the local and the nonlocal elements of the conductance matrix have pairwise identical antisymmetric components. Moreover, we find that the nonlocal elements are directly related to the local BCS charges of the bound states close to the normal probes and we show how the BCS charge of overlapping Majorana bound states can be extracted from experiments.
•  Jeroen Danon, Anna Birk Hellenes, Esben Bork Hansen, Lucas Casparis, Andrew P. Higginbotham, Karsten Flensberg Journal reference: Phys. Rev. Lett. 124, 036801 (2020) [pdf] DOI: 10.1103/PhysRevLett.124.036801

• Flux-induced topological superconductivity in full-shell nanowires - Abstract
• We consider a new model system supporting Majorana zero modes based on semiconductor nanowires with a full superconducting shell. We demonstrate that, in the presence of spin-orbit coupling in the semiconductor induced by a radial electric field, the winding of the superconducting order parameter leads to a topological phase supporting Majorana zero modes. The topological phase persists over a large range of chemical potentials and can be induced by a predictable and weak magnetic field piercing the cylinder. The system can be readily realized in semiconductor nanowires covered by a full superconducting shell, opening a pathway for realizing topological quantum computing proposals.
•  Roman M. Lutchyn, Georg W. Winkler, Bernard van Heck, Torsten Karzig, Karsten Flensberg, Leonid I. Glazman, Chetan Nayak Journal reference: Science 367, eaav3392 (2020) [pdf] DOI: 10.1126/science.aav3392

• 2019
• Current-Induced Gap Opening in Interacting Topological Insulator Surfaces - Abstract
• Two-dimensional topological insulators (TIs) host gapless helical edge states that are predicted to support a quantized two-terminal conductance. Quantization is protected by time-reversal symmetry, which forbids elastic backscattering. Paradoxically, the current-carrying state itself breaks the time-reversal symmetry that protects it. Here we show that the combination of electron-electron interactions and momentum-dependent spin polarization in helical edge states gives rise to feedback through which an applied current opens a gap in the edge state dispersion, thereby breaking the protection against elastic backscattering. Current-induced gap opening is manifested via a nonlinear contribution to the system's $I-V$ characteristic, which persists down to zero temperature. We discuss prospects for realizations in recently discovered large bulk band gap TIs, and an analogous current-induced gap opening mechanism for the surface states of three-dimensional TIs.
•  Ajit C. Balram, Karsten Flensberg, Jens Paaske, Mark S. Rudner Journal reference: Phys. Rev. Lett. 123, 246803 (2019) [pdf] DOI: 10.1103/PhysRevLett.123.246803

• Fidelity and visibility loss in Majorana qubits by entanglement with environmental modes - Abstract
• We study the dynamics and readout of topological qubits encoded by zero-energy Majorana bound states in a topological superconductor. We take into account bosonic modes due to the electromagnetic environment which couple the Majorana manifold to above-gap continuum quasi-particles. This coupling causes the degenerate ground state of the topological superconductor to be dressed in a polaron-like manner by quasi-particle states and bosons, and the system to become gapless. Topological protection and hence full coherence is only maintained if the qubit is operated and read out within the low-energy spectrum of the dressed states. We discuss reduction of fidelity and/or visibility if this condition is violated by a quantum-dot readout that couples to the bare (undressed) Majorana modes. For a projective measurement of the bare Majorana basis, we formulate a Bloch-Redfield approach that is valid for weak Majorana-environment coupling and takes into account constraints imposed by fermion-number-parity conservation. Within the Markovian approximation, our results essentially confirm earlier theories of finite-temperature decoherence based on Fermi's golden rule. However, the full non-Markovian dynamics reveals, in addition, the fidelity reduction by a projective measurement. Using a spinless nanowire model with $p$-wave pairing, we provide quantitative results characterizing these effects.
•  Morten I. K. Munk, Reinhold Egger, Karsten Flensberg Journal reference: Phys. Rev. B 99, 155419 (2019) [pdf] DOI: 10.1103/PhysRevB.99.155419

• Coulomb-interaction-induced Majorana edge modes in nanowires - Abstract
• We show that Majorana edge modes appear in a strongly correlated phase of semiconducting nanowires with discrete rotational symmetry in the cross section. These modes exist in the absence of spin-orbit coupling, magnetic fields and superconductivity. They appear purely due to the combination of the three-dimensional Coulomb interaction and orbital physics, which generates a fermionic condensate exhibiting a topological ground state degeneracy in a sector of the spectrum which is gapped to continuum modes. The gap can be comparable in magnitude to the topological superconducting gap in other solid-state candidate systems for Majorana edge modes, and may similarly be probed via tunnel spectroscopy.
•  Tommy Li, Michele Burrello, Karsten Flensberg Journal reference: Phys. Rev. B 100, 045305 (2019) [pdf] DOI: 10.1103/PhysRevB.100.045305

• Coulomb Blockade of a Nearly Open Majorana Island - Abstract
• We consider the ground-state energy and the spectrum of the low-energy excitations of a Majorana island formed of topological superconductors connected by a single-mode junction of arbitrary transmission. Coulomb blockade results in $e$-periodic modulation of the energies with the gate-induced charge. We find the amplitude of modulation as a function of reflection coefficient ${\cal R}$. The amplitude scales as $\sqrt{\cal R}$ in the limit ${\cal R}\to 0$. At larger ${\cal R}$, the dependence of the amplitude on the Josephson and charging energies is similar to that of a conventional-superconductor Cooper-pair box. The crossover value of ${\cal R}$ is small and depends on the ratio of the charging energy to superconducting gap.
•  Dmitry I. Pikulin, Karsten Flensberg, Leonid I. Glazman, Manuel Houzet, Roman M. Lutchyn Journal reference: Phys. Rev. Lett. 122, 016801 (2019) [pdf] DOI: 10.1103/PhysRevLett.122.016801

• 2018
• Hybridization at Superconductor-Semiconductor Interfaces - Abstract
• Hybrid superconductor-semiconductor devices are currently one of the most promising platforms for realizing Majorana zero modes. Their topological properties are controlled by the band alignment of the two materials, as well as the electrostatic environment, which are currently not well understood. Here, we pursue to fill in this gap and address the role of band bending and superconductor-semiconductor hybridization in such devices by analyzing a gated single Al-InAs interface using a self-consistent Schrodinger-Poisson approach. Our numerical analysis shows that the band bending leads to an interface quantum well, which localizes the charge in the system near the superconductor-semiconductor interface. We investigate the hybrid band structure and analyze its response to varying the gate voltage and thickness of the Al layer. This is done by studying the hybridization degrees of the individual subbands, which determine the induced pairing and effective $g$-factors. The numerical results are backed by approximate analytical expressions which further clarify key aspects of the band structure. We find that one can obtain states with strong superconductor-semiconductor hybridization at the Fermi energy, but this requires a fine balance of parameters, with the most important constraint being on the width of the Al layer. In fact, in the regime of interest, we find an almost periodic dependence of the hybridization degree on the Al width, with a period roughly equal to the thickness of an Al monolayer. This implies that disorder and shape irregularities, present in realistic devices, may play an important role for averaging out this sensitivity and, thus, may be necessary for stabilizing the topological phase.
•  August E. G. Mikkelsen, Panagiotis Kotetes, Peter Krogstrup, Karsten Flensberg Journal reference: Phys. Rev. X 8, 031040 (2018) [pdf] DOI: 10.1103/PhysRevX.8.031040

• Four-Majorana qubit with charge readout: Dynamics and decoherence - Abstract
• We present a theoretical analysis of a Majorana-based qubit consisting of two topological superconducting islands connected via a Josephson junction. The qubit is operated by electrostatic gates which control the coupling of two of the four Majorana zero modes. At the end of the operation, readout is performed in the charge basis. Even though the operations are not topologically protected, the proposed experiment can potentially shed light on the coherence of the parity degree of freedom in Majorana devices and serve as a first step towards topological Majorana qubits. We discuss in detail the charge-stability diagram and its use for characterizing the parameters of the devices, including the overlap of the Majorana edge states. We describe the multi-level spectral properties of the system and present a detailed study of its controlled coherent oscillations, as well as decoherence resulting from coupling to a non-Markovian environment. In particular, we study a gate-controlled protocol where conversion between Coulomb-blockade and transmon regimes generates coherent oscillations of the qubit state due to the overlap of Majorana modes. We show that, in addition to fluctuations of the Majorana coupling, considerable measurement errors may be accumulated during the conversion intervals when electrostatic fluctuations in the superconducting islands are present. These results are also relevant for several proposed implementations of topological qubits which rely on readout based on charge detection.
•  Tommy Li, William A. Coish, Michael Hell, Karsten Flensberg, Martin Leijnse Journal reference: Phys. Rev. B 98, 205403 (2018) [pdf] DOI: 10.1103/PhysRevB.98.205403

• Hybridization of Subgap States in One-Dimensional Superconductor-Semiconductor Coulomb Islands - Abstract
• We present measurements of one-dimensional superconductor-semiconductor Coulomb islands, fabricated by gate confinement of a two-dimensional InAs heterostructure with an epitaxial Al layer. When tuned via electrostatic side gates to regimes without sub-gap states, Coulomb blockade reveals Cooper-pair mediated transport. When sub-gap states are present, Coulomb peak positions and heights oscillate in a correlated way with magnetic field and gate voltage, as predicted theoretically, with (anti) crossings in (parallel) transverse magnetic field indicating Rashba-type spin-orbit coupling. Overall results are consistent with a picture of overlapping Majorana zero modes in finite wires.
•  E. C. T. O'Farrell, A. C. C. Drachmann, M. Hell, A. Fornieri, A. M. Whiticar, E. B. Hansen, S. Gronin, G. C. Gardener, C. Thomas, M. J. Manfra, K. Flensberg, C. M. Marcus, F. Nichele Journal reference: Phys. Rev. Lett. 121, 256803 (2018) [pdf] DOI: 10.1103/PhysRevLett.121.256803

• Distinguishing Majorana bound states from localized Andreev bound states by interferometry - Abstract
• Experimental evidence for Majorana bound states (MBSs) is so far mainly based on the robustness of a zero-bias conductance peak. However, similar features can also arise due to Andreev bound states (ABSs) localized at the end of an island. We show that these two scenarios can be distinguished by an interferometry experiment based on embedding a Coulomb-blockaded island into an Aharonov-Bohm ring. For two ABSs, when the ground state is nearly degenerate, cotunneling can change the state of the island and interference is suppressed. By contrast, for two MBSs the ground state is nondegenerate and cotunneling has to preserve the island state, which leads to $h / e$-periodic conductance oscillations with magnetic flux. Such interference setups can be realized with semiconducting nanowires or two-dimensional electron gases with proximity-induced superconductivity and may also be a useful spectroscopic tool for parity-flip mechanisms.
•  Michael Hell, Karsten Flensberg, Martin Leijnse Journal reference: Phys. Rev. B 97, 161401 (2018) [pdf] DOI: 10.1103/PhysRevB.97.161401

• Probing electron-hole components of subgap states in Coulomb blockaded Majorana islands - Abstract
• Recent tunneling spectroscopy experiments in semiconducting nanowires with proximity-induced superconductivity have reported robust zero-bias conductance peaks. Such a feature can be compatible with the existence of topological Majorana bound states (MBSs) and with a trivial Andreev bound state (ABS) near zero energy. Here, we argue that additional information, that can distinguish between the two cases, can be extracted from Coulomb-blockade experiments of Majorana islands. The key is the ratio of peak heights of consecutive conductance peaks give information about the electron and hole components of the lowest-energy subgap state. In the MBS case, this ratio goes to one half for long wires, while for short wires with finite MBS overlap it oscillates a function of Zeeman energy with the same period as the MBS energy splitting. We explain how the additional information might help to distinguish a trivial ABS at zero energy from a true MBS and show case examples.
•  Esben Bork Hansen, Jeroen Danon, Karsten Flensberg Journal reference: Phys. Rev. B 97, 041411 (2018) [pdf] DOI: 10.1103/PhysRevB.97.041411

• Anharmonicity of a superconducting qubit with a few-mode Josephson junction - Abstract
• Coherent operation of gate-voltage-controlled hybrid transmon qubits (gatemons) based on semiconductor nanowires was recently demonstrated. Here we experimentally investigate the anharmonicity in epitaxial InAs-Al Josephson junctions, a key parameter for their use as a qubit. Anharmonicity is found to be reduced by roughly a factor of two compared to conventional metallic junctions, and dependent on gate voltage. Experimental results are consistent with a theoretical model, indicating that Josephson coupling is mediated by a small number of highly transmitting modes in the semiconductor junction.
•  A. Kringhøj, L. Casparis, M. Hell, T. W. Larsen, F. Kuemmeth, M. Leijnse, K. Flensberg, P. Krogstrup, J. Nygård, K. D. Petersson, C. M. Marcus Journal reference: Phys. Rev. B 97, 060508 (2018) [pdf] DOI: 10.1103/PhysRevB.97.060508

• 2017
• Scaling of Majorana Zero-Bias Conductance Peaks - Abstract
• We report an experimental study of the scaling of zero-bias conductance peaks compatible with Majorana zero modes as a function of magnetic field, tunnel coupling, and temperature in one-dimensional structures fabricated from an epitaxial semiconductor-superconductor heterostructure. Results are consistent with theory, including a peak conductance that is proportional to tunnel coupling, saturates at $2e^2/h$, decreases as expected with field-dependent gap, and collapses onto a simple scaling function in the dimensionless ratio of temperature and tunnel coupling.
•  Fabrizio Nichele, Asbjorn C. C. Drachmann, Alexander M. Whiticar, Eoin C. T. O'Farrell, Henri J. Suominen, Antonio Fornieri, Tian Wang, Geoffrey C. Gardner, Candice Thomas, Anthony T. Hatke, Peter Krogstrup, Michael J. Manfra, Karsten Flensberg, Charles M. Marcus Journal reference: Phys. Rev. Lett. 119, 136803 (2017) [pdf] DOI: 10.1103/PhysRevLett.119.136803

• Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes - Abstract
• We present designs for scalable quantum computers composed of qubits encoded in aggregates of four or more Majorana zero modes, realized at the ends of topological superconducting wire segments that are assembled into superconducting islands with significant charging energy. Quantum information can be manipulated according to a measurement-only protocol, which is facilitated by tunable couplings between Majorana zero modes and nearby semiconductor quantum dots. Our proposed architecture designs have the following principal virtues: (1) the magnetic field can be aligned in the direction of all of the topological superconducting wires since they are all parallel; (2) topological $T$-junctions are not used, obviating possible difficulties in their fabrication and utilization; (3) quasiparticle poisoning is abated by the charging energy; (4) Clifford operations are executed by a relatively standard measurement: detection of corrections to quantum dot energy, charge, or differential capacitance induced by quantum fluctuations; (5) it is compatible with strategies for producing good approximate magic states.
•  Torsten Karzig, Christina Knapp, Roman M. Lutchyn, Parsa Bonderson, Matthew B. Hastings, Chetan Nayak, Jason Alicea, Karsten Flensberg, Stephan Plugge, Yuval Oreg, Charles M. Marcus, Michael H. Freedman Journal reference: Phys. Rev. B 95, 235305 (2017) [pdf] DOI: 10.1103/PhysRevB.95.235305

• Conductance spectroscopy on Majorana wires and the inverse proximity effect - Abstract
• Recent experimental searches for signatures of Majorana-like excitations in proximitized semiconducting nanowires involve conductance spectroscopy, where the evidence sought after is a robust zero-bias peak (in longer wires) and its characteristic field-dependent splitting (in shorter wires). Although experimental results partially confirm the theoretical predictions, commonly observed discrepancies still include (i) a zero-bias peak that is significantly lower than the predicted value of $2e^2/h$ and (ii) the absence of the expected "Majorana oscillations" of the lowest-energy modes at higher magnetic fields. Here, we investigate how the inevitable presence of a normal drain lead connected to the hybrid wire can affect the conductance spectrum of the hybrid wire. We present numerical results using a one-band model for the proximitized nanowire, where the superconductor is considered to be in the diffusive regime, described by semi-classical Green functions. We show how the presence of the normal drain could (at least partially) account for the observed discrepancies, and we complement this with analytic results providing more insights in the underlying physics.
•  Jeroen Danon, Esben Bork Hansen, Karsten Flensberg Journal reference: Phys. Rev. B 96, 125420 (2017) [pdf] DOI: 10.1103/PhysRevB.96.125420

• Coupling and braiding Majorana bound states in networks defined in two-dimensional electron gases with proximity-induced superconductivity - Abstract
• Two-dimensional electron gases with strong spin-orbit coupling covered by a superconducting layer offer a flexible and potentially scalable platform for Majorana networks. We predict Majorana bound states (MBSs) to appear for experimentally achievable parameters and realistic gate potentials in two designs: either underneath a narrow stripe of a superconducting layer (S-stripes) or where a narrow stripe has been removed from a uniform layer (N-stripes). The coupling of the MBSs can be tuned for both types in a wide range (<1 nev to>10 $\mu$eV) using gates placed adjacent to the stripes. For both types, we numerically compute the local density of states for two parallel Majorana-stripe ends as well as Majorana trijunctions formed in a tuning-fork geometry. The MBS coupling between parallel Majorana stripes can be suppressed below 1 neV for potential barriers in the meV range for separations of about 200 nm. We further show that the MBS couplings in a trijunction can be gate-controlled in a range similar to the intra-stripe coupling while maintaining a sizable gap to the excited states (tens of $\mu$eV). Altogether, this suggests that braiding can carried out on a time scale of 10-100 ns.
•  Michael Hell, Karsten Flensberg, Martin Leijnse Journal reference: Phys. Rev. B 96, 035444 (2017) [pdf] DOI: 10.1103/PhysRevB.96.035444

• Majorana box qubits - Abstract
• Quantum information protected by the topology of the storage medium is expected to exhibit long coherence times. Another feature are topologically protected gates generated through braiding of Majorana bound states. However, braiding requires structures with branched topological segments which have inherent difficulties in the semiconductor-superconductor heterostructures now believed to host Majorana bound states. In this paper, we construct quantum bits taking advantage of the topological protection and non-local properties of Majorana bound states in a network of parallel wires, but without relying on braiding for quantum gates. The elementary unit is made from three topological wires, two wires coupled by a trivial superconductor and the third acting as an interference arm. Coulomb blockade of the combined wires spawns a fractionalized spin, non-locally addressable by quantum dots used for single-qubit readout, initialization, and manipulation. We describe how the same tools allow for measurement-based implementation of the Clifford gates, in total making the architecture universal. Proof-of-principle demonstration of topologically protected qubits using existing techniques is therefore within reach.
•  Stephan Plugge, Asbjørn Rasmussen, Reinhold Egger, Karsten Flensberg Journal reference: New J. Phys 19, 012001 (2017) [pdf] DOI: 10.1088/1367-2630/aa54e1

• Transport Signatures of Quasiparticle Poisoning in a Majorana Island - Abstract
• We investigate effects of quasiparticle poisoning in a Majorana island with strong tunnel coupling to normal-metal leads. In addition to the main Coulomb blockade diamonds, "shadow" diamonds appear, shifted by 1e in gate voltage, consistent with transport through an excited (poisoned) state of the island. Comparison to a simple model yields an estimate of parity lifetime for the strongly coupled island (~ 1 {\mu}s) and sets a bound for a weakly coupled island (> 10 {\mu}s). Fluctuations in the gate-voltage spacing of Coulomb peaks at high field, reflecting Majorana hybridization, are enhanced by the reduced lever arm at strong coupling. In energy units, fluctuations are consistent with previous measurements.
•  S. M. Albrecht, E. B. Hansen, A. P. Higginbotham, F. Kuemmeth, T. S. Jespersen, J. Nygård, P. Krogstrup, J. Danon, K. Flensberg, C. M. Marcus Journal reference: Phys. Rev. Lett. 118, 137701 (2017) [pdf] DOI: 10.1103/PhysRevLett.118.137701

• Two-Dimensional Platform for Networks of Majorana Bound States - Abstract
• We model theoretically a two-dimensional electron gas (2DEG) covered by a superconductor and demonstrate that topological superconducting channels are formed when stripes of the superconducting layer are removed. As a consequence, Majorana bound states (MBS) are created at the ends of the stripes. We calculate the topological invariant and energy gap of a single stripe, using realistic values for an InAs 2DEG proximitized by an epitaxial Al layer. We show that the topological gap is enhanced when the structure is made asymmetric. This can be achieved by either imposing a phase difference (by driving a supercurrent or using a magnetic-flux loop) over the strip or by replacing one superconductor by a metallic gate. Both strategies also enable control over the MBS splitting, thereby facilitating braiding and readout schemes based on controlled fusion of MBS. Finally, we outline how a network of Majorana stripes can be designed.
•  Michael Hell, Martin Leijnse, Karsten Flensberg Journal reference: Phys. Rev. Lett. 118, 107701 (2017) [pdf] DOI: 10.1103/PhysRevLett.118.107701

• Anomalous Fraunhofer interference in epitaxial superconductor-semiconductor Josephson junctions - Abstract
• We investigate patterns of critical current as a function of perpendicular and in-plane magnetic fields in superconductor-semiconductor-superconductor (SNS) junctions based on InAs/InGaAs heterostructures with an epitaxial Al layer. This material system is of interest due to its exceptionally good superconductor-semiconductor coupling, as well as large spin-orbit interaction and g-factor in the semiconductor. Thin epitaxial Al allows the application of large in-plane field without destroying superconductivity. For fields perpendicular to the junction, flux focusing results in aperiodic node spacings in the pattern of critical currents known as Fraunhofer patterns by analogy to the related interference effect in optics. Adding an in-plane field yields two further anomalies in the pattern. First, higher order nodes are systematically strengthened, indicating current flow along the edges of the device, as a result of confinement of Andreev states driven by an induced flux dipole; second, asymmetries in the interference appear that depend on the field direction and magnitude. A model is presented, showing good agreement with experiment, elucidating the roles of flux focusing, Zeeman and spin-orbit coupling, and disorder in producing these effects.
•  H. J. Suominen, J. Danon, M. Kjaergaard, K. Flensberg, J. Shabani, C. J. Palmstrøm, F. Nichele, C. M. Marcus Journal reference: Phys. Rev. B 95, 035307 (2017) [pdf] DOI: 10.1103/PhysRevB.95.035307

• 2016
• Majorana bound state in a coupled quantum-dot hybrid-nanowire system - Abstract
• Hybrid nanowires combining semiconductor and superconductor materials appear well suited for the creation, detection, and control of Majorana bound states (MBSs). We demonstrate the emergence of MBSs from coalescing Andreev bound states (ABSs) in a hybrid InAs nanowire with epitaxial Al, using a quantum dot at the end of the nanowire as a spectrometer. Electrostatic gating tuned the nanowire density to a regime of one or a few ABSs. In an applied axial magnetic field, a topological phase emerges in which ABSs move to zero energy and remain there, forming MBSs. We observed hybridization of the MBS with the end-dot bound state, which is in agreement with a numerical model. The ABS/MBS spectra provide parameters that are useful for understanding topological superconductivity in this system.
•  M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, C. M. Marcus Journal reference: Science 354, 1557-1562 (2016) [pdf] DOI: 10.1126/science.aaf3961

• Quantized conductance doubling and hard gap in a two-dimensional semiconductor–superconductor heterostructure - Abstract
• The prospect of coupling a two-dimensional (2D) semiconductor heterostructure to a superconductor opens new research and technology opportunities, including fundamental problems in mesoscopic superconductivity, scalable superconducting electronics, and new topological states of matter. For instance, one route toward realizing topological matter is by coupling a 2D electron gas (2DEG) with strong spin-orbit interaction to an s-wave superconductor. Previous efforts along these lines have been hindered by interface disorder and unstable gating. Here, we report measurements on a gateable InGaAs/InAs 2DEG with patterned epitaxial Al, yielding multilayer devices with atomically pristine interfaces between semiconductor and superconductor. Using surface gates to form a quantum point contact (QPC), we find a hard superconducting gap in the tunneling regime, overcoming the soft-gap problem in 2D superconductor-semiconductor hybrid systems. With the QPC in the open regime, we observe a first conductance plateau at 4e^2/h, as expected theoretically for a normal-QPC-superconductor structure. The realization of a hard-gap semiconductor-superconductor system that is amenable to top-down processing provides a means of fabricating scalable multicomponent hybrid systems for applications in low-dissipation electronics and topological quantum information.
•  M. Kjaergaard, F. Nichele, H. J. Suominen, M. P. Nowak, M. Wimmer, A. R. Akhmerov, J. A. Folk, K. Flensberg, J. Shabani, C. J. Palmstrom, C. M. Marcus Journal reference: Nat. Commun. 7, 12841 (2016) [pdf] DOI: 10.1038/ncomms12841

• Spiral magnetic order and topological superconductivity in a chain of magnetic adatoms on a two-dimensional superconductor - Abstract
• We study the magnetic and electronic phases of a 1D magnetic adatom chain on a 2D superconductor. In particular, we confirm the existence of a `self-organized' 1D topologically non-trivial superconducting phase within the set of subgap Yu-Shiba-Rusinov (YSR) states formed along the magnetic chain. This phase is stabilized by incommensurate spiral correlations within the magnetic chain that arise from the competition between short-range ferromagnetic and long-range antiferromagnetic electron-induced exchange interactions, similar to a recent study for a 3D superconductor [M. Schecter et al. Phys. Rev. B 93, 140503(R) 2016]. The exchange interaction along diagonal directions are also considered and found to display behavior similar to a 1D substrate when close to half filling. We show that the topological phase diagram is robust against local superconducting order parameter suppression and weak substrate spin-orbit coupling. Lastly, we study the effect of a direct ferromagnetic exchange coupling between the adatoms, and find the region of spiral order in the phase diagram to be significantly enlarged in a wide range of the direct exchange coupling.
•  M. H. Christensen, M. Schecter, K. Flensberg, B. M. Andersen, J. Paaske Journal reference: Phys. Rev. B 94, 144509 (2016) [pdf] DOI: 10.1103/PhysRevB.94.144509

• No-go theorem for a time-reversal invariant topological phase in noninteracting systems coupled to conventional superconductors - Abstract
• We prove that a system of non-interacting electrons proximity coupled to a conventional s-wave superconductor cannot realize a time reversal invariant topological phase. This is done by showing that for such a system, in either one or two dimensions, the topological invariant of the corresponding symmetry class (DIII) is always trivial. Our results suggest that the pursuit of Majorana bound states in time-reversal invariant systems should be aimed at interacting systems or at proximity to unconventional superconductors.
•  Arbel Haim, Erez Berg, Karsten Flensberg, Yuval Oreg Journal reference: Phys. Rev. B 94, 161110 (2016) [pdf] DOI: 10.1103/PhysRevB.94.161110

• Interaction-driven topological superconductivity in one dimension - Abstract
• We study one-dimensional topological superconductivity in the presence of time-reversal symmetry. This phase is characterized by having a bulk gap, while supporting a Kramers' pair of zero-energy Majorana bound states at each of its ends. We present a general simple model which is driven into this topological phase in the presence of repulsive electron-electron interactions. We further propose two experimental setups and show that they realize this model at low energies. The first setup is a narrow two-dimensional topological insulator partially covered by a conventional s-wave superconductor, and the second is a semiconductor wire in proximity to an s-wave superconductor. These systems can therefore be used to realize and probe the time-reversal invariant topological superconducting phase. The effect of interactions is studied using both a mean-field approach and a renormalization group analysis.
•  Arbel Haim, Konrad Wölms, Erez Berg, Yuval Oreg, Karsten Flensberg Journal reference: Phys. Rev. B 94, 115124 (2016) [pdf] DOI: 10.1103/PhysRevB.94.115124

• Signatures of Majorana Kramers pairs in superconductor-Luttinger liquid and superconductor-quantum dot-normal lead junctions - Abstract
• Time-reversal invariant topological superconductors are characterized by the presence of Majorana Kramers pairs localized at defects. One of the transport signatures of Majorana Kramers pairs is the quantized differential conductance of $4e^2/h$ when such a one-dimensional superconductor is coupled to a normal-metal lead. The resonant Andreev reflection, responsible for this phenomenon, can be understood as the boundary condition change for lead electrons at low energies. In this paper, we study the stability of the Andreev reflection fixed point with respect to electron-electron interactions in the Luttinger liquid. We first calculate the phase diagram for the Luttinger liquid-Majorana Kramers pair junction and show that its low-energy properties are determined by Andreev reflection scattering processes in the spin-triplet channel, i.e. the corresponding Andreev boundary conditions are similar to that in a spin-triplet superconductor - normal lead junction. We also study here a quantum dot coupled to a normal lead and a Majorana Kramers pair and investigate the effect of local repulsive interactions leading to an interplay between Kondo and Majorana correlations. Using a combination of renormalization group analysis and slave-boson mean-field theory, we show that the system flows to a new fixed point which is controlled by the Majorana interaction rather than the Kondo coupling. This Majorana fixed point is characterized by correlations between the localized spin and the fermion parity of each spin sector of the topological superconductor. We investigate the stability of the Majorana phase with respect to Gaussian fluctuations.
•  Younghyun Kim, Dong E. Liu, Erikas Gaidamauskas, Jens Paaske, Karsten Flensberg, Roman M. Lutchyn Journal reference: Phys. Rev. B 94, 075439 (2016) [pdf] DOI: 10.1103/PhysRevB.94.075439

• Roadmap to Majorana surface codes - Abstract
• Surface codes offer a very promising avenue towards fault-tolerant quantum computation. We argue that two-dimensional interacting networks of Majorana bound states in topological superconductor/semiconductor heterostructures hold several distinct advantages in that direction, both concerning the hardware realization and the actual operation of the code. We here discuss how topologically protected logical qubits in this Majorana surface code architecture can be defined, initialized, manipulated, and read out. All physical ingredients needed to implement these operations are routinely used in topologically trivial quantum devices. In particular, we show that by means of quantum interference terms in linear conductance measurements, composite single-electron pumping protocols, and gate-tunable tunnel barriers, the full set of quantum gates required for universal quantum computation can be implemented.
•  S. Plugge, L. A. Landau, E. Sela, A. Altland, K. Flensberg, R. Egger Journal reference: Phys. Rev. B 94, 174514 (2016) [pdf] DOI: 10.1103/PhysRevB.94.174514

• Quantum charge fluctuations of a proximitized nanowire - Abstract
• Motivated by recent experiment, we consider charging of a nanowire which is proximitized by a superconductor and connected to a normal-state lead by a single-channel junction. The charge $Q$ of the nanowire is controlled by gate voltage $e{\cal N}_g/C$. A finite conductance of the contact allows for quantum charge fluctuations, making the function $Q(\mathcal{N}_g)$ continuous. It depends on the relation between the superconducting gap $\Delta$ and the effective charging energy $E^*_C$. The latter is determined by the junction conductance, in addition to the geometrical capacitance of the proximitized nanowire. We investigate $Q(\mathcal{N}_g)$ at zero magnetic field $B$, and at fields exceeding the critical value $B_c$ corresponding to the topological phase transition. Unlike the case of $\Delta = 0$, the function $Q(\mathcal{N}_g)$ is analytic even in the limit of negligible level spacing in the nanowire. At $B=0$ and $\Delta>E^*_C$, the maxima of $dQ/d\mathcal{N}_g$ are smeared by $2e$-fluctuations described by a single-channel "charge Kondo" physics, while the $B=0$, $\Delta<e^*_c$ case="case" is="is" described="described" by="by" a="a" crossover="crossover" between="between" the="the" kondo="Kondo" and="and" mixed-valence="mixed-valence" regimes="regimes" of="of" the="the" anderson="Anderson" impurity="impurity" model.="model." in="In" the="the" topological="topological" phase,="phase," $q(\mathcal{n}_g)$="$Q(\mathcal{N}_g)$" is="is" analytic="analytic" function="function" of="of" the="the" gate="gate" voltage="voltage" with="with" $e$-periodic="$e$-periodic" steps.="steps." in="In" the="the" weak="weak" tunneling="tunneling" limit,="limit," $dq/d\mathcal{n}_g$="$dQ/d\mathcal{N}_g$" has="has" peaks="peaks" corresponding="corresponding" to="to" breit-wigner="Breit-Wigner" resonances,="resonances," whereas="whereas" in="In" the="the" strong="strong" tunneling="tunneling" limit="limit" (i.e.,="(i.e.," small="small" reflection="reflection" amplitude="amplitude" $r$="$r$" )=")" these="these" resonances="resonances" are="are" broadened,="broadened," and="and" $dq/d\mathcal{n}_g-e="$dQ/d\mathcal{N}_g-e" \propto="\propto" r\cos(2\pi="r\cos(2\pi" \mathcal{n}_g)$. •  Roman M. Lutchyn, Karsten Flensberg, Leonid I. Glazman Journal reference: Phys. Rev. B 94, 125407 (2016) [pdf] DOI: 10.1103/PhysRevB.94.125407 • Time scales for Majorana manipulation using Coulomb blockade in gate-controlled superconducting nanowires - Abstract • We numerically compute the low-energy spectrum of a gate-controlled superconducting topological nanowire segmented into two islands, each Josephson-coupled to a bulk superconductor. This device may host two pairs of Majorana bound states and could provide a platform for testing Majorana fusion rules. We analyze the crossover between (i) a charge-dominated regime utilizable for initialization and readout of Majorana bound states, (ii) a single-island regime for dominating inter-island Majorana coupling, (iii) a Josephson-plasmon regime for large coupling to the bulk superconductors, and (iv) a regime of four Majorana bound states allowing for topologically protected Majorana manipulations. From the energy spectrum, we derive conservative estimates for the time scales of a fusion-rule testing protocol proposed recently [arXiv:1511.05153]. We also analyze the steps needed for basic Majorana braiding operations in branched nanowire structures. •  Michael Hell, Jeroen Danon, Karsten Flensberg, Martin Leijnse Journal reference: Phys. Rev. B 94, 035424 (2016) [pdf] DOI: 10.1103/PhysRevB.94.035424 • Self-organized topological superconductivity in a Yu-Shiba-Rusinov chain - Abstract • We study a chain of magnetic moments exchange coupled to a conventional three dimensional superconductor. In the normal state the chain orders into a collinear configuration, while in the superconducting phase we find that ferromagnetism is unstable to the formation of a magnetic spiral state. Beyond weak exchange coupling the spiral wavevector greatly exceeds the inverse superconducting coherence length as a result of the strong spin-spin interaction mediated through the subgap band of Yu-Shiba-Rusinov states. Moreover, the simple spin-spin exchange description breaks down as the subgap band crosses the Fermi energy, wherein the spiral phase becomes stabilized by the spontaneous opening of a$p-$wave superconducting gap within the band. This leads to the possibility of electron-driven topological superconductivity with Majorana boundary modes using magnetic atoms on superconducting surfaces. •  M. Schecter, K. Flensberg, M. H. Christensen, B. M. Andersen, J. Paaske Journal reference: Phys. Rev. B 93, 140503 (2016) [pdf] DOI: 10.1103/PhysRevB.93.140503 • Milestones Toward Majorana-Based Quantum Computing - Abstract • We introduce a scheme for preparation, manipulation, and readout of Majorana zero modes in semiconducting wires with mesoscopic superconducting islands. Our approach synthesizes recent advances in materials growth with tools commonly used in quantum-dot experiments, including gate-control of tunnel barriers and Coulomb effects, charge sensing, and charge pumping. We outline a sequence of milestones interpolating between zero-mode detection and quantum computing that includes (1) detection of fusion rules for non-Abelian anyons using either proximal charge sensors or pumped current; (2) validation of a prototype topological qubit; and (3) demonstration of non-Abelian statistics by braiding in a branched geometry. The first two milestones require only a single wire with two islands, and additionally enable sensitive measurements of the system's excitation gap, quasiparticle poisoning rates, residual Majorana zero-mode splittings, and topological-qubit coherence times. These pre-braiding experiments can be adapted to other manipulation and readout schemes as well. •  David Aasen, Michael Hell, Ryan V. Mishmash, Andrew Higginbotham, Jeroen Danon, Martin Leijnse, Thomas S. Jespersen, Joshua A. Folk, Charles M. Marcus, Karsten Flensberg, Jason Alicea Journal reference: Phys. Rev. X 6, 031016 (2016) [pdf] DOI: 10.1103/PhysRevX.6.031016 • Phase-tunable Majorana bound states in a topological N-SNS junction - Abstract • We theoretically study the differential conductance of a one-dimensional normal-superconductor-normal-superconductor (N-SNS) junction with a phase bias applied between the two superconductors. We consider specifically a junction formed by a spin-orbit coupled semiconducting nanowire with regions of the nanowire having superconducting pairing induced by a bulk$s$-wave superconductor. When the nanowire is tuned into a topologically non-trivial phase by a Zeeman field, it hosts zero-energy Majorana modes at its ends as well as at the interface between the two superconductors. The phase-dependent splitting of the Majorana modes gives rise to features in the differential conductance that offer a clear distinction between the topologically trivial and non-trivial phases. We calculate the transport properties of the junction numerically and also present a simple analytical model that captures the main properties of the predicted tunneling spectroscopy. •  Esben Bork Hansen, Jeroen Danon, Karsten Flensberg Journal reference: Phys. Rev. B 93, 094501 (2016) [pdf] DOI: 10.1103/PhysRevB.93.094501 • Effects of spin-orbit coupling and spatial symmetries on the Josephson current in SNS junctions - Abstract • We present an analysis of the symmetries of the interference pattern of critical currents through a two-dimensional superconductor-semiconductor-superconductor junction, taking into account Rashba and Dresselhaus spin-orbit interaction, an arbitrarily oriented magnetic field, disorder, and structural asymmetries. We relate the symmetries of the pattern to the absence or presence of symmetries in the Hamiltonian, which provides a qualitative connection between easily measurable quantities and the spin-orbit coupling and other symmetries of the junction. We support our analysis with numerical calculations of the Josephson current based on a perturbative expansion up to eighth order in tunnel coupling between the normal region and the superconductors. •  Asbjørn Rasmussen, Jeroen Danon, Henri Suominen, Fabrizio Nichele, Morten Kjaergaard, Karsten Flensberg Journal reference: Phys. Rev. B 93, 155406 (2016) [pdf] DOI: 10.1103/PhysRevB.93.155406 • Braiding properties of Majorana Kramers pairs - Abstract • We consider the braiding of Kramers pairs of Majorana bound states. We derive the most general transformation on the many-body ground state that is applied as the result of such a braiding process. The result is derived in the context of a simple toy model, but we will show that it has the most general form that is compatible with local and global conservation of electron parity. In accordance with earlier work the resulting transformation turns out to be path dependent, which shows that Kramers pairs of Majorana bound states cannot be used for topological quantum computation. We also discuss under which conditions the result is path independent and corresponds to two independent exchanges of pairs of Majorana bound states. •  Konrad Wölms, Ady Stern, Karsten Flensberg Journal reference: Phys. Rev. B 93, 045417 (2016) [pdf] DOI: 10.1103/PhysRevB.93.045417 • 2015 • Yu-Shiba-Rusinov states in phase-biased superconductor–quantum dot–superconductor junctions - Abstract • We study the effects of a phase difference on Yu-Shiba-Rusinov (YSR) states in a spinful Coulomb-blockaded quantum dot contacted by a superconducting loop. In the limit where charging energy is larger than the superconducting gap, we determine the subgap excitation spectrum, the corresponding supercurrent, and the differential conductance as measured by a normal-metal tunnel probe. In absence of a phase difference only one linear combination of the superconductor lead electrons couples to the spin, which gives a single YSR state. With finite phase difference, however, it is effectively a two-channel scattering problem and therefore an additional state emerges from the gap edge. The energy of the phase-dependent YSR states depend on the gate voltage and one state can cross zero energy twice inside the valley with odd occupancy. These crossings are shifted by the phase difference towards the charge degeneracy points, corresponding to larger exchange couplings. Moreover, the zero-energy crossings give rise to resonant peaks in the differential conductance with magnitude$4e^2/h$. Finally, we demonstrate that the quantum fluctuations of the dot spin do not alter qualitatively any of the results. •  Gediminas Kiršanskas, Moshe Goldstein, Karsten Flensberg, Leonid I. Glazman, Jens Paaske Journal reference: Phys. Rev. B 92, 235422 (2015) [pdf] DOI: 10.1103/PhysRevB.92.235422 • Environmental Coulomb blockade of topological superconductor-normal metal junctions - Abstract • We study charge transport of a topological superconductor connected to different electromagnetic environments using a low-energy description where only the Majorana bound states in the superconductor are included. Extending earlier findings who found a crossover between perfect Andreev reflection with conductance$2e^2/h$to a regime with blocked transport when the resistance of the environment is larger than$2e^2/h$, we consider Majorana bound states coupled to metallic dots. in particular, we study two topological superconducting leads connected by a metallic quantum dot in both the weak tunneling and strong tunneling regimes. For weak tunneling, we project onto the most relevant charge states. For strong tunneling, we start from the Andreev fixed point and integrate out charge fluctuations which gives an effective low-energy model for the non-perturbative gate-voltage modulated cotunneling current. In both regimes and in contrast to cotunneling with normal leads, the conductance is temperature independent because of the resonant Andreev reflections, which are included to all orders. •  Konrad Wölms, Karsten Flensberg Journal reference: Phys. Rev. B 92, 165428 (2015) [pdf] DOI: 10.1103/PhysRevB.92.165428 • Nonlocal damping of helimagnets in one-dimensional interacting electron systems - Abstract • We investigate the magnetization relaxation of a one-dimensional helimagnetic system coupled to interacting itinerant electrons. The relaxation is assumed to result from the emission of plasmons, the elementary excitations of the one-dimensional interacting electron system, caused by slow changes of the magnetization profile. This dissipation mechanism leads to a highly nonlocal form of magnetization damping that is strongly dependent on the electron-electron interaction. Forward scattering processes lead to a spatially constant damping kernel, while backscattering processes produce a spatially oscillating contribution. Due to the nonlocal damping, the thermal fluctuations become spatially correlated over the entire system. We estimate the characteristic magnetization relaxation times for magnetic quantum wires and nuclear helimagnets. •  Kjetil M. D. Hals, Karsten Flensberg, Mark S. Rudner Journal reference: Phys. Rev. B 92, 094403 (2015) [pdf] DOI: 10.1103/PhysRevB.92.094403 • Quantum transport in carbon nanotubes - Abstract • Carbon nanotubes are a versatile material in which many aspects of condensed matter physics come together. Recent discoveries, enabled by sophisticated fabrication, 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 conventional semiconductors, electrons in nanotubes have two angular momentum quantum numbers, arising from spin and from valley freedom. We focus on the interplay between the two. In single quantum dots defined in short lengths of nanotube, the energy levels associated with each degree of freedom, and the spin-orbit coupling between them, are revealed by Coulomb blockade spectroscopy. 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 behaviour. 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, we also state clearly open questions for the field. 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 very low level. •  E. A. Laird, F. Kuemmeth, G. Steele, K. Grove-Rasmussen, J. Nygård, K. Flensberg, L. P. Kouwenhoven Journal reference: Rev. Mod. Phys. 87, 703 (2015) [pdf] DOI: 10.1103/RevModPhys.87.703 • Interaction effects on proximity-induced superconductivity in semiconducting nanowires - Abstract • We investigate the effect of electron-electron interactions on proximity-induced$s$-wave superconductivity in one-dimensional nanowires. We treat the interactions on a self-consistent mean-field level, and find an analytic expression for the effective pairing potential in the presence of interactions, valid for a weakly tunnel coupled wire. We show that for a set of two nanowires placed in parallel on a superconducting substrate, the interaction-induced reduction of the pairing energy could result in the effective interwire pairing potential exceeding the intrawire potential, which is one of the requirements for creating a time-reversal symmetric topological superconducting state in such a two-wire system. •  Jeroen Danon, Karsten Flensberg Journal reference: Phys. Rev. B 91, 165425 (2015) [pdf] DOI: 10.1103/PhysRevB.91.165425 • Spin-Lattice Order in One-Dimensional Conductors: Beyond the RKKY Effect - Abstract • We investigate magnetic order in a lattice of classical spins coupled to an isotropic gas of one-dimensional (1d) conduction electrons via local exchange interactions. The frequently discussed Ruderman-Kittel-Kasuya-Yosida (RKKY) effective exchange model for this system predicts that spiral order is always preferred. Here we consider the problem nonperturbatively, and find that such order vanishes above a critical value of the exchange coupling that depends strongly on the lattice spacing. The critical coupling tends to zero as the lattice spacing becomes commensurate with the Fermi wave vector, signalling the breakdown of the perturbative RKKY picture, and spiral order, even at weak coupling. We provide the exact phase diagram for arbitrary exchange coupling and lattice spacing, and discuss its stability. Our results shed new light on the problem of utilizing a spiral spin-lattice state to drive a one-dimensional superconductor into a topological phase. •  Michael Schecter, Mark S. Rudner, Karsten Flensberg Journal reference: Phys. Rev. Lett. 114, 247205 (2015) [pdf] DOI: 10.1103/PhysRevLett.114.247205 • Parity lifetime of bound states in a proximitized semiconductor nanowire - Abstract • Quasiparticle excitations can compromise the performance of superconducting devices, causing high frequency dissipation, decoherence in Josephson qubits, and braiding errors in proposed Majorana-based topological quantum computers. Quasiparticle dynamics have been studied in detail in metallic superconductors 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. •  A. P. Higginbotham, S. M. Albrecht, G. Kirsanskas, W. Chang, F. Kuemmeth, P. Krogstrup, T. S. Jespersen, J. Nygard, K. Flensberg, C. M. Marcus Journal reference: Nature Physics 11, 1017 (2015) [pdf] DOI: 10.1038/nphys3461 • 2014 • Local Adiabatic Mixing of Kramers Pairs of Majorana Bound States - Abstract • We consider Kramers pairs of Majorana bound states under adiabatic time evolution. This is important for the prospects of using such bound states as parity qubits. We show that local adiabatic perturbations can cause a rotation in the space spanned by the Kramers pair. Hence the quantum information is unprotected against local perturbations, in contrast to the case of single localized Majorana bound states in systems with broken time reversal symmetry. We give an analytical and a numerical example for such a rotation, and specify sufficient conditions under which a rotation is avoided. We give a general scheme for determining when these conditions are satisfied, and exemplify it with a general model of a quasi 1D time reversal symmetric topological superconductor. •  Konrad Wölms, Ady Stern, Karsten Flensberg Journal reference: Phys. Rev. Lett. 113, 246401 (2014) [pdf] DOI: 10.1103/PhysRevLett.113.246401 • Tunnel spectroscopy of Majorana bound states in topological superconductor/quantum dot Josephson junctions - Abstract • We theoretically investigate electronic transport through a junction where a quantum dot (QD) is tunnel coupled on both sides to semiconductor nanowires with strong spin-orbit interaction and proximity-induced superconductivity. The results are presented as stability diagrams, i.e., the differential conductance as a function of the bias voltage applied across the junction and the gate voltage used to control the electrostatic potential on the QD. A small applied magnetic field splits and modifies the resonances due to the Zeeman splitting of the QD level. Above a critical field strength, Majorana bound states (MBS) appear at the interfaces between the two superconducting nanowires and the QD, resulting in a qualitative change of the entire stability diagram, suggesting this setup as a promising platform to identify MBS. Our calculations are based on a nonequilibrium Green's function description and is exact when Coulomb interactions on the QD can be neglected. In addition, we develop a simple pictorial view of the involved transport processes, which is equivalent to a description in terms of multiple Andreev reflections, but provides an alternative way to understand the role of the QD level in enhancing transport for certain gate and bias voltages. We believe that this description will be useful in future studies of interacting QDs coupled to superconducting leads (with or without MBS), where it can be used to develop a perturbation expansion in the tunnel coupling. •  Guang-Yao Huang, Martin Leijnse, Karsten Flensberg, Hongqi Xu Journal reference: Phys. Rev. B 90, 214507 (2014) [pdf] DOI: 10.1103/PhysRevB.90.214507 • Designing - Abstract • We propose and analyze a new way of using$\pi\$ stacking to design molecular junctions that either enhance or suppress a phononic heat current, but at the same time remain conductors for an electric current. Such functionality is highly desirable in thermoelectric energy converters, as well as in other electronic components where heat dissipation should be minimized or maximized. We suggest a molecular design consisting of two masses coupled to each other with one mass coupled to each lead. By having a small coupling (spring constant) between the masses, it is possible to either reduce, or perhaps more surprisingly enhance the phonon conductance. We investigate a simple model system to identify optimal parameter regimes and then use first principle calculations to extract model parameters for a number of specific molecular realizations, confirming that our proposal can indeed be realized using standard molecular building blocks.
•  Gediminas Kiršanskas, Qian Li, Karsten Flensberg, Gemma C. Solomon, Martin Leijnse Journal reference: Appl. Phys. Lett. 105, 233102 (2014) [pdf] DOI: 10.1063/1.4903340

• Majorana Bound States in Two-Channel Time-Reversal-Symmetric Nanowire Systems - Abstract
• We consider time-reversal-symmetric two-channel semiconducting quantum wires proximity coupled to an s-wave superconductor. We analyze the requirements for a nontrivial topological phase and find that necessary conditions are 1) the determinant of the pairing matrix in channel space must be negative, 2) inversion symmetry must be broken, and 3) the two channels must have different spin-orbit couplings. The first condition can be implemented in semiconducting nanowire systems where interactions suppress intra-channel pairing, while the inversion symmetry can be broken by tuning the chemical potentials of the channels. For the case of collinear spin-orbit directions, we find a general expression for the topological invariant by block diagonalization into two blocks with chiral symmetry only. By projection to the low-energy sector, we solve for the zero modes explicitly and study the details of the gap closing, which in the general case happens at finite momenta.
•  Erikas Gaidamauskas, Jens Paaske, Karsten Flensberg Journal reference: Phys. Rev. Lett. 122, 126402 (2014) [pdf] DOI: 10.1103/PhysRevLett.112.126402

• 2013
• Coupling Spin Qubits via Superconductors - Abstract
• We show how superconductors can be used to couple, initialize, and read out spatially separated spin qubits. When two single-electron quantum dots are tunnel coupled to the same superconductor, the singlet component of the two-electron state partially leaks into the superconductor via crossed Andreev reflection. This induces a gate-controlled singlet-triplet splitting which, with an appropriate superconductor geometry, remains large for dot separations within the superconducting coherence length. Furthermore, we show that when two double-dot singlet-triplet qubits are tunnel coupled to a superconductor with finite charging energy, crossed Andreev reflection enables a strong two-qubit coupling over distances much larger than the coherence length.
•  Martin Leijnse, Karsten Flensberg Journal reference: Phys. Rev. Lett. 111, 060501 (2013) [pdf] DOI: 10.1103/PhysRevLett.111.060501

• 2012
• Introduction to topological superconductivity and Majorana fermions - Abstract
• This short review article provides a pedagogical introduction to the rapidly growing research field of Majorana fermions in topological superconductors. We first discuss in some details the simplest "toy model" in which Majoranas appear, namely a one-dimensional tight-binding representation of a p-wave superconductor, introduced more than ten years ago by Kitaev. We then give a general introduction to the remarkable properties of Majorana fermions in condensed matter systems, such as their intrinsically non-local nature and exotic exchange statistics, and explain why these quasiparticles are suspected to be especially well suited for low-decoherence quantum information processing. We also discuss the experimentally promising (and perhaps already successfully realized) possibility of creating topological superconductors using semiconductors with strong spin-orbit coupling, proximity-coupled to standard s-wave superconductors and exposed to a magnetic field. The goal is to provide an introduction to the subject for experimentalists or theorists who are new to the field, focusing on the aspects which are most important for understanding the basic physics. The text should be accessible for readers with a basic understanding of quantum mechanics and second quantization, and does not require knowledge of quantum field theory or topological states of matter.
•  Martin Leijnse, Karsten Flensberg Journal reference: Semicond. Sci. Technol. 27, 124003 (2012) [pdf] DOI: 10.1088/0268-1242/27/12/124003

• Parity qubits and poor man's Majorana bound states in double quantum dots - Abstract
• We study a double quantum dot connected via a common superconducting lead and show that this system can be tuned to host one Majorana bound state (MBS) on each dot. We call them "poor man's Majorana bound states" since they are not topologically protected, but otherwise share the properties of MBS formed in topological superconductors. We describe the conditions for the existence of the two spatially separated MBS, which include breaking of spin degeneracy in the two dots, with the spins polarized in different directions. Therefore, we propose to use a magnetic field configuration where the field directions on the two dot form an angle. By control of this angle the cross Andreev reflection and the tunnel amplitudes can be tuned to be approximately equal, which is a requirement for the formation of the MBS. We show that the fermionic state encoded in the two Majoranas constitutes a parity qubit, which is non-local and can only be measured by probing both dots simultaneously. Using a many-particle basis for the MBS, we discuss the role of interactions and show that inter-dot interactions always lift the degeneracy. We also show how the MBS can be probed by transport measurements and discuss how the combination of several such double dot systems allows for entanglement of parity qubits and measurement of their dephasing times.
•  Martin Leijnse, Karsten Flensberg Journal reference: Phys. Rev. B. 86, 134528 (2012) [pdf] DOI: 10.1103/PhysRevB.86.134528

• Finite-bias conductance anomalies at a singlet-triplet crossing - Abstract
• Quantum dots and single-molecule transistors may exhibit level crossings induced by tuning external parameters such as magnetic field or gate voltage. For Coulomb blockaded devices, this shows up as an inelastic cotunneling threshold in the differential conductance, which can be tuned to zero at the crossing. Here we show that, in addition, level crossings can give rise to a nearly vertical step-edge, ridge or even a Fano-like ridge-valley feature in the differential conductance inside the relevant Coulomb diamond. We study a gate-tunable quasidegeneracy between singlet and triplet ground states, and demonstrate how these different shapes may result from a competition between nonequilibrium occupations and weak (spin-orbit) mixing of the states. Our results are shown to be in qualitative agreement with recent transport measurements on a Mn complex [E. A. Osorio, et al., Nano Lett. 10, 105 (2010)]. The effect remains entirely general and should be observable in a wide range of Coulomb blockaded devices.
•  Chiara Stevanato, Martin Leijnse, Karsten Flensberg, Jens Paaske Journal reference: Phys. Rev. B 86, 165427 (2012) [pdf] DOI: 10.1103/PhysRevB.86.165427

• Hybrid topological-spin qubit systems for two-qubit-spin gates - Abstract
• We investigate a hybrid quantum system involving spin qubits, based on the spins of electrons confined in quantum dots, and topological qubits, based on Majorana fermions. In such a system, gated control of the charge on the quantum dots allows transfer of quantum information between the spin and topological qubits, and the topological system can be used to facilitate transfer of spin qubits between spatially separated quantum dots and to initialize entangled spin-qubit pairs. Here, we show that the coupling to the topological system also makes it possible to perform entangling two-qubit gates on spatially separated spin qubits. The two-qubit gates are based on a combination of topologically protected braiding operations, gate-controlled charge transfer between the dots and edge Majorana modes, and measurements of the state of the topological qubits.
•  Martin Leijnse, Karsten Flensberg Journal reference: Phys. Rev. B 86, 104511 (2012) [pdf] DOI: 10.1103/PhysRevB.86.104511

• Emerging Dirac and Majorana fermions for carbon nanotubes with proximity-induced pairing and spiral magnetic field - Abstract
• We study the low-energy bandstructure of armchair and small-bandgap semiconducting carbon nanotubes with proximity-induced superconducting pairing when a spiral magnetic field creates strong effective spin-orbit interactions from the Zeeman term and a periodic potential from the orbital part. We find that gapless Dirac fermions can be generated by variation of a single parameter. For a semiconducting tube with the field in the same plane, a non-degenerate zero mode at momentum k=0 can be induced, allowing for the generation of topologically protected Majorana fermion end states.
•  Reinhold Egger, Karsten Flensberg Journal reference: Physical Review B 85, 235462 (2012) [pdf] DOI: 10.1103/PhysRevB.85.235462