Center for Quantum Devices > Research > Publications > Karsten Flensberg
Publications by Karsten Flensberg
 2019

Fidelity and visibility reduction in Majorana qubits by entanglement
with environmental modes 
Abstract
 We study the dynamics and readout of topological qubits encoded by zeroenergy Majorana bound states in a topological superconductor. We take into account bosonic modes due to the electromagnetic environment which couple the Majorana manifold to abovegap continuum quasiparticles. This coupling causes the degenerate ground state of the topological superconductor to be dressed in a polaronlike manner by quasiparticle 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 lowenergy spectrum of the dressed states. We discuss reduction of fidelity and/or visibility if this condition is violated by a quantumdot readout that couples to the bare (undressed) Majorana modes. For a projective measurement of the bare Majorana basis, we formulate a BlochRedfield approach that is valid for weak Majoranaenvironment coupling and takes into account constraints imposed by fermionnumberparity conservation. Within the Markovian approximation, our results essentially confirm earlier theories of finitetemperature decoherence based on Fermi's golden rule. However, the full nonMarkovian 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 1812.00254v2 [pdf]

Fidelity and visibility reduction in Majorana qubits by entanglement
with environmental modes 
Abstract
 2018

Coulombinteractioninduced 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 spinorbit coupling, magnetic fields and superconductivity. They appear purely due to the combination of the threedimensional 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 solidstate candidate systems for Majorana edge modes, and may similarly be probed via tunnel spectroscopy.
Tommy Li, Michele Burrello, Karsten Flensberg 1809.09564v1 [pdf]

Topological superconductivity in full shell proximitized 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 spinorbit 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 1809.05512v1 [pdf]

Coulomb Blockade of a Nearly Open Majorana Island 
Abstract
 We consider the groundstate energy and the spectrum of the lowenergy excitations of a Majorana island formed of topological superconductors connected by a singlemode junction of arbitrary transmission. Coulomb blockade results in $e$periodic modulation of the energies with the gateinduced 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 conventionalsuperconductor Cooperpair 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) [ 1809.01660v1 ] DOI: 10.1103/PhysRevLett.122.016801

Hybridization at SuperconductorSemiconductor Interfaces 
Abstract
 Hybrid superconductorsemiconductor 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 superconductorsemiconductor hybridization in such devices by analyzing a gated single AlInAs interface using a selfconsistent SchrodingerPoisson approach. Our numerical analysis shows that the band bending leads to an interface quantum well, which localizes the charge in the system near the superconductorsemiconductor 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 superconductorsemiconductor 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) [ 1801.03439v3 ] DOI: 10.1103/PhysRevX.8.031040

FourMajorana qubit with charge readout: Dynamics and decoherence 
Abstract
 We present a theoretical analysis of a Majoranabased 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 chargestability diagram and its use for characterizing the parameters of the devices, including the overlap of the Majorana edge states. We describe the multilevel spectral properties of the system and present a detailed study of its controlled coherent oscillations, as well as decoherence resulting from coupling to a nonMarkovian environment. In particular, we study a gatecontrolled protocol where conversion between Coulombblockade 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) [ 1807.05839v1 ] DOI: 10.1103/PhysRevB.98.205403

Hybridization of Subgap States in OneDimensional SuperconductorSemiconductor Coulomb Islands 
Abstract
 We present measurements of onedimensional superconductorsemiconductor Coulomb islands, fabricated by gate confinement of a twodimensional InAs heterostructure with an epitaxial Al layer. When tuned via electrostatic side gates to regimes without subgap states, Coulomb blockade reveals Cooperpair mediated transport. When subgap 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 Rashbatype spinorbit 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) [ 1804.09676v1 ] 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 zerobias 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 Coulombblockaded island into an AharonovBohm 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 twodimensional electron gases with proximityinduced superconductivity and may also be a useful spectroscopic tool for parityflip mechanisms.
Michael Hell, Karsten Flensberg, Martin Leijnse Journal reference: Phys. Rev. B 97, 161401 (2018) [ 1710.05294v2 ] DOI: 10.1103/PhysRevB.97.161401

Probing electronhole components of subgap states in Coulomb blockaded Majorana islands 
Abstract
 Recent tunneling spectroscopy experiments in semiconducting nanowires with proximityinduced superconductivity have reported robust zerobias 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 Coulombblockade 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 lowestenergy 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) [ 1712.07972v2 ] DOI: 10.1103/PhysRevB.97.041411

Coulombinteractioninduced Majorana edge modes in nanowires 
Abstract
 2017

Scaling of Majorana ZeroBias Conductance Peaks 
Abstract
 We report an experimental study of the scaling of zerobias conductance peaks compatible with Majorana zero modes as a function of magnetic field, tunnel coupling, and temperature in onedimensional structures fabricated from an epitaxial semiconductorsuperconductor 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 fielddependent 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) [ 1706.07033v2 ] DOI: 10.1103/PhysRevLett.119.136803

Scalable designs for quasiparticlepoisoningprotected 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 measurementonly 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) [ 1610.05289v4 ] DOI: 10.1103/PhysRevB.95.235305

Conductance spectroscopy on Majorana wires and the inverse proximity effect 
Abstract
 Recent experimental searches for signatures of Majoranalike excitations in proximitized semiconducting nanowires involve conductance spectroscopy, where the evidence sought after is a robust zerobias peak (in longer wires) and its characteristic fielddependent splitting (in shorter wires). Although experimental results partially confirm the theoretical predictions, commonly observed discrepancies still include (i) a zerobias peak that is significantly lower than the predicted value of $2e^2/h$ and (ii) the absence of the expected "Majorana oscillations" of the lowestenergy 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 oneband model for the proximitized nanowire, where the superconductor is considered to be in the diffusive regime, described by semiclassical 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) [ 1706.03587v1 ] DOI: 10.1103/PhysRevB.96.125420

Coupling and braiding Majorana bound states in networks defined in twodimensional electron gases with proximityinduced superconductivity 
Abstract
 Twodimensional electron gases with strong spinorbit 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 (Sstripes) or where a narrow stripe has been removed from a uniform layer (Nstripes). 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 Majoranastripe ends as well as Majorana trijunctions formed in a tuningfork 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 gatecontrolled in a range similar to the intrastripe 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 10100 ns.
Michael Hell, Karsten Flensberg, Martin Leijnse Journal reference: Phys. Rev. B 96, 035444 (2017) [ 1704.06427v1 ] DOI: 10.1103/PhysRevB.96.035444

Anharmonicity of a superconducting qubit with a fewmode Josephson junction 
Abstract
 Coherent operation of gatevoltagecontrolled hybrid transmon qubits (gatemons) based on semiconductor nanowires was recently demonstrated. Here we experimentally investigate the anharmonicity in epitaxial InAsAl 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) [ 1703.05643v1 ] DOI: 10.1103/PhysRevB.97.060508

Majorana bound state in a coupled quantumdot hybridnanowire 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 enddot 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, 15571562 (2016) [ 1612.07989v2 ] DOI: 10.1126/science.aaf3961

Scaling of Majorana ZeroBias Conductance Peaks 
Abstract
 2016

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 semiconductorsuperconductor heterostructures now believed to host Majorana bound states. In this paper, we construct quantum bits taking advantage of the topological protection and nonlocal 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, nonlocally addressable by quantum dots used for singlequbit readout, initialization, and manipulation. We describe how the same tools allow for measurementbased implementation of the Clifford gates, in total making the architecture universal. Proofofprinciple 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) [ 1609.01697v2 ] DOI: 10.1088/13672630/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 normalmetal 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 gatevoltage 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) [ 1612.05748v1 ] DOI: 10.1103/PhysRevLett.118.137701

TwoDimensional Platform for Networks of Majorana Bound States 
Abstract
 We model theoretically a twodimensional 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 magneticflux 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) [ 1608.08769v2 ] DOI: 10.1103/PhysRevLett.118.107701

Quantized conductance doubling and hard gap in a twodimensional semiconductor–superconductor heterostructure 
Abstract
 The prospect of coupling a twodimensional (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 spinorbit interaction to an swave 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 softgap problem in 2D superconductorsemiconductor hybrid systems. With the QPC in the open regime, we observe a first conductance plateau at 4e^2/h, as expected theoretically for a normalQPCsuperconductor structure. The realization of a hardgap semiconductorsuperconductor system that is amenable to topdown processing provides a means of fabricating scalable multicomponent hybrid systems for applications in lowdissipation 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) [ 1603.01852v2 ] DOI: 10.1038/ncomms12841

Anomalous Fraunhofer interference in epitaxial superconductorsemiconductor Josephson junctions 
Abstract
 We investigate patterns of critical current as a function of perpendicular and inplane magnetic fields in superconductorsemiconductorsuperconductor (SNS) junctions based on InAs/InGaAs heterostructures with an epitaxial Al layer. This material system is of interest due to its exceptionally good superconductorsemiconductor coupling, as well as large spinorbit interaction and gfactor in the semiconductor. Thin epitaxial Al allows the application of large inplane 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 inplane 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 spinorbit 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) [ 1611.00190v1 ] DOI: 10.1103/PhysRevB.95.035307

Spiral magnetic order and topological superconductivity in a chain of magnetic adatoms on a twodimensional 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 `selforganized' 1D topologically nontrivial superconducting phase within the set of subgap YuShibaRusinov (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 shortrange ferromagnetic and longrange antiferromagnetic electroninduced 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 spinorbit 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) [ 1607.08190v2 ] DOI: 10.1103/PhysRevB.94.144509

Nogo theorem for a timereversal invariant topological phase in noninteracting systems coupled to conventional superconductors 
Abstract
 We prove that a system of noninteracting electrons proximity coupled to a conventional swave 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 timereversal 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) [ 1605.07179v2 ] DOI: 10.1103/PhysRevB.94.161110

Interactiondriven topological superconductivity in one dimension 
Abstract
 We study onedimensional topological superconductivity in the presence of timereversal symmetry. This phase is characterized by having a bulk gap, while supporting a Kramers' pair of zeroenergy 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 electronelectron interactions. We further propose two experimental setups and show that they realize this model at low energies. The first setup is a narrow twodimensional topological insulator partially covered by a conventional swave superconductor, and the second is a semiconductor wire in proximity to an swave superconductor. These systems can therefore be used to realize and probe the timereversal invariant topological superconducting phase. The effect of interactions is studied using both a meanfield approach and a renormalization group analysis.
Arbel Haim, Konrad Wölms, Erez Berg, Yuval Oreg, Karsten Flensberg Journal reference: Phys. Rev. B 94, 115124 (2016) [ 1605.09385v2 ] DOI: 10.1103/PhysRevB.94.115124

Signatures of Majorana Kramers pairs in superconductorLuttinger liquid and superconductorquantum dotnormal lead junctions 
Abstract
 Timereversal 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 onedimensional superconductor is coupled to a normalmetal 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 electronelectron interactions in the Luttinger liquid. We first calculate the phase diagram for the Luttinger liquidMajorana Kramers pair junction and show that its lowenergy properties are determined by Andreev reflection scattering processes in the spintriplet channel, i.e. the corresponding Andreev boundary conditions are similar to that in a spintriplet 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 slaveboson meanfield 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) [ 1605.02073v2 ] DOI: 10.1103/PhysRevB.94.075439

Roadmap to Majorana surface codes 
Abstract
 Surface codes offer a very promising avenue towards faulttolerant quantum computation. We argue that twodimensional 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 singleelectron pumping protocols, and gatetunable 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) [ 1606.08408v1 ] 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 normalstate lead by a
singlechannel 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 singlechannel "charge Kondo"
physics, while the $B=0$, $\Delta

Motivated by recent experiment, we consider charging of a nanowire which is
proximitized by a superconductor and connected to a normalstate lead by a
singlechannel 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 singlechannel "charge Kondo"
physics, while the $B=0$, $\Delta
Roman M. Lutchyn, Karsten Flensberg, Leonid I. Glazman Journal reference: Phys. Rev. B 94, 125407 (2016) [ 1606.06756v1 ] DOI: 10.1103/PhysRevB.94.125407

Time scales for Majorana manipulation using Coulomb blockade in gatecontrolled superconducting nanowires 
Abstract
 We numerically compute the lowenergy spectrum of a gatecontrolled superconducting topological nanowire segmented into two islands, each Josephsoncoupled 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 chargedominated regime utilizable for initialization and readout of Majorana bound states, (ii) a singleisland regime for dominating interisland Majorana coupling, (iii) a Josephsonplasmon 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 fusionrule 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) [ 1601.07369v2 ] DOI: 10.1103/PhysRevB.94.035424

Selforganized topological superconductivity in a YuShibaRusinov 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 spinspin interaction mediated through the subgap band of YuShibaRusinov states. Moreover, the simple spinspin 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 electrondriven 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

YuShibaRusinov states in phasebiased superconductor–quantum dot–superconductor junctions 
Abstract
 We study the effects of a phase difference on YuShibaRusinov (YSR) states in a spinful Coulombblockaded 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 normalmetal 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 twochannel scattering problem and therefore an additional state emerges from the gap edge. The energy of the phasedependent 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 zeroenergy 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

Milestones Toward MajoranaBased 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 quantumdot experiments, including gatecontrol of tunnel barriers and Coulomb effects, charge sensing, and charge pumping. We outline a sequence of milestones interpolating between zeromode detection and quantum computing that includes (1) detection of fusion rules for nonAbelian anyons using either proximal charge sensors or pumped current; (2) validation of a prototype topological qubit; and (3) demonstration of nonAbelian 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 zeromode splittings, and topologicalqubit coherence times. These prebraiding 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) [ 1511.05153v2 ] DOI: 10.1103/PhysRevX.6.031016

Phasetunable Majorana bound states in a topological NSNS junction 
Abstract
 We theoretically study the differential conductance of a onedimensional normalsuperconductornormalsuperconductor (NSNS) junction with a phase bias applied between the two superconductors. We consider specifically a junction formed by a spinorbit 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 nontrivial phase by a Zeeman field, it hosts zeroenergy Majorana modes at its ends as well as at the interface between the two superconductors. The phasedependent splitting of the Majorana modes gives rise to features in the differential conductance that offer a clear distinction between the topologically trivial and nontrivial 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 spinorbit 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 twodimensional superconductorsemiconductorsuperconductor junction, taking into account Rashba and Dresselhaus spinorbit 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 spinorbit 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) [ 1510.05251v2 ] DOI: 10.1103/PhysRevB.93.155406

Majorana box qubits 
Abstract
 2015

Environmental Coulomb blockade of topological superconductornormal metal junctions 
Abstract
 We study charge transport of a topological superconductor connected to different electromagnetic environments using a lowenergy 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 lowenergy model for the nonperturbative gatevoltage 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 onedimensional interacting electron systems 
Abstract
 We investigate the magnetization relaxation of a onedimensional helimagnetic system coupled to interacting itinerant electrons. The relaxation is assumed to result from the emission of plasmons, the elementary excitations of the onedimensional 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 electronelectron 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

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

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 manybody 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

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 spinorbit 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 electronelectron 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 wellunderstood 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. GroveRasmussen, 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 proximityinduced superconductivity in semiconducting nanowires 
Abstract
 We investigate the effect of electronelectron interactions on proximityinduced $s$wave superconductivity in onedimensional nanowires. We treat the interactions on a selfconsistent meanfield 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 interactioninduced 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 timereversal symmetric topological superconducting state in such a twowire system.
Jeroen Danon, Karsten Flensberg Journal reference: Phys. Rev. B 91, 165425 (2015) [pdf] DOI: 10.1103/PhysRevB.91.165425

SpinLattice Order in OneDimensional Conductors: Beyond the RKKY Effect 
Abstract
 We investigate magnetic order in a lattice of classical spins coupled to an isotropic gas of onedimensional (1d) conduction electrons via local exchange interactions. The frequently discussed RudermanKittelKasuyaYosida (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 spinlattice state to drive a onedimensional 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 Majoranabased topological quantum computers. Quasiparticle dynamics have been studied in detail in metallic superconductors but remain relatively unexplored in semiconductorsuperconductor 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 gateconfined semiconductor nanowire with an epitaxially grown superconductor layer, yielding an isolated, proximitized nanowire segment. We identify Andreevlike 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

Environmental Coulomb blockade of topological superconductornormal metal junctions 
Abstract
 2014

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 spinorbit interaction and proximityinduced 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.
GuangYao Huang, Martin Leijnse, Karsten Flensberg, Hongqi Xu Journal reference: Phys. Rev. B 90, 214507 (2014) [pdf] DOI: 10.1103/PhysRevB.90.214507

Designing $π$stacked molecular structures to control heat transport
through molecular junctions 
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 fermions in superconducting nanowires without spinorbit coupling 
Abstract
 We show that confined Majorana fermions can exist in nanowires with proximity induced swave superconducting pairing if the direction of an external magnetic field rotates along the wire. The system is equivalent to nanowires with Rashbatype spinorbit coupling, with strength proportional to the derivative of the field angle. For realistic parameters, we demonstrate that a set of permanent magnets can bring a nearby nanowire into the topologically nontrivial phase with localized Majorana modes at its ends. Without the requirement of spinorbit coupling this opens up for a new route for demonstration and design of Majorana fermion systems.
Morten Kjaergaard, Konrad Wölms, Karsten Flensberg Journal reference: Phys. Rev. B 85, 020503(R) (2012) [pdf] DOI: 10.1103/PhysRevB.85.020503

Majorana Bound States in TwoChannel TimeReversalSymmetric Nanowire Systems 
Abstract
 We consider timereversalsymmetric twochannel semiconducting quantum wires proximity coupled to an swave 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 spinorbit couplings. The first condition can be implemented in semiconducting nanowire systems where interactions suppress intrachannel pairing, while the inversion symmetry can be broken by tuning the chemical potentials of the channels. For the case of collinear spinorbit directions, we find a general expression for the topological invariant by block diagonalization into two blocks with chiral symmetry only. By projection to the lowenergy 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

Tunnel spectroscopy of Majorana bound states in topological superconductor/quantum dot Josephson junctions 
Abstract
 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 singleelectron quantum dots are tunnel coupled to the same superconductor, the singlet component of the twoelectron state partially leaks into the superconductor via crossed Andreev reflection. This induces a gatecontrolled singlettriplet splitting which, with an appropriate superconductor geometry, remains large for dot separations within the superconducting coherence length. Furthermore, we show that when two doubledot singlettriplet qubits are tunnel coupled to a superconductor with finite charging energy, crossed Andreev reflection enables a strong twoqubit 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

Coupling Spin Qubits via Superconductors 
Abstract
 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 onedimensional tightbinding representation of a pwave 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 nonlocal nature and exotic exchange statistics, and explain why these quasiparticles are suspected to be especially well suited for lowdecoherence quantum information processing. We also discuss the experimentally promising (and perhaps already successfully realized) possibility of creating topological superconductors using semiconductors with strong spinorbit coupling, proximitycoupled to standard swave 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: 10.1088/02681242/27/12/124003

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 nonlocal and can only be measured by probing both dots simultaneously. Using a manyparticle basis for the MBS, we discuss the role of interactions and show that interdot 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

Finitebias conductance anomalies at a singlettriplet crossing 
Abstract
 Quantum dots and singlemolecule 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 stepedge, ridge or even a Fanolike ridgevalley feature in the differential conductance inside the relevant Coulomb diamond. We study a gatetunable quasidegeneracy between singlet and triplet ground states, and demonstrate how these different shapes may result from a competition between nonequilibrium occupations and weak (spinorbit) 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 topologicalspin qubit systems for twoqubitspin 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 spinqubit pairs. Here, we show that the coupling to the topological system also makes it possible to perform entangling twoqubit gates on spatially separated spin qubits. The twoqubit gates are based on a combination of topologically protected braiding operations, gatecontrolled 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

Cotunneling renormalization in carbon nanotube quantum dots 
Abstract
 We determine the levelshifts induced by cotunneling in a Coulomb blockaded carbon nanotube quantum dot using leading order quasidegenerate perturbation theory within a single nanotube quartet. It is demonstrated that otherwise degenerate and equally tunnelcoupled $K$ and $K'$ states are mixed by cotunneling and therefore split up in energy except at the particle/holesymmetric midpoints of the Coulomb diamonds. In the presence of an external magnetic field, we show that cotunneling induces a gatedependent $g$factor renormalization, and we outline different scenarios which might be observed experimentally, depending on the values of both intrinsic $KK'$ splitting and spinorbit coupling.
Gediminas Kiršanskas, Jens Paaske, Karsten Flensberg Journal reference: Phys. Rev. B 86, 075452 (2012) [pdf] DOI: 10.1103/PhysRevB.86.075452

Emerging Dirac and Majorana fermions for carbon nanotubes with proximityinduced pairing and spiral magnetic field 
Abstract
 We study the lowenergy bandstructure of armchair and smallbandgap semiconducting carbon nanotubes with proximityinduced superconducting pairing when a spiral magnetic field creates strong effective spinorbit 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 nondegenerate 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

SpinOrbitInduced Strong Coupling of a Single Spin to a Nanomechanical Resonator 
Abstract
 We theoretically investigate the deflectioninduced coupling of an electron spin to vibrational motion due to spinorbit coupling in suspended carbon nanotube quantum dots. Our estimates indicate that, with current capabilities, a quantum dot with an odd number of electrons can serve as a realization of the JaynesCummings model of quantum electrodynamics in the strongcoupling regime. A quantized flexural mode of the suspended tube plays the role of the optical mode and we identify two distinct twolevel subspaces, at small and large magnetic field, which can be used as qubits in this setup. The strong intrinsic spinmechanical coupling allows for detection, as well as manipulation of the spin qubit, and may yield enhanced performance of nanotubes in sensing applications.
András Pályi, P. R. Struck, Mark Rudner, Karsten Flensberg, Guido Burkard Journal reference: Phys. Rev. Lett. 108, 206811 (2012) [pdf] DOI: 10.1103/PhysRevLett.108.206811

Dephasing and hyperfine interaction in carbon nanotubes double quantum dots: Disordered case 
Abstract
 We study theoretically the \emph{return probability experiment}, used to measure the dephasing time $T_2^*$, in a double quantum dot (DQD) in semiconducting carbon nanotubes (CNTs) with spinorbit coupling and disorder induced valley mixing. Dephasing is due to hyperfine interaction with the spins of the ${}^{13}$C nuclei. Due to the valley and spin degrees of freedom four bounded states exist for any given longitudinal mode in the quantum dot. At zero magnetic field the spinorbit coupling and the valley mixing split those four states into two Kramers doublets. The valley mixing term for a given dot is determined by the intradot disorder and therefore the states in the Kramers doublets belonging to different dots are different. We show how nonzero singleparticle interdot tunneling amplitudes between states belonging to different doublets give rise to new avoided crossings, as a function of detuning, in the relevant two particle spectrum, crossing over from the two electrons in one dot states configuration, $(0,2)$, to the one electron in each dot configuration, $(1,1)$. In contrast to the clean system, multiple LandauZener processes affect the separation and the joining stages of each singleshot measurement and they affect the outcome of the measurement in a way that strongly depends on the initial state. We find that a welldefined return probability experiment is realized when, at each singleshot cycle, the (0,2) ground state is prepared. In this case, valley mixing increases the saturation value of the measured return probability, whereas the probability to return to the (0,2) ground state remains unchanged. Finally, we study the effect of the valley mixing in the high magnetic field limit; for a parallel magnetic field the predictions coincide with a clean nanotube, while the disorder effect is always relevant with a magnetic field perpendicular to the nanotube axis.
Andres A. Reynoso, Karsten Flensberg Journal reference: Phys. Rev. B 85, 195441 (2012) [pdf] DOI: 10.1103/PhysRevB.85.195441

MagneticField Dependence of Tunnel Couplings in Carbon Nanotube Quantum Dots 
Abstract
 By means of sequential and cotunneling spectroscopy, we study the tunnel couplings between metallic leads and individual levels in a carbon nanotube quantum dot. The levels are ordered in shells consisting of two doublets with strong and weaktunnel couplings, leading to gatedependent level renormalization. By comparison to a one and twoshell model, this is shown to be a consequence of disorderinduced valley mixing in the nanotube. Moreover, a parallel magnetic field is shown to reduce this mixing and thus suppress the effects of tunnel renormalization.
K. GroveRasmussen, S. Grap, J. Paaske, K. Flensberg, S. Andergassen, V. Meden, H. I. Jørgensen, K. Muraki, T. Fujisawa Journal reference: Phys. Rev. Lett. 108, 176802 (2012) [pdf] DOI: 10.1103/PhysRevLett.108.176802

Introduction to topological superconductivity and Majorana fermions 
Abstract
 2011

Quantum Information Transfer between Topological and Spin Qubit Systems 
Abstract
 We propose a method to coherently transfer quantum information, and to create entanglement, between topological qubits and conventional spin qubits. Our suggestion uses gated control to transfer an electron (spin qubit) between a quantum dot and edge Majorana modes in adjacent topological superconductors. Because of the spin polarization of the Majorana modes, the electron transfer translates spin superposition states into superposition states of the Majorana system, and vice versa. Furthermore, we show how a topological superconductor can be used to facilitate longdistance quantum information transfer and entanglement between spatially separated spin qubits.
Martin Leijnse, Karsten Flensberg Journal reference: Phys. Rev. Lett. 107, 210502 (2011) [pdf] DOI: 10.1103/PhysRevLett.107.210502

Number conserving theory for topologically protected degeneracy in onedimensional fermions 
Abstract
 Semiconducting nanowires in proximity to superconductors are among promising candidates to search for Majorana fermions and topologically protected degeneracies which may ultimately be used as building blocks for topological quantum computers. The prediction of neutral Majorana fermions in the proximityinduced superconducting systems ignores numberconservation and thus leaves open the conceptual question of how a topological degeneracy that is robust to all local perturbations arises in a numberconserving system. In this work, we study how local attractive interactions generate a topological groundstate neardegeneracy in a quasi onedimensional superfluid using bosonization of the fermions. The local attractive interactions opens a topological quasiparticle gap in the odd channel wires (with more than one channel) with end Majorana modes associated with a topological neardegeneracy. We explicitly study the robustness of the topological degeneracy to local perturbations and find that such local perturbations result in quantum phase slips which split of the topological degeneracy by an amount that does not decrease exponentially with the length of the wire, but still decreases rapidly if the number of channels is large. Therefore a bulk superconductor with a large number of channels is crucial for true topological degeneracy.
Jay D. Sau, B. I. Halperin, K. Flensberg, S. Das Sarma Journal reference: Phys. Rev. B 84, 144509 (2011) [pdf] DOI: 10.1103/PhysRevB.84.144509

Scheme to measure Majorana fermion lifetimes using a quantum dot 
Abstract
 We propose a setup to measure the lifetime of the parity of a pair of Majorana bound states. The proposed experiment has one edge Majorana state tunnel coupled to a quantum dot, which in turn is coupled to a metallic electrode. When the Majorana Fermions overlap, even a small relaxation rate qualitatively changes the nonlinear transport spectrum, and for strong overlap the lifetime can be read off directly from the height of a current peak. This is important for the usage of Majorana Fermions as a platform for topological quantum computing, where the parity relaxation is a limiting factor.
Martin Leijnse, Karsten Flensberg Journal reference: Phys. Rev. B 84, 140501(R) (2011) [pdf] DOI: 10.1103/PhysRevB.84.140501

Nonequilibrium Transport through a Spinful Quantum Dot with Superconducting Leads 
Abstract
 We study the nonlinear cotunneling current through a spinful quantum dot contacted by two superconducting leads. Applying a general nonequilibrium Green function formalism to an effective Kondo model, we study the rich variation in the IVcharacteristics with varying asymmetry in the tunnel coupling to source and drain electrodes. The current is found to be carried respectively by multiple Andreev reflections in the symmetric limit, and by spininduced YuShibaRussinov bound states in the strongly asymmetric limit. The interplay between these two mechanisms leads to qualitatively different IVcharacteristics in the crossover regime of intermediate symmetry, consistent with recent experimental observations of negative differential conductance and repositioned conductance peaks in subgap cotunneling spectroscopy.
B. M. Andersen, K. Flensberg, V. Koerting, J. Paaske Journal reference: Phys. Rev. Lett. 107, 256802 (2011) [pdf] DOI: 10.1103/PhysRevLett.107.256802

Image charge effects in singlemolecule junctions: Breaking of symmetries and negativedifferential resistance in a benzene singleelectron transistor 
Abstract
 Both experiments and theoretical studies have demonstrated that the interaction between the current carrying electrons and the induced polarization charge in singlemolecule junctions leads to a strong renormalization of molecular charging energies. However, the effect on electronic excitations and molecular symmetries remain unclear. Using a theoretical framework developed for semiconductor nanostructure based singleelectron transistors (SETs), we demonstrate that the image charge interaction breaks the molecular symmetries in a benzene based singlemolecule transistor operating in the Coulomb blockade regime. This results in the appearance of a socalled blocking state, which gives rise to negative differential resistance (NDR). We show that the appearance of NDR and its magnitude in the symmetrybroken benzene SET depends in a complicated way on the interplay between the manybody matrix elements, the lead tunnel coupling asymmetry, and the bias polarity. In particular, the current reducing property of the blocking state causing the NDR, is shown to vanish under strongly asymmetric tunnel couplings, when the molecule is coupled stronger to the drain electrode. The calculated IV characteristic may serve as an indicator for image charge broken molecular symmetries in experimental situations.
K. Kaasbjerg, K. Flensberg Journal reference: Phys. Rev. B 84, 115457 (2011) [pdf] DOI: 10.1103/PhysRevB.84.115457

Dephasing and hyperfine interaction in carbon nanotube double quantum dots: The clean limit 
Abstract
 We consider theoretically ${}^{13}$Chyperfine interaction induced dephasing in carbon nanotubes double quantum dots with curvature induced spinorbit coupling. For two electrons initially occupying a single dot, we calculate the average return probability after separation into the two dots, which have random nuclearspin configurations. We focus on the long time saturation value of the return probability, $P_\infty$. Because of the valley degree of freedom, the analysis is more complex than in, for example, GaAs quantum dots, which have two distinct $P_\infty$ values depending on the magnetic field. Here the prepared state and the measured state is nonunique because two electrons in the same dot are allowed in six different states. Moreover, for one electron in each dot sixteen states exist and therefore are available for being mixed by the hyperfine field. The return probability experiment is found to be strongly dependent on the prepared state, on the external magnetic fieldboth Zeeman and orbital effects  and on the spinorbit splitting. The lowest saturation value, being $P_\infty$=1/3, occurs at zero magnetic field for nanotubes with spinorbit coupling and the initial state being the groundstate, this situation is equivalent to double dots without the valley degree of freedom. In total, we report nine dynamically different situations that give $P_\infty$=1/3, 3/8, 2/5, 1/2 and for valley antisymmetric prepared states in an axial magnetic field, $P_\infty$=1. When the groundstate is prepared the ratio between the spinorbit splitting and the Zeeman energy due to a perpendicular magnetic field can tune the effective hyperfine field continuously from being three dimensional to two dimensional giving saturation values from $P_\infty$=1/3 to 3/8.
Andres A. Reynoso, Karsten Flensberg Journal reference: Phys. Rev. B 84, 205449 (2011) [pdf] DOI: 10.1103/PhysRevB.84.205449

Quantum Information Transfer between Topological and Spin Qubit Systems 
Abstract
 2010

Interactioninduced negative differential resistance in asymmetric molecular junctions 
Abstract
 Combining insights from quantum chemistry calculations with master equations, we discuss a mechanism for negative differential resistance (NDR) in molecular junctions, operated in the regime of weak tunnel coupling. The NDR originates from an interplay of orbital spatial asymmetry and strong electronelectron interaction, which causes the molecule to become trapped in a nonconducting state above a voltage threshold. We show how the desired asymmetry can be selectively introduced in individual orbitals in e.g., OPEtype molecules by functionalization with a suitable side group, which is in linear conjugation to one end of the molecule and crossconjugated to the other end.
Martin Leijnse, Wei Sun, Mogens Brøndsted Nielsen, Per Hedegård, Karsten Flensberg Journal reference: J. Chem. Phys. 134, 104107 (2011) [pdf] DOI: 10.1063/1.3560474

NonAbelian Operations on Majorana Fermions via SingleCharge Control 
Abstract
 We demonstrate that nonAbelian rotations within the degenerate groundstate manifold of a set of Majorana fermions can be realized by the addition or removal of single electrons, and propose an implementation using Coulomb blockaded quantum dots. The exchange of electrons generates rotations similar to braiding, though not in real space. Unlike braiding operations, rotations by a continuum of angles are possible, while still being partially robust against perturbations. The quantum dots can also be used for readout of the state of the Majorana system via a charge measurement.
Karsten Flensberg Journal reference: Phys. Rev. Lett. 106, 090503 (2011) [pdf] DOI: 10.1103/PhysRevLett.106.090503

Nonequilibrium transport via spininduced subgap states in superconductor/quantum dot/normal metal cotunnel junctions 
Abstract
 We study lowtemperature transport through a Coulomb blockaded quantum dot (QD) contacted by a normal (N), and a superconducting (S) electrode. Within an effective cotunneling model the conduction electron self energy is calculated to leading order in the cotunneling amplitudes and subsequently resummed to obtain the nonequilibrium Tmatrix, from which we obtain the nonlinear cotunneling conductance. For even occupied dots the system can be conceived as an effective S/Ncotunnel junction with subgap transport mediated by Andreev reflections. The net spin of an odd occupied dot, however, leads to the formation of subgap resonances inside the superconducting gap which gives rise to a characteristic peakdip structure in the differential conductance, as observed in recent experiments.
V. Koerting, B. M. Andersen, K. Flensberg, J. Paaske Journal reference: Phys. Rev. B 82, 245108 (2010) [pdf] DOI: 10.1103/PhysRevB.82.245108

Spinorbit effects in carbonnanotube double quantum dots 
Abstract
 We study the energy spectrum of symmetric double quantum dots in narrowgap carbon nanotubes with one and two electrostatically confined electrons in the presence of spinorbit and Coulomb interactions. Compared to GaAs quantum dots, the spectrum exhibits a much richer structure because of the spinorbit interaction that couples the electron's isospin to its real spin through two independent coupling constants. In a single dot, both constants combine to split the spectrum into two Kramers doublets, while the antisymmetric constant solely controls the difference in the tunneling rates of the Kramers doublets between the dots. For the twoelectron regime, the detailed structure of the spinorbit split energy spectrum is investigated as a function of detuning between the quantum dots in a 22dimensional Hilbert space within the framework of a singlelongitudinalmode model. We find a competing effect of the tunneling and Coulomb interaction. The former favors a leftright symmetric twoparticle ground state, while in the regime where the Coulomb interaction dominates over tunneling, a leftright antisymmetric ground state is found. As a result, ground states on both sides of the $(11)$$(02)$ degeneracy point may possess opposite leftright symmetry, and the electron dynamics when tuning the system from one side of the $(11)$$(02)$ degeneracy point to the other is controlled by three selection rules (in spin, isospin, and leftright symmetry). We discuss implications for the spindephasing and Pauli blockade experiments.
S. Weiss, E. I. Rashba, F. Kuemmeth, H. O. H Churchill, K. Flensberg Journal reference: Physical Review B 82, 165427 (2010) [pdf] DOI: 10.1103/PhysRevB.82.165427

Tunneling characteristic of a chain of Majorana bound states 
Abstract
 We consider theoretically tunneling characteristic of a junction between a normal metal and a chain of coupled Majorana bound states generated at crossings between topological and nontopological superconducting sections, as a result of, for example, disorder in nanowires. While an isolated Majorana state supports a resonant Andreev process, yielding a zero bias differential conductance peak of height 2e^2/h, the situation with more coupled Majorana states is distinctively different with both zeros and 2e^2/h peaks in the differential conductance. We derive a general expression for the current between a normal metal and a network of coupled Majorana bound states and describe the differential conductance spectra for a generic set of situations, including regular, disordered, and infinite chains of bound states.
Karsten Flensberg DOI: 10.1103/PhysRevB.82.180516 1009.3533v1 [pdf]

Gatedependent spinorbit coupling in multielectron carbon nanotubes 
Abstract
 Understanding how the orbital motion of electrons is coupled to the spin degree of freedom in nanoscale systems is central for applications in spinbased electronics and quantum computation. We demonstrate this coupling of spin and orbit in a carbon nanotube quantum dot in the general multielectron regime in presence of finite disorder. Further, we find a strong systematic dependence of the spinorbit coupling on the electron occupation of the quantum dot. This dependence, which even includes a sign change is not demonstrated in any other system and follows from the curvatureinduced spinorbit split Diracspectrum of the underlying graphene lattice. Our findings unambiguously show that the spinorbit coupling is a general property of nanotube quantum dots which provide a unique platform for the study of spinorbit effects and their applications.
Thomas Sand Jespersen, Kasper GroveRasmussen, Jens Paaske, Koji Muraki, Toshimasa Fujisawa, Jesper Nygård, Karsten Flensberg 1008.1600v2 [pdf]

Exchange cotunneling through quantum dots with spinorbit coupling 
Abstract
 We investigate the effects of spinorbit interaction (SOI) on the exchange cotunneling through a spinful Coulomb blockaded quantum dot. In the case of zero magnetic field, Kondo effect is shown to take place via a Kramers doublet and the SOI will merely affect the Kondo temperature. In contrast, we find that the breaking of timereversal symmetry in a finite field has a marked influence on the effective Anderson, and Kondo models for a single level. The nonlinear conductance can now be asymmetric in bias voltage and may depend strongly on direction of the magnetic field. A measurement of the angle dependence of finitefield cotunneling spectroscopy thus provides valuable information about orbital, and spin degrees of freedom and their mutual coupling.
J. Paaske, A. Andersen, K. Flensberg DOI: 10.1103/PhysRevB.82.081309 1006.2371v1 [pdf]

Bends in nanotubes allow electric spin control and coupling 
Abstract
 We investigate combined effects of spinorbit coupling and magnetic field in carbon nanotubes containing one or more bends along their length. We show how bends can be used to provide electrical control of confined spins, while spins confined in straight segments remain insensitive to electric fields. Device geometries that allow general rotation of single spins are presented and analyzed. In addition, capacitive coupling along bends provides coherent spinspin interaction, including between otherwise disconnected nanotubes, completing a universal set of one and twoqubit gates.
K. Flensberg, C. M. Marcus Journal reference: Phys. Rev. B 81, 195418 (2010) [pdf] DOI: 10.1103/PhysRevB.81.195418

Nonlinear thermoelectric properties of molecular junctions with vibrational coupling 
Abstract
 We present a detailed study of the nonlinear thermoelectric properties of a molecular junction, represented by a dissipative AndersonHolstein model. A single orbital level with strong Coulomb interaction is coupled to a localized vibrational mode and we account for both electron and phonon exchange with both electrodes, investigating how these contribute to the heat and charge transport. We calculate the efficiency and power output of the device operated as a heat to electric power converter and identify the optimal operating conditions, which are found to be qualitatively changed by the presence of the vibrational mode. Based on this study of a generic model system, we discuss the desirable properties of molecular junctions for thermoelectric applications.
M. Leijnse, M. R. Wegewijs, K. Flensberg Journal reference: Phys. Rev. B 82, 045412 (2010) [pdf] DOI: 10.1103/PhysRevB.82.045412

Transport via coupled states in a C60 peapod quantum dot 
Abstract
 We have measured systematic repetitions of avoided crossings in low temperature threeterminal transport through a carbon nanotube with encapsulated C60 molecules. We show that this is a general effect of the hybridization of a host quantum dot with an impurity. The welldefined nanotube allows identification of the properties of the impurity, which we suggest to be a chain of C60 molecules inside the nanotube. This electronic coupling between the two subsystems opens the interesting and potentially useful possibility of contacting the encapsulated molecules via the tube.
Anders Eliasen, Jens Paaske, Karsten Flensberg, Sebastian Smerat, Martin Leijnse, Maarten R. Wegewijs, Henrik I. Jørgensen, Marc Monthioux, Jesper Nygård DOI: 10.1103/PhysRevB.81.155431 1002.0477v1 [pdf]

Interactioninduced negative differential resistance in asymmetric molecular junctions 
Abstract
 2009

Electrical manipulation of spin states in a single electrostatically
gated transitionmetal complex 
Abstract
 We demonstrate an electrically controlled highspin (S=5/2) to lowspin (S=1/2) transition in a threeterminal device incorporating a single Mn2+ ion coordinated by two terpyridine ligands. By adjusting the gatevoltage we reduce the terpyridine moiety and thereby strengthen the ligandfield on the Mnatom. Adding a single electron thus stabilizes the lowspin configuration and the corresponding sequential tunnelling current is suppressed by spinblockade. From lowtemperature inelastic cotunneling spectroscopy, we infer the magnetic excitation spectrum of the molecule and uncover also a strongly gatedependent singlettriplet splitting on the lowspin side. The measured biasspectroscopy is shown to be consistent with an exact diagonalization of the Mncomplex, and an interpretation of the data is given in terms of a simplified effective model.
Edgar A. Osorio, Kasper MothPoulsen, Herre S. J. van der Zant, Jens Paaske, Per Hedegard, Karsten Flensberg, Jesper Bendix, Thomas Bjornholm DOI: 10.1021/nl9029785 0912.2640v1 [pdf]

Mesoscopic conductance fluctuations in InAs nanowirebased SNS junctions 
Abstract
 We report a systematic experimental study of mesoscopic conductance fluctuations in superconductor/normal/superconductor (SNS) devices Nb/InAsnanowire/Nb. These fluctuations far exceed their value in the normal state and strongly depend on temperature even in the lowtemperature regime. This dependence is attributed to high sensitivity of perfectly conducting channels to dephasing and the SNS fluctuations thus provide a sensitive probe of dephasing in a regime where normal transport fails to detect it. Further, the conductance fluctuations are strongly nonlinear in bias voltage and reveal subgap structure. The experimental findings are qualitatively explained in terms of multiple Andreev reflections in chaotic quantum dots with imperfect contacts.
T. S. Jespersen, M. L. Polianski, C. B. Soerensen, K. Flensberg, J. Nygaard Journal reference: New J. Phys. 11, 113025 (2009) [pdf] DOI: 10.1088/13672630/11/11/113025

Superconductivityenhanced bias spectroscopy in carbon nanotube quantum dots 
Abstract
 We study lowtemperature transport through carbon nanotube quantum dots in the Coulomb blockade regime coupled to niobiumbased superconducting leads. We observe pronounced conductance peaks at finite sourcedrain bias, which we ascribe to elastic and inelastic cotunneling processes enhanced by the coherence peaks in the density of states of the superconducting leads. The inelastic cotunneling lines display a marked dependence on the applied gate voltage which we relate to different tunnelingrenormalizations of the two subbands in the nanotube. Finally, we discuss the origin of an especially pronounced subgap structure observed in every fourth Coulomb diamond.
K. GroveRasmussen, H. I. Jørgensen, B. M. Andersen, J. Paaske, T. S. Jespersen, J. Nygård, K. Flensberg, P. E. Lindelof Journal reference: Phys. Rev. B 79, 134518 (2009) [pdf] DOI: 10.1103/PhysRevB.79.134518

Electron–electron interaction effects in quantum point contacts 
Abstract
 We consider electronelectron interaction effects in quantum point contacts on the first quantization plateau, taking into account all scattering processes. We compute the lowtemperature linear and nonlinear conductance, shot noise, and thermopower, by perturbation theory and a selfconsistent nonperturbative method. On the conductance plateau, the lowtemperature corrections are solely due to momentumnonconserving processes that change the relative number of left and rightmoving electrons. This leads to a suppression of the conductance for increasing temperature or voltage. The size of the suppression is estimated for a realistic saddlepoint potential, and is largest in the beginning of the conductance plateau. For large magnetic field, interaction effects are strongly suppressed by the Pauli principle, and hence the first spinsplit conductance plateau has a much weaker interaction correction. For the nonperturbative calculations, we use a selfconsistent nonequilibrium Green's function approach, which suggests that the conductance saturates at elevated temperatures. These results are consistent with many experimental observations related to the socalled 0.7 anomaly.
A. M. Lunde, A. De Martino, A. Schulz, R. Egger, K. Flensberg Journal reference: New J. Phys. 11, 023031 (2009) [pdf] DOI: 10.1088/13672630/11/2/023031

Electrical manipulation of spin states in a single electrostatically
gated transitionmetal complex 
Abstract
 2008

Critical and excess current through an open quantum dot: Temperature and magneticfield dependence 
Abstract
 We present measurements of temperature and magnetic field dependence of the critical current and excess current in a carbon nanotube Josephson quantum dot junction. The junction is fabricated in a controlled environment which allows for extraction of the full critical current. The measurements are performed in the open quantum dot regime, and fitted to theory with good qualitative agreement. We also show how to extract level spacing, level broadening, and charging energy of an open quantum dot from a bias spectroscopy plot.
H. Ingerslev Jørgensen, K. GroveRasmussen, K. Flensberg, P. E. Lindelof Journal reference: Phys. Rev. B 79, 155441 (2009) [pdf] DOI: 10.1103/PhysRevB.79.155441

Relaxation and Dephasing in a Twoelectron 13C Nanotube Double Quantum
Dot 
Abstract
 We use charge sensing of Pauli blockade (including spin and isospin) in a twoelectron 13C nanotube double quantum dot to measure relaxation and dephasing times. The relaxation time, T1, first decreases with parallel magnetic field then goes through a minimum in a field of 1.4 T. We attribute both results to the spinorbitmodified electronic spectrum of carbon nanotubes, which suppresses hyperfine mediated relaxation and enhances relaxation due to soft phonons. The inhomogeneous dephasing time, T2*, is consistent with previous data on hyperfine coupling strength in 13C nanotubes.
H. O. H. Churchill, F. Kuemmeth, J. W. Harlow, A. J. Bestwick, E. I. Rashba, K. Flensberg, C. H. Stwertka, T. Taychatanapat, S. K. Watson, C. M. Marcus Journal reference: Phys. Rev. Lett. 102, 166802 (2009). [pdf] DOI: 10.1103/PhysRevLett.102.166802

Interplay between interference and Coulomb interaction in the ferromagnetic Anderson model with applied magnetic field 
Abstract
 We study the competition between interference due to multiple singleparticle paths and Coulomb interaction in a simple model of an Andersonlike impurity with localmagneticfieldinduced level splitting coupled to ferromagnetic leads. The model along with its potential experimental relevance in the field of spintronics serves as a nontrivial benchmark system where various quantum transport approaches can be tested and compared. We present results for the linear conductance obtained by a spindependent implementation of the density matrix renormalization group scheme which are compared with a meanfield solution as well as a seemingly more advanced HubbardI approximation. We explain why meanfield yields nearly perfect results, while the more sophisticated HubbardI approach fails, even at a purely conceptual level since it breaks hermiticity of the related density matrix. Furthermore, we study finite bias transport through the impurity by the meanfield approach and recently developed higherorder density matrix equations. We find that the meanfield solution fails to describe the plausible results of the higherorder density matrix approach both quantitatively and qualitatively as it does not capture some essential features of the currentvoltage characteristics such as negative differential conductance.
Jonas Nyvold Pedersen, Dan Bohr, Andreas Wacker, Tomas Novotny, Peter Schmitteckert, Karsten Flensberg Journal reference: Phys. Rev. B 79, 125403 (2009) [pdf] DOI: 10.1103/PhysRevB.79.125403

Strong PolarizationInduced Reduction of Addition Energies in SingleMolecule Nanojunctions 
Abstract
 We address polarizationinduced renormalization of molecular levels in solidstate based singlemolecule transistors and focus on an organic conjugate molecule where a surprisingly large reduction of the addition energy has been observed. We have developed a scheme that combines a selfconsistent solution of a quantum chemical calculation with a realistic description of the screening environment. Our results indeed show a large reduction, and we explain this to be a consequence of both (a) a reduction of the electrostatic molecular charging energy and (b) polarization induced level shifts of the HOMO and LUMO levels. Finally, we calculate the charge stability diagram and explain at a qualitative level general features observed experimentally.
Kristen Kaasbjerg, Karsten Flensberg Journal reference: Nano Letters 8, 3809 (2008) [pdf] DOI: 10.1021/nl8021708

Gatedependent tunnelinginduced level shifts observed in carbon nanotube quantum dots 
Abstract
 We have studied electron transport in clean singlewalled carbon nanotube quantum dots. Because of the large number of Coulomb blockade diamonds simultaneously showing both shell structure and Kondo effect, we are able to perform a detailed analysis of tunneling renormalization effects. Thus determining the environment induced level shifts of this artificial atom. In shells where only one of the two orbitals is coupled strongly, we observe a marked asymmetric gatedependence of the inelastic cotunneling lines together with a systematic gate dependence of the size (and shape) of the Coulomb diamonds. These effects are all given a simple explanation in terms of secondorder perturbation theory in the tunnel coupling.
J. V. Holm, H. I. Jørgensen, K. GroveRasmussen, J. Paaske, K. Flensberg, P. E. Lindelof Journal reference: Phys. Rev. B 77, 161406(R) (2008) [pdf] DOI: 10.1103/PhysRevB.77.161406

Critical and excess current through an open quantum dot: Temperature and magneticfield dependence 
Abstract
 2007

Singlet–triplet physics and shell filling in carbon nanotube double quantum dots 
Abstract
 An artifcial twoatomic molecule, also called a double quantum dot (DQD), is an ideal system for exploring few electron physics. Spinentanglement between just two electrons can be explored in such systems where singlet and triplet states are accessible. These two spinstates can be regarded as the two states in a quantum twostate system, a socalled singlettriplet qubit. A very attractive material for realizing spin based qubits is the carbon nanotube (CNT), because it is expected to have a very long spin coherence time. Here we show the existence of a gatetunable singlettriplet qubit in a CNT DQD. We show that the CNT DQD has clear shell structures of both four and eight electrons, with the singlettriplet qubit present in the fourelectron shells. We furthermore observe inelastic cotunneling via the singlet and triplet states, which we use to probe the splitting between singlet and triplet, in good agreement with theory.
H. Ingerslev Jørgensen, K. GroveRasmussen, K. Y. Wang, A. M. Blackburn, K. Flensberg, P. E. Lindelof, D. A. Williams Journal reference: Nature Physics 4, 536  539 (2008) [pdf] DOI: 10.1038/nphys987

Coulomb Blockade of a Threeterminal Quantum Dot 
Abstract
 We study an interacting singlelevel quantum dot weakly coupled to three electrodes. When two electrodes are biased by voltages with opposite polarities, while keeping the third lead (the stem) grounded, the current through the stem is a measure of electronhole asymmetry of the dot. In this setup we calculate the stem current for both metallic and ferromagnetic (collinearly polarized) leads and discuss how the threeterminal device gives additional information compared to the usual twoterminal setup. We calculate both the sequential and cotunneling contribution for the currents. For the latter part we include a regularization procedure for the cotunneling current, which enables us to also describe the behavior at the charge degeneracy points.
Robert Andrzej Zak, Karsten Flensberg DOI: 10.1103/PhysRevB.77.045329 0711.1787v1 [pdf]

Critical Current 0−π Transition in Designed Josephson Quantum Dot Junctions 
Abstract
 We report on quantum dot based Josephson junctions designed specifically for measuring the supercurrent. From highaccuracy fitting of the currentvoltage characteristics we determine the full magnitude of the supercurrent (critical current). Strong gate modulation of the critical current is observed through several consecutive Coulomb blockade oscillations. The critical current crosses zero close to, but not at, resonance due to the socalled 0$\pi$ transition in agreement with a simple theoretical model.
H. Ingerslev Jørgensen, T. Novotný, K. GroveRasmussen, K. Flensberg, P. E. Lindelof Journal reference: NANO LETTERS 7 (8): 24412445 AUG 2007 [pdf] DOI: 10.1021/nl071152w

Electronelectron interaction effects in quantum point contacts 
Abstract
 We consider interaction effects in quantum point contacts on the first quantization plateau, taking into account all non momentumconserving processes. We compute lowtemperature linear and nonlinear conductance, shot noise, and thermopower by perturbation theory, and show that they are consistent with experimental observations on the socalled "0.7 anomaly". The full temperaturedependent conductance is obtained from selfconsistent secondorder perturbation theory and approaches ~ e^2/h at higher temperatures, but still smaller than the Fermi temperature.
Anders Mathias Lunde, Alessandro De Martino, Reinhold Egger, Karsten Flensberg 0707.1989v1 [pdf]

KondoEnhanced Andreev Tunneling in InAs Nanowire Quantum Dots 
Abstract
 We report measurements of the nonlinear conductance of InAs nanowire quantum dots coupled to superconducting leads. We observe a clear alternation between odd and even occupation of the dot, with subgappeaks at $V_{sd}=\Delta/e$ markedly stronger(weaker) than the quasiparticle tunneling peaks at $V_{sd}=2\Delta/e$ for odd(even) occupation. We attribute the enhanced $\Delta$peak to an interplay between Kondocorrelations and Andreev tunneling in dots with an odd number of spins, and substantiate this interpretation by a poor man's scaling analysis.
T. SandJespersen, J. Paaske, B. M. Andersen, K. GroveRasmussen, H. I. Jørgensen, M. Aagesen, C. Sørensen, P. E. Lindelof, K. Flensberg, J. Nygård Journal reference: Phys. Rev. Lett. 99, 126603 (2007) [ condmat/0703264v1 ] DOI: 10.1103/PhysRevLett.99.126603

Threeparticle collisions in quantum wires: Corrections to thermopower and conductance 
Abstract
 We consider the effect of electronelectron interaction on the electron transport through a finite length singlemode quantum wire with reflectionless contacts. The twoparticle scattering events cannot alter the electric current and therefore we study the effect of threeparticle collisions. Within the Boltzmann equation framework, we calculate corrections to the thermopower and conductance to the leading order in the interaction and in the length of wire $L$. We check explicitly that the threeparticle collision rate is identically zero in the case of several integrable interaction potentials. In the general (nonintegrable) case, we find a positive contribution to the thermopower to leading order in $L$. The processes giving rise to the correction involve electron states deep in the Fermi sea. Therefore the correction follows an activation law with the characteristic energy of the order of the Fermi energy for the electrons in the wire.
Anders Mathias Lunde, Karsten Flensberg, Leonid I. Glazman Journal reference: Phys. Rev. B 75, 245418 (2007) [ condmat/0703042v1 ] DOI: 10.1103/PhysRevB.75.245418

Singlet–triplet physics and shell filling in carbon nanotube double quantum dots 
Abstract
 2006

Spinorbit induced spinqubit control in nanowires 
Abstract
 We elaborate on a number of issues concerning our recent proposal for spinqubit manipulation in nanowires using the spinorbit coupling. We discuss the experimental status and describe in further detail the scheme for singlequbit rotations. We present a derivation of the effective twoqubit coupling which can be extended to higher orders in the Coulomb interaction. The analytic expression for the coupling strength is shown to agree with numerics.
Christian Flindt, Anders S. Sorensen, Karsten Flensberg Journal reference: J. Phys.: Conf. Ser. 61, 302 (2007) [ condmat/0612293v1 ] DOI: 10.1088/17426596/61/1/061

SpinOrbit Mediated Control of Spin Qubits 
Abstract
 We propose to use the spinorbit interaction as a means to control electron spins in quantum dots, enabling both single qubit and two qubit operations. Very fast single qubit operations may be achieved by temporarily displacing the electrons. For two qubit operations the coupling mechanism is based on a combination of the spinorbit coupling and the mutual longranged Coulomb interaction. Compared to existing schemes using the exchange coupling, the spinorbit induced coupling is less sensitive to random electrical fluctuations in the electrodes defining the quantum dots.
Christian Flindt, Anders S. Sorensen, Karsten Flensberg Journal reference: Phys. Rev. Lett. 97, 240501 (2006) [ condmat/0603559v2 ] DOI: 10.1103/PhysRevLett.97.240501

InteractionInduced Resonance in Conductance and Thermopower of Quantum Wires 
Abstract
 We study the effect of electronelectron interaction on the transport properties of short clean quantum wires adiabatically connected to reservoirs. Interactions lead to resonances in a multichannel wire at particular values of the Fermi energy. We investigate in detail the resonance in a twochannel wire. The (negative) conductance correction peaks at the resonance, and decays exponentially as the Fermi energy is tuned away; the resonance width being given by the temperature. Likewise, the thermopower shows a characteristic structure, which is surprisingly well approximated by the socalled Mott formula. Finally, fourfold splitting of the resonance in a magnetic field provides a unique signature of the effect.
Anders Mathias Lunde, Karsten Flensberg, Leonid I. Glazman Journal reference: Phys. Rev. Lett. 97, 256802 (2006) [ condmat/0609228v1 ] DOI: 10.1103/PhysRevLett.97.256802

Electron Transport in SingleWall Carbon Nanotube Weak Links in the FabryPerot Regime 
Abstract
 We fabricated reproducible high transparency superconducting contacts consisting of superconducting Ti/Al/Ti trilayers to gated singlewalled carbon nanotubes (SWCNTs). The reported semiconducting SWCNT have normal state differential conductance up to $3e^2/h$ and exhibit clear FabryPerot interference patterns in the bias spectroscopy plot. We observed subharmonic gap structure in the differential conductance and a distinct peak in the conductance at zero bias which is interpreted as a manifestation of a supercurrent. The gate dependence of this supercurrent as well as the excess current are examined and compared to a coherent theory of superconducting point contacts with good agreement.
H. I. Jørgensen, K. GroveRasmussen, T. Novotný, K. Flensberg, P. E. Lindelof Journal reference: Phys. Rev. Lett. 96, 207003 (2006) [ condmat/0510200v4 ] DOI: 10.1103/PhysRevLett.96.207003

Spinorbit induced spinqubit control in nanowires 
Abstract
 2005

Josephson current through a molecular transistor in a dissipative environment 
Abstract
 We study the Josephson coupling between two superconductors through a single correlated molecular level, including Coulomb interaction on the level and coupling to a bosonic environment. All calculations are done to the lowest, i.e., the fourth, order in the tunneling coupling and we find a suppression of the supercurrent due to the combined effect of the Coulomb interaction and the coupling to environmental degrees of freedom. Both analytic and numerical results are presented.
Tomas Novotny, Alessandra Rossini, Karsten Flensberg Journal reference: Phys. Rev. B 72, 224502 (2005) [ condmat/0508071v3 ] DOI: 10.1103/PhysRevB.72.224502

Nonequilibrium theory of Coulomb blockade in open quantum dots 
Abstract
 We develop a nonequilibrium theory to describe weak Coulomb blockade effects in open quantum dots. Working within the bosonized description of electrons in the point contacts, we expose deficiencies in earlier applications of this method, and address them using a 1/N expansion in the inverse number of channels. At leading order this yields the selfconsistent potential for the charging interaction. Coulomb blockade effects arise as quantum corrections to transport at the next order. Our approach unifies the phase functional and bosonization approaches to the problem, as well as providing a simple picture for the conductance corrections in terms of renormalization of the dot's elastic scattering matrix, which is obtained also by elementary perturbation theory. For the case of ideal contacts, a symmetry argument immediately allows us to conclude that interactions give no signature in the averaged conductance. Nonequilibrium applications to the pumped current in a quantum pump are worked out in detail.
Piet W. Brouwer, Austen Lamacraft, Karsten Flensberg Journal reference: Phys. Rev. B 72, 075316 (2005) [ condmat/0502518v2 ] DOI: 10.1103/PhysRevB.72.075316

Weak Coulomb Blockade Effect in Quantum Dots 
Abstract
 We develop the general nonequilibrium theory of transport through a quantum dot, including Coulomb Blockade effects via a 1/N expansion, where N is the number of scattering channels. At lowest order we recover the Landauer formula for the current plus a selfconsistent equation for the dot potential. We obtain the leading corrections and compare with earlier approaches. Finally, we show that to leading and next leading order in 1/N there is no interaction correction to the weak localization, in contrast to previous theories, but consistent with experiments by Huibers et al. [Phys. Rev. Lett. 81, 1917 (1998)], where N=4.
Piet W. Brouwer, Austen Lamacraft, Karsten Flensberg Journal reference: Phys. Rev. Lett. 94, 136801 (2005) [ condmat/0409244v2 ] DOI: 10.1103/PhysRevLett.94.136801

Electron–vibron coupling in suspended nanotubes 
Abstract
 We consider the electronvibron coupling in suspended nanotube quantum dots. Modelling the tube as an elastic medium, we study the possible coupling mechanism for exciting the stretching mode in a singleelectrontransistor setup. Both the forces due to the longitudinal and the transverse fields are included. The effect of the longitudinal field is found to be too small to be seen in experiment. In contrast, the transverse field which couple to the stretching mode via the bending of the tube can in some cases give sizeable FranckCondon factors. However, the length dependence is not compatible with recent experiments [Sapmaz et al. condmat/0508270].
Karsten Flensberg Journal reference: New Journal of Physics 8, 5 (2006) [ condmat/0510044v1 ] DOI: 10.1088/13672630/8/1/005

On the Mott formula for the thermopower of noninteracting electrons in quantum point contacts 
Abstract
 We calculate the linear response thermopower S of a quantum point contact using the Landauer formula and therefore assume noninteracting electrons. The purpose of the paper, is to compare analytically and numerically the linear thermopower S of noninteracting electrons to the low temperature approximation, S^1=(pi^2/3e)k^2 T d(ln G(mu,T=0))/dmu, and the socalled Mott expression, S^M=(pi^2/3e)k^2 T d(ln G(mu,T))/dmu, where G(mu,T) is the (temperature dependent) conductance. This comparison is important, since the Mott formula is often used to detect deviations from singleparticle behavior in the thermopower of a point contact.
Anders Mathias Lunde, Karsten Flensberg Journal reference: J. Phys.: Condens. Matter 17 (2005) 38793884. [ condmat/0506050v1 ] DOI: 10.1088/09538984/17/25/014

Noncollinear magnetoconductance of a quantum dot 
Abstract
 We study theoretically the linear conductance of a quantum dot connected to ferromagnetic leads. The dot level is split due to a noncollinear magnetic field or intrinsic magnetization. The system is studied in the noninteracting approximation, where an exact solution is given, and, furthermore, with Coulomb correlations in the weak tunneling limit. For the noninteracting case, we find an antiresonance for a particular direction of the applied field, noncollinear to the parallel magnetization directions of the leads. The antiresonance is destroyed by the correlations, giving rise to an interaction induced enhancement of the conductance. The angular dependence of the conductance is thus distinctly different for the interacting and noninteracting cases when the magnetizations of the leads are parallel. However, for antiparallel lead magnetizations the interactions do not alter the angle dependence significantly.
Jonas N. Pedersen, Jesper Q. Thomassen, Karsten Flensberg Journal reference: Phys. Rev. B 72, 045341 (2005) [ condmat/0412145v2 ] DOI: 10.1103/PhysRevB.72.045341

Rectification in single molecular dimers with strong polaron effect 
Abstract
 We study theoretically the transport properties of a molecular two level system with large electronvibron coupling in the Coulomb blockade regime. We show that when the electronvibron coupling induces polaron states, the currentvoltage characteristic becomes strongly asymmetric because, in one current direction, one of the polaron state blocks the current through the other. This situation occurs when the coupling between the polaron states is smaller than the coupling to the leads. We discuss the relevance of our calculation for experiments on C_140 molecules.
Gregers A. Kaat, Karsten Flensberg Journal reference: Phys. Rev. B 71, 155408 (2005) [ condmat/0411173v2 ] DOI: 10.1103/PhysRevB.71.155408

Josephson current through a molecular transistor in a dissipative environment 
Abstract
 2004

Vibrational Sidebands and the Kondo Effect in Molecular Transistors 
Abstract
 Electron transport through molecular quantum dots coupled to a single vibrational mode is studied in the Kondo regime. We apply a generalized SchriefferWolff transformation to determine the effective lowenergy spinspinvibroninteraction. From this model we calculate the nonlinear conductance and find Kondo sidebands located at biasvoltages equal to multiples of the vibron frequency. Due to selection rules, the sidepeaks are found to have strong gatevoltage dependences, which can be tested experimentally. In the limit of weak electronvibron coupling, we employ a perturbative renormalization group scheme to calculate analytically the nonlinear conductance.
Jens Paaske, Karsten Flensberg Journal reference: Phys. Rev. Lett. 94, 176801 (2005) [ condmat/0409158v1 ] DOI: 10.1103/PhysRevLett.94.176801

Intershell resistance in multiwall carbon nanotubes: A Coulomb drag study 
Abstract
 We calculate the intershell resistance R_{21} in a multiwall carbon nanotube as a function of temperature T and Fermi level (e.g. a gate voltage), varying the chirality of the inner and outer tubes. This is done in a socalled Coulomb drag setup, where a current I_1 in one shell induces a voltage drop V_2 in another shell by the screened Coulomb interaction between the shells neglecting the intershell tunnelling. We provide benchmark results for R_{21}=V_2/I_1 within the Fermi liquid theory using Boltzmann equations. The band structure gives rise to strongly chirality dependent suppression effects for the Coulomb drag between different tubes due to selection rules combined with mismatching of wave vector and crystal angular momentum conservation near the Fermi level. This gives rise to orders of magnitude changes in R_{21} and even the sign of R_{21} can change depending on the chirality of the inner and outer tube and misalignment of inner and outer tube Fermi levels. However for any tube combination, we predict a dip (or peak) in R_{21} as a function of gate voltage, since R_{21} vanishes at the electronhole symmetry point. As a byproduct, we classified all metallic tubes into either zigzaglike or armchairlike, which have two different nonzero crystal angular momenta m_a, m_b and only zero angular momentum, respectively.
A. M. Lunde, K. Flensberg, A. P. Jauho Journal reference: Phys. Rev. B 71, 125408 (2005). [ condmat/0408112v1 ] DOI: 10.1103/PhysRevB.71.125408

Dissipative tunneling and orthogonality catastrophe in molecular transistors 
Abstract
 Transport through molecular devices with weak tunnel coupling to the leads but with strong coupling to a single vibrational mode is considered in the case where the vibration is damped by coupling to the environment. In particular, we investigate what influence the electrostatic coupling of the charge on the molecule to the vibrational modes of the environment has on the $I$$V$ characteristics. We find that, for comparable characteristic length scales of the vanderWaals and electrostatic interaction of the molecule with the environmental vibrational modes, the $I$$V$ characteristics are qualitatively changed from what one would expect from orthogonality catastrophe and develop a steplike discontinuity at the onset of conduction. For the case of very different length scales, we recover dissipation consistent with modeling the electrostatic forces as an external influence on the system comprised of molecule and substrate, which implies the appearance of orthogonality catastrophe, in accord with previous results.
Stephan Braig, Karsten Flensberg Journal reference: Phys. Rev. B 70, 085317 (2004) [ condmat/0401347v2 ] DOI: 10.1103/PhysRevB.70.085317

Vibrational Sidebands and the Kondo Effect in Molecular Transistors 
Abstract
 2003

Tunneling broadening of vibrational sidebands in molecular transistors 
Abstract
 Transport through molecular quantum dots coupled to a single vibration mode is studied in the case with strong coupling to the leads. We use an expansion in the correlation between electrons on the molecule and electrons in the leads and show that the tunneling broadening is strongly suppressed by the combination of the Pauli principle and the quantization of the oscillator. As a consequence the first FrankCondon step is sharper than the higher order ones, and its width, when compared to the bare tunneling strength, is reduced by the overlap between the groundstates of the displaced and the nondisplaced oscillator.
Karsten Flensberg Journal reference: Phys. Rev. B 68, 205323 (2003) [ condmat/0302193v3 ] DOI: 10.1103/PhysRevB.68.205323

Vibrational sidebands and dissipative tunneling in molecular transistors 
Abstract
 Transport through molecular devices with strong coupling to a single vibrational mode is considered in the case where the vibration is damped by coupling to the environment. We focus on the weak tunneling limit, for which a rate equation approach is valid. The role of the environment can be characterized by a frictional damping term $\mysig(\omega)$ and corresponding frequency shift. We consider a molecule that is attached to a substrate, leading to frequencydependent frictional damping of the single oscillator mode of the molecule, and compare it to a reference model with frequencyindependent damping featuring a constant quality factor $Q$. For large values of $Q$, the transport is governed by tunneling between displaced oscillator states giving rise to the wellknown series of the FrankCondon steps, while at small $Q$, there is a crossover to the classical regime with an energy gap given by the classical displacement energy. Using realistic values for the elastic properties of the substrate and the size of the molecule, we calculate $I$$V$ curves and find qualitative agreement between our theory and recent experiments on $C_{60}$ singlemolecule devices.
Stephan Braig, Karsten Flensberg Journal reference: Phys. Rev. B 68, 205324 (2003) [ condmat/0303236v3 ] DOI: 10.1103/PhysRevB.68.205324

Tunneling broadening of vibrational sidebands in molecular transistors 
Abstract
 2002

Sign reversal of drag in bilayer systems with inplane periodic potential modulation 
Abstract
 We develop a theory for describing frictional drag in bilayer systems with inplane periodic potential modulations, and use it to investigate the drag between bilayer systems in which one of the layers is modulated in one direction. At low temperatures, as the density of carriers in the modulated layer is changed, we show that the transresistivity component in the direction of modulation can change its sign. We also give a physical explanation for this behavior.
Audrius Alkauskas, Karsten Flensberg, Ben YuKuang Hu, AnttiPekka Jauho Journal reference: Phys. Rev. B 66, 201304 (2002) [ condmat/0206340v1 ] DOI: 10.1103/PhysRevB.66.201304

Sign reversal of drag in bilayer systems with inplane periodic potential modulation 
Abstract
 2001

Conductance of Rashba spinsplit systems with ferromagnetic contacts 
Abstract
 We study theoretically the conductance of heterostructures with ferromagnetic conductors (F) and a two dimensional electron gas with Rashba spinorbit interaction (R) using the LandauerB\"{u}ttiker formalism. Assuming a onedimensional model, we first find the $S$matrix for the FR interface. This result is then applied to different devices such as a FRF structure, first suggested by Datta and Das[Appl. Phys. Lett. 56, 665 (1990)]. We find analytic results for the conductance for the case of collinear magnetization.
Morten Hogsbro Larsen, A. Mathias Lunde, Karsten Flensberg Journal reference: Phys. Rev. B 66, 033304 (2002) [ condmat/0112175v1 ] DOI: 10.1103/PhysRevB.66.033304

Mesoscopic fluctuations of Coulomb drag between quasiballistic onedimensional wires 
Abstract
 Quasiballistic 1D quantum wires are known to have a conductance of the order of 2e^2/h, with small sampletosample fluctuations. We present a study of the transconductance G_12 of two Coulombcoupled quasiballistic wires, i.e., we consider the Coulomb drag geometry. We show that the fluctuations in G_12 differ dramatically from those of the diagonal conductance G_ii: the fluctuations are large, and can even exceed the mean value, thus implying a possible reversal of the induced drag current. We report extensive numerical simulations elucidating the fluctuations, both for correlated and uncorrelated disorder. We also present analytic arguments, which fully account for the trends observed numerically.
Niels Asger Mortensen, Karsten Flensberg, AnttiPekka Jauho Journal reference: Phys. Rev. B 65, 085317 (2002) . [ condmat/0108263v2 ] DOI: 10.1103/PhysRevB.65.085317

Coulomb drag in phasecoherent mesoscopic structures  Numerical study
of disordered 1Dwires 
Abstract
 We study Coulomb drag between two parallel disordered mesoscopic 1Dwires. By numerical ensemble averaging we calculate the statistical properties of the transconductance G_21 including its distribution. For wires with mutually uncorrelated disorder potentials we find that the mean value is finite, but with comparable fluctuations so that signreversal is possible. For identical disorder potentials the mean value and the fluctuations nare enhanced compared to the case of uncorrelated disorder.
Niels Asger Mortensen, Karsten Flensberg, AnttiPekka Jauho Journal reference: Springer Proceedings in Physics 87, 1347 (2001) [ condmat/0108204v1 ]

Coulomb Drag in the Mesoscopic Regime 
Abstract
 We present a theory for Coulomb drag between two mesoscopic systems which expresses the drag in terms of scattering matrices and wave functions. The formalism can be applied to both ballistic and disordered systems and the consequences can be studied either by numerical simulations or analytic means such as perturbation theory or random matrix theory. The physics of Coulomb drag in the mesoscopic regime is very different from Coulomb drag between extended electron systems. In the mesoscopic regime we in general find fluctuations of the drag comparable to the mean value. Examples are the vanishing average drag for chaotic 2Dsystems and the dominating fluctuations of drag between quasiballistic wires with almost ideal transmission.
Niels Asger Mortensen, Karsten Flensberg, AnttiPekka Jauho Journal reference: Physica Scripta T101, 177 (2002). [ condmat/0108203v1 ] DOI: 10.1238/Physica.Topical.101a00177

Diffusion equation and spin drag in spinpolarized transport 
Abstract
 We study the role of electronelectron interactions for spin polarized transport using the Boltzmann equation and derive a set of coupled transport equations. For spin polarized transport the electronelectron interactions are important, because they tend to equilibrate the momentum of the two spin species. This ``spin drag'' effect enhances the resistivity of the system. The enhancement is stronger the lower the dimension and should be measurable in for example a two dimensional electron gas with ferromagnetic contacts. We also include spin flip scattering which has two effects: it equilibrates the spin density imbalance and, provided it has a non swave component, also the current imbalance.
Karsten Flensberg, Thomas Stibius Jensen, Niels Asger Mortensen Journal reference: Phys. Rev. B 64, 245308 (2001). [ condmat/0107149v1 ] DOI: 10.1103/PhysRevB.64.245308

The anomalous 0.5 and 0.7 conductance plateaus in quantum point contacts 
Abstract
 The anomalous 0.5 and 0.7 conductance plateaus in quantum point contacts in zero magnetic field are analyzed within a phenomenological model. The model utilizes the LandauerButtiker formalism and involves enhanced spin correlations and thermal depopulation of spin subbands. In particular we can account for the plateau values 0.5 and 0.7, as well as the unusual temperature and magnetic field dependences of the 0.7 plateau. Finally, the model predicts the possibility of coexisting 0.5 and 0.7 plateaus.
Henrik Bruus, Vadim V. Cheianov, Karsten Flensberg Journal reference: Physica E 10, 97 (2001) [ condmat/0106504v1 ] DOI: 10.1016/S13869477(01)000613

Conductance of Rashba spinsplit systems with ferromagnetic contacts 
Abstract
 2000

Coulomb Drag in Coherent Mesoscopic Systems 
Abstract
 We present a theory for Coulomb drag between two mesoscopic systems. Our formalism expresses the drag in terms of scattering matrices and wave functions, and its range of validity covers both ballistic and disordered systems. The consequences can be worked out either by analytic means, such as the random matrix theory, or by numerical simulations. We show that Coulomb drag is sensitive to localized states, which usual transport measurements do not probe. For chaotic 2Dsystems we find a vanishing average drag, with a nonzero variance. Disordered 1Dwires show a finite drag, with a large variance, giving rise to a possible sign change of the induced current.
Niels Asger Mortensen, Karsten Flensberg, AnttiPekka Jauho Journal reference: Phys. Rev. Lett. 86, 1841 (2001). [ condmat/0007046v2 ] DOI: 10.1103/PhysRevLett.86.1841

Dephasing in semiconductor–superconductor structures by coupling to a voltage probe 
Abstract
 We study dephasing in semiconductorsuperconductor structures caused by coupling to a voltage probe. We consider structures where the semiconductor consists of two scattering regions between which partial dephasing is possible. As a particular example we consider a situation with a doublebarrier junction in the normal region. For a singlemode system we study the conductance both as a function of the position of the Fermi level and as a function of the barrier transparency. At resonance, where the doublebarrier is fully transparent, we study the suppression of the ideal factoroftwo enhancement of the conductance when a finite coupling to the voltage probe is taken into account.
Niels Asger Mortensen, AnttiPekka Jauho, Karsten Flensberg Journal reference: Superlattice Microstr. 28, 67 (2000). [ condmat/9909029v2 ] DOI: 10.1006/spmi.2000.0890

From mesoscopic magnetism to the anomalous 0.7 conductance plateau 
Abstract
 We present a simple phenomenological model which offers a unifying interpretation of the experimental observations on the 0.7 conductance anomaly of quantum point contacts. The model utilizes the LandauerButtiker formalism and involves enhanced spin correlations and thermal depopulation of spin subbands. In particular our model can account for the plateau value 0.7 and the unusual temperature and magnetic field dependence. Furthermore it predicts an anomalous suppression of shot noise at the 0.7 plateau.
Henrik Bruus, Vadim V. Cheianov, Karsten Flensberg condmat/0002338v1 [pdf]

Coulomb Drag in Coherent Mesoscopic Systems 
Abstract
 1999

Andreev scattering and conductance enhancement in mesoscopic
semiconductorsuperconductor junctions 
Abstract
 An inherent difficulty in studying mesoscopic effects in semiconductorsuperconductor hybrid structures is the large Schottky barrier which often forms at the interface. A large technological effort has been invested in in improving the contact between the superconductor and the twodimensional electron gas (2DEG) of a semiconductor heterostructure, and in recent years this has become possible for e.g. GaAsAl, GaAsIn, and InAsNb junctions. This development motivates quantitative theoretical modeling of samplespecific transport properties. The aim of our work is to model the conducting properties of a ballistic 2DEGS interfaces with a QPC in the normal region and also to take into account scattering due to a weak Schottky barrier and/or nonmatching Fermi properties of the semiconductor and superconductor.
Niels Asger Mortensen, AnttiPekka Jauho, Karsten Flensberg Journal reference: Extended abstracts of Electron Transport in Mesoscopic Systems, pp. 120121 (1999) [ condmat/9911372v1 ]

Contact resistance of quantum tubes 
Abstract
 We consider the conductance of a quantum tube connected to a metallic contact. The number of angular momentum states that the tube can support depends on the strength of the radial confinement. We calculate the transmission coefficients which yield the conductance via the Landauer formula. We relate our results to armchair carbon nanotubes embedded in a metal. For Al and Au contacts and tubes with a realistic radial confinement we find that the transmission can be close to unity corresponding to a contact resistance close to h/2e^2 per band at the Fermi level in the carbon nanotube.
Niels Asger Mortensen, Kristinn Johnsen, AnttiPekka Jauho, Karsten Flensberg Journal reference: Superlattices and Microstructures 26, 351 (1999). [ condmat/9907026v2 ] DOI: 10.1006/spmi.1999.0812

Screening, nonadiabaticity, and quantized acoustoelectric current 
Abstract
 Quantized singleelectron transport driven by surface acoustic waves (SAW) through a pinchedoff narrow constriction is studied theoretically. Longrange Coulomb interaction causes the tunneling coupling between the twodimensional electron gas (2DEG) and the moving minimum of the SAWinduced potential to decay rapidly with time. The energy scale, associated with the characteristic time of this decay, controls both the width of the transition regions between the plateaus and the slope of the plateaus. This sets a limit for the accuracy of the quantization of acoustoelectric current at low temperature.
M. Pustilnik, K. Flensberg, Q. Niu Journal reference: J. Low Temp. Phys. 118, 571 (2000) [ condmat/9909187v2 ]

Nonadiabaticity and singleelectron transport driven by surface acoustic waves 
Abstract
 Singleelectron transport driven by surface acoustic waves (SAW) through a narrow constriction, formed in twodimensional electron gas, is studied theoretically. Due to longrange Coulomb interaction, the tunneling coupling between the electron gas and the moving minimum of the SAWinduced potential rapidly decays with time. As a result, nonadiabaticiy sets a limit for the accuracy of the quantization of acoustoelectric current.
Karsten Flensberg, Qian Niu, Michael Pustilnik Journal reference: Phys. Rev. B 60, R16291 (1999) [ condmat/9908096v2 ] DOI: 10.1103/PhysRevB.60.R16291

TOWARDS SINGLEELECTRON METROLOGY 
Abstract
 We review the status of the understanding of singleelectron transport (SET) devices with respect to their applicability in metrology. Their envisioned role as the basis of a highprecision electrical standard is outlined and is discussed in the context of other standards. The operation principles of single electron transistors, turnstiles and pumps are explained and the fundamental limits of these devices are discussed in detail. We describe the various physical mechanisms that influence the device uncertainty and review the analytical and numerical methods needed to calculate the intrinsic uncertainty and to optimise the fabrication and operation parameters. Recent experimental results are evaluated and compared with theoretical predictions. Although there are discrepancies between theory and experiments, the intrinsic uncertainty is already small enough to start preparing for the first SETbased metrological applications.
Karsten Flensberg, Arkadi A. Odintsov, Feike Liefrink, Paul Teunissen Journal reference: Int. J. Mod. Phys. B 13, 26512687 (1999) [ condmat/9908219v1 ] DOI: 10.1142/S0217979299002587

Conductance enhancement in quantumpointcontact semiconductorsuperconductor devices 
Abstract
 We present numerical calculations of the conductance of an interface between a phasecoherent twodimensional electron gas and a superconductor with a quantum point contact in the normal region. Using a scattering matrix approach we reconsider the geometry of De Raedt, Michielsen, and Klapwijk [Phys. Rev. B, 50, 631 (1994)] which was studied within the timedependent Bogoliubovde Gennes formalism. We find that the factoroftwo enhancement of the conductance G_NS compared to the normal state conductance G_N for ideal interfaces may be suppressed for interfaces with a quantum point contact with only a few propagating modes. The suppression is found to depend strongly on the position of the Fermi level. We also study the suppression due to a barrier at the interface and find an anomalous behavior caused by quasiparticle interference. Finally, we consider the limit of sequential tunneling and find a suppression of the factoroftwo enhancement which may explain the absence of conductance enhancement in experiments on metalsuperconductor structures.
Niels Asger Mortensen, AnttiPekka Jauho, Karsten Flensberg, Henning Schomerus Journal reference: Phys. Rev. B 60, 13762 (1999). [ condmat/9903205v2 ] DOI: 10.1103/PhysRevB.60.13762

Andreev scattering and conductance enhancement in mesoscopic
semiconductorsuperconductor junctions 
Abstract
 1998

Angle dependence of Andreev scattering at semiconductor–superconductor interfaces 
Abstract
 We study the angle dependence of the Andreev scattering at a semiconductorsuperconductor interface, generalizing the onedimensional theory of Blonder, Tinkham and Klapwijk. An increase of the momentum parallel to the interface leads to suppression of the probability of Andreev reflection and increase of the probability of normal reflection. We show that in the presence of a Fermi velocity mismatch between the semiconductor and the superconductor the angles of incidence and transmission are related according to the wellknown Snell's law in optics. As a consequence there is a critical angle of incidence above which only normal reflection exists. For two and threedimensional interfaces a lower excess current compared to ballistic transport with perpendicular incidence is found. Thus, the onedimensional BTK model overestimates the barrier strength for two and threedimensional interfaces.
Niels Asger Mortensen, Karsten Flensberg, AnttiPekka Jauho Journal reference: Phys. Rev. B 59, 10176 (1999). [ condmat/9807049v3 ] DOI: 10.1103/PhysRevB.59.10176

Localized plasmons in point contacts 
Abstract
 Using a hydrodynamic model of the electron fluid in a point contact geometry we show that localized plasmons are likely to exist near the constriction. We attempt to relate these plasmons with the recent experimental observation of deviations of the quantum point contact conductance from ideal integer quantization. As a function of temperature this deviation exhibits an activated behavior, exp(T_a/T), with a density dependent activation temperature T_a of the order of 2 K. We suggest that T_a can be identified with the energy needed to excite localized plasmons, and we discuss the conductance deviations in terms of a simple theoretical model involving quasiparticle lifetime broadening due to coupling to the localized plasmons.
Henrik Bruus, Karsten Flensberg Journal reference: Semicond. Sci. Technol. vol 13, A30A32 (1998) [ condmat/9807342v1 ]

Coulomb Drag of Luttinger Liquids and Quantum Hall Edges 
Abstract
 We study the transconductance for two coupled onedimensional wires or edge states described by Luttinger liquid models. The wires are assumed to interact over a finite segment. We find for the interaction parameter $g=1/2$ that the drag rate is finite at zero temperature, which cannot occur in a Fermiliquid system. The zero temperature drag is, however, cut off at low temperature due to the finite length of the wires. We also consider edge states in the fractional quantum Hall regime, and we suggest that the low temperature enhancement of the drag effect might be seen in the fractional quantum Hall regime.
Karsten Flensberg Journal reference: Phys. Rev. Lett. 81, 184 (1998) [ condmat/9802220v2 ] DOI: 10.1103/PhysRevLett.81.184

Angle dependence of Andreev scattering at semiconductor–superconductor interfaces 
Abstract
 1997

Frictional drag between quantum wells mediated by phonon exchange 
Abstract
 We use the Kubo formalism to evaluate the contribution of acoustic phonon exchange to the frictional drag between nearby twodimensional electron systems. In the case of free phonons, we find a divergent drag rate ($\tau_{D}^{1}$). However, $\tau_{D}^{1}$ becomes finite when phonon scattering from either lattice imperfections or electronic excitations is accounted for. In the case of GaAs quantum wells, we find that for a phonon mean free path $\ell_{ph}$ smaller than a critical value, imperfection scattering dominates and the drag rate varies as $ln (\ell_{ph}/d)$ over many orders of magnitude of the layer separation $d$. When $\ell_{ph}$ exceeds the critical value, the drag rate is dominated by coupling through an electronphonon collective mode localized in the vicinity of the electron layers. We argue that the coupled electronphonon mode may be observable for realistic parameters. Our theory is in good agreement with experimental results for the temperature, density, and $d$dependence of the drag rate.
Martin Bonsager, Karsten Flensberg, Ben YuKuang Hu, Allan H. MacDonald Journal reference: Phys. Rev. B 57, 7085 (1998). [ condmat/9707111v1 ] DOI: 10.1103/PhysRevB.57.7085

Frictional Coulomb drag in strong magnetic fields 
Abstract
 A treatment of frictional Coulomb drag between two 2dimensional electron layers in a strong perpendicular magnetic field, within the independent electron picture, is presented. Assuming fully resolved Landau levels, the linear response theory expression for the transresistivity $\rho_{21}$ is evaluated using diagrammatic techniques. The transresistivity is given by an integral over energy and momentum transfer weighted by the product of the screened interlayer interaction and the phasespace for scattering events. We demonstrate, by a numerical analysis of the transresistivity, that for wellresolved Landau levels the interplay between these two factors leads to characteristic features in both the magnetic field and the temperature dependence of $\rho_{21}$. Numerical results are compared with recent experiments.
M. C. Bonsager, K. Flensberg, B. YK. Hu, A. P. Jauho Journal reference: Phys. Rev. B 56, 10314 (1997). [ condmat/9704232v1 ] DOI: 10.1103/PhysRevB.56.10314

Correlation Effects on the Coupled Plasmon Modes of a Double Quantum Well 
Abstract
 At temperatures comparable to the Fermi temperature, we have measured a plasmon enhanced Coulomb drag in a GaAs/AlGaAs double quantum well electron system. This measurement provides a probe of the manybody corrections to the coupled plasmon modes, and we present a detailed comparison between experiment and theory testing the validity of local field theories. Using a perpendicular magnetic field to raise the magnetoplasmon energy we can induce a crossover to singleparticle Coulomb scattering.
N P R Hill, J. T. Nicholls, E. H. Linfield, M. Pepper, D. A. Ritchie, G. A. C. Jones, Ben YuKuang Hu, Karsten Flensberg Journal reference: Phys. Rev. Lett. 78, 2204 (1997). [ condmat/9703066v1 ] DOI: 10.1103/PhysRevLett.78.2204

Coherentphotonassisted cotunneling in a Coulomb blockade device 
Abstract
 We study cotunneling in a double junction Coulomb blockade device under the influence of time dependent potentials. It is shown that the acbias leads to photon assisted cotunneling which in some cases may dominate the transport. We derive a general nonperturbative expression for the tunneling current in the presence of oscillating potentials and give a perturbative expression for the photon assisted cotunneling current.
Karsten Flensberg Journal reference: Phys. Rev. B 55, 13118 (1997). [ condmat/9607139v3 ] DOI: 10.1103/PhysRevB.55.13118

Frictional drag between quantum wells mediated by phonon exchange 
Abstract
 1996

Observation of scaling behavior in a Coulomb blockade system 
Abstract
 We describe two experiments to study the influence of fluctuations in the electron charge on the transport properties of a quantum dot. First, we scan a device from single to double quantumdot behavior by varying the conductance of a connecting point contact. Second, we measure the dependence of the charging energy on the conductance of the barriers. The experiments are compared with traces obtained from a theory based on a Luttinger liquid type description. This theory predicts a scaling behaviour of the charging energy, in good agreement with our experiments.
K. Flensberg, L. W. Molenkamp condmat/9607138v1 [pdf]

MagnetoCoulomb Drag: Interplay of ElectronElectron Interactions and Landau Quantization 
Abstract
 We use the Kubo formalism to calculate the transresistivity $\rho_{21}$ for carriers in coupled quantum wells in a large perpendicular magnetic field $B$. We find that $\rho_{21}$ is enhanced by approximately 50100 times over that of the B=0 case in the interplateau regions of the integer quantum Hall effect. The presence of both electronelectron interactions and Landau quantization results in (i) a twinpeaked structure of $\rho_{21}(B)$ in the interplateau regions at low temperatures, and, (ii) for the chemical potential at the center of a Landau level band, a peaked temperature dependence of $\rho_{21}(T)/T^2$.
M. C. Bonsager, K. Flensberg, B. Y. K. Hu, A. P. Jauho Journal reference: Phys. Rev. Lett. 77, 1366 (1996) [ condmat/9603110v1 ] DOI: 10.1103/PhysRevLett.77.1366

Observation of scaling behavior in a Coulomb blockade system 
Abstract
 1995

Electronelectron scattering in linear transport in twodimensional systems 
Abstract
 We describe a method for numerically incorporating electronelectron scattering in quantum wells for small deviations of the distribution function from equilibrium, within the framework of the Boltzmann equation. For a given temperature $T$ and density $n$, a symmetric matrix needs to be evaluated only once, and henceforth it can be used to describe electronelectron scattering in any Boltzmann equation linearresponse calculation for that particular $T$ and $n$. Using this method, we calculate the distribution function and mobility for electrons in a quantumwell, including full finitetemperature dynamic screening effects. We find that at some parameters which we investigated, electronelectron scattering reduces mobility by approximately 40\%.
Ben YuKuang Hu, Karsten Flensberg Journal reference: Phys. Rev. B 53 (1996) 10072 [ condmat/9510044v1 ] DOI: 10.1103/PhysRevB.53.10072

Plasmon enhancement of Coulomb drag in doublequantumwell systems 
Abstract
 We derive an expression for the drag rate (i.e., interlayer momentum transfer rate) for carriers in two coupled twodimensional gases to lowest nonvanishing order in the screened interlayer electronelectron interaction, valid for {\sl arbitrary} intralayer scattering mechanisms, using the Boltzmann transport equation. We calculate the drag rate for experimentally relevant parameters, and show that for moderately high temperatures ($T\gtrsim 0.2 T_F$, where $T_F$ is the Fermi temperature) the dynamical screening of the interlayer results in a large enhancement of the drag rate due to the presence of coupled plasmon modes. This plasmon enhancement causes the scaled drag rate to have a peak (i) as a function of temperature at $T \approx 0.5 T_F$, and (ii) as a function of the ratio of densities of the carriers in the two layers when their Fermi velocities are equal. We also show that the drag rate can be significantly affected by the {\sl intralayer} scattering mechanisms; in particular, the drag rate changes approximately by a factor of 2 when the dopant layer modulation doped structures are moved in from 400~\AA to 100~\AA.
Karsten Flensberg, Ben YuKuang Hu Journal reference: Phys. Rev. B 52 (1995) 14796 [ condmat/9507059v1 ] DOI: 10.1103/PhysRevB.52.14796

Linearresponse theory of Coulomb drag in coupled electron systems 
Abstract
 We report a fully microscopic theory for transconductivity, or, equivalently, momentum transfer rate, of Coulomb coupled electron systems. We use the Kubo linear response formalism, and our main formal result expresses the transconductivity in terms of two fluctuation diagrams, which are topologically related, but not equivalent to, the AslamazovLarkin diagrams known for superconductivity. Previously reported results are shown to be special cases of our general expression; specifically, for constant impurity scattering rates, we recover the Boltzmann equation results in the semiclassical clean limit, and the memory function results in dirty systems. Furthermore, we show that for energy dependent relaxation times, the final result is not expressible in terms of standard densityresponse functions. Other new results include: (i) at T=0, the frequency dependence of the transfer rate is found to be proportional to $\Omega$ and $\Omega^2$ for frequencies below and above the impurity scattering rate, respectively and (ii) the weak localization correction to the transconductivity is given by $\delta\sigma^{WL}_{21} \propto \delta\sigma^{WL}_{11} +\delta\sigma^{WL}_{22}$.
Karsten Flensberg, Ben YuKuang Hu, AnttiPekka Jauho, Jari M. Kinaret Journal reference: Phys. Rev. B 52, 14761 (1995) [ condmat/9504092v1 ] DOI: 10.1103/PhysRevB.52.14761

Scaling of the Coulomb Energy Due to Quantum Fluctuations in the Charge on a Quantum Dot 
Abstract
 The charging energy of a quantum dot is measured through the effect of its potential on the conductance of a second dot. This technique allows a measurement of the scaling of the dot's charging energy with the conductance of the tunnel barriers leading to the dot. We find that the charging energy scales quadratically with the reflection probability of the barriers. In a second experiment we study the transition from a single to a doubledot which exhibits a scaling behavior linear in the reflection probability. The observed powerlaws agree with a recent theory.
L. W. Molenkamp, K. Flensberg, M. Kemerink Journal reference: Phys. Rev. Lett. 75, 4282 (1995) [ condmat/9503135v1 ] DOI: 10.1103/PhysRevLett.75.4282

Electronelectron scattering in linear transport in twodimensional systems 
Abstract
 1994

Coulomb Drag as a Probe of Coupled Plasmon Modes in Parallel Quantum Wells 
Abstract
 We show theoretically that the Coulomb drag rate between two parallel quasitwodimensional electron gases is substantially enhanced by the coupled acoustic and optic plasmon modes of the system at temperatures $T \gtrsim 0.2T_F$ (where $T_F$ is the Fermi temperature) for experimentally relevant parameters. The acoustic mode causes a sharp upturn in the scaled drag rate as a function of temperature at $T \approx 0.2 T_F$. Other experimental signatures of plasmondominated drag are a $d^{3}$ dependence on the well separation $d$, and a peak in the drag rate as a function of relative carrier densities at matched Fermi velocities.
Karsten Flensberg, Ben YuKuang Hu Journal reference: Phys. Rev. Lett. 73, 3572 (1994). [ condmat/9406052v1 ] DOI: 10.1103/PhysRevLett.73.3572

Coulomb Drag as a Probe of Coupled Plasmon Modes in Parallel Quantum Wells 
Abstract
 1993

Magnetoconductivity of quantum wires with elastic and inelastic
scattering 
Abstract
 We use a Boltzmann equation to determine the magnetoconductivity of quantum wires. The presence of a confining potential in addition to the magnetic field removes the degeneracy of the Landau levels and allows one to associate a group velocity with each singleparticle state. The distribution function describing the occupation of these singleparticle states satisfies a Boltzmann equation, which may be solved exactly in the case of impurity scattering. In the case where the electrons scatter against both phonons and impurities we solve numerically  and in certain limits analytically  the integral equation for the distribution function, and determine the conductivity as a function of temperature and magnetic field. The magnetoconductivity exhibits a maximum at a temperature, which depends on the relative strength of the impurity and electronphonon scattering, and shows oscillations when the Fermi energy or the magnetic field is varied.
Henrik Bruus, Karsten Flensberg, Henrik Smith DOI: 10.1103/PhysRevB.48.11144 condmat/9303028v1 [pdf]

Magnetoconductivity of quantum wires with elastic and inelastic
scattering 
Abstract