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For the distinction of the Andreev bound states and Majorana bound states, we study the waiting time distributions (WTDs) for heterostructures, based on one dimensional edge states of a two dimensional topological insulators (TI) in combination with an proximitized s-wave superconductor (SC) and an applied magnetic field. We show for the time reversal symmetric (TRS) situation of a Josephson junction details of the WTD. This includes different transport processes, different numbers of Andreev bound states and the phase difference of the SC. We further consider a Zeeman field in the normal part of the junctions revealing novel features in the WTD along the phase transition between trivial bound states and Majorana bound states. We finally discuss clear signatures to discriminate between them., Comment: 10 pages, 6 figures

Sub-gap states in semiconducting-superconducting nanowire hybrid devices are controversially discussed as potential topologically non-trivial quantum states. One source of ambiguity is the lack of an energetically and spatially well defined tunnel spectrometer. Here, we use quantum dots directly integrated into the nanowire during the growth process to perform tunnel spectroscopy of discrete sub-gap states in a long nanowire segment. In addition to sub-gap states with a standard magnetic field dependence, we find topologically trivial sub-gap states that are independent of the external magnetic field, i.e. that are pinned to a constant energy as a function of field. We explain this effect qualitatively and quantitatively by taking into account the strong spin-orbit interaction in the nanowire, which can lead to a decoupling of Andreev bound states from the field due to a spatial spin texture of the confined eigenstates.

The superconducting proximity effect has been the focus of significant research efforts over many years and has recently attracted renewed interest as the basis of topologically non-trivial states in materials with a large spin orbit interaction, with protected boundary states useful for quantum information technologies. However, spectroscopy of these states is challenging because of the limited spatial and energetic control of conventional tunnel barriers. Here, we report electronic spectroscopy measurements of the proximity gap in a semiconducting indium arsenide (InAs) nanowire (NW) segment coupled to a superconductor (SC), using a spatially separated quantum dot (QD) formed deterministically during the crystal growth. We extract the characteristic parameters describing the proximity gap which is suppressed for lower electron densities and fully developed for larger ones. This gate-tunable transition of the proximity effect can be understood as a transition from the long to the short junction regime of subgap bound states in the NW segment. Our device architecture opens up the way to systematic, unambiguous spectroscopy studies of subgap bound states, such as Majorana bound states.

We consider a biased Normal-Superconducting junction with various types of superconductivity. Depending on the class of superconductivity, a Majorana bound state may appear at the interface. We show that this has important consequences on the distribution of waiting times of electrons flowing out of such an interface. Therefore, the waiting time distribution is shown to be a clear fingerprint of Majorana bound state physics and may be considered as an experimental signature of its presence.

We consider a normal-superconducting junction in order to investigate the effect of new physical ingredients on waiting times. First, we study the interplay between Andreev and specular scattering at the interface on the distribution of waiting times of electrons or holes separately. In that case the distribution is not altered dramatically compared to the case of a single quantum channel with a quantum point contact since the interface acts as an Andreev mirror for holes. We then consider a fully entangled state originating from spliting of Cooper pairs at the interface and demonstrate a significant enhancement of the probability to detect two consecutive electrons in a short time interval. Finally, we discuss the electronic waiting time distribution in the more realistic situation of partial entanglement.

A Josephson junction may be driven through a transition where the superconducting condensate favors an odd over an even number of electrons. At this switch in the ground-state fermion parity, an Andreev bound state crosses through the Fermi level, producing a zero-mode that can be probed by a point contact to a grounded metal. We calculate the time-dependent charge transfer between superconductor and metal for a linear sweep through the transition. One single quasiparticle is exchanged with charge $Q$ depending on the coupling energies $\gamma_1,\gamma_2$ of the metal to the Majorana operators of the zero-mode. For a single-channel point contact, $Q$ equals the electron charge $e$ in the adiabatic limit of slow driving, while in the opposite quenched limit $Q=2e\sqrt{\gamma_1\gamma_2}/(\gamma_1+\gamma_2)$ varies between $0$ and $e$. This provides a method to produce single charge-neutral quasiparticles on demand., Comment: 10 pages, 5 figures, v. 3: numerical simulation and a derivation of charge transfer in adiabatic regime added, v. 4: multi-channel probe case added

We have measured the lifetime of spin imbalances in the quasiparticle population of a superconductor ($\tau_s$) in the frequency domain. A time-dependent spin imbalance is created by injecting spin-polarised electrons at finite excitation frequencies into a thin-film mesoscopic superconductor (Al) in an in-plane magnetic field (in the Pauli limit). The time-averaged value of the spin imbalance signal as a function of excitation frequency, $f_{RF}$ shows a cut-off at $f_{RF} \approx 1/(2\pi\tau_s)$. The spin imbalance lifetime is relatively constant in the accessible ranges of temperatures, with perhaps a slight increase with increasing magnetic field. Taking into account sample thickness effects, $\tau_s$ is consistent with previous measurements and of the order of the electron-electron scattering time $\tau_{ee}$. Our data are qualitatively well-described by a theoretical model taking into account all quasiparticle tunnelling processes from a normal metal into a superconductor., Comment: Includes Supplementary Information

We study the spin and charge accumulation in junctions between a superconductor and a ferromagnet or a normal metal in the presence of a Zeeman magnetic field, when the junction is taken out of equilibrium by applying a voltage bias. We write down the most general form for the spin and charge current in such junctions, taking into account all spin-resolved possible tunneling processes. We make use of these forms to calculate the spin accumulation in NS junctions subjected to a DC bias, and to an AC bias, sinusoidal or rectangular. We observe that in the limit of negligeable changes on the superconducting gap, the NS dynamical conductance is insensitive to spin imbalance. Therefore to probe the spin accumulation in the superconductor, one needs to separate the injection and detection point, i. e. the electrical spin detection must be non-local. We address also the effect of the spin accumulation induced in the normal leads by driving a spin current and its effects on the detection of the spin accumulation in the superconductor. Finally, we investigate the out-of-equilibrium spin susceptibility of the SC, and we show that it deviates drastically from it's equilibrium value.

We study the formation of Majorana fermions in honeycomb-lattice structures in the presence of a Zeeman field, Rashba spin-orbit coupling, and in the proximity of an s-wave superconductor. We show that an exact mapping exists between an anisotropic hexagonal-lattice nanoribbon at k = 0 and a one-dimensional chain, for which the existence of Majorana fermions has been extensively discussed. Consequently we can predict the conditions for the emergence of Majorana fermions at the edges of such ribbon, and relate the existence of Majoranas to a band inversion in the bulk band structure. Moreover we find that similar situations arise in isotropic lattices and we give some examples which show the formation of Majorana fermions in these structures., Comment: 7 pages, 9 figures

We study the proximity effect in a topological nanowire tunnel coupled to an s-wave superconducting substrate. We use a general Green's function approach that allows us to study the evolution of the Andreev bound states in the wire into Majorana fermions. We show that the strength of the tunnel coupling induces a topological transition in which the Majorana fermionic states can be destroyed when the coupling is very strong. Moreover, we provide a phenomenologial study of the effects of disorder in the superconductor on the formation of Majorana fermions. We note a non-trivial effect of a quasiparticle broadening term which can take the wire from a topological into a non-topological phase in certain ranges of parameters. Our results have also direct consequences for a nanowire coupled to an inhomogenous superconductor.

We study the effect of a finite proximity superconducting (SC) coherence length in SN and SNS junctions consisting of a semiconducting topological insulating wire whose ends are connected to either one or two s-wave superconductors. We find that such systems behave exactly as SN and SNS junctions made from a single wire for which some regions are sitting on top of superconductors, the size of the topological SC region being determined by the SC coherence length. We also analyze the effect of a non-perfect transmission at the NS interface on the spatial extension of the Majorana fermions. Moreover, we study the effects of continuous phase gradients in both an open and closed (ring) SNS junction. We find that such phase gradients play an important role in the spatial localization of the Majorana fermions.

What happens to spin-polarised electrons when they enter a superconductor? Superconductors at equilibrium and at finite temperature contain both paired particles (of opposite spin) in the condensate phase as well as unpaired, spin-randomised quasiparticles. Injecting spin-polarised electrons into a superconductor thus creates both spin and charge imbalances [1, 2, 3, 4, 5, 6, 7] (respectively Q* and S*, cf. Ref. [4]). These must relax when the injection stops, but not necessarily over the same time (or length) scale as spin relaxation requires spin-dependent interactions while charge relaxation does not. These different relaxation times can be probed by creating a dynamic equilibrium between continuous injection and relaxation, which leads to constant-in-time spin and charge imbalances. These scale with their respective relaxation times and with the injection current. While charge imbalances in superconductors have been studied in great detail both theoretically [8] and experimentally [9], spin imbalances have not received much experimental attention [6, 10] despite intriguing theoretical predictions of spin-charge separation effects [11, 12]. These could occur e.g. if the spin relaxation time is longer than the charge relaxation time, i.e. Q* relaxes faster than S*. Fundamentally, spin-charge decoupling in superconductors is possible because quasiparticles can have any charge between e and -e, and also because the condensate acts as a particle reservoir [13, 11, 12]. Here we present evidence for an almost-chargeless spin imbalance in a mesoscopic superconductor.

We study one-dimensional topological SN and SNS long junctions obtained by placing a topological insulating nanowire in the proximity of either one or two SC finite-size leads. Using the Majorana Polarization order parameter (MP) introduced in Phys. Rev. Lett. 108, 096802 (2012)(arxiv:1109.5697) we find that the extended Andreev bound states (ABS) of the normal part of the wire acquire a finite MP: for a finite-size SN junction the ABS spectrum exhibits a zero-energy extended state which carries a full Majorana fermion, while the ABS of long SNS junctions with phase difference $\pi$ transform into two zero-energy states carrying two Majorana fermions with the same MP. Given their extended character inside the whole normal link, and not only close to an interface, these Majorana-Andreev states can be directly detected in tunneling spectroscopy experiments.

We propose an approach allowing the computation of currents and their correlations in interacting multiterminal mesoscopic systems involving quantum dots coupled to normal and/or superconducting leads. The formalism relies on the expression of branching currents and noise crossed correlations in terms of one- and two-particle Green's functions for the dots electrons, which are then evaluated self-consistently within a conserving approximation. We then apply this to the Cooper-pair beam-splitter setup recently proposed [L. Hofstetter et al. Nature (London) 461 960 (2009); Phys. Rev. Lett. 107 136801 (2011); L. G. Herrmann et al. Phys. Rev. Lett. 104 026801 (2010)], which we model as a double quantum dot with weak interactions, connected to a superconducting lead and two normal ones. Our method not only enables us to take into account a local repulsive interaction on the dots, but also to study its competition with the direct tunneling between dots. Our results suggest that even a weak Coulomb repulsion tends to favor positive current cross correlations in the antisymmetric regime (where the dots have opposite energies with respect to the superconducting chemical potential).

We study the Josephson effect through a magnetic molecule with anisotropic properties. Performing calculations in the tunneling regime, we show that the exchange coupling between the electron spin on the molecule and the molecular spin can trigger a transition from the $\pi$ state to the 0 state, and we study how the spin anisotropy affects this transition. We show that the behavior of the critical current as a function of an external magnetic field can give access to valuable information about the spin anisotropy of the molecule.

We consider a double quantum dot coupled to two normal leads and one superconducting lead, modeling the Cooper pair beam splitter studied in two recent experiments. Starting from a microscopic Hamiltonian we derive a general expression for the branching current and the noise crossed correlations in terms of single and two-particle Green's function of the dot electrons. We then study numerically how these quantities depend on the energy configuration of the dots and the presence of direct tunneling between them, isolating the various processes which come into play. In absence of direct tunneling, the antisymmetric case (the two levels have opposite energies with respect to the superconducting chemical potential) optimizes the Crossed Andreev Reflection (CAR) process while the symmetric case (the two levels have the same energies) favors the Elastic Cotunneling (EC) process. Switching on the direct tunneling tends to suppress the CAR process, leading to negative noise crossed correlations over the whole voltage range for large enough direct tunneling.