Majorana Wires Research Articles

Nonlocality of local Andreev conductances as a probe for topological Majorana wires

Nonlocality of local Andreev conductances as a probe for topological Majorana wires

Spectral and Transport Properties of Andreev Molecule Coupled to Majorana Wire

AbstractThe quasiparticle spectrum and transport properties of the double quantum dot (DQD) deposited on a superconducting substrate (Andreev molecule) and side‐coupled to a nanowire hosting Majorana zero modes (MZMs) are studied. Placing a DQD on the superconducting substrate induces the trivial Andreev‐bound states (ABSs) in quantum dots. However, coupling of DQD with a nanowire causes the leakage of the MZM from the topological nanowire into quantum dots. The relationship between the Andreev states and the Majorana mode for different values of the coupling parameters is analyzed. Additionally, it is shown that the connection point of a metallic tip, treated as an scanning tunneling microscope (STM) tip, affects the measured results of the differential conductance.

Topological phases induced by charge fluctuations in Majorana wires

Topological phases induced by charge fluctuations in Majorana wires

Coulomb-assisted nonlocal electron transport between two pairs of Majorana bound states in a superconducting island

We investigate the nonlocal transport modulated by Coulomb interactions in devices comprising two interacting Majorana wires, where both nanowires are in proximity to a mesoscopic superconducting (SC) island. Each Majorana bound state (MBS) is coupled to one lead via a quantum dot with resonant levels. In this device, the nonlocal correlations can be induced in the absence of Majorana energy splitting. We find that the negative differential conductance and giant current noise cross correlation could be induced, due to the interplay between nonlocality of MBSs and dynamical Coulomb blockade effect. This feature may provide a signature for the existence of the MBSs.

Conductance based machine learning of optimal gate voltages for disordered Majorana wires

Conductance based machine learning of optimal gate voltages for disordered Majorana wires

Bogoliubov-de Gennes equation on graphs: A model for tree-branched Majorana wire network

We consider Bogoliubov-de Gennes equation on a metric tree graph. Formulation of the problem for arbitrary graph topology is provided. Self-adjoint vertex boundary conditions are derived. Exact solutions of the problem is obtained for quantum tree graph. A quantum graph based model for tree-branched Majorana wire network is proposed.

Magnetic field effect on transport properties of double quantum dot — Majorana wire system in cross-shaped configuration

We theoretically study the influence of an external magnetic field on the transport properties of a double quantum dot (DQD)-Majorana nanowire system in a cross-shaped configuration with central dot coupled to a superconductor and normal metallic leads. In such a system, we can observe a subtle interplay between generation of Andreev bound states, Majorana mode leaking into quantum dots and interdots Fano-like interference processes. We show that properly selected side-coupled quantum dot energy can slow down/activate the process of Majorana mode leaking into DQD and additionally can strongly affect the differential and linear conductance in the considered system. We also study the influence of spin polarization of quantum dot–Majorana mode coupling on the transport properties of the considered system, especially on its linear conductance.

Spin effects on transport and zero-bias anomaly in a hybrid Majorana wire-quantum dot system

We examine the impact of spin effects on the nonequilibrium transport properties of a nanowire hosting Majorana zero-energy modes at its ends, coupled to a quantum dot junction with ferromagnetic leads. Using the real-time diagrammatic technique, we determine the current, differential conductance and current cross-correlations in the nonlinear response regime. We also explore transport in different magnetic configurations of the system, which can be quantified by the tunnel magnetoresistance. We show that the presence of Majorana quasiparticles gives rise to unique features in all spin-resolved transport characteristics, in particular, to zero-bias anomaly, negative differential conductance, negative tunnel magnetoresistance, and it is also reflected in the current cross-correlations. Moreover, we study the dependence of the zero-bias anomaly on various system parameters and demonstrate its dependence on the magnetic configuration of the system as well as on the degree of spin polarization in the leads. A highly nontrivial behavior is also found for the tunnel magnetoresistance, which exhibits regions of enhanced or negative values—new features resulting from the coupling to Majorana wire.

Weakly coupled Majorana wire arrays under tilted magnetic fields

An array of Rashba-coupled semiconducting nanowires lying in proximity to an s-wave superconducting substrate, with weak inter-wire coupling, in the presence of an external magnetic field shows even–odd effect in the differential conductance over a chosen parameter space. Such an effect is a direct consequence of end Majoranas in each nanowire hybridizing into bonding and anti-bonding states. In the present work, we study in detail the impact of tilting of external magnetic field on the differential conductance of an array of both uncoupled and weakly coupled wires. The phase diagram evolution with various control parameters including the tilt angle of the magnetic field has also been presented. From detailed analysis of the field-angle dependence of the odd–even effect, and the evolution of the same over a large parameter space we summarize that the results can be used to exploit magnetic-field angle in an array of Rashba-coupled semiconducting nanowires on a superconducting substrate as an important tuning parameter to investigate zero-bias conductance peak arising from Majorana edge modes vis-a-vis that arising from a non-topological origin.

Bound states in nonuniform superconductors

This paper shows that the WKB quantization rule is suitable for the Andreev bound states in nonuniform superconductors. We consider nonhomogeneous superconductivity gap functions $\mathrm{\ensuremath{\Delta}}(x)$ in superconductors with the Bogoliubov quasiparticle energy $E$, the Fermi level ${E}_{F}$, and the total momentum $\mathbf{p}$ at ${E}_{F}+E$. The Andreev bound states in the well of slowly varying $|\mathrm{\ensuremath{\Delta}}(x)|$ are studied, and the well may also be induced by the phase variation of $\mathrm{\ensuremath{\Delta}}(x)$ for massless Dirac fermions. By applying the WKB method to the Bogoliubov--de Gennes equation, we obtain two main results: (i) For ${E}_{F}\ensuremath{\sim}0$, the bound states are determined by ${\ensuremath{\int}}_{{L}_{E}}^{{R}_{E}}|\mathbf{p}\phantom{\rule{0.16em}{0ex}}d\mathbf{x}|=(\frac{1}{2}+n)\ensuremath{\pi}\ensuremath{\hbar}$, where $n\ensuremath{\in}{\mathbb{N}}_{0}$ and ${L}_{E}$ and ${R}_{E}$ are the boundary points between the classically allowed region and forbidden regions, and (ii) for ${E}_{F}\ensuremath{\gg}E$ and $|\mathrm{\ensuremath{\Delta}}(x)|$, the bound states are given by ${\ensuremath{\int}}_{{L}_{E}}^{{R}_{E}}|\mathbf{p}\ifmmode\pm\else\textpm\fi{}{\mathbf{p}}_{F}|dx=(\frac{1}{2}+n)\ensuremath{\pi}\ensuremath{\hbar}$ with small ${p}_{y(z)}$. Empirical quantization conditions are provided for broader parameter regions. In addition to applying the traditional WKB method, we also develop a generalized WKB method to tackle semimetals with parabolic dispersion relationships. The applications of our results are discussed, for example, Dirac $\ensuremath{\pi}$ junctions or nonchiral Majorana wires, SNS junctions, the excitation threshold, and the tunneling rate in NSN junctions. In the $\ensuremath{\pi}$ junction, the Majorana zero modes correspond to the zero-point energy in the WKB formalism. This observation may provide insights into the Majorana bound state in a vortex and the Majorana fermions in high-energy physics.

Machine Learning Optimization of Majorana Hybrid Nanowires.

As the complexity of quantum systems such as quantum bit arrays increases, efforts to automate expensive tuning are increasingly worthwhile. We investigate machine learning based tuning of gate arrays using the covariance matrix adaptation evolution strategy algorithm for the case study of Majorana wires with strong disorder. We find that the algorithm is able to efficiently improve the topological signatures, learn intrinsic disorder profiles, and completely eliminate disorder effects. For example, with only 20 gates, it is possible to fully recover Majorana zero modes destroyed by disorder by optimizing gate voltages.

Topological states of generalized dissipative Majorana wires

We study the generalized one-dimensional (1D) quantum dissipative models corresponding to a Majorana wire which can possess more than one Majorana bound state at each end. The system consists of a 1D fermionic open quantum system whose dynamics is governed by a quadratic Lindblad equation. Using the adjoint Lindblad equation for the fermionic two-point correlations, we find the gaps in the damping and purity spectra of a generic 1D model. Then, using the symmetry-based classification, we show that a winding number as the topological invariant can be defined which distinguishes different steady states of the system in the presence of damping and purity gaps. Then we focus on certain models with different Lindblad quantum jump terms and explore their phase diagrams by calculating the damping and the purity gaps as well as the winding number. In particular, we show that by inclusion of quantum jumps between next-nearest-neighbor sites, higher winding numbers and equivalently more Majorana bound states can be achieved. Also, by introducing imbalanced couplings we can switch between states with negative and positive winding numbers. Finally, we should mention that since our formulation is based on the fermionic correlations rather than the Majorana operators, it can be easily extended to the dissipative topological phases belonging to other symmetry classes.

4π -periodic anomalous Josephson effect between Majorana zero modes

Usually the Josephson current is driven by the superconducting phase difference $\ensuremath{\varphi}$ between two superconducting leads. Here we propose an anomalous Josephson current between two Majorana nanowires, which can be driven by both the superconducting phase difference $\ensuremath{\varphi}$ and the orientation difference $\ensuremath{\phi}$ between the Zeeman fields in two Majorana wires. We show that the orientation of the Zeeman field acts as an effective superconducting phase under appropriate conditions, which provides a new signature of Majorana zero modes. Due to the presence of Majorana zero modes, the anomalous Josephson current is $4\ensuremath{\pi}$ periodic in both $\ensuremath{\varphi}$ and $\ensuremath{\phi}$. When both $\ensuremath{\varphi}$ and $\ensuremath{\phi}$ are periodically varying, we predict the existence of peaks in the average Josephson current, which are the analog to Shapiro steps in the ac Josephson effect. Similarly, the existence of Majorana zero modes can be confirmed by the missing of odd peaks. The findings provide a scheme to experimentally verify the $4\ensuremath{\pi}$-periodic Josephson effect, as well as a platform to realize the anomalous Josephson junction.

Tunneling characteristics of weakly coupled Majorana wire arrays

The Majorana modes in solid-state systems have the potential to be applied in fault tolerant quantum computing. It is well known that a semiconducting nanowire with strong Rashba coupling and in the proximity of a superconductor hosts Majorana edge modes. An array of such nanowires with inter-wire coupling gives an approximate description of a two-dimensional topological superconductor, where depending on the strength of the magnetic field and the chemical potential, a rich phase diagram hosting trivial and different types of non-trivial phases can be achieved. In this work, we theoretically consider such a two-dimensional assembly of spin–orbit coupled superconducting nanowires and calculate the collective tunneling conductance between normal electrodes and the wires in the topological regime. When the number of wires in the assembly is N, as a consequence of the way the Majorana bonding and anti-bonding states form, we find that N conductance peaks symmetric about the bias V=0 appear for even N. When N is odd, a zero-bias conductance peak (ZBCP) also appears. We have identified the region in the parameter space where this odd–even rule applies. The effects of finite temperature, finite dissipation, and the barrier potential on the conductance profiles have also been discussed in detail. A device comprising of an array of weakly coupled Majorana wires can be realized by standard nano-fabrication techniques where individual nanowires can be turned ON or OFF by using a mechanical switch (or local top gating) to make N either even or odd—thereby switching the ZBCP OFF or ON, respectively, modulo the control parameter being in the desired range. Hence, in principle, our results can be used to realize and detect topological superconductivity efficiently and in a controlled manner.

Bath-induced decoherence in finite-size Majorana wires at non-zero temperature

Braiding Majorana zero-modes around each other is a promising route toward topological quantum computing. Yet, two competing maxims emerge when implementing Majorana braiding in real systems: on the one hand, perfect braiding should be conducted adiabatically slowly to avoid non-topological errors. On the other hand, braiding must be conducted fast such that decoherence effects introduced by the environment are negligible, which are generally unavoidable in finite-size systems. This competition results in an intermediate time scale for Majorana braiding that is optimal, but generally not error-free. Here, we calculate this intermediate time scale for a T-junction of short one-dimensional topological superconductors coupled to a bosonic bath that generates fluctuations in the local electric potential, which stem from, e.g. environmental photons or phonons of the substrate. We thereby obtain boundaries for the speed of Majorana braiding with a predetermined gate fidelity. Our results emphasize the general susceptibility of Majorana-based information storage in finite-size systems and can serve as a guide for determining the optimal braiding times in future experiments.

Transmission amplitude through a Coulomb blockaded Majorana wire

We study coherent electronic transport through a Coulomb blockaded superconducting Rashba wire in the co-tunneling regime between conductance resonances. By varying an external Zeeman field the wire can be tuned into a topological regime, where non-local transport through Majorana zero modes is the dominant mechanism. We model coherent transport in the co-tunneling regime by using a scattering matrix formalism, and find that the transmission amplitude has a maximum as a function of Zeeman field, whose height is proportional to the wire length. We relate the transmission amplitude to the Majorana correlation length, and argue that the Zeeman field and length dependence of the transmission amplitude are unique signatures for the presence of Majorana zero modes.

Closing of the induced gap in a hybrid superconductor-semiconductor nanowire

Hybrid superconductor-semiconductor nanowires are predicted to undergo a field-induced phase transition from a trivial to a topological superconductor, marked by the closure and re-opening of the excitation gap, followed by the emergence of Majorana bound states at the nanowire ends. Many local density-of-states measurements have reported signatures of the topological phase, however this interpretation has been challenged by alternative explanations. Here, by measuring nonlocal conductance, we identify the closure of the excitation gap in the bulk of the semiconductor before the emergence of zero-bias peaks. This observation is inconsistent with scenarios where zero-bias peaks occur due to end-states with a trivially gapped bulk, which have been extensively considered in the theoretical and experimental literature. We observe that after the gap closes, nonlocal signals fluctuate strongly and persist irrespective of the presence of local-conductance zero-bias peaks. Thus, our observations are also incompatible with a simple picture of clean topological superconductivity. This work presents a new experimental approach for probing the spatial extent of states in Majorana wires, and reveals the presence of a regime with a continuum of spatially extended states and uncorrelated zero-bias peaks.

Magnetization dynamics in a Majorana-wire–quantum-dot setup

We theoretically study the quench dynamics of the local magnetization in a hybrid Majorana-wire--quantum-dot system coupled to external leads. In order to thoroughly understand the origin of the dot magnetization dynamics, we consider either normal metal or ferromagnetic electrodes. In the first case, the magnetization arises exclusively from the proximity to the topological superconductor hosting Majorana zero-energy modes and the associated development of an induced exchange field. We predict a nonmonotonic dependence of the dot's magnetization in the odd-occupation regime and show that the dynamics is governed by the magnitude of the coupling to Majorana wire. However, when the system is coupled to ferromagnetic leads, the ferromagnet and Majorana contributions to the effective exchange field are competing with each other and reveal a nontrivial dynamical behavior. As a result, the time-dependent magnetization can undergo multiple sign changes preceding the relaxation to a new thermal value. We also identify the transport regime, where fine tuning of the coupling to Majorana wire within a narrow range allows one to manipulate the magnetic state of the system. The effect of spin polarization of the leads and influence of the finite overlap between the Majorana edge modes are also examined. Moreover, we analyze the quench in the energy of the quantum dot orbital level and demonstrate that the rather straightforward charge dynamics can disguise nontrivial time evolution of the magnetization. Finally, we compare predicted dynamics with results obtained for quantum dot coupled to spin-polarized fermionic bound state instead of Majorana zero-energy mode.

Dissipative Floquet Majorana Modes in Proximity-Induced Topological Superconductors.

We study a realistic Floquet topological superconductor, a periodically driven nanowire proximitized to an equilibrium s-wave superconductor. Because of the strong energy and density fluctuations caused by the superconducting proximity effect, the Floquet Majorana wire becomes dissipative. We show that the Floquet band structure is still preserved in this dissipative system. In particular, we find that the Floquet Majorana zero and π modes can no longer be simply described by the Floquet topological band theory. We also propose an effective model to simplify the calculation of the lifetime of these Floquet Majoranas and find that the lifetime can be engineered by the external driving field.

Spontaneous Currents and Topologically Protected States in Superconducting Hybrid Structures with the Spin\u2013Orbit Coupling (Brief Review)

The results of recent theoretical studies of features of superconducting states in hybrid structures whose properties are significantly determined by the spin–orbit effects have been reported. The two main phenomena appearing in such systems in the presence of additional spin splitting caused either by the Zeeman effect in a magnetic field or by the exchange field: (i) the generation of spontaneous currents and (ii) the appearance of topologically nontrivial superconducting phases. It has been shown that the spin–orbit coupling can be a key mechanism that allows implementing new inhomogeneous phase structures, in particular, the so-called “phase batteries.” The effect of geometric factors on the properties of topologically nontrivial superconducting states has been analyzed. New types of topological transitions in vortex states of Majorana wires have been proposed.