Majorana bound states are spatially localized superpositions of electron and hole excitations in the middle of a superconducting energy gap. A single qubit can be encoded nonlocally in a pair of spatially separated Majorana bound states. Such Majorana qubits are in demand as building blocks of a topological quantum computer, but direct experimental tests of the nonlocality remain elusive. Here we propose a method to probe the nonlocality by means of crossed Andreev reflection, which is the injection of an electron into one bound state followed by the emission of a hole by the other bound state (equivalent to the splitting of a Cooper pair over the two states). We have found that, at sufficiently low excitation energies, this nonlocal scattering process dominates over local Andreev reflection involving a single bound state. As a consequence, the low-temperature and low-frequency fluctuations $δI_{i}$ of currents into the two bound states $i=1,2$ are maximally correlated: $\overline{δI_{1}δI_{2}}=\overline{δI_{i}^{2}}$.
We study a Cooper pair transistor realized by a mesoscopic superconductor island that couples to a pair of $s$-wave superconducting leads. For a trivial island, the critical supercurrent between the leads exhibits a well-known $2e$-periodicity in the island-gate charge. Here, we show that for an island with spatially separated zero-energy Majorana or Andreev bound states the periodicity of the magnitude of the critical supercurrent transitions to $1e$ in the island-gate charge. Moreover, for Andreev bound states the current-phase relation displays a sign reversal when the parity of the charge …
A semiconducting nanowire with strong Rashba spin-orbit coupling and coupled to a superconductor can be tuned by an external Zeeman field into a topological phase with Majorana zero modes. Here we theoretically investigate how this exotic topological superconductor phase manifests in Josephson junctions based on such proximitized nanowires. In particular, we focus on critical currents in the short junction limit ($L_{\rm N}\ll ξ$, where $L_{\rm N}$ is the junction length and $ξ$ is the superconducting coherence length) and show that they contain important information about nontrivial topology …
We show that the topologically trivial zero bias peak (ZBP) emerging in semiconductor Majorana wires due to soft confinement exhibits correlated splitting oscillations as a function of the applied Zeeman field, similar to the correlated splitting of the Majorana ZBP. Also, we find that the presence of a strong impurity can effectively cut the wire in two and destroy the correlated splitting in both the trivial and the Majorana regimes. We identify a strong nonlocal effect that operates only in the topologically trivial regime and demonstrate that the dependence of the ZBP on the confining gate…
A superconductor connected to normal leads allows to generate Einstein-Podolsky-Rosen pairs by Cooper pair splitting. It has been realized with quantum dots either defined in carbon nanotubes or InAs nanowires. After establishing the presence of Cooper pair splitting in such devices new works have invesigated the effects of a finite potential difference between the quantum dots to improve and characterize the efficiency of Cooper pair splitting. In this paper we present a generic model for Cooper pair splitting and develop two minimal models specifically for the two experimental realisations a…
We consider a two-dimensional px+i py superconductor in the presence of multiple vortices, which support zero-energy Majorana fermion states in their cores. Intervortex tunnelings of the Majorana fermions lift the topological state degeneracy. Using the Bogoliubov-de Gennes equation, we calculate splitting of the zero-energy modes due to these tunneling events. We also discuss superconducting fluctuations and, in particular, their effect on the energy splitting.
Motivated by the recent experiments [Scientific reports 6, 23051 (2016), Phys. Rev. Lett. 114, 096602 (2015)], we theoretically investigate Cooper pair splitting current in a graphene based Cooper pair beam splitter geometry. By considering the graphene based superconductor as an entangler device, instead of normal (2D) BCS superconductor, we show that the Cooper pair splitting current mediated by Crossed Andreev process is amplified compared to its normal superconductor counterpart. This amplification is attributed to the strong suppression of local normal Andreev reflection process (arising …
This letter is devoted to the optimization of the Cooper pair splitting efficiency in a Y-shaped junction. The latter consists of two quantum dots, one superconducting and two normal leads. We tailor the bias in the two normal leads such that the Cooper pairs leaving the superconductor are split up resulting in entangled electrons, one on each quantum dot. We are able to achieve a splitting efficiency of more than 99% which is significantly better than the efficiencies obtained in experiments so far.
Controlled generation and detection of quantum entanglement between spatially separated particles constitute an essential prerequisite both for testing the foundations of quantum mechanics and for realizing future quantum technologies. Splitting of Cooper pairs from a superconductor provides entangled electrons at separate locations. However, experimentally accessing the individual split Cooper pairs constitutes a major unresolved issue as they mix together with electrons from competing processes. Here, we overcome this challenge with the first real-time observation of the splitting of individ…
We study both analytically and numerically the role of orbital effects caused by a magnetic field applied along the axis of a semiconducting Rashba nanowire in the topological regime hosting Majorana fermions. We demonstrate that the orbital effects can be effectively taken into account in a one-dimensional model by shifting the chemical potential, and, thus modifying the topological criterion. We focus on the energy splitting between two Majorana fermions in a finite nanowire and find a striking interplay between orbital and Zeeman effects on this splitting. In the limit of strong spin-orbit …
We measure an aluminum superconducting double quantum dot and find that its electrical impedance, specifically its quantum capacitance, depends on whether or not it contains a single broken Cooper pair. In this way we are able to observe, in real time, the thermally activated breaking and recombination of Cooper pairs. Furthermore, we apply external microwave light and break single Cooper pairs by the absorption of single microwave photons.
We propose a path for constructing diverse interacting spin systems from topological nanowires in Cooper Boxes. The wires are grouped into a three-wire building block called an 'hexon', consisting of six Majorana zero modes. In the presence of a strong charging energy, the hexon becomes a Cooper box equivalent to two spin-$1/2$ degrees of freedom. By considering arrays of hexons and controlling the distances between the various wires, one can tune the Hamiltonian governing the low-energy spins, thus providing a route for controllably constructing interacting spin systems in one- and two-dimens…
We study theoretically how electron-phonon interaction affects the energies and level broadening (inverse lifetime) of Majorana bound states (MBSs) in a clean topological nanowire at low temperatures. At zero temperature, the energy splitting between the right and left MBSs remains exponentially small with increasing nanowire length $L$. At finite temperatures, however, the absorption of thermal phonons leads to the broadening of energy levels of the MBSs that does not decay with system length, and the coherent absorption/emission of phonons at opposite ends of the nanowire results in MBSs ene…
Split Cooper pair is a natural source for entangled electrons which is a basic ingredient for quantum information in solid state. We report an experiment on a superconductor-graphene double quantum dot (QD) system, in which we observe Cooper pair splitting (CPS) up to a CPS efficiency of ~ 10%. With bias on both QDs, we are able to detect a positive conductance correlation across the two distinctly decoupled QDs. Furthermore, with bias only on one QD, CPS and elastic co-tunneling can be distinguished by tuning the energy levels of the QDs to be asymmetric or symmetric with respect to the Fermi…
Whereas identification of Cooper pair symmetry is the first and crucial step in the investigation of unconventional superconductors, only a few have been established so far because of its own difficulties. To solve this problem, we develop a theory for identification of pairing symmetry using knowledge of topological superconductivity. Establishing the multipole theory of emergent Majorana fermions in time-reversal-invariant topological superconductors, we discover a one-to-one correspondence between the electromagnetic response of Majorana fermions and Cooper pair symmetry. The emergent Major…
The two electrons of a Cooper pair in a conventional superconductor form a singlet and therefore a maximally entangled state. Recently, it was demonstrated that the two particles can be extracted from the superconductor into two spatially separated contacts via two quantum dots (QDs) in a process called Cooper pair splitting (CPS). Competing transport processes, however, limit the efficiency of this process. Here we demonstrate efficiencies up to 90%, significantly larger than required to demonstrate interaction-dominated CPS, and on the right order to test Bell's inequality with electrons. We…
Recent observations of a zero bias conductance peak in tunneling transport measurements in superconductor--semiconductor nanowire devices provide evidence for the predicted zero--energy Majorana modes, but not the conclusive proof for their existence. We establish that direct observation of a splitting of the zero bias conductance peak can serve as the smoking gun evidence for the existence of the Majorana mode. We show that the splitting has an oscillatory dependence on the Zeeman field (chemical potential) at fixed chemical potential (Zeeman field). By contrast, when the density is constant …
Entanglement, being at the heart of the Einstein-Podolsky-Rosen (EPR) paradox, is a necessary ingredient in processing quantum information. Cooper pairs in superconductors - being composites of two fully entangled electrons - can be split adiabatically, thus forming entangled electrons. We fabricated such electron splitter by contacting an aluminum superconductor strip at the center of a suspended InAs nanowire; terminated at both ends with two normal metallic drains. Intercepting each half of the nanowire by gate - induced Coulomb blockaded quantum dot strongly impeded the flow of Cooper pair…
Splitting of Cooper pairs has recently been realized experimentally for s-wave Cooper pairs. A split Cooper pair represents an entangled two-electron pair state which has possible application in on-chip quantum computation. Likewise the spin-activity of interfaces in nanoscale tunnel junctions has been investigated theoretically and experimentally in recent years. However, the possible implications of spin-active interfaces in Cooper pair splitters so far have not been investigated. We analyse the current and the cross correlation of currents in a superconductor ferromagnet beamsplitter includ…
In three-terminal NSN hybrid structures the influence of additional barriers on the nonlocal conductance and on current cross-correlations is studied within a scattering theory. In metallic systems with additional barriers and phase averaging, which simulate disordered regions, local processes can be enhanced by reflectionless tunneling but this mechanism has little influence on nonlocal processes and on current cross-correlations. Therefore Cooper pair splitting cannot be enhanced by reflectionless tunneling. On the contrary, in ballistic systems, additional barriers lead to Fabry-Perot reson…
We consider a model of ballistic quasi-one dimensional semiconducting wire with intrinsic spin-orbit interaction placed on the surface of a bulk s-wave superconductor (SC), in the presence of an external magnetic field. This setup has been shown to give rise to a topological superconducting state in the wire, characterized by a pair of Majorana-fermion (MF) bound states formed at the two ends of the wire. Here we demonstrate that, besides the well-known direct overlap-induced energy splitting, the two MF bound states may hybridize via elastic correlated tunneling processes through virtual quas…
By solving the Bogoliubov--De Gennes equations analytically, we derive the fermionic zero-modes satisfying the Majorana property that exist in vortices of a two-dimensional $s$-wave Fermi superfluid with spin-orbit coupling and Zeeman spin-splitting. The Majorana zero-mode becomes normalisable and exponentially localised to the vicinity of the vortex core when the superfluid is topologically non-trivial. We calculate the energy splitting due to Majorana hybridisation and identify that the $s$-wave Majorana vortices obey non-Abelian statistics.
The toric code based on Majorana fermions on mesoscopic superconducting islands is a promising candidate for quantum information processing. In the limit of vanishing Cooper-pair tunneling, it has been argued that the phase transition separating the topologically ordered phase of the toric code from the trivial one is in the universality class of (2+1)D-XY. On the other hand, in the limit of infinitely large Cooper-pair tunneling, the phase transition is in the universality class of (2+1)D-Ising. In this work, we treat the case of finite Cooper-pair tunneling and address the question of how th…
Nanodevices consisting of a quantum dot tunnel coupled to one superconducting and two normal electrodes may serve as a source of entangled electrons. As a result of crossed Andreev reflection the Cooper pair of s-wave character may be split into two electrons and each of them goes into a distinct normal electrode, preserving entanglement. Efficiency of the process depends on the specific system and is tunable by electric means. Our calculations show that in the studied device this efficiency may attain values as large as $80 \%$.
Over the years, supersymmetric quantum mechanics has evolved from a toy model of high energy physics to a field of its own. Although various examples of supersymmetric quantum mechanics have been found, systems that have a natural realization are scarce. Here, we show that the extension of the conventional Cooper-pair box by a 4pi-periodic Majorana-Josephson coupling realizes supersymmetry for certain values of the ratio between the conventional Josephson and the Majorana- Josephson coupling strength. The supersymmetry we find is a "hidden" minimally bosonized supersymmetry that provides a non…
We theoretically study the spin-resolved subgap transport properties of a Cooper pair splitter based on a triple quantum dot attached to superconducting and ferromagnetic leads. Using the Keldysh Green's function formalism, we analyze the dependence of the Andreev conductance, Cooper pair splitting efficiency, and tunnel magnetoresistance on the gate and bias voltages applied to the system. We show that the system's transport properties are strongly affected by spin dependence of tunneling processes and quantum interference between different local and nonlocal Andreev reflections. We also stud…
A single Majorana Kramers pair hosts only one component of the magnetic multipole. This can be used to determine the bulk Cooper-pair symmetry through surface-sensitive spectroscopic measurements, either by applying a magnetic field or by using a ferromagnet/superconductor junction. This paper proposes that the electric response, which is free from the Meissner effect, can be used an alternative method to measure the bulk Cooper-pair symmetry in time-reversal-invariant superconductors with double Majorana Kramers pairs. The relationships among electric multipoles, strain tensors and supercondu…
Electron transport through Majorana nanowire with strongly asymmetric couplings to normal metal leads is considered. In three terminal geometry (electrically grounded nanowire) it is shown that the presence of unbiased electrode restores zero-bias anomaly even for strong Majorana energy splitting. For effectively two-terminal geometry we show that electrical current through asymmetric Majorana junction is qualitatively different from the analogous current through a resonant (Breit-Wigner) level.
Hybrid quantum dot-oscillator systems have become attractive platforms to inspect quantum coherence effects at the nanoscale. Here, we investigate a Cooper-pair splitter setup consisting of two quantum dots, each linearly coupled to a local resonator. The latter can be realized either by a microwave cavity or a nanomechanical resonator. Focusing on the subgap regime, we demonstrate that cross-Andreev reflection, through which Cooper pairs are split into both dots, can efficiently cool down simultaneously both resonators into their ground state. Moreover, we show that a nonlocal heat transfer b…
Since the proposal of monopole Cooper pairing in [Phys. Rev. Lett. 120, 067003 (2018)], considerable research efforts have been dedicated to the study of Cooper pairing order parameters constrained (or obstructed) by the nontrivial normal-state band topology at Fermi surfaces in 3D systems. In the current work, we generalize the topologically obstructed pairings between Chern states (like the monopole Cooper pairing) by proposing Euler obstructed Cooper pairing in 3D systems. The Euler obstructed Cooper pairing widely exists between two Fermi surfaces with nontrivial band topology characterize…
We study a time-reversal-invariant topological superconductor island hosting spatially separated Majorana Kramers pairs, with weak tunnel couplings to two s-wave superconducting leads. When the topological superconductor island is in the Coulomb blockade regime, we predict that a Josephson current flows between the two leads due to a non-local transfer of Cooper pairs mediated by the Majorana Kramers pairs. Interestingly, we find that the sign of the Josephson current is controlled by the joint parity of all four Majorana bound states on the island. Consequently, this parity-controlled Josephs…
Charge transport through a small topological superconducting island in contact with a normal and a superconducting electrode occurs through a cycle that involves coherent oscillations of Cooper pairs and tunneling in/out the normal electrode through a Majorana bound state, the Josephson-Majorana cycle. We illustrate this mechanism by studying the current-voltage characteristics of a superconductor-topological superconductor-normal metal single-electron transistor. At low bias and temperature the Josephson-Majorana cycle is the dominant mechanism for transport. We discuss a three-terminal confi…
Semiconductor-superconductor heterostructures represent a promising platform for the detection of Majorana zero modes and subsequently the processing of quantum information using their exotic non-Abelian statistics. Theoretical modeling of such low-dimensional semiconductors is generally based on phenomenological effective models. However, a more microscopic understanding of their band structure and, especially, of the spin-orbit coupling of electrons in these devices is important for optimizing their parameters for applications in quantum computing. In this paper, we approach this problem by …
We study transport through a Coulomb blockaded topologically nontrivial superconducting wire (with Majorana end states) contacted by metallic leads. An exact formula for the current through this interacting Majorana single-charge transistor is derived in terms of wire spectral functions. A comprehensive picture follows from three different approaches. We find Coulomb oscillations with universal halving of the finite-temperature peak conductance under strong blockade conditions, where the valley conductance mainly comes from elastic cotunneling. The nonlinear conductance exhibits finite-voltage…
We report on the fabrication and electrical characterization of an InAs double - nanowire (NW) device consisting of two closely placed parallel NWs coupled to a common superconducting electrode on one side and individual normal metal leads on the other. In this new type of device we detect Cooper-pair splitting (CPS) with a sizeable efficiency of correlated currents in both NWs. In contrast to earlier experiments, where CPS was realized in a single NW, demonstrating an intrawire electron pairing mediated by the superconductor (SC), our experiment demonstrates an inter- wire interaction mediate…
Cooper pair splitting (CPS) is a way to create spatially separated, entangled electron pairs. To this day, CPS is often identified in experiments as a spatial current correlation. However, such correlations can arise even in the absence of CPS, when a quantum dot is strongly coupled to the superconductor, and a subgap Shiba state is formed. Here, we present a detailed experimental characterization of those spatial current correlations, as the tunnel barrier strength between the quantum dot and the neighboring normal electrode is tuned. The correlation of the non-local signal and the barrier st…
Thermoelectric effect is exploited to optimize the Cooper pair splitting efficiency in a Y-shaped junction, which consists of two normal leads coupled to an $s$-wave superconductor via double noninteracting quantum dots. Here, utilizing temperature difference rather than bias voltage between the two normal leads, and tuning the two dot levels such that the transmittance of elastic cotunneling process is particle-hole symmetric, we find currents flowing through the normal leads are totally contributed from the splitting of Cooper pairs emitted from the superconductor. Such a unitary splitting e…
Electric weak links, the term used for those parts of an electrical circuit that provide most of the resistance against the flow of an electrical current, are important elements of many nanodevices. Quantum dots, nanowires and nano-constrictions that bridge two bulk conductors (or superconductors) are examples of such weak links. Here we consider nanostructures where the electronic spin-orbit interaction is strong in the weak link but is unimportant in the bulk conductors, and explore theoretically the role of the spin-orbit active weak link (which we call a "Rashba spin splitter") as a source…
We construct a minimal circuit, based on the top-transmon design, to rotate a qubit formed out of four Majorana zero-modes at the edge of a two-dimensional topological insulator. Unlike braiding operations, generic rotations have no topological protection, but they do allow for a full characterization of the coherence times of the Majorana qubit. The rotation is controlled by variation of the flux through a pair of split Josephson junctions in a Cooper pair box, without any need to adjust gate voltages. The Rabi oscillations of the Majorana qubit can be monitored via oscillations in the resona…
We investigate theoretically the properties of a weak link between two superconducting leads, which has the form of a non-superconducting nanowire with a strong Rashba spin-orbit coupling caused by an electric field. In the Coulomb blockade regime of single-electron tunneling, we find that such a weak link acts as a "spin splitter" of the spin states of Cooper pairs tunneling through the link, to an extent that depends on the direction of the electric field. We show that the Josephson current is sensitive to interference between the resulting two transmission channels, one where the spins of b…
We describe a Josephson device composed of two superconductors separated by two interacting quantum dots in parallel, as a probe for Cooper pair splitting. In addition to sequential tunneling of electrons through each dot, an additional transport channel exists in this system: crossed Andreev reflection, where a Cooper pair from the source is split between the two dots and recombined in the drain superconductor. Unlike non-equilibrium scenarios for Cooper pair splitting which involves superconducting/normal metal "forks", our proposal relies on an Aharonov-Bohm measurement of the DC Josephson …
We theoretically consider the observed soft gap in the proximity-induced superconducting state of semiconductor nanowires in the presence of spin-orbit coupling, Zeeman splitting, and tunneling leads, but in the absence of any extrinsic disorder. We critically consider the effects of three distinct intrinsic physical mechanisms (tunnel barrier to normal leads, temperature, and dissipation) on the phenomenology of the gap softness in the tunnel spectroscopy of the normal-superconductor junction as a function of spin splitting and chemical potential. We find that all three mechanisms individuall…
Motivated by a recent experimental report[1] claiming the likely observation of the Majorana mode in a semiconductor-superconductor hybrid structure[2,3,4,5], we study theoretically the dependence of the zero bias conductance peak associated with the zero-energy Majorana mode in the topological superconducting phase as a function of temperature, tunnel barrier potential, and a magnetic field tilted from the direction of the wire for realistic wires of finite lengths. We find that higher temperatures and tunnel barriers as well as a large magnetic field in the direction transverse to the wire l…
Motivated by an important recent experiment [Deng et al., Science 354, 1557 (2016)], we theoretically consider the interplay between Andreev bound states(ABSs) and Majorana bound states(MBSs) in quantum dot-nanowire semiconductor systems with proximity-induced superconductivity(SC), spin-orbit coupling and Zeeman splitting. The dot induces ABSs in the SC nanowire which show complex behavior as a function of Zeeman splitting and chemical potential, and the specific question is whether two such ABSs can come together forming a topological MBS. We consider physical situations involving the dot be…
We study the spin-resolved subgap transport in a triple quantum-dot system coupled to one superconducting and two ferromagnetic leads. We examine the Andreev processes in the parallel and antiparallel alignments of ferromagnets magnetic moments in both the linear and nonlinear response regimes. The emphasis is put on the analysis of the current cross-correlations between the currents flowing through the left and right arms of the device and relevant electron waiting time distributions. We show that both quantities can give an important insight into the subgap transport processes and their anal…
This work discusses theoretically the behavior of a microwave cavity and a Cooper pair beam splitter (CPS) coupled non-resonantly. The cavity frequency pull is modified when the CPS is resonant with a microwave excitation. This provides a direct way to probe the coherence of the Cooper pair splitting process. More precisely, the cavity frequency pull displays an anticrossing whose specificities can be attributed unambiguously to coherent Cooper pair injection. This work illustrates that microwave cavities represent a powerful tool to investigate current transport in complex nanocircuits.
In a device with a superconductor coupled to two parallel quantum dots (QDs) the electrical tunability of the QD levels can be used to exploit non-classical current correlations due to the splitting of Cooper pairs. We experimentally investigate the effect of a finite potential difference across one quantum dot on the conductance through the other completely grounded QD in a Cooper pair splitter fabricated on an InAs nanowire. We demonstrate that the electrical transport through the device can be tuned by electrical means to be dominated either by Cooper pair splitting (CPS), or by elastic co-…
We calculate the dynamical structure factor of the ν=5/2 non-abelian quantum Hall state in the dipole approximation, valid for large momenta. Due to the fact that both quasi-particles (qps) and quasi-holes (qhs) have an internal Majorana degree of freedom, a qp-qh pair has a fermionic degree of freedom which can be either empty or occupied, and leads to a splitting of the roton mode. Observation of this splitting by means of finite wavelength optical spectroscopy could provide evidence for Majorana modes in the ν=5/2 quantum Hall state.
Majorana zero modes are quasiparticle excitations in condensed matter systems that have been proposed as building blocks of fault-tolerant quantum computers [1]. They are expected to exhibit non-Abelian particle statistics, in contrast to the usual statistics of fermions and bosons, enabling quantum operations to be performed by braiding isolated modes around one another. Quantum braiding operations are topologically protected insofar as these modes are pinned near zero energy, and the pinning is predicted to be exponential as the modes become spatially separated. Following theoretical proposa…
The electrons forming a Cooper pair in a superconductor can be spatially separated preserving their spin entanglement by means of quantum dots coupled to both the superconductor and independent normal leads. We investigate the thermoelectric properties of such a Cooper pair splitter and demonstrate that cooling of a reservoir is an indication of non-local correlations induced by the entangled electron pairs. Moreover, we show that the device can be operated as a non-local thermoelectric heat engine. Both as a refrigerator and as a heat engine, the Cooper pair splitter reaches efficiencies clos…
Perfect Cooper pair splitting is proposed, based on crossed Andreev reflection (CAR) in a p-type semiconductor - superconductor - n-type semiconductor (pSn) junction. The ideal splitting is caused by the energy filtering that is enforced by the bandstructure of the electrodes. The pSn junction is modeled by the Bogoliubov-de Gennes equations and an extension of the Blonder-Tinkham-Klapwijk theory beyond the Andreev approximation. Despite a large momentum mismatch, the CAR current is predicted to be large. The proposed straightforward experimental design and the 100% degree of pureness of the n…
We demonstrate high efficiency Cooper pair splitting in a graphene-based device. We utilize a true Y-shape design effectively placing the splitting channels closer together: graphene is used as the central superconducting electrode as well as QD output channels, unlike previous designs where a conventional superconductor was used with tunnel barriers to the quantum dots (QD) of a different material. Superconductivity in graphene is induced via the proximity effect, thus resulting in both a large measured superconducting gap $Δ=0.5$meV, and a long coherence length $ξ=200$nm. The graphene-graphe…
