Cooper pair splitting by means of graphene quantum dots

Crossed Andreev Reflection and Elastic Cotunneling in Three Quantum Dots Coupled by Superconductors

The formation of a topological superconducting phase in a quantum-dot-based Kitaev chain requires nearest neighbor crossed Andreev reflection and elastic cotunneling. Here, we report on a hybrid InSb nanowire in a three-site Kitaev chain geometry-the smallest system with well-defined bulk and edge-where two superconductor-semiconductor hybrids separate three quantum dots. We demonstrate pairwise crossed Andreev reflection and elastic cotunneling between both pairs of neighboring dots and show sequential tunneling processes involving all three quantum dots. These results are the next step toward the realization of topological superconductivity in long Kitaev chain devices with many coupled quantum dots.

Controllable Single Cooper Pair Splitting in Hybrid Quantum Dot Systems

Cooper pair splitters hold utility as a platform for investigating the entanglement of electrons in Cooper pairs, but probing splitters with voltage-biased Ohmic contacts prevents the retention of electrons from split pairs since they can escape to the drain reservoirs. We report the ability to controllably split and retain single Cooper pairs in a multi-quantum-dot device isolated from lead reservoirs, and separately demonstrate a technique for detecting the electrons emerging from a split pair. First, we identify a coherent Cooper pair splitting charge transition using dispersive gate sensing at GHz frequencies. Second, we utilize a double quantum dot as an electron parity sensor to detect parity changes resulting from electrons emerging from a superconducting island.

Triplet correlations in Cooper pair splitters realized in a two-dimensional electron gas

Cooper pairs occupy the ground state of superconductors and are typically composed of maximally entangled electrons with opposite spin. In order to study the spin and entanglement properties of these electrons, one must separate them spatially via a process known as Cooper pair splitting (CPS). Here we provide the first demonstration of CPS in a semiconductor two-dimensional electron gas (2DEG). By coupling two quantum dots to a superconductor-semiconductor hybrid region we achieve efficient Cooper pair splitting, and clearly distinguish it from other local and non-local processes. When the spin degeneracy of the dots is lifted, they can be operated as spin-filters to obtain information about the spin of the electrons forming the Cooper pair. Not only do we observe a near perfect splitting of Cooper pairs into opposite-spin electrons (i.e. conventional singlet pairing), but also into equal-spin electrons, thus achieving triplet correlations between the quantum dots. Importantly, the exceptionally large spin-orbit interaction in our 2DEGs results in a strong triplet component, comparable in amplitude to the singlet pairing. The demonstration of CPS in a scalable and flexible platform provides a credible route to study on-chip entanglement and topological superconductivity in the form of artificial Kitaev chains.

Particle-hole symmetry protects spin-valley blockade in graphene quantum dots

Particle-hole symmetry plays an important role in the characterization of topological phases in solid-state systems¹. It is found, for example, in free-fermion systems at half filling and it is closely related to the notion of antiparticles in relativistic field theories². In the low-energy limit, graphene is a prime example of a gapless particle-hole symmetric system described by an effective Dirac equation^3,4 in which topological phases can be understood by studying ways to open a gap by preserving (or breaking) symmetries^5,6. An important example is the intrinsic Kane-Mele spin-orbit gap of graphene, which leads to a lifting of the spin-valley degeneracy and renders graphene a topological insulator in a quantum spin Hall phase⁷ while preserving particle-hole symmetry. Here we show that bilayer graphene allows the realization of electron-hole double quantum dots that exhibit near-perfect particle-hole symmetry, in which transport occurs via the creation and annihilation of single electron-hole pairs with opposite quantum numbers. Moreover, we show that particle-hole symmetric spin and valley textures lead to a protected single-particle spin-valley blockade. The latter will allow robust spin-to-charge and valley-to-charge conversion, which are essential for the operation of spin and valley qubits.

Tunable Superconducting Coupling of Quantum Dots via Andreev Bound States in Semiconductor-Superconductor Nanowires

Semiconductor quantum dots have proven to be a useful platform for quantum simulation in the solid state. However, implementing a superconducting coupling between quantum dots mediated by a Cooper pair has so far suffered from limited tunability and strong suppression. This has limited applications such as Cooper pair splitting and quantum dot simulation of topological Kitaev chains. In this Letter, we propose how to mediate tunable effective couplings via Andreev bound states in a semiconductor-superconductor nanowire connecting two quantum dots. We show that in this way it is possible to individually control both the coupling mediated by Cooper pairs and by single electrons by changing the properties of the Andreev bound states with easily accessible experimental parameters. In addition, the problem of coupling suppression is greatly mitigated. We also propose how to experimentally extract the coupling strengths from resonant current in a three-terminal junction. Our proposal will enable future experiments that have not been possible so far.

Singlet and triplet Cooper pair splitting in hybrid superconducting nanowires

In most naturally occurring superconductors, electrons with opposite spins form Cooper pairs. This includes both conventional s-wave superconductors such as aluminium, as well as high-transition-temperature, d-wave superconductors. Materials with intrinsic p-wave superconductivity, hosting Cooper pairs made of equal-spin electrons, have not been conclusively identified, nor synthesized, despite promising progress^1-3. Instead, engineered platforms where s-wave superconductors are brought into contact with magnetic materials have shown convincing signatures of equal-spin pairing^4-6. Here we directly measure equal-spin pairing between spin-polarized quantum dots. This pairing is proximity-induced from an s-wave superconductor into a semiconducting nanowire with strong spin-orbit interaction. We demonstrate such pairing by showing that breaking a Cooper pair can result in two electrons with equal spin polarization. Our results demonstrate controllable detection of singlet and triplet pairing between the quantum dots. Achieving such triplet pairing in a sequence of quantum dots will be required for realizing an artificial Kitaev chain^7-9.

Spin cross-correlation experiments in an electron entangler

Correlations are fundamental in describing many-body systems. However, in experiments, correlations are notoriously difficult to assess on a microscopic scale, especially for electron spins. Even though it is firmly established theoretically that the electrons in a Cooper pair¹ of a superconductor form maximally spin-entangled singlet states with opposite spin projections^2-4, no spin correlation experiments have been demonstrated so far. Here we report the direct measurement of the spin cross-correlations between the currents of a Cooper pair splitter^5-13, an electronic device that emits electrons originating from Cooper pairs. We use ferromagnetic split-gates^14,15, compatible with nearby superconducting structures, to individually spin polarize the transmissions of the quantum dots in the two electronic paths, which act as tunable spin filters. The signals are detected in standard transport and in highly sensitive transconductance experiments. We find that the spin cross-correlation is negative, consistent with spin singlet emission, and deviates from the ideal value mostly due to the overlap of the Zeeman split quantum dot states. Our results demonstrate a new route to perform spin correlation experiments in nano-electronic devices, especially suitable for those relying on magnetic field sensitive superconducting elements, like triplet or topologically non-trivial superconductors^16-18, or to perform Bell tests with massive particles^19,20.

Local and Nonlocal Two-Electron Tunneling Processes in a Cooper Pair Splitter

We measure the rates and coupling coefficients for local Andreev, nonlocal Andreev, and elastic cotunneling processes. The nonlocal Andreev process, giving rise to Cooper pair splitting, exhibits the same coupling coefficient as the elastic cotunneling whereas the local Andreev process is more than 2 orders of magnitude stronger than the corresponding nonlocal one. Theory estimates describe the findings and explain the large difference in the nonlocal and local coupling arising from competition between electron diffusion in the superconductor and tunnel junction transparency.

Entanglement of Nambu Spinors and Bell Inequality Test without Beam Splitters

The identification of electronic entanglement in solids remains elusive so far, which is owed to the difficulty of implementing spinor-selective beam splitters with tunable polarization direction. Here, we propose to overcome this obstacle by producing and detecting a particular type of entanglement encoded in the Nambu spinor or electron-hole components of quasiparticles excited in quantum Hall edge states. Because of the opposite charge of electrons and holes, the detection of the Nambu spinor translates into a charge-current measurement, which eliminates the need for beam splitters and assures a high detection rate. Conveniently, the spinor correlation function at fixed effective polarizations derives from a single current-noise measurement, with the polarization directions of the detector easily adjusted by coupling the edge states to a voltage gate and a superconductor, both having been realized in experiments. We show that the violation of Bell inequality occurs in a large parameter region. Our Letter opens a new route for probing quasiparticle entanglement in solid-state physics exempt from traditional beam splitters.

Dynamic Cooper Pair Splitter

Cooper pair splitters are promising candidates for generating spin-entangled electrons. However, the splitting of Cooper pairs is a random and noisy process, which hinders further synchronized operations on the entangled electrons. To circumvent this problem, we here propose and analyze a dynamic Cooper pair splitter that produces a noiseless and regular flow of spin-entangled electrons. The Cooper pair splitter is based on a superconductor coupled to quantum dots, whose energy levels are tuned in and out of resonance to control the splitting process. We identify the optimal operating conditions for which exactly one Cooper pair is split per period of the external drive and the flow of entangled electrons becomes noiseless. To characterize the regularity of the Cooper pair splitter in the time domain, we analyze the g^{(2)} function of the output currents and the distribution of waiting times between split Cooper pairs. Our proposal is feasible using current technology, and it paves the way for dynamic quantum information processing with spin-entangled electrons.

Real-time observation of Cooper pair splitting showing strong non-local correlations

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 individual Cooper pairs, enabling direct access to the time-resolved statistics of Cooper pair splitting. We determine the correlation statistics arising from two-electron processes and find a pronounced peak that is two orders of magnitude larger than the background. Our experiment thereby allows to unambiguously pinpoint and select split Cooper pairs with 99% fidelity. These results open up an avenue for performing experiments that tap into the spin-entanglement of split Cooper pairs.

The splitting of Cooper pairs in superconductors has been challenging to detect experimentally. Here, the authors observe the real-time splitting of individual Cooper pairs in a superconducting device.

Electrically Controlled Crossed Andreev Reflection in Two-Dimensional Antiferromagnets

We report generic and tunable crossed Andreev reflection (CAR) in a superconductor sandwiched between two antiferromagnetic layers. We consider recent examples of two-dimensional magnets with hexagonal lattices, where gate voltages control the carrier type and density, and predict a robust signature of perfect CAR in the nonlocal differential conductance with one electron-doped and one hole-doped antiferromagnetic lead. The magnetic field-free and spin-degenerate CAR signal is electrically controlled and visible over a large voltage range, showing promise for solid-state quantum entanglement applications.

Ballistic Graphene Cooper Pair Splitter

We report an experimental study of a Cooper pair splitter based on ballistic graphene multiterminal junctions. In a two transverse junction geometry, namely the superconductor-graphene-superconductor and the normal metal-graphene-normal metal, we observe clear signatures of Cooper pair splitting in the local as well as nonlocal electronic transport measurements. Our experimental data can be very well described by our beam splitter model. These results open up possibilities to design new entangled state detection experiments using ballistic Cooper pair splitters.

Thermoelectric current in a graphene Cooper pair splitter

Generation of electric voltage in a conductor by applying a temperature gradient is a fundamental phenomenon called the Seebeck effect. This effect and its inverse is widely exploited in diverse applications ranging from thermoelectric power generators to temperature sensing. Recently, a possibility of thermoelectricity arising from the interplay of the non-local Cooper pair splitting and the elastic co-tunneling in the hybrid normal metal-superconductor-normal metal structures was predicted. Here, we report the observation of the non-local Seebeck effect in a graphene-based Cooper pair splitting device comprising two quantum dots connected to an aluminum superconductor and present a theoretical description of this phenomenon. The observed non-local Seebeck effect offers an efficient tool for producing entangled electrons.

Thermoelectricity due to the interplay of the nonlocal Cooper pair splitting and the elastic co-tunneling in normal metal-superconductor-normal metal structure is predicted. Here, the authors observe the non-local Seebeck effect in a graphene-based Cooper pair splitting device.

Engineering Crossed Andreev Reflection in Double-Bilayer Graphene

Crossed Andreev reflection (CAR) is a nonlocal process that converts an incoming electron (hole) from one normal electrode to an out-going hole (electron) in another normal electrode through a superconductor (SC). CAR corresponds to the inverse process of Cooper pair splitting, which generates a quantum-entangled electron pair with spatial separation. Here, we fabricated vertically stacked double bilayer graphene (BLG) connected via a superconducting electrode and achieved a spacing between BLG sheets of ∼14 nm, which is far shorter than the superconducting coherence length. We confirm the highly efficient CAR effect by observing strong negative differential resistance in a nonlocal configuration and demonstrate that the competing processes against the CAR can be effectively suppressed by separately tuning the chemical potential of each BLG. The dependence of nonlocal signals on bias voltage, temperature, and chemical potential is consistent with the predicted CAR process. Our results provide a new pathway to a novel SC-based quantum entangler with the in situ tunability of the correlated-pair-splitting efficiency.

Dominant nonlocal superconducting proximity effect due to electron-electron interaction in a ballistic double nanowire

Cooper pair splitting (CPS) can induce nonlocal correlation between two normal conductors that are coupled to a superconductor. CPS in a double one-dimensional electron gas is an appropriate platform for extracting a large number of entangled electron pairs and is one of the key ingredients for engineering Majorana fermions with no magnetic field. In this study, we investigated CPS by using a Josephson junction of a gate-tunable ballistic InAs double nanowire. The measured switching current into the two nanowires is significantly larger than the sum of the switching current into the respective nanowires, indicating that interwire superconductivity is dominant compared with intrawire superconductivity. From its dependence on the number of propagating channels in the nanowires, the observed CPS is assigned to one-dimensional electron-electron interaction. Our results will pave the way for the utilization of one-dimensional electron-electron interaction to reveal the physics of high-efficiency CPS and to engineer Majorana fermions in double nanowire systems via CPS.

Enhancement of crossed Andreev reflection in a Kitaev ladder connected to normal metal leads

We study nonlocal transport in a two-leg Kitaev ladder connected to two normal metals. The coupling between the two legs of the ladder when the legs are maintained at a (large) superconducting phase difference, results in the creation of subgap Andreev states. These states in turn are responsible for the enhancement of crossed Andreev reflection (CAR). We find that tuning the different parameters of the system suitably leads to enhancement of CAR signalled by transconductance acquiring the most negative value possible. Furthermore, subgap states cause oscillations of the transconductance as a function of various system parameters such as chemical potential and ladder length, which are seen to be a consequence of Fabry-Pérot resonance.

Perfect Crossed Andreev Reflection in Dirac Hybrid Junctions in the Quantum Hall Regime

Perfect crossed Andreev reflection (CAR) is striking for high-efficiency Cooper pair splitting, which bears promising applications in quantum communication. Recent experimental advances have disclosed the way to explore CAR in Dirac fermion systems under ultrastrong magnetic fields. We develop a scattering approach to study quantum-Hall-superconductor-quantum-Hall junctions formed by a two-dimensional time-reversal symmetric Dirac semimetal. We propose two different setups of the hybrid junction in the quantum limit, where only zeroth Landau levels are involved in transport to exploit perfect CAR. In both setups, the CAR probability can reach unity without applying bias voltage and is controllable by the magnetic field strength, the junction width, the length, and the doping of the superconductor. CAR dominates the nonlocal transport and is directly measurable by the differential conductances. We also identify a quantized spin injection per CAR event in one of the two setups. Our proposal is experimentally feasible and will be helpful for exploring high-efficiency Cooper pair splitting and spin injection in Dirac materials.

Transport signatures of an Andreev molecule in a quantum dot-superconductor-quantum dot setup

Hybrid devices combining quantum dots with superconductors are important building blocks of conventional and topological quantum-information experiments. A requirement for the success of such experiments is to understand the various tunneling-induced non-local interaction mechanisms that are present in the devices, namely crossed Andreev reflection, elastic co-tunneling, and direct interdot tunneling. Here, we provide a theoretical study of a simple device that consists of two quantum dots and a superconductor tunnel-coupled to the dots, often called a Cooper-pair splitter. We study the three special cases where one of the three non-local mechanisms dominates, and calculate measurable ground-state properties, as well as the zero-bias and finite-bias differential conductance characterizing electron transport through this device. We describe how each non-local mechanism controls the measurable quantities, and thereby find experimental fingerprints that allow one to identify and quantify the dominant non-local mechanism using experimental data. Finally, we study the triplet blockade effect and the associated negative differential conductance in the Cooper-pair splitter, and show that they can arise regardless of the nature of the dominant non-local coupling mechanism. Our results should facilitate the characterization of hybrid devices, and their optimization for various quantum-information-related experiments and applications.

Electron waiting times in hybrid junctions with topological superconductors

We investigate the waiting time distributions (WTDs) of superconducting hybrid junctions, considering both conventional and topologically nontrivial superconductors hosting Majorana bound states at their edges. To this end, we employ a scattering matrix formalism that allows us to evaluate the waiting times between the transmissions and reflections of electrons or holes. Specifically, we analyze normal-metal–superconductor (NIS) junctions and NISIN junctions, where Cooper pairs are spatially split into different leads. The distribution of waiting times is sensitive to the simultaneous reflection of electrons and holes, which is enhanced by the zero-energy state in topological superconductors. For the NISIN junctions, the WTDs of trivial superconductors feature a sharp dependence on the applied voltage, while for topological ones they are mostly independent of it. This particular voltage dependence is again connected to the presence of topological edge states, showing that WTDs are a promising tool for identifying topological superconductivity.

Proximity coupling in superconductor-graphene heterostructures

This review discusses the electronic properties and the prospective research directions of superconductor-graphene heterostructures. The basic electronic properties of graphene are introduced to highlight the unique possibility of combining two seemingly unrelated physics, superconductivity and relativity. We then focus on graphene-based Josephson junctions, one of the most versatile superconducting quantum devices. The various theoretical methods that have been developed to describe graphene Josephson junctions are examined, together with their advantages and limitations, followed by a discussion on the advances in device fabrication and the relevant length scales. The phase-sensitive properties and phase-particle dynamics of graphene Josephson junctions are examined to provide an understanding of the underlying mechanisms of Josephson coupling via graphene. Thereafter, microscopic transport of correlated quasiparticles produced by Andreev reflections at superconducting interfaces and their phase-coherent behaviors are discussed. Quantum phase transitions studied with graphene as an electrostatically tunable 2D platform are reviewed. The interplay between proximity-induced superconductivity and the quantum-Hall phase is discussed as a possible route to study topological superconductivity and non-Abelian physics. Finally, a brief summary on the prospective future research directions is given.

Current cross-correlations in double quantum dot based Cooper pair splitters with ferromagnetic leads

We investigate the current cross-correlations in a double quantum dot based Cooper pair splitter coupled to one superconducting and two ferromagnetic electrodes. The analysis is performed by assuming a weak coupling between the double dot and ferromagnetic leads, while the coupling to the superconductor is arbitrary. Employing the perturbative real-time diagrammatic technique, we study the Andreev transport properties of the device, focusing on the Andreev current cross-correlations, for various parameters of the model, both in the linear and nonlinear response regimes. Depending on parameters and transport regime, we find both positive and negative current cross-correlations. Enhancement of the former type of cross-correlations indicates transport regimes, in which the device works with high Cooper pair splitting efficiency, contrary to the latter type of correlations, which imply negative influence on the splitting. The processes and mechanisms leading to both types of current cross-correlations are thoroughly examined and discussed, giving a detailed insight into the Andreev transport properties of the considered device.

Universality of Schmidt decomposition and particle identity

Schmidt decomposition is a widely employed tool of quantum theory which plays a key role for distinguishable particles in scenarios such as entanglement characterization, theory of measurement and state purification. Yet, its formulation for identical particles remains controversial, jeopardizing its application to analyze general many-body quantum systems. Here we prove, using a newly developed approach, a universal Schmidt decomposition which allows faithful quantification of the physical entanglement due to the identity of particles. We find that it is affected by single-particle measurement localization and state overlap. We study paradigmatic two-particle systems where identical qubits and qutrits are located in the same place or in separated places. For the case of two qutrits in the same place, we show that their entanglement behavior, whose physical interpretation is given, differs from that obtained before by different methods. Our results are generalizable to multiparticle systems and open the way for further developments in quantum information processing exploiting particle identity as a resource.

Low-noise correlation measurements based on software-defined-radio receivers and cooled microwave amplifiers

We present a microwave correlation measurement system based on two low-cost USB-connected software defined radio dongles modified to operate as coherent receivers by using a common local oscillator. Existing software is used to obtain I/Q samples from both dongles simultaneously at a software tunable frequency. To achieve low noise, we introduce an easy low-noise solution for cryogenic amplification at 600-900 MHz based on single discrete HEMT with 21 dB gain and 7 K noise temperature. In addition, we discuss the quantization effects in a digital correlation measurement and determination of optimal integration time by applying Allan deviation analysis.

High Efficiency CVD Graphene-lead (Pb) Cooper Pair Splitter

Generation and manipulation of quantum entangled electrons is an important concept in quantum mechanics, and necessary for advances in quantum information processing; but not yet established in solid state systems. A promising device is a superconductor-two quantum dots Cooper pair splitter. Early nanowire based devices, while efficient, are limited in scalability and further electron manipulation. We demonstrate an optimized, high efficiency, CVD grown graphene-based Cooper pair splitter. Our device is designed to induce superconductivity in graphene via the proximity effect, resulting in both a large superconducting gap Δ = 0.5 meV, and coherence length ξ = 200 nm. The flat nature of the device lowers parasitic capacitance, increasing charging energy EC. Our design also eases geometric restrictions and minimizes output channel separation. As a result we measure a visibility of up to 86% and a splitting efficiency of up to 62%. This will pave the way towards near unity efficiencies, long distance splitting, and post-splitting electron manipulation.

Quantum entanglement of identical particles by standard information-theoretic notions

Quantum entanglement of identical particles is essential in quantum information theory. Yet, its correct determination remains an open issue hindering the general understanding and exploitation of many-particle systems. Operator-based methods have been developed that attempt to overcome the issue. Here we introduce a state-based method which, as second quantization, does not label identical particles and presents conceptual and technical advances compared to the previous ones. It establishes the quantitative role played by arbitrary wave function overlaps, local measurements and particle nature (bosons or fermions) in assessing entanglement by notions commonly used in quantum information theory for distinguishable particles, like partial trace. Our approach furthermore shows that bringing identical particles into the same spatial location functions as an entangling gate, providing fundamental theoretical support to recent experimental observations with ultracold atoms. These results pave the way to set and interpret experiments for utilizing quantum correlations in realistic scenarios where overlap of particles can count, as in Bose-Einstein condensates, quantum dots and biological molecular aggregates.

Magnetic Field Tuning and Quantum Interference in a Cooper Pair Splitter

Cooper pair splitting (CPS) is a process in which the electrons of the naturally occurring spin-singlet pairs in a superconductor are spatially separated using two quantum dots. Here, we investigate the evolution of the conductance correlations in an InAs CPS device in the presence of an external magnetic field. In our experiments the gate dependence of the signal that depends on both quantum dots continuously evolves from a slightly asymmetric Lorentzian to a strongly asymmetric Fano-type resonance with increasing field. These experiments can be understood in a simple three-site model, which shows that the nonlocal CPS leads to symmetric line shapes, while the local transport processes can exhibit an asymmetric shape due to quantum interference. These findings demonstrate that the electrons from a Cooper pair splitter can propagate coherently after their emission from the superconductor and how a magnetic field can be used to optimize the performance of a CPS device. In addition, the model calculations suggest that the estimate of the CPS efficiency in the experiments is a lower bound for the actual efficiency.