The control of nonlocal entanglement in solid state systems is a crucial ingredient of quantum technologies. We investigate a Cooper-pair splitter based on a double quantum dot realised in a semiconducting nanowire. In the presence of interdot tunnelling the system provides a simple mechanism to develop spatial entanglement even in absence of nonlocal coupling with the superconducting lead. We discuss the possibility to control the symmetry (singlet or triplet) of spatially separated, entangled electron pairs taking advantage of the spin-orbit coupling of the nanowire. We also demonstrate that the spin-orbit coupling does not impact over the entanglement purity of the nonlocal state generated in the double quantum dot system.
We investigate the nonlocal thermoelectric transport in a Cooper-pair splitter based on a double-quantum-dot-superconductor three-terminal hybrid structure. We find that the nonlocal coupling between the superconductor and the quantum dots gives rise to nonlocal thermoelectric effects which originate from the nonlocal particle-hole breaking of the system. We show that Cooper-pair splitting induces the generation of a thermo-current in the superconducting lead without any transfer of charge between the two normal metal leads. Conversely, we show that a nonlocal heat exchange between the normal …
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 conditio…
We consider the sub-gap physics of a hybrid double-quantum dot Cooper-pair splitter with large single-level spacings, in the presence of tunnelling between the dots and finite Coulomb intra- and inter-dot Coulomb repulsion. In the limit of a large superconducting gap, we treat the coupling of the dots to the superconductor exactly. We employ a generalized master-equation method which easily yields currents, noise and cross-correlators. In particular, for finite inter- and intra-dot Coulomb interaction, we investigate how the transport properties are determined by the interplay between local an…
We report an experimental study of Cooper pair splitting in an encapsulated graphene based multiterminal junction in the ballistic transport regime. Our device consists of two transverse junctions, namely the superconductor/graphene/superconductor and the normal metal/graphene/normal metal junctions. In this case, the electronic transport through one junction can be tuned by an applied bias along the other. 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 using a modified…
The Andreev transport properties of double quantum dot based Cooper pair splitters with one superconducting and two normal leads are studied theoretically in the Kondo regime. The influence of the superconducting pairing correlations on the local density of states, Andreev transmission coefficient and Cooper pair splitting efficiency is thoroughly analyzed. It is shown that finite superconducting pairing potential quickly suppresses the SU(2) Kondo effect, which can however reemerge for relatively large values of coupling to superconductor. In the SU(4) Kondo regime, a crossover from the SU(4)…
We measure the tunneling rates and coupling coefficients for local Andreev, non-local Andreev and elastic cotunneling processes. The non-local Andreev process, giving rise to Cooper pair splitting, exhibits the same coupling coefficient as the elastic co-tunneling whereas the local Andreev process is more than two orders of magnitude stronger than the corresponding non-local one. Theory estimates describe the findings and explain the large difference in the non-local and local coupling arising from competition between electron diffusion in the superconductor and tunnel junction transparency.
We investigate theoretically the noise and the full counting statistics of electrons that are emitted from a superconductor into two spatially separated quantum dots by the splitting of Cooper pairs and further on collected in two normal-state electrodes. With negatively-biased drain electrodes and a large superconducting gap, the dynamics of the Cooper pair splitter can be described by a Markovian quantum master equation. Using techniques from full counting statistics, we evaluate the electrical currents, their noise power spectra, and the power-power correlations in the output leads. The cur…
A Cooper pair splitter consists of a central superconducting contact, S, from which electrons are injected into two parallel, spatially separated quantum dots (QDs). This geometry and electron interactions can lead to correlated electrical currents due to the spatial separation of spin-singlet Cooper pairs from S. We present experiments on such a device with a series of bottom gates, which allows for spatially resolved tuning of the tunnel couplings between the QDs and the electrical contacts and between the QDs. Our main findings are gate-induced transitions between positive conductance corre…
Electron waiting times are an important concept in the analysis of quantum transport in nano-scale conductors. Here we show that the statistics of electron waiting times can be used to characterize Cooper pair splitters that create spatially separated spin-entangled electrons. A short waiting time between electrons tunneling into different leads is associated with the fast emission of a split Cooper pair, while long waiting times are governed by the slow injection of Cooper pairs from a superconductor. Experimentally, the waiting time distributions can be measured using real-time single-electr…
The generation of spin-entangled electrons is an important prerequisite for future solid-state quantum technologies. Cooper pairs in a superconductor can be split into separate electrons in a spin-singlet state, however, detecting their entanglement remains an open experimental challenge. Proposals to detect the entanglement by violating a Bell inequality typically require a large number of current cross-correlation measurements, and not all entangled states can be detected in this way. Here, we instead formulate an entanglement witness that can detect the spin-entanglement using only three cr…
A Cooper pair splitter consists of two quantum dots side-coupled to a conventional superconductor. Usually, the quantum dots are assumed to have a large charging energy compared to the superconducting gap, in order to suppress processes other than the coherent splitting of Cooper pairs. In this work, in contrast, we investigate the limit in which the charging energy is smaller than the superconducting gap. This allows us, in particular, to study the effect of a Zeeman field comparable to the charging energy. We find analytically that in this parameter regime the superconductor mediates an inte…
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 …
We analyze a model of double quantum dot Cooper pair splitter coupled to two ferromagnetic detectors, and demonstrate the possibility of determination of spin correlation by current measurements. We use perturbation theory taking account of the exchange interaction with the detectors, which leads to complex spin dynamics in the dots. This affects the measured spin and restricts the use of ferromagnetic detectors to the nonlinear current-voltage characteristic regime at current plateau, where the relevant spin projection is conserved, in contrast to the linear current-voltage characteristic reg…
We simulated the radiative response of the cavity quantum electrodynamics (QED) coupled to the double quantum dot Cooper pair splitter and analyzed its spectral dependence to get insight into dynamics of the Cooper pair transfers. The model is confined to the energy subspace where two entangled electrons are transferred to two normal electrodes through the inter-dot singlet state on two proximitized quantum dots. Our research is focused on the Andreev scatterings in the subgap regime, for which the local charge susceptibility $Π(ω_p)$ is derived, by means of Keldysh Green functions, in a whole…
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…
New generation of Cooper pair splitters defined on hybrid nanostructures are devices with high tunable coupling parameters. Transport measurements through these devices revealed clear signatures of interference effects and motivated us to introduce a new model, called the 3-sites model. These devices provide an ideal playground to tune the Cooper pair splitting (CPS) efficency on demand, and displays a rich variety of physical phenomena. In the present work we analyze theoretically the conductance of the 3-sites model in the linear and non-linear regimes and characterize the most representativ…
Exact analytical formulas are derived, by means of Keldysh Green functions, for currents and current correlation functions in a Cooper pair splitter modelled on a double quantum dot system coherently coupled to a superconductor and two normal metallic electrodes. Confining to the subspace with the inter-dot singlet we show perfect entanglement of split electrons in two separated crossed Andreev reflection processes. The studies are focused on the noise power spectrum in a whole bias voltage range. In particular, in the large voltage limit shot noise dominates and its spectrum exhibits two extr…
Thermoelectric effect generating electricity from thermal gradient and vice versa appears in numerous generic applications. Recently, an original prospect of thermoelectricity arising from the nonlocal Cooper pair splitting (CPS) and the elastic co-tunneling (EC) in hybrid normal metal-superconductor-normal metal (NSN) structures was foreseen. Here we demonstrate experimentally the existence of non-local Seebeck effect in a graphene-based CPS device comprising two quantum dots connected to an aluminum superconductor and theoretically validate the observations. This non-local Seebeck effect off…
We investigate Andreev transport through a quantum dot attached to two external ferromagnetic leads and one superconducting electrode. The transport properties of the system are studied by means of the real-time diagrammatic technique in the sequential tunneling regime. To distinguish various contributions to Andreev current we calculate the current cross-correlations, i.e. correlations between currents flowing through two junctions with normal leads. We analyze dependence of current cross-correlations on various parameters of the considered model, both in linear and nonlinear transport regime…
Current correlations in the interacting Cooper-pair beam-splitter
我们提出了一种方法，允许在相互作用的多终端介观系统中的电流和它们的相关性的计算涉及耦合到正常和/或超导引线的量子点。的形式主义依赖于分支电流和噪声交叉相关的表达在一个和两个粒子的点电子，然后评估自洽内的保守近似格林函数。然后，我们将其应用到最近提出的库珀对分束器设置[L。Hofstetter等人Nature（London）461 960（2009）; Phys.Rev.Lett. 107 136801（2011）; L. G. Herrmann等人，Phys. Rev. Lett. 104 026801（2010）]，我们将其建模为具有弱相互作用的双量子点，连接到超导引线和两个正常引线。我们的方法不仅使我们能够考虑到一个本地的排斥相互作用的点，但也研究它的竞争与点之间的直接隧穿。我们的研究结果表明，即使是一个弱的库仑排斥倾向于支持反对称制度（其中的点有相反的能量相对于超导化学势）的正电流交叉相关。
We present a proof-of-principle of how electronic transport measurements permits the observation of the Autler-Townes doublet, an optical property of nanodevices. The quantum physical system consists of one optically pumped quantum dot, a second auxiliary quantum dot, and a superconductor lead which provides an effective coupling between the dots via crossed Andreev reflection. Electrodes, working as source and drain, acts as nonequilibrium electronic reservoirs. Our calculations of the photocurrent at both, transient and stationary regimes, obtained using a density matrix formalism for open q…
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…
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…
Hybrid nanostructures consisting of two parallel InAs nanowires connected by an epitaxially grown superconductor (SC) shell recently became available. Due to the defect-free SC-semiconductor interface and the two quasi-one-dimensional channels being close by, these novel platforms can be utilized to spatially separate entangled pairs of electrons by using quantum dots (QD) in the so-called Cooper pair splitting (CPS) process. The minimized distance between the QDs overcomes the limitations of single-wire-based geometries and can boost the splitting efficiency. Here we investigate CPS in such a…
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, namely, crossed Andreev reflection, elastic cotunneling, and direct interdot tunneling, that are present in the device. Here, we provide a theoretical study of a simple device which consists of two quantum dots and a superconductor tunnel-coupled to the dots, often called a Cooper-pair splitter. We study the three…
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…
We propose an all-electrical experimental setup to detect and manipulate the amplitude of odd-frequency pairing in a double quantum dot. Odd-frequency pair amplitude is induced from the breakdown of orbital symmetry when Cooper pairs are injected in the double dot with electrons in different dots. When the dot levels are aligned with the Fermi energy, i.e., on resonance, nonlocal Andreev processes are directly connected to the presence of odd-frequency pairing. Therefore, their amplitude can be manipulated by tuning the level positions. Detection of nonlocal Andreev processes by conductance me…
In this work, we demonstrate the on-chip two-photon interference between single photons emitted by a single self-assembled InGaAs quantum dot and an external laser. The quantum dot is embedded within one arm of an air-clad directional coupler which acts as a beam-splitter for incoming light. Photons originating from an attenuated external laser are coupled to the second arm of the beam-splitter and then combined with the quantum dot photons, giving rise to two-photon quantum interference between dissimilar sources. We verify the occurrence of on-chip Hong-Ou-Mandel interference by cross-correl…
We demonstrate electro-mechanical control of an on-chip GaAs optical beam splitter containing a quantum dot single-photon source. The beam splitter consists of two nanobeam waveguides, which form a directional coupler (DC). The splitting ratio of the DC is controlled by varying the out-of-plane separation of the two waveguides using electro-mechanical actuation. We reversibly tune the beam splitter between an initial state, with emission into both output arms, and a final state with photons emitted into a single output arm. The device represents a compact and scalable tuning approach for use i…
