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arXiv:2203.07970 · Nature

Spin cross-correlation experiments in an electron entangler

Arunav Bordoloi, Valentina Zannier, Lucia Sorba, Christian Schönenberger, Andreas Baumgartner

Published Nov 23, 2022• 53 citations
arXivHTMLPDF

Abstract

Abstract Correlations are fundamental in describing many body systems - not only in natural sciences. However, in experiments, correlations are notoriously difficult to assess on the microscopic scale, especially for electron spins. Here, we demonstrate a direct measurement of the spin cross-correlations between the currents of a Cooper pair splitter, an electronic device that emits electrons originating from Cooper pairs in a superconductor. While it is firmly established theoretically that these electron pairs form maximally spin-entangled singlet states with opposite spin projections, no spin correlation experiments have been demonstrated so far. We use ferromagnetic sidegates, compatible with superconducting electronic structures, to individually spin polarize the transmissions of two quantum dots fabricated 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, compatible with spin singlet emission, and deviates from the ideal value mostly due to a finite overlap of the Zeeman split quantum dot states. Our results demonstrate a new route to perform spin auto- and cross correlation experiments in nanometer scaled electronic devices, especially suitable for those relying on magnetic field sensitive superconducting elements, like unconventional, triplet or topologically non-trivial superconductors, or to perform Bell tests with massive particles, like electrons.

DOI
10.1038/s41586-022-05436-z

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Abstract Correlations are fundamental in describing many body systems - not only in natural sciences. However, in experiments, correlations are notoriously difficult to assess on the microscopic scale, especially for electron spins. Here, we demonstrate a direct measurement of the spin cross-correlations between the currents of a Cooper pair splitter, an electronic device that emits electrons originating from Cooper pairs in a superconductor. While it is firmly established theoretically that these electron pairs form maximally spin-entangled singlet states with opposite spin projections, no spin correlation experiments have been demonstrated so far. We use ferromagnetic sidegates, compatible with superconducting electronic structures, to individually spin polarize the transmissions of two quantum dots fabricated 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, compatible with spin singlet emission, and deviates from the ideal value mostly due to a finite overlap of the Zeeman split quantum dot states. Our results demonstrate a new route to perform spin auto- and cross correlation experiments in nanometer scaled electronic devices, especially suitable for those relying on magnetic field sensitive superconducting elements, like unconventional, triplet or topologically non-trivial superconductors, or to perform Bell tests with massive particles, like electrons.
Entanglement is a valuable resource for quantum computing and information technologies. A promising way to obtain spin-entangled electrons is the splitting of Cooper pairs residing in superconductors. This is realized by a nanodevice called Cooper pair splitter. A Cooper pair splitter is realized by tunnel-coupling a superconductor to two quantum dots, which are each further tunnelcoupled to separate terminal leads. This setup enables the extraction and splitting of Cooper pairs from the superconductor into the terminals via the quantum dots. The quantum dots are commonly fabricated from semiconducting materials like InAs and InSb, that can manifest spin-orbit interaction. This thesis studies the effect of spin-orbit interaction on the spin state of the electron pair in the Cooper pair splitter. Activating the spin-orbit effects in a Cooper pair splitter requires a static magnetic field to be applied to the quantum dots. Together with the gate electrodes controlling the electrostatic energy of the quantum dots, the external magnetic field provides a handle for addressing different two-particle spin states. The spin-orbit interaction can be activated by various combinations of the gate voltage and Zeeman magnetic field, that create resonances between different states of the system. The most interesting coherent evolution, that can be invoked in this way, involves the simultaneous activation of the two spin-polarized triplet states. This gives a rise to an entangled Bell state, (|↑↑>+|↓↓>) / √2, that moreover exhibits spin blockade and thus stabilizes entanglement in the Cooper pair splitter. Secondly, the spin-orbit interaction can be used to bypass suppressed Cooper pair tunneling at a high superconductor-quantum dots detuning. Introducing a large Zeeman magnetic field allows Cooper pairs to tunnel in to the dots given that their spin state flips to the triplet |↓↓>. Although this regime lacks spin-entanglement, it is of interest for creating triplet Cooper pairs and spin-polarized superconducting currents.
A superconductor is a natural source of spin-entangled spatially separated electron pairs. Although the first Cooper-pair splitter devices have been realized recently, an experimental confirmation of the spin state and the entanglement of the emitted electron pairs has been lacking up to now. In this paper, a method is proposed to confirm the spin-singlet character of individual split Cooper pairs. Two quantum dots (QDs), each of them holding one spin-prepared electron, serve as the detector of the spin state of a single Cooper pair that is forced to split when it tunnels out from the superconductor to the QDs. The number of charges on the QDs, measured at the end of the procedure, carries information on the spin state of the extracted Cooper pair. The method relies on the experimentally established toolkit of QD-based spin qubits: resonant spin manipulation, Pauli blockade, and charge measurement.
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 cross-correlation measurements of the currents in the outputs of a Cooper pair splitter. We identify the optimal measurement settings for witnessing the entanglement, and we illustrate the use of our entanglement witness with a realistic model of a Cooper pair splitter for which we evaluate the cross-correlations of the output currents. Specifically, we find that the entanglement of the spins can be detected even with a moderate level of decoherence. Our work thereby paves the way for an experimental detection of the entanglement produced by Cooper pair splitters.
Abstract Entanglement of electrons is studied by means of current–current correlations in two Cooper pair splitter devices: with one and two proximized quantum dots (1QD and 2QD), in presence of intra- and inter-dot Coulomb interactions, and weakly coupled with metallic electrodes. The 1QD system, where Cooper pairs can be transmitted to the same or split to different normal electrodes, is contrasted with the 2QD device, where double occupancy of a single quantum dot is forbidden and transport is only through an inter-dot singlet due to non-local crossed Andreev reflection processes deep in the superconducting energy gap. Separating the current correlation function into components for partial currents of electrons and holes through various Andreev bound states, one can see bunching and antibunching of split particles: inter-level components between electron and hole currents flowing to different electrodes are positive, while intra-level electron–electron or hole–hole components are negative, respectively. Spectral decomposition of the frequency-dependent current cross-correlation is performed to get better insight into mechanisms of entanglement and dynamics of split Cooper pairs, and to extract various charge fluctuation processes with different relaxation times, related to electron and hole currents flowing through the Andreev bound states. Only low frequency polarization fluctuations are seen in the current cross-correlations, while various negative and positive high frequency (charge fluctuations) components compensate each other in the symmetric system.
New strategies to generate and to detect spin-entangled pairs of electrons in mesoscopic solid-state heterostructures are proposed. A bilayer-graphene device is designed in which Cooper pairs from an s-wave superconductor are injected into an electrostatically-defined topological channel. Due to a particular band structure, two electrons of a Cooper pair propagate in opposite directions and are hence spatially separated if atomic-scale defects are sparse. Neither energy filtering nor Coulomb repulsion are required. The device can be interpreted as a normal/superconducting/normal junction, in which emission of nonlocal singlets is equivalent to local Andreev reflection, in contrast to the widespread identification of Cooper pair splitting with crossed Andreev reflection. To detect entanglement, the Josephson current through two parallel single-level quantum dots is investigated at all occupations. Combining exact diagonalization and perturbation theory, signatures of nonlocal Cooper pair transport are identified in the critical current, a macroscopic quantity. The model reproduces recent experimental observations and predicts a nonlocal triplet ground state on the quantum dots due to a tunable superconductor-mediated exchange coupling, which, too, is visible in the critical current exactly if nonlocal Cooper pairs are transported. Entanglement detection is pursued, too, by converting spin-entangled Cooper pairs into polarization-entangled photons, which are subsequently probed by a Bell-type measurement. A closed emission cycle is constructed, such that no correlations between the photons and the electrons remain, which are detrimental for entanglement transfer. Inevitable side channels and imperfections are identified and accounted for by a particular measurement protocol. Furthermore, it is discussed how spin-charge separation can be used to extract signatures of nonlocal spin singlets from the average current through an electronic beam splitter. This complements a known mechanism based on exchange statistics by which only the more difficult-to-access noise is sensitive for entanglement. The device can be realized by two crossed nanowires or by quantum-Hall edge states in Corbino geometry. The thesis also contains a review on entanglement and on recent theoretical and experimental progress in solid-state systems towards entanglement detection and generation, focusing on Cooper pair splitting.
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 interdot tunneling term with a spin symmetry determined by the Zeeman field. Together with electrostatically tunable quantum dots, we show that this makes it possible to engineer a spin triplet state shared between the quantum dots. Compared to previous works, we thus extend the capabilities of the Cooper pair splitter to create entangled nonlocal electron pairs.