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arXiv:2110.04783 · Physical review. B./Physical review. B

Optimal entanglement witness for Cooper pair splitters

Minh Tam, Christian Flindt, Fredrik Brange

Published Dec 22, 2021• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.104.245425

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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 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.
We present theoretical studies on models for experimental determination of spin correlations and detection of quantum entanglement of spatially separated electrons of Cooper pairs. These models are based on both electric current cross-correlations and simpler dc current measurements. By proper optimization of the entanglement witness method, we demonstrate that entanglement detection is possible with any magnetic detectors regardless of their efficiency. This allows us to break the relatively high previously established limit of detector efficiency (spin polarization), $p>1/\sqrt{3}\ensuremath{\approx}58%$.
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.
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.
Abstract 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.
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.