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arXiv:quant-ph/0406013 · Physical Review B

Entanglement of electron spins in superconductors

Sangchul Oh, Jaewan Kim

Published Apr 29, 2005• 35 citations
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Abstract

We investigate entanglement of two electron spins forming Cooper pairs in an $s$-wave superconductor. The two-electron space-spin density matrix is obtained from the BCS ground state using a two-particle Green's function. It is demonstrated that a two spin state is not given by a spin singlet state but by a Werner state. It is found that the entanglement length, within which two spins are entangled, is not of the order of the coherence length but of the order of the Fermi wave length.

DOI
10.1103/physrevb.71.144523

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We investigate entanglement of two electron spins forming Cooper pairs in an sss-wave superconductor. The two-electron space-spin density matrix is obtained from the BCS ground state using a two-particle Green's function. It is demonstrated that a two spin state is not given by a spin singlet state but by a Werner state. It is found that the entanglement length, within which two spins are entangled, is not of the order of the coherence length but of the order of the Fermi wave length.
Bipartite quantum entanglement of electron spins of a Cooper pair in a dirty sss-wave superconductor is investigated at absolute zero temperature in terms of the exchange interaction and potential scattering as well as the relative distance between the two electrons. In the case of transition-metal impurities, we utilize the T-matrix approach to obtain the relevant Green's functions. We employ the two-electron spin-space density matrix, which is associated with the perturbed Green's functions. We show that a two-spin state can be described by the Werner state for dirty superconductors as well as for clean ones. We find that both the first and second quantum phase transitions (QPTs) are generated by the competition between potential scattering and other interactions. Meanwhile, it is not possible to determine the allowable value for each interaction individually. In the case of rare-earth impurities, concurrence, as a measure of bipartite entanglement, is discussed in terms of the relative distance and collision times for all finite and infinite Debye frequencies. The results confirm that quantum correlation and QPTs can be tuned by impurities.
We study entanglement of electron spins in many-body systems based on the Green's function approach. As an application we obtain the two-particle density matrix of a noninteracting electron gas and identify its two-spin density matrix as a Werner state. We calculate entanglement measures, a classical correlation, mutual information, and a pair-distribution function of two electrons at zero and finite temperatures. We find that changes of entanglement measures are proportional to T2{T}^{2}T2 at low temperatures.
Based on Yosida's ground state of the single-impurity Kondo Hamiltonian, we study three kinds of entanglement between an impurity and conduction electron spins. First, it is shown that the impurity spin is maximally entangled with all the conduction electrons. Second, a two-spin density matrix of the impurity spin and one conduction electron spin is given by a Werner state. We find that the impurity spin is not entangled with one conduction electron spin even within the Kondo screening length ${\ensuremath{\xi}}_{K}$, although there is the spin-spin correlation between them. Third, we show the density matrix of two conduction electron spins is nearly the same as that of a free electron gas. The single impurity does not change the entanglement structure of the conduction electrons in contrast to the dramatic change in electrical resistance.
We consider an s-wave superconductor (SC) which is tunnel coupled to two spatially separated Luttinger-liquid (LL) leads. We demonstrate that such a setup acts as an entangler, i.e., it creates spin singlets of two electrons which are spatially separated, thereby providing a source of electronic Einstein-Podolsky-Rosen pairs. We show that in the presence of a bias voltage, which is smaller than the energy gap in the SC, a stationary current of spin-entangled electrons can flow from the SC to the LL leads due to Andreev tunneling events. We discuss two competing transport channels for Cooper pairs to tunnel from the SC into the LL leads. On the one hand, the coherent tunneling of two electrons into the same LL lead is shown to be suppressed by strong LL correlations compared to single-electron tunneling into a LL. On the other hand, the tunneling of two spin-entangled electrons into different leads is suppressed by the initial spatial separation of the two electrons coming from the same Cooper pair. We show that the latter suppression depends crucially on the effective dimensionality of the SC. We identify a regime of experimental interest in which the separation of two spin-entangled electrons is favored. We determine the decay of the singlet state of two electrons injected into different leads caused by the LL correlations. Although the electron is not a proper quasiparticle of the LL, the spin information can still be transported via the spin-density fluctuations produced by the injected spin-entangled electrons.
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.