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arXiv:cond-mat/0001011 · Physical review. B, Condensed matter

Spin-dependent Josephson current through double quantum dots and measurement of entangled electron states

Mahn‐Soo Choi, Christoph Bruder, Daniel Loss

Published Nov 15, 2000• 163 citations
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Abstract

We study a double quantum dot, each dot of which is tunnel coupled to superconducting leads. In the Coulomb blockade regime, a spin-dependent Josephson coupling between two superconductors is induced, as well as an antiferromagnetic Heisenberg exchange coupling between the spins on the double dot that can be tuned by the superconducting phase difference. We show that the correlated spin states---singlet or triplets---on the double dot can be probed via the Josephson current in a dc-superconducting quantum interference device setup.

DOI
10.1103/physrevb.62.13569

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We study a double quantum dot, each dot of which is tunnel coupled to superconducting leads. In the Coulomb blockade regime, a spin-dependent Josephson coupling between two superconductors is induced, as well as an antiferromagnetic Heisenberg exchange coupling between the spins on the double dot that can be tuned by the superconducting phase difference. We show that the correlated spin states---singlet or triplets---on the double dot can be probed via the Josephson current in a dc-superconducting quantum interference device setup.
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Serial double quantum dots created in semiconductor nanostructures provide a versatile platform for investigating two-electron spin quantum states, which can be tuned by electrostatic gating and an external magnetic field. In this Rapid Communication, we directly measure the supercurrent reversal between adjacent charge states of an InAs nanowire double quantum dot with superconducting leads, in good agreement with theoretical models. In the even charge parity sector, we observe a supercurrent blockade with increasing magnetic field, corresponding to the spin singlet to triplet transition. Our results demonstrate a direct spin to supercurrent conversion, the superconducting equivalent of the Pauli spin blockade. This effect can be exploited in hybrid quantum architectures coupling the quantum states of spin systems and superconducting circuits.
Addressing the feasibility of quantum communication with electrons we consider entangled spin states of electrons in a double-dot which is weakly coupled to leads. We show that the entanglement of two electrons in the double-dot can be detected in mesoscopic transport and noise measurements. In the Coulomb blockade and cotunneling regime the singlet and triplet states lead to phase-coherent current and noise contributions of opposite signs and to Aharonov-Bohm and Berry phase oscillations. These oscillations are a genuine two-particle effect and provide a direct measure of nonlocality in entangled states. We show that the ratio of zero-frequency noise to current is equal to the electron charge.
We propose and analyze a spin-entangler for electrons based on an s-wave superconductor coupled to two quantum dots, each of which is coupled to normal Fermi leads. We show that in the presence of a voltage bias and in the Coulomb blockade regime two correlated electrons provided by the Andreev process can coherently tunnel from the superconductor via different dots into different leads. The spin singlet coming from the Cooper pair remains preserved in this process, and the setup provides a source of mobile and nonlocal spin-entangled electrons. The transport current is calculated and shown to be dominated by a two-particle Breit-Wigner resonance that allows the injection of two spin-entangled electrons into different leads at exactly the same orbital energy, which is a crucial requirement for the detection of spin entanglement via noise measurements. The coherent tunneling of both electrons into the same lead is suppressed by the on-site Coulomb repulsion and/or the superconducting gap, while the tunneling into different leads is suppressed through the initial separation of the tunneling electrons. In the regime of interest the particle-hole excitations of the leads are shown to be negligible. The Aharonov-Bohm oscillations in the current are shown to contain single- and two-electron periods with amplitudes that both vanish with increasing Coulomb repulsion albeit differently fast.
We study the critical Josephson current flowing through a double quantum dot weakly coupled to two superconducting leads. We use analytical as well as numerical methods to investigate this setup in the limit of small and large bandwidth leads in all possible charging states, where we account for on-site interactions exactly. Our results provide clear signatures of nonlocal spin-entangled pairs, which support interpretations of recent experiments [R. S. Deacon, A. Oiwa, J. Sailer, S. Baba, Y. Kanai, K. Shibata, K. Hirakawa, and S. Tarucha, Nat. Commun. 6, 7446 (2015)]. In addition, we find that the ground state with one electron on each quantum dot can undergo a tunable singlet-triplet phase transition in the regime where the superconducting gap in the leads is not too large, which gives rise to an additional new signature of nonlocal Cooper-pair transport.
We show how superconductors can be used to couple, initialize, and read out spatially separated spin qubits. When two single-electron quantum dots are tunnel coupled to the same superconductor, the singlet component of the two-electron state partially leaks into the superconductor via crossed Andreev reflection. This induces a gate-controlled singlet-triplet splitting which, with an appropriate superconductor geometry, remains large for dot separations within the superconducting coherence length. Furthermore, we show that when two double-dot singlet-triplet qubits are tunnel coupled to a superconductor with finite charging energy, crossed Andreev reflection enables a strong two-qubit coupling over distances much larger than the coherence length.