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# Spin cross-correlation experiments in an electron entangler

## Abstract

Correlations are fundamental in describing many-body systems. However, in experiments, correlations are notoriously difficult to assess on a microscopic scale, especially for electron spins. Even though it is firmly established theoretically that the electrons in a Cooper pair1 of a superconductor form maximally spin-entangled singlet states with opposite spin projections2,3,4, no spin correlation experiments have been demonstrated so far. Here we report the direct measurement of the spin cross-correlations between the currents of a Cooper pair splitter5,6,7,8,9,10,11,12,13, an electronic device that emits electrons originating from Cooper pairs. We use ferromagnetic split-gates14,15, compatible with nearby superconducting structures, to individually spin polarize the transmissions of the quantum dots 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, consistent with spin singlet emission, and deviates from the ideal value mostly due to the overlap of the Zeeman split quantum dot states. Our results demonstrate a new route to perform spin correlation experiments in nano-electronic devices, especially suitable for those relying on magnetic field sensitive superconducting elements, like triplet or topologically non-trivial superconductors16,17,18, or to perform Bell tests with massive particles19,20.

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## Data availability

All data in the publication are available in numerical form at https://doi.org/10.5281/zenodo.7140767.

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## Acknowledgements

This work has received funding from the Swiss National Science Foundation, the Swiss Nanoscience Institute, the Swiss NCCR QSIT, the FlagERA project, the QuantERA SuperTop project network and the FET Open project AndQC. C.S. has received funding from the European Research Council under the European Union’s Horizons 2020 research and innovation programme.

## Author information

Authors

### Contributions

A. Bordoloi fabricated the devices, performed the measurements, analysed and interpreted the data. V.Z. and L.S. grew the NWs. A. Baumgartner helped with the measurements, data analysis and interpretation. A. Bordoloi and A. Baumgartner wrote the paper. C.S. and A. Baumgartner initiated and supervised the project. All authors discussed the results and contributed to the manuscript.

### Corresponding authors

Correspondence to Arunav Bordoloi or Andreas Baumgartner.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Extended data figures and tables

### Extended Data Fig. 1 Spin dependence for competing two electron transport processes in a CPS device.

Here, we list all possible two electron transport processes relevant in a Cooper pair splitter, with a focus on the effects of spin filters placed at the two output ports. We note that LPT + SET (process 4) may mimick the CPS charge signal, but can be distinguished using spin filtering.

### Extended Data Fig. 2 $${\boldsymbol{\Delta }}{{\boldsymbol{G}}}_{{\bf{1}}}^{{\bf{m}}}$$ for the QD resonance A1 in Fig. 1d of the main text.

Maximum conductance $${G}_{1}^{{\rm{m}}}$$ as a function of gate voltage Vg2 for resonance A1 of QD1, showing peaks whenever QD2 is tuned across any of the Coulomb blockade resonances A2-D2.

### Extended Data Fig. 3 Normal-state measurements at B = + 150 mT.

a,b Differential conductances G1 and G2 respectively, measured simultaneously as a function of Vg1 and Vg2 at a bias voltage of Vdc = 0 and an external magnetic field of B = + 150 mT. c Maximum conductance $${G}_{1}^{{\rm{m}}}$$ as a function of gate voltage Vg2 at B = + 150 mT (red curve), showing a much smaller modulation and no obvious correlation to G2, when the latter (black curve) is tuned through Coulomb blockade peaks using Vg2. d Maximum conductance $${G}_{1}^{{\rm{m}}}$$ as a function of gate voltage Vg2 at B = 0 for the same resonances as in c, showing peaks when G2 is tuned across Coulomb blockade resonances by Vg2. We note that the scale of $${G}_{1}^{{\rm{m}}}$$ is adjusted to show the same conductance span.

### Extended Data Fig. 4 Data supporting Fig. 2 of the main text.

a,b Differential conductances G1 and G2 respectively, as a function of Vg1 and Vg2 at zero bias voltage, Vdc = 0 and zero external magnetic field, B = 0, for the resonance crossing (R1, R2) described in Fig. 2 in the main text.

### Extended Data Fig. 5 Transconductance Measurements for the four magnetization states at B = 0 and B = + 200 mT.

a,b,c,d Transconductance $${G}_{12}^{{\rm{(tr)}}}=\frac{{I}_{1}^{{\rm{(ac2)}}}}{{V}_{{\rm{g2}}}^{{\rm{(ac)}}}}$$ (a,c) and $${G}_{21}^{{\rm{(tr)}}}=\frac{{I}_{2}^{{\rm{(ac1)}}}}{{V}_{{\rm{g1}}}^{{\rm{(ac)}}}}$$ (b,d) measured as a function of Vg1 and Vg2 with a bias of Vdc = 25μV applied to S, for each magnetization state (j, k) indicated in each figure, at B = 0 (a,b) and B = + 200 mT (c,d) for the resonance crossings M1 and N2 in Fig. 3 of the main text. We do not observe any modulation of the transconductance if S is in the normal state (B = + 200 mT).

### Extended Data Fig. 6 Conductance maxima modulation at finite magnetic fields for main text Fig. 4.

a,b Differential conductances G1 and G2 respectively, measured as a function of Vg1 and Vg2 at zero bias Vdc = 0 and B = 0. c Maximum conductance $${G}_{1}^{{\rm{m}}}$$ as a function of the gate voltage Vg2 for the resonances in a and b, showing peaks when G2 is tuned across Coulomb blockade resonances by Vg2. d,e,f Modulation of the conductance maximum, $$\Delta {G}_{1}^{{\rm{m}}}$$, for all four magnetization states (j, k) measured at B = 0 (d), B = ± 20 mT (e), and B = ± 45 mT (f) for the resonance crossing (X1,X2).

### Extended Data Fig. 7 Background conductance Gbg versus the external magnetic field B.

Here we plot the background conductance at the respective resonance position, extracted from the parabolic fits discussed in the main text. This background is most probably dominated by LPT processes. We note that Gbg decreases with increasing B, as expected for the local Cooper pair related processes (see Extended Data Fig. 1) for an increasing QD spin polarization, while we would not expect such a decay for normal-state single electron processes on such small field scales.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–12 and references.

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Bordoloi, A., Zannier, V., Sorba, L. et al. Spin cross-correlation experiments in an electron entangler. Nature (2022). https://doi.org/10.1038/s41586-022-05436-z

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• DOI: https://doi.org/10.1038/s41586-022-05436-z