Quantum Entanglement in Superconducting

Beamsplitters

Henning SollerCapri, 16.4.2012

Definition of EntanglementThe state of the system cannot be written as a product state.

R. F. Werner, Phys. Rev. A 40, 4277, 1989J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108,1175 (1957)

spin states

ground state of BCS superconductor

L. Hofstetter, S. Csonka, J. Nygard, and C. Schönenberger, Nature 461, 960, 2009 L. Herrmann, F. Portier, P. Roche, A. Levy Yeyati, T. Kontos, and C. Strunk Phys. Rev. Lett. 104, 026801, 2010L. Hofstetter, S. Csonka, A. Baumgartner, G. Fülöp, S. d’Hollosy, J. Nygård, and C. Schönenberger Phys. Rev. Lett. 107, 136801, 2011J. Wei and V. Chandrasekhar Nature Physics 6, 494–498 (2010)

C. Schönenberger, Physik in unserer Zeit 2, 58-59, 2010

How to detect?

Result: Result:

Now let us consider just tunnel contacts:

A. Di Lorenzo and Yu. V. Nazarov, Phys. Rev. Lett. 94, 210601, 2005

for chosen axes m and m‘

Nice in principle, but…

• What happens for setups closer to actual experiments?

• What about interaction effects?• Polarisation > 84 % is needed• Time-resolved detection scheme

Other entanglement detection schemes:D. Loss and E. V. Sukhorukov, Phys. Rev. Lett. 84, 1035, 2000G. Burkard, D. Loss, and E. V. Sukhorukov, Phys. Rev. B 61, R16303, 2000N. M. Chtchelkatchev, G. Blatter, G. B. Lesovik, and T. Martin, Phys. Rev. B 66, 161320, 2002A. Bednorz and W. Belzig, Phys. Rev. B 83, 125304, 2011

Setups closer to experimentChaotic cavity instead of tunnel contacts:J. P. Morten, D. Huertas-Hernando, W. Belzig, and A. Brataas, Europhys. Lett. 81, 40002, 2008

Result for the Bell parameter stays the same!

Quantum dots instead of tunnel contacts:H. Soller and A. Komnik, Eur. Phys. J. D 63, 3, 2011

Result for the Bell parameter stays the same!

Interaction effects1. Geometric suppression factors or usage of a topological insulator lead to length dependence of the nonlocal conductances

Result for the Bell parameter stays the same!

2. Onsite interaction (phonons, Coulomb interaction,… )

Result for the Bell parameter stays the same!

Polarisation > 84 % neededEntanglement: The state of the system cannot be written as product state.

Bell inequality: ε > 2 means that local reality is violated!

Entanglement and Bell violation does not mean the same and generically it is easier to verify the presence of entanglement than the violation of local reality.

S. M. Roy, Phys. Rev. Lett. 94, 010402, 2005J. Uffink and M. Seevinck, Physics Letters A 372, 1205, 2008

If Alice‘s and Bob‘s measurement directions for spin are orthogonal we can prove that themaximal value for ε for a separable state is only √2

We only need polarisation P > 70% for an entanglement witness

Only need ε > √2 to verify the presence of entanglement

Experimental SetupG1

G2F2

F1

G4

G3 F3

F4

S

InAs QD 1QD 2

QD 3QD 4

measure nonlocal conductances directly

eliminates the need for time-resolved measurement

L. Hofstetter, S. Csonka, A. Baumgartner, G. Fülöp, S. d'Hollosy, J. Nygård and C. Schönenberger, Phys. Rev. Lett. 107, 136801, 2011

Kondo effect in superconductor-ferromagnet hybrids enhances the spin polarisation to ≈70%

H. Soller, L. Hofstetter, S. Csonka, A. Levy Yeyati, C. Schönenberger and A. Komnik, in preparation

Synthetic antiferromagnets allow for small switchablemagnets

C. Wang, Y. Cui, J. A. Katine, R. A. Buhrman and D. C. Ralph, Nat. Phys. 7, 496–501, 2011

Conclusion

•Scheme for Bell measurements

•Theoretical Improvements

•Experimental Realisation