Topological phase and quantum criptography with spin-orbit entanglement of the photon

1. Universidade Federal Fluminense Instituto de Física - Niterói – RJ - Brasil Topological phase and quantum criptography with spin-orbit entanglement of the photon Antonio Zelaquett Khoury Financial Support: CNPq - CAPES – FAPERJ INSTITUTO DO MILÊNIO DE INFORMAÇÃO QUÂNTICA

2. Outline • Geom. phase for a spin ½ in a magnetic field • Geometric quantum computation • The Pancharatnam phase • Beams carrying OAM • Topological phase for entangled states • BB84 QKD without a shared reference frame • Conclusions

3. Geometric phase of a spin 1/2 in a magnetic field

4. Spin 1/2 in a time dependent magnetic field BERRY PHASE

5. Geometric quantum computation

6. Geometric conditional phase gate Conditional phase gate J.A. Jones, V. Vedral, A. Ekert, G. Castagnoll, NATURE V.403, 869 (2000) L.-M. Duan, J.I. Cirac, P.Zoller SCIENCE V.292, 1695 (2001)

7. The Pancharatnam phase

8. Pancharatnam phase S. Pancharatnam, Proc. Indian Acad. Sci. Sect. A, V.44, 247 (1956) Collected Works of S. Pancharatnam, Oxford Univ. Press, London (1975).

9. Beams carrying orbital angular momentum

10. Angular momentum Hermite-Gauss (HG) Laguerre-Gauss (LG) (Paraxial Wave Equation) Rectangular Cylindrical Gauss-Laguerre beams carrying OAM

11. Astigmatic mode converter Cylindrical lenses at 45o Poincaré representation for beams carrying OAM Poincaré representation of first order Gaussian modes

12. Geometric phase from astigmatic mode conversion E.J. Galvez, P.R. Crawford, H.I. Sztul, M.J. Pysher, P.J. Haglin, R.E. Williams, Physical Review Letters V.90, 203901 (2003)

13. Topological phase for entangled states C. E. Rodrigues de Souza, J. A. O. Huguenin and A. Z. Khoury IF-UFF P. Milman LMPQ – Jussieu - France

15. Bloch sphere (or Poincaré sphere) ONE QUBIT  TWO QUBITS  Two Bloch spheres?? Geometric representation for two-qubit states Only for product states!!!

16. Two-qubit PURE STATES  Geometric representation for two-qubit PURE states P. Milman and R. Mosseri, Phys. Rev. Lett. 90, 230403 (2003). P. Milman, Phys. Rev. A 73, 062118 (2006). Bloch ball SO(3) sphere (opposite points identified) (Concurrence) Bloch ball colapses to a point!!!! Maximally entangled state 

17. SO(3) sphere 0-type trajectories  π-type trajectories  Topological phase for maximally entangled states Cyclic evolutions preserveing maximal entanglement (“Closed” trajectories)  Two homotopy classes:

18. Separable polarization-OAM modes

19. 4 1 3 2 Nonseparable polarization-OAM modes Geometric representation on the SO(3) sphere

20. Holographic preparation of the LG modes PBS Nonseparable mode preparation

21. 4 1 3 2 4’ Interferometric measurement CCD θ = 0 0 θ = 45 0 θ = - 45 0 θ = 0 0, 22.5 0, 45 0, 67.5 0, 90 0 1 4 3 2 4 1 (θ = 00) / 4’ 1 (θ = 900) 41 θ = 0 0

22. θ=00 θ=22.50 θ=450 θ=67.50 θ=900 θ=00 θ=22.50 θ=450 θ=67.50 θ=900 Experimental results Unseparable mode Separable mode

23. Unseparable mode Separable mode Theoretical expressions

24. Calculated images Unseparable mode Separable mode

25. Interference pattern (θ=450) CONCURRENCE Partial separability and concurrence Partially separable mode

26. BB84 Quantum key distribution without a shared reference frame C. E. Rodrigues de Souza, C. V. S. Borges, J. A. O. Huguenin and A. Z. Khoury IF-UFF L. Aolita and S. P. Walborn IF-UFRJ

27. 0 0 V H 45o - 45o 1 1 0 0 Photons 1 1 The BB84 protocol Bennett and Brassard 1984 Polarizers Polarizers HV HV +/- +/- ALICE BOB

28. Basis • HV • +/- • +/- • +/- • HV • +/- • HV • HV • HV • Result • 0 • 0 • 1 • 1 • 1 • 1 • 0 • 1 • 0 • Basis • +/- • +/- • HV • +/- • HV • HV • HV • +/- • HV • Result • 1 • 0 • 1 • 1 • 1 • 1 • 0 • 0 • 0 Alice and Bob check their basis, but not their results ! ALICE BOB      1 0 0 0 1

29. Logic basis 0/1 Logic basis +/- Spin-orbit entanglement L. Aolita and S. P. Walborn PRL 98, 100501 (2007) Invariant under rotations ! ! ! !

30. , , , , , , , BB84 without frame alignment BASIS BASIS Photons ALICE BOB Robust against alignment noise ! ! ! !

31. BOB ALICE CNOT R(θ) X X R(φ) Procedure sketch 0 1 + -

32. Experimental setup

33. , State sent by Alice Bob’s detector 1 Bob’s detector 0 Bob’s detector 1 Alice sends 1 Bob’s detector 0 Rotation of Alice’s setup Experimental results Bob`s detection basis:

34. Conclusions

35. Conclusions • Spin-orbit entanglement • Topological phase for spin-orbit transformations • Potential applications to conditional gates • Quantum criptography without frame alignment