The document discusses spatial control of photons and optical vortices. It describes how photons can be put into Laguerre-Gaussian modes using mode converters or holograms. These modes carry orbital angular momentum and can be used to encode quantum information. Precise control over generation, manipulation, and measurement of photon states enables applications in quantum communication.

2. What are the limits of the spatial control of photons? Gabriel Molina-Terriza gabriel.molina@icfo.es

6. Spatial Information: Laguerre-Gaussian modes LG mp Radial number (number of radial nodes) Azimuthal number (number of twists of the phase) p 0 1 2 m 0 1 2

7. Optical vortices They appear like dark spots in the optical beam Molina-Terriza et al Optics Letters, 25, 1135 2000 Molina-Terriza et al PRL, 87, 023902 2001 Propagation direction

9. Concepts: Orbital Angular Momentum The LG modes can be decomposed in multipole electromagnetic modes (rotationally symmetric modes): Molina-Terriza PRA 2008

11. Topological defects

12. Superpositions + - = = +i = - i = Molina-Terriza et al, PRL 88, 013601 (2002)

14. Control of the photons We have the technology and knowledge to control the state of the generated photons. We can implement any communication protocol using high dimensions Molina-Terriza et al, Phys. Rev. Lett. 92, 167903 (2004)

16. The standard solution is: Heads Tails Heads Tails

18. Security in QCT The trace density and the fidelity of pairs of matrices are related. The curves relating F and D depend on the dimension of the state No restrictions on density matrices Two dimensional states Spekkens and Rudolph PRA 65, 012310 (2002)

19. There is a trade-off between the security of the sender and that of the receiver You can cheat in quantum coin tossing, but in the end you will be caught The optimal security is obtained with 3D states Spekkens & Rudolph (01) Ambainis (02)

20. The alignment of 10 fibers and 8 beam-splitters is difficult, but as a result the communication protocol stands alone We designed the electronics, so that we could store the detection and the correlations of photons individually

21. We proved that you can cheat in QCT, but you can be caught Head Tail Failure Bob wins 100% Molina-Terriza et al, PRL, 94, 040501 (2005)

22. The space-momentum is indeed represented by a very large Hilbert space Infinite dimensional Hilbert spaces are best represented by Wigner functions What are the limits of the spatial control of photons? This is the experimental spatial Wigner function from a single photon from an entangled pair in a Hermite-Gaussian mode You would find the same Wigner function in the quadratures of the electromagnetic field of a single photon

23. What are the limits of the spatial control of photons? You would think that the limit for packing spatial information in one photon is the wavelength BUT: Plasmons in metals Gold Nanoparticles Nanoantennaes Colloidal Quantum dots

24. What are the limits of the spatial control of photons? Gabriel Molina-Terriza gabriel.molina@icfo.es

25. Definition Optical vortices are spots in the transverse beam profile where the amplitude vanishes and the wavefront is singular

27. Entanglement: Non-local information 1 1 2 2 +

28. Entanglement: Non-local information 1 1 2 2 + Thanks to this sign if one photon is projected onto one state, the other automatically is in the corresponding state

30. Control of entanglement

32. Effect of the geometry z x z s x s z i x i  s  i Pump beam Signal photons Idler photons Observation plane Observation plane  s  i

33. Torres et al., Opt. Lett., 29 , 1939 (2004) Effect of the geometry

34. l pump = 4 crystal pump signal idler

35. Experimental results If you want to compensate the geometrical effect on the OAM, use large pump beams Molina-Terriza et al, Phys. Rev. A 72, 065802 (2005)

37. Effect of the walk-off The walk-off breaks the azimuthal symmetry Schmidt Number 0 90 180 270 360 12 1 angle x y walk-off direction

38. Walk-off effect is non-negligible if

39. Experimental Setup Instead of rotating the detectors, we rotated the polarization of the pump and the crystal CW diode laser

40. Experimental Setup: Counting coincidences zoom in coincidences In 10 seconds 0.3mm pinhole 50 steps (appr. 11mm) 50 steps Bidimensional Coincidences Scan

41. We have experimentally demonstrated that the walk-off allows us to control the momentum state x y walk-off direction y coordinate (mm) x coordinate (mm) experimental theoretical Bidimensional Coincidences Scan 0.3mm pinhole

42. Control of the photons In order to know the state which arrives to Bob, we have to perform a tomography. To implement the tomography we need to control the rotations and projections of the state Molina-Terriza et al, Phys. Rev. Lett. 92, 167903 (2004)

45. We can theoretically reproduce the effect of the holograms, which allows us to perfectly control any rotation or projection Control of the photons

46. Control of the photons

47. We used for the first time controlled 3D quantum states for a complete communication task Each detector consists of a hologram, a single mode optical fiber and a single photon detector

48. Implemented protocol Ambainis quant-ph/0204022