1. Experiments With Entangled Photons Paulo Henrique Souto Ribeiro Instituto de Física - UFRJ Summer School of Optics Concépcion January/2010

2. Quantum Optics Group at IF/UFRJ

3. Group members Experiments: Prof. Paulo Henrique Souto Ribeiro Prof. Stephen Patrick Walborn Theory: Prof. Luiz Davidovich Prof. Nicim Zagury Prof. Ruynet Matos Filho Prof. Fabricio Toscano Msc and PhD students: Adriana Auyuanet Larrieu, Adriano H. de Oliveira Aragão, Bruno de Moura Escher, Bruno Taketani, Daniel Schneider Tasca, Gabriel Horacio Aguilar, Osvaldo Jimenez farias, Gabriela Barreto Lemos, Rafael Chaves.

4. UFRJ UFMG USP-SÃO PAULO UFAL UFF

5. Outline: Part I -Simultaneity in parametric down- conversion -Violation of a classical inequality -Consequences of simultaneity: i)localized one-photon state; ii)the Hong-Ou-Mandel interferometer iii) measurement of the tunneling time Part II -Polarization entanglement -Bell’s inequalities -Entanglement measurement Part III -Entanglement dynamics -Kraus operators -Entanglement sudden death -Process tomography -Evolution of entanglement Part VI -Spatial correlations -The transfer of the angular spectrum -Continuous variables etanglement- EPR paradox -Non-gaussian entanglement -Non-local optical vortex

6. Part I - Simultaneity in parametric down-conversion - Violation of a classical inequality - Consequences of simultaneity: i) localized one-photon state; ii) the Hong-Ou-Mandel interferometer iii) measurement of the tunneling time

7. Parametric Down-conversion Espontaneous emission Stimulated emission Twin Photons

8. Parametric Down-conversion

9. Observation of simultaneity

11. Parametric down-conversion: quantum state Time evolution Time evolution operator Time integral

12. Simultaneity in parametric down-conversion Quantum state for weak interaction

13. Simultaneity in parametric down-conversion Quantum state including some approximations

14. Simultaneity in parametric down-conversion Calculation of expectation values Electric field operator Intensity Coincidence

15. Simultaneity in parametric down-conversion: very simple view

16. Simultaneity in parametric down-conversion: very simple view Quantum state: simple version Electric field operator: plane wave, almost monochromatic Coincidence

17. Simultaneity in parametric down-conversion: very simple view Plane wave pumping field

18. Coincidence detection

20. Measurement of time delays =168ps =185ps

21. Simultaneity in parametric down-conversion: very simple view + detection filters Plane wave pumping field

22. Simultaneity in parametric down-conversion: very simple view + detection filters Interference filter: typical  = 10nm,  = 3.8 x 10 13 Hz, t = 82 fs << 100ps

23. Simultaneity in parametric down-conversion: very simple view + timing resolution

24. Localized one photon state

26. Violation of a classical inequality

28. Hong, Ou and Mandel Interferometer

29. Hong, Ou and Mandel Interferometer: single mode approach Beam splitter Input-output relations

30. Hong, Ou and Mandel Interferometer: single mode approach Beam splitter Two-photon input state Coincidence probability

31. Hong, Ou and Mandel Interferometer

32. Single-photon tunneling time

33. Part II - Polarization entanglement - Bell’s inequalities - Entanglement measurement

34. Polarization entanglement: generation Kwiat et al. PRL 75, 4337 (1995)

35. Kwiat et al. PRA 60, R773 (1999) White et al. PRL 83, 3103 (1999) Polarization entanglement: generation

36. Kwiat et al. PRA 60, R773 (1999) White et al. PRL 83, 3103 (1999) Polarization entanglement: generation

37. Mixed state Pure entangled state Mixed states and entangled states

38. Detection of entanglement: violation of the Bell inequality

39. Bell-CHSH inequality Bell inequality and Bell states

40. Bell states for the photon polarization Coincidence rate for  + : Bell inequality and Bell states

41. Bell states for the photon polarization Bell inequality and Bell states Coincidence rate for  + :

42. Maximal violation Bell inequality and Bell states

43. Maximal violation Bell inequality and Bell states

44. Maximal violation Bell inequality and Bell states

45. Violation of a Bell inequality - Detects but does not quantify the entanglement properly - Some entangled states do not violate the Bell inequality - Valid for dichotomic or dichotomized systems Bell inequality and entanglement

46. Take a set of measurements : Reconstruction of the density matrix Quantum state tomography

49. With   one can compute all quantities related to the system

50. Concurrency: Direct measurement of entanglement

51. Mintert, Kus, and Buchleitner, Phys. Rev. Lett. 95 260502 (2005). Direct measurement of entanglement using copies of states

52. Direct measurement of entanglement: pure states Pure state Two copies Maximally entangled state Two copies

53. Experiment with entangled photons

54. Two copies of a state in a single photon Polarization state

55. Linear momentum state Two copies of a state in a single photon

56. Simultaneous entanglement in polarization and linear momentum Two copies of a state in a single photon

57. Bell state projection Bell states combining momentum and polarization

58. C-NOT with a SAGNAC interferometer

59. Spatial rotations with cilyndrical lenses

61. Direct measurement of entangled with two copies

62. S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, F. Mintert, A. Buchleitner, Nature 440 1022 (2006) Direct measurement of entangled with two copies

63. S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, F. Mintert, A. Buchleitner, Nature 440 1022 (2006) Direct measurement of entangled with two copies

64. Part III -Entanglement dynamics -Kraus operators -Entanglement sudden death -Process tomography -Evolution of entanglement

65. Entanglement dynamics T. Yu, J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004). T. Yu, J. H. Eberly, Phys. Rev. Lett. 97, 140403 (2006).

66. Amplitude decay channel Quantum channel and Kraus map

67. Operadores de Kraus para o canal de amplitude Quantum channel and Kraus operators

68. Amplitude decay channel for one photon polarization

88. Kwiat et al. PRA 60, R773 (1999) White et al. PRL 83, 3103 (1999) Polarization entangled state

89. M. P. Almeida et al., Science 316, 579 (2007) Experimental observation of the entanglement sudden death

90. M. P. Almeida et al., Science 316, 579 (2007) Experimental observation of the entanglement sudden death

91. Process tomography

92. Reconstruction of the Kraus operators

93. T. Konrad et al., Nature Physics 4, 99 (2008). A dynamical law for the entanglement

98. O. Farias et al., Science 324, 1414 (2009) A dynamical law for the entanglement

99. O. Farias et al., Science 324, 1414 (2009) A dynamical law for the entanglement: experimental test

100. A dynamical law for the entanglement: generalization for mixed states T. Konrad et al., Nature Physics 4, 99 (2008).

101. A dynamical law for the entanglement: generalization for mixed states

102. A dynamical law for the entanglement: generalization for mixed states

103. A. Jamiołkowski, Rep. Math. Phys. 3, 275 (1972) How to find $' A dynamical law for the entanglement: generalization for mixed states

104. O. Farias et al., Science 324, 1414 (2009) A dynamical law for the entanglement: generalization for mixed states experimental test

105. Part VI -Spatial correlations -The transfer of the angular spectrum -Continuous variables etanglement- EPR paradox -Non-gaussian entanglement -Non-local optical vortex

106. Spatial correlations in the far field

110. Spatial anti-bunching: non-classical behavior Cauchy-Swartz inequality Homogeneity and stationarity

111. Spatial anti-bunching: non-classical behavior

112. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi Phys. Rev. Lett. 88, 120401 (2002). Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller Phys. Rev. Lett. 84, 2722 (2000). Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller Phys. Rev. Lett. 84, 2722 (2000). Inseparability DGCZ criterion MGVT criterion

113. Inseparability

114. Inseparability:proof

118. Inseparability criterion DGCZ criterion

119. Inseparability

120. Inseparability:proof

124. MGVT criterion

125. Inseparability

129. Non-gaussian entanglement Gaussian states are completely characterized by the second order momenta: Then, DGCZ, MGVT and other criteria based on second order momenta are non optimal for non-gaussian states.

130. Higher order criterion E. Shchukin and W. Vogel Inseparability criteria for continuous bipartite quantum states. Phys Rev Lett. 95, 230502 (2005) To the second order: a and b are annihillation operators for modes a and b.

131. Higher order criterion E. Shchukin and W. Vogel Inseparability criteria for continuous bipartite quantum states. Phys Rev Lett. 95, 230502 (2005) The state has a positive partial transpose, if and only if all principal minors are non-negative.

132. Gaussian and non-gaussian states Production of a gaussian state with parametric down-conversion

133. Gaussian and non-gaussian states Production of a non-gaussian state with parametric down-conversion

134. Higher order criterion We found a non-gaussian state that does not violate any second order criterion: According to R. Simon Phys. Rev. Lett. 84, 2726 (2000), if is satisfied, no second order criterion is violated. For 0.57 < s/t < 1.73  satisfies the inequality.

135. Higher order criterion However it gives the negative minor below for the higher order criterion

136. Isomorphism between a multimode single photon field and a single mode multiphoton field The inequality is violated for r=1/t and 0.68 < s/t < 1.53

137. Experimental observation of genuine non-gaussian entanglement Quantum entanglement beyond Gaussian criteria R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro and S. P. Walborn Proc. Nat. Acad. Sci. 106, 21517-21520(2009)

138. Experimental observation of genuine non-gaussian entanglement Quantum entanglement beyond Gaussian criteria R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro and S. P. Walborn Proc. Nat. Acad. Sci. 106, 21517-21520(2009)

139. Experimental observation of genuine non-gaussian entanglement Quantum entanglement beyond Gaussian criteria R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro and S. P. Walborn Proc. Nat. Acad. Sci. 106, 21517-21520(2009)