1. Two-Photon Fields: Coherence, Interference and Entanglement Anand Kumar Jha Indian Institute of Technology, Kanpur QIPA 2103, HRI, Allahabad

2. Introduction to one-photon coherence Parametric down-conversion Two-photon interference: Temporal, spatial and angular Two-photon coherence and entanglement Outline

3. Necessary condition for interference: One-Photon Interference: “A photon interferes with itself ” - Dirac Mandel and Wolf, Optical Coherence and Quantum Optics ( when I 1 = I 2 =C/2 )

4. One-Photon Interference: “A photon interferes with itself ” - Dirac Mandel and Wolf, Optical Coherence and Quantum Optics Necessary condition for interference:

5. Parametric down-conversion two-photon field Leads to second-order nonlinear optical effects

6. Entanglement in time and energy “Temporal” two-photon coherence Entanglement in position and momentum “Spatial” two-photon coherence Entanglement in angular position and orbital angular momentum “Angular” two-photon coherence Conservation laws and entanglement Coherence length of pump laser: ~ 10 cms. Coherence length of signal-idler field: ~ c  100  m.

7. Hong-Ou-Mandel effect C. K. Hong et al., PRL 59, 2044 (1987) Frustrated two-photon Creation T. J. Herzog et al., PRL 72, 629 (1994) Induced Coherence X. Y. Zou et al., PRL 67, 318 (1991) Postponed Compensation Experiment T. B. Pittman, PRL 77, 1917 (1996) Two-Photon Interference

8. T. J. Herzog et al., PRL 72, 629 (1994) Two-Photon Interference: A two-photon interferes with itself

9. T. J. Herzog et al., PRL 72, 629 (1994)

10. Two-Photon Interference: A two-photon interferes with itself Jha, O’Sullivan, Chan, and Boyd et al., PRA 77, 021801(R) (2008) two-photon path-length difference

11. Two-Photon Interference: A two-photon interferes with itself Jha, O’Sullivan, Chan, and Boyd et al., PRA 77, 021801(R) (2008) two-photon path-asymmetry length difference two-photon path-length difference

12. Two-Photon Interference: A two-photon interferes with itself two-photon path-asymmetry length difference two-photon path-length difference Jha, O’Sullivan, Chan, and Boyd et al., PRA 77, 021801(R) (2008)

13. Two-Photon Interference: A two-photon interferes with itself Jha, O’Sullivan, Chan, and Boyd et al., PRA 77, 021801(R) (2008) two-photon path-asymmetry length difference two-photon path-length difference R. J. Glauber, Phys. Rev. 130, 2529 (1963)

14. Necessary conditions for two-photon interference: Two-Photon Interference: A two-photon interferes with itself Jha, O’Sullivan, Chan, and Boyd et al., PRA 77, 021801(R) (2008) ( when R 1 = R 2 =C/2 ) two-photon path-asymmetry length difference two-photon path-length difference

15. Two-Photon Interference: A two-photon interferes with itself ΔL plays the same role in two-photon interference as Δl does in one-photon interference Case I : = 0 two-photon path-asymmetry length difference two-photon path-length difference

16. Two-Photon Interference: A two-photon interferes with itself ΔL ’ has no one-photon analog The curve represents how coherence is lost due to an increase in the two-photon path-length asymmetry difference ΔL ’ Case I : = 0 two-photon path-asymmetry length difference two-photon path-length difference

17. 1 2 Hong-Ou-Mandel (HOM) Effect

18. ΔL = 0 ; ΔL ’ = 2x 1 2 2x x Hong-Ou-Mandel (HOM) Effect

19. Postponed Compensation Experiment 1(t-t) 2 (r-r) D DD D

20. 1(t-t) 2 (r-r) x D DD ΔL = D ; ΔL ’ = 2x x D Postponed Compensation Experiment

21. Experimental Verification Jha, O’Sullivan, Chan, and Boyd et al., PRA 77, 021801(R) (2008)

22. Experimental Verification Jha, O’Sullivan, Chan, and Boyd et al., PRA 77, 021801(R) (2008) Hong-Ou-Mandel effect C. K. Hong et al., PRL 59, 2044 (1987)

23. Spatial Two-photon Interference Coincidence count rate: Two-photon transverse position vector : Two-photon position-asymmetry vector : (Gaussian Schell- model pump) A. K. Jha and R.W. Boyd, PRA 81, 013828 (2010)

24. Coherence and entanglement

25. Angular Fourier Relationship Barnett and Pegg, PRA 41, 3427 (1990) Franke-Arnold et al., New J. Phys. 6, 103 (2004) Forbes, Alonso, and Siegman J. Phys. A 36, 707 (2003) Allen et al., PRA 45, 8185 (1992) Angular position l=0 l=2 l=1 Laguerre-Gauss basis

26. Angular One-Photon Interference OAM-mode distribution: E. Yao et al., Opt. Express 14, 13089 (2006) A. K. Jha, et al., PRA 78, 043810 (2008) Jha, Agarwal, and Boyd, PRA84, 063847 (2011)

27. Angular Two-Photon Interference State of the two photons produced by PDC:

28. Angular Two-Photon Interference State of the two photons after the aperture: State of the two photons produced by PDC:

29. Angular Two-Photon Interference State of the two photons after the aperture: State of the two photons produced by PDC: Visibility: Coincidence count rate: W. K. Wootters, PRL 80, 2245 (1998) Concurrence

30. Angular Two-Photon Interference State of the two photons after the aperture: State of the two photons produced by PDC: Visibility: Concurrence of the two-qubit state: Coincidence count rate: A. K. Jha et al., PRL 104, 010501 (2010) W. K. Wootters, PRL 80, 2245 (1998) Concurrence

31. Angular Two-Photon Interference State of the two photons after the aperture: Coincidence count rate: State of the two photons produced by PDC: Visibility: Concurrence of the two-qubit state: A. K. Jha et al., PRL 104, 010501 (2010)

32. Angular Two-Photon Interference State of the two photons after the aperture: Coincidence count rate: State of the two photons produced by PDC: Visibility: Concurrence of the two-qubit state: A. K. Jha et al., PRL 104, 010501 (2010)

33. Angular Two-Photon Interference Coincidence count rate: Visibility: Concurrence of the two-qubit state: A. K. Jha et al., PRL 104, 010501 (2010)

34. Angular Two-Photon Interference Coincidence count rate: Visibility: Concurrence of the two-qubit state: A. K. Jha et al., PRL 104, 010501 (2010) Jha, Agarwal, and Boyd, PRA84, 063847 (2011)

35. Summary A unified description of two-photon interference effects in terms of two-photon path length difference (  L) and two-photon path-asymmetry length difference (  L ’ ). Coherence properties of the source (pump) field get completely transferred to the down-converted entangled photons.  Does similar transfer occur in other nonlinear optical processes, such as four-wave mixing, that produce entangled photons? (working in collaboration with Dr. Saikat Ghosh) The coherence measure of two-photon fields provide a physically intuitive way of quantifying two-photon entanglement.  Do the coherence measures of multi-photon field, in general, provide a physically intuitive way of quantifying multi-photon, high-dimensional entanglement?

36. Acknowledgments Robert Boyd, University of Rochester, USA Emil Wolf, University of Rochester, USA Miles Padgett University of Glasgow, Scotland, UK Steve Barnett, University of Glasgow, Scotland, UK Jonathan Leach, Heriott-Watt University, Scotland, UK Collaborators at IIT Kanpur: Saikat Ghosh, Harshwardhan Wanare, V Subrahmanyam Group members: Niharika Singh (postdoc), Umesh Kumar and Prashant Kumar (undergraduates).

37. Angular One-Photon Interference

39. Some One-Photon Interference Effects Jha, O’Sullivan, Chan, and Boyd et al., PRA 77, 021801(R) (2008)

40. Some One-Photon Interference Effects P.W. Milonni et al., PRA 53, 4556 (1996) Induced Coherence X. Y. Zou et al., PRL 67, 318 (1991)

41. Some One-Photon Interference Effects P.W. Milonni et al., PRA 53, 4556 (1996) Set of all possible mutually exclusive events One-photon interference profile at a given detection point is the sum of two-photon interference profiles Induced Coherence X. Y. Zou et al., PRL 67, 318 (1991)

42. Some One-Photon Interference Effects P.W. Milonni et al., PRA 53, 4556 (1996) Induced Coherence X. Y. Zou et al., PRL 67, 318 (1991) Fringes observed at both the detectors Fringes observed at detector But only uniform intensity is observed at detector because of the cancellation due to out-of-phase fringe patterns: R si and R ti

43. Angular Two-Photon Interference Concurrence of the two-qubit state: A. K. Jha et al., PRL 104, 010501 (2010) Jha, Agarwal, and Boyd, PRA83, 053829 (2011)

44. Angular Two-Photon Interference Concurrence of the two-qubit state: A. K. Jha et al., PRL 104, 010501 (2010) Jha, Agarwal, and Boyd, PRA83, 053829 (2011)