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Two-Photon Fields: Coherence, Interference and Entanglement

Two-Photon Fields: Coherence, Interference and Entanglement. Anand Kumar Jha Indian Institute of Technology, Kanpur QIPA 2103, HRI, Allahabad. Outline. Introduction to one-photon coherence Parametric down-conversion Two-photon interference: Temporal, spatial and angular

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Two-Photon Fields: Coherence, Interference and Entanglement

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

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

  3. One-Photon Interference: “A photon interferes with itself ” - Dirac Necessary condition for interference: ( when I1= I2=C/2) Mandel and Wolf, Optical Coherence and Quantum Optics

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

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

  6. Conservation laws and entanglement 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 Coherence length of pump laser:~ 10 cms. Coherence length of signal-idler field:~ c/Dw ~ 100 mm.

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

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

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

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

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

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

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

  14. Two-Photon Interference: A two-photon interferes with itself two-photon path-length difference two-photon path-asymmetry length difference Necessary conditions for two-photon interference: ( when R1= R2=C/2) Jha, O’Sullivan, Chan, and Boyd et al., PRA 77, 021801(R) (2008)

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

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

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

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

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

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

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

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

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

  24. Coherence and entanglement

  25. Laguerre-Gauss basis Angular Fourier Relationship Angular position l=2 l=0 l=1 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)

  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 produced by PDC: State of the two photons after the aperture:

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

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

  31. Angular Two-Photon Interference State of the two photons produced by PDC: State of the two photons after the aperture: Coincidence count rate: 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 produced by PDC: State of the two photons after the aperture: Coincidence count rate: 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: Jha, Agarwal, and Boyd, PRA84, 063847 (2011) A. K. Jha et al., PRL 104, 010501 (2010)

  35. Summary • A unified description of two-photon interference effects in terms of two-photon • path length difference (DL) and two-photon path-asymmetry length difference (DL’). • 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. SaikatGhosh) • 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

  38. Angular One-Photon Interference

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

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

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

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

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

  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)

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