Quantum imaging with N-photon states in position space |
Optics Express, Vol. 19, Issue 24, pp. 24228-24240 (2011)
http://dx.doi.org/10.1364/OE.19.024228
Acrobat PDF (1043 KB)
Abstract
We investigate the physics of quantum imaging with N > 2 entangled photons in position space. It is shown that, in paraxial approximation, the space-time propagation of the quantum state can be described by a generalized Huygens-Fresnel principle for the N-photon wave function. The formalism allows the initial conditions to be set on multiple reference planes, which is very convenient to describe the generation of multiple photon pairs in separate thin crystals. Applications involving state shaping and spatial entanglement swapping are developed.
© 2011 OSA
1. Introduction
1. C. Lu, X. Zhou, O. Guhne, W. Gao, J. Zhang, Z. Yuan, A. Goebel, T. Yang, and J. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007). [CrossRef]
2. H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010). [CrossRef] [PubMed]
3. E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4–8 (2006). [CrossRef]
4. W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. Gühne, A. Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state,” Nat. Phys. 6, 331–335 (2010). [CrossRef]
5. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the Standard Quantum Limit with Four-Entangled Photons,” Science 316, 726–729 (2007). [CrossRef] [PubMed]
6. R. Okamoto, H. F. Hofmann, T. Nagata, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit: phase super-sensitivity of N-photon interferometers,” New J. Phys. 10, 073033 (2008). [CrossRef]
8. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87–139 (2010). [CrossRef]
9. L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A 69, 042305 (2004). [CrossRef]
14. C. Bonato, S. Bonora, A. Chiuri, P. Mataloni, G. Milani, G. Vallone, and P. Villoresi, “Phase control of a path-entangled photon state by a deformable membrane mirror,” J. Opt. Soc. Am. B 27, A175–A180 (2010). [CrossRef]
15. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000). [CrossRef] [PubMed]
16. P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001). [CrossRef]
17. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995). [CrossRef] [PubMed]
19. T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996). [CrossRef] [PubMed]
20. T. E. Keller, M. H. Rubin, Y. Shih, and L.-A. Wu, “Theory of the three-photon entangled state,” Phys. Rev. A 57, 2076–2079 (1998). [CrossRef]
21. J. Wen, E. Oh, and S. Du, “Tripartite entanglement generation via four-wave mixings: narrowband triphoton W state,” J. Opt. Soc. Am. B 27, A11–A20 (2010). [CrossRef]
22. J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007). [CrossRef]
25. J. Wen, S. Du, and M. Xiao, “Improving spatial resolution in quantum imaging beyond the Rayleigh diffraction limit using multiphoton W entangled states,” Phys. Lett. A 374, 3908 – 3911 (2010). [CrossRef]
22. J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007). [CrossRef]
23. J. Wen, P. Xu, M. H. Rubin, and Y. Shih, “Transverse correlations in triphoton entanglement: Geometrical and physical optics,” Phys. Rev. A 76, 023828 (2007). [CrossRef]
11. R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74, 013801 (2006). [CrossRef]
12. W. H. Peeters, J. J. Renema, and M. P. van Exter, “Engineering of two-photon spatial quantum correlations behind a double slit,” Phys. Rev. A 79, 043817 (2009). [CrossRef]
26. B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000). [CrossRef]
28. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19, 1174–1184 (2002). [CrossRef]
1. C. Lu, X. Zhou, O. Guhne, W. Gao, J. Zhang, Z. Yuan, A. Goebel, T. Yang, and J. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007). [CrossRef]
2. Photon wave functions in position representation
29. M. Hawton, “Photon position operator with commuting components,” Phys. Rev. A 59, 954–959 (1999). [CrossRef]
29. M. Hawton, “Photon position operator with commuting components,” Phys. Rev. A 59, 954–959 (1999). [CrossRef]
30. M. Hawton and W. E. Baylis, “Photon position operators and localized bases,” Phys. Rev. A 64, 012101 (2001). [CrossRef]
31. M. Hawton, “Photon wave functions in a localized coordinate space basis,” Phys. Rev. A 59, 3223–3227 (1999). [CrossRef]
34. J. E. Sipe, “Photon wave functions,” Phys. Rev. A 52, 1875–1883 (1995). [CrossRef] [PubMed]
34. J. E. Sipe, “Photon wave functions,” Phys. Rev. A 52, 1875–1883 (1995). [CrossRef] [PubMed]
36. B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006). [CrossRef]
37. B. J. Smith and M. G. Raymer, “Photon wave functions, wave-packet quantization of light, and coherence theory,” New J. Phys. 9, 414 (2007). [CrossRef]
36. B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006). [CrossRef]
36. B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006). [CrossRef]
3. Generalized Huygens-Fresnel principle
26. B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000). [CrossRef]
28. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19, 1174–1184 (2002). [CrossRef]
40. T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004). [CrossRef] [PubMed]
41. J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989). [CrossRef] [PubMed]
4. Detection process and wave function reduction
27. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001). [CrossRef] [PubMed]
5. Shaping the wave function through the detection process: application to super-resolution imaging
20. T. E. Keller, M. H. Rubin, Y. Shih, and L.-A. Wu, “Theory of the three-photon entangled state,” Phys. Rev. A 57, 2076–2079 (1998). [CrossRef]
42. V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009). [CrossRef]
43. E. Brainis, C. Muldoon, L. Brandt, and A. Kuhn, “Coherent imaging of extended objects,” Opt. Commun. 282, 465–472 (2009). [CrossRef]
44. G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008). [CrossRef]
45. M. A. Solis-Prosser and L. Neves, “Remote state preparation of spatial qubits,” Phys. Rev. A 84, 012330 (2011). [CrossRef]
46. S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007). [CrossRef]
6. Entangled images
18. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). [CrossRef] [PubMed]
47. P. L. Saldanha and C. H. Monken, “Interaction between light and matter: a photon wave function approach,” New J. Phys. 13, 073015 (2011). [CrossRef]
18. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). [CrossRef] [PubMed]
7. Conclusion
Acknowledgments
References and links
1. | C. Lu, X. Zhou, O. Guhne, W. Gao, J. Zhang, Z. Yuan, A. Goebel, T. Yang, and J. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007). [CrossRef] |
2. | H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010). [CrossRef] [PubMed] |
3. | E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4–8 (2006). [CrossRef] |
4. | W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. Gühne, A. Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state,” Nat. Phys. 6, 331–335 (2010). [CrossRef] |
5. | T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the Standard Quantum Limit with Four-Entangled Photons,” Science 316, 726–729 (2007). [CrossRef] [PubMed] |
6. | R. Okamoto, H. F. Hofmann, T. Nagata, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit: phase super-sensitivity of N-photon interferometers,” New J. Phys. 10, 073033 (2008). [CrossRef] |
7. | J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multi-photon entanglement and interferometry,” to appear in Rev. Mod. Phys. (2011). |
8. | S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87–139 (2010). [CrossRef] |
9. | L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A 69, 042305 (2004). [CrossRef] |
10. | L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005). [CrossRef] [PubMed] |
11. | R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74, 013801 (2006). [CrossRef] |
12. | W. H. Peeters, J. J. Renema, and M. P. van Exter, “Engineering of two-photon spatial quantum correlations behind a double slit,” Phys. Rev. A 79, 043817 (2009). [CrossRef] |
13. | G. Lima, A. Vargas, L. Neves, R. Guzmán, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express 17, 10688–10696 (2009). [CrossRef] [PubMed] |
14. | C. Bonato, S. Bonora, A. Chiuri, P. Mataloni, G. Milani, G. Vallone, and P. Villoresi, “Phase control of a path-entangled photon state by a deformable membrane mirror,” J. Opt. Soc. Am. B 27, A175–A180 (2010). [CrossRef] |
15. | A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000). [CrossRef] [PubMed] |
16. | P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001). [CrossRef] |
17. | D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995). [CrossRef] [PubMed] |
18. | T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). [CrossRef] [PubMed] |
19. | T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996). [CrossRef] [PubMed] |
20. | T. E. Keller, M. H. Rubin, Y. Shih, and L.-A. Wu, “Theory of the three-photon entangled state,” Phys. Rev. A 57, 2076–2079 (1998). [CrossRef] |
21. | J. Wen, E. Oh, and S. Du, “Tripartite entanglement generation via four-wave mixings: narrowband triphoton W state,” J. Opt. Soc. Am. B 27, A11–A20 (2010). [CrossRef] |
22. | J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007). [CrossRef] |
23. | J. Wen, P. Xu, M. H. Rubin, and Y. Shih, “Transverse correlations in triphoton entanglement: Geometrical and physical optics,” Phys. Rev. A 76, 023828 (2007). [CrossRef] |
24. | J. Wen and M. H. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A 79, 025802 (2009). [CrossRef] |
25. | J. Wen, S. Du, and M. Xiao, “Improving spatial resolution in quantum imaging beyond the Rayleigh diffraction limit using multiphoton W entangled states,” Phys. Lett. A 374, 3908 – 3911 (2010). [CrossRef] |
26. | B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000). [CrossRef] |
27. | A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001). [CrossRef] [PubMed] |
28. | A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19, 1174–1184 (2002). [CrossRef] |
29. | M. Hawton, “Photon position operator with commuting components,” Phys. Rev. A 59, 954–959 (1999). [CrossRef] |
30. | M. Hawton and W. E. Baylis, “Photon position operators and localized bases,” Phys. Rev. A 64, 012101 (2001). [CrossRef] |
31. | M. Hawton, “Photon wave functions in a localized coordinate space basis,” Phys. Rev. A 59, 3223–3227 (1999). [CrossRef] |
32. | I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol. A 86, 97–116 (1994). |
33. | I. Bialynicki-Birula, “Photon wave function,” in Progress in Optics, vol. 36, E. Wolf, ed. (North-Holland, Elsevier, Amsterdam, 1996), chap. 5, pp. 248–294. |
34. | J. E. Sipe, “Photon wave functions,” Phys. Rev. A 52, 1875–1883 (1995). [CrossRef] [PubMed] |
35. | Note that the quantum mechanical scalar product |
36. | B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006). [CrossRef] |
37. | B. J. Smith and M. G. Raymer, “Photon wave functions, wave-packet quantization of light, and coherence theory,” New J. Phys. 9, 414 (2007). [CrossRef] |
38. | J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005), 3rd ed. |
39. | In [26,28], time is only introduced to account for the bandwidth of the continuous biphoton stream and compute coincidence rates in the slow detector limit. |
40. | T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004). [CrossRef] [PubMed] |
41. | J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989). [CrossRef] [PubMed] |
42. | V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009). [CrossRef] |
43. | E. Brainis, C. Muldoon, L. Brandt, and A. Kuhn, “Coherent imaging of extended objects,” Opt. Commun. 282, 465–472 (2009). [CrossRef] |
44. | G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008). [CrossRef] |
45. | M. A. Solis-Prosser and L. Neves, “Remote state preparation of spatial qubits,” Phys. Rev. A 84, 012330 (2011). [CrossRef] |
46. | S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007). [CrossRef] |
47. | P. L. Saldanha and C. H. Monken, “Interaction between light and matter: a photon wave function approach,” New J. Phys. 13, 073015 (2011). [CrossRef] |
48. | J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998). [CrossRef] |
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OCIS Codes
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(270.0270) Quantum optics : Quantum optics
ToC Category:
Quantum Optics
History
Original Manuscript: September 21, 2011
Revised Manuscript: October 29, 2011
Manuscript Accepted: November 1, 2011
Published: November 14, 2011
Citation
E. Brainis, "Quantum imaging with N-photon states in position space," Opt. Express 19, 24228-24240 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24228
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References
- C. Lu, X. Zhou, O. Guhne, W. Gao, J. Zhang, Z. Yuan, A. Goebel, T. Yang, and J. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys.3, 91–95 (2007). [CrossRef]
- H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature466, 601–603 (2010). [CrossRef] [PubMed]
- E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys.8, 4–8 (2006). [CrossRef]
- W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. Gühne, A. Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state,” Nat. Phys.6, 331–335 (2010). [CrossRef]
- T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the Standard Quantum Limit with Four-Entangled Photons,” Science316, 726–729 (2007). [CrossRef] [PubMed]
- R. Okamoto, H. F. Hofmann, T. Nagata, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit: phase super-sensitivity of N-photon interferometers,” New J. Phys.10, 073033 (2008). [CrossRef]
- J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multi-photon entanglement and interferometry,” to appear in Rev. Mod. Phys. (2011).
- S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep.495, 87–139 (2010). [CrossRef]
- L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A69, 042305 (2004). [CrossRef]
- L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005). [CrossRef] [PubMed]
- R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A74, 013801 (2006). [CrossRef]
- W. H. Peeters, J. J. Renema, and M. P. van Exter, “Engineering of two-photon spatial quantum correlations behind a double slit,” Phys. Rev. A79, 043817 (2009). [CrossRef]
- G. Lima, A. Vargas, L. Neves, R. Guzmán, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express17, 10688–10696 (2009). [CrossRef] [PubMed]
- C. Bonato, S. Bonora, A. Chiuri, P. Mataloni, G. Milani, G. Vallone, and P. Villoresi, “Phase control of a path-entangled photon state by a deformable membrane mirror,” J. Opt. Soc. Am. B27, A175–A180 (2010). [CrossRef]
- A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000). [CrossRef] [PubMed]
- P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001). [CrossRef]
- D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett.74, 3600–3603 (1995). [CrossRef] [PubMed]
- T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A52, R3429–R3432 (1995). [CrossRef] [PubMed]
- T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A53, 2804–2815 (1996). [CrossRef] [PubMed]
- T. E. Keller, M. H. Rubin, Y. Shih, and L.-A. Wu, “Theory of the three-photon entangled state,” Phys. Rev. A57, 2076–2079 (1998). [CrossRef]
- J. Wen, E. Oh, and S. Du, “Tripartite entanglement generation via four-wave mixings: narrowband triphoton W state,” J. Opt. Soc. Am. B27, A11–A20 (2010). [CrossRef]
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