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

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    [CrossRef]
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  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).
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  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).
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  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).
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    [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]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  31. M. Hawton, “Photon wave functions in a localized coordinate space basis,” Phys. Rev. A 59, 3223–3227 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
  35. Note that the quantum mechanical scalar product 〈Ψ(2)|Ψ(1)〉=∑h=±∫d3k[f±(2)(k)]*f±(1)(k)=∫d3r1∫d3r2[Ψ(2)(r2)]*⋅Ψ(1)(r1)𝒲(r1−r2) is evaluate using a double integral in the position representation with a non local kernel 𝒲 (ρ) = (h̄c2π2|ρ|2)−1.
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  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.
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  41. J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989).
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  43. E. Brainis, C. Muldoon, L. Brandt, and A. Kuhn, “Coherent imaging of extended objects,” Opt. Commun. 282, 465–472 (2009).
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  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).
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  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).
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  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]
  49. N. J. Cerf, M. Lévy, and G. V. Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
    [CrossRef]
  50. M. P. Almeida, S. P. Walborn, and P. H. Souto Ribeiro, “Experimental investigation of quantum key distribution with position and momentum of photon pairs,” Phys. Rev. A 72, 022313 (2005).
    [CrossRef]
  51. L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
    [CrossRef] [PubMed]
  52. D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Continuous-variable quantum computation with spatial degrees of freedom of photons,” Phys. Rev. A 83, 052325 (2011).
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  54. B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
    [CrossRef]

2011 (4)

M. A. Solis-Prosser and L. Neves, “Remote state preparation of spatial qubits,” Phys. Rev. A 84, 012330 (2011).
[CrossRef]

P. L. Saldanha and C. H. Monken, “Interaction between light and matter: a photon wave function approach,” New J. Phys. 13, 073015 (2011).
[CrossRef]

D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Continuous-variable quantum computation with spatial degrees of freedom of photons,” Phys. Rev. A 83, 052325 (2011).
[CrossRef]

B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[CrossRef]

2010 (6)

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]

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]

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]

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]

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]

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]

2009 (5)

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]

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]

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]

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[CrossRef]

E. Brainis, C. Muldoon, L. Brandt, and A. Kuhn, “Coherent imaging of extended objects,” Opt. Commun. 282, 465–472 (2009).
[CrossRef]

2008 (3)

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]

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[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]

2007 (7)

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]

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]

J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007).
[CrossRef]

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]

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]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

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]

2006 (3)

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[CrossRef]

E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4–8 (2006).
[CrossRef]

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]

2005 (2)

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]

M. P. Almeida, S. P. Walborn, and P. H. Souto Ribeiro, “Experimental investigation of quantum key distribution with position and momentum of photon pairs,” Phys. Rev. A 72, 022313 (2005).
[CrossRef]

2004 (2)

L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A 69, 042305 (2004).
[CrossRef]

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]

2002 (1)

2001 (4)

M. Hawton and W. E. Baylis, “Photon position operators and localized bases,” Phys. Rev. A 64, 012101 (2001).
[CrossRef]

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]

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]

N. J. Cerf, M. Lévy, and G. V. Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[CrossRef]

2000 (2)

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]

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]

1999 (2)

M. Hawton, “Photon wave functions in a localized coordinate space basis,” Phys. Rev. A 59, 3223–3227 (1999).
[CrossRef]

M. Hawton, “Photon position operator with commuting components,” Phys. Rev. A 59, 954–959 (1999).
[CrossRef]

1998 (2)

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]

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]

1996 (1)

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]

1995 (3)

J. E. Sipe, “Photon wave functions,” Phys. Rev. A 52, 1875–1883 (1995).
[CrossRef] [PubMed]

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. A 52, R3429–R3432 (1995).
[CrossRef] [PubMed]

1994 (1)

I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol. A 86, 97–116 (1994).

1989 (1)

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989).
[CrossRef] [PubMed]

’t Hooft, G. W.

B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[CrossRef]

Abouraddy, A. F.

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]

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]

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]

Abrams, D. S.

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]

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]

Aguirre Gómez, J. G.

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]

Almeida, M. P.

M. P. Almeida, S. P. Walborn, and P. H. Souto Ribeiro, “Experimental investigation of quantum key distribution with position and momentum of photon pairs,” Phys. Rev. A 72, 022313 (2005).
[CrossRef]

Assche, G. V.

N. J. Cerf, M. Lévy, and G. V. Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[CrossRef]

Baylis, W. E.

M. Hawton and W. E. Baylis, “Photon position operators and localized bases,” Phys. Rev. A 64, 012101 (2001).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol. A 86, 97–116 (1994).

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.

Bonato, C.

Bonora, S.

Boto, A. N.

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]

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]

Bouwmeester, D.

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]

Brainis, E.

E. Brainis, C. Muldoon, L. Brandt, and A. Kuhn, “Coherent imaging of extended objects,” Opt. Commun. 282, 465–472 (2009).
[CrossRef]

Brandt, L.

E. Brainis, C. Muldoon, L. Brandt, and A. Kuhn, “Coherent imaging of extended objects,” Opt. Commun. 282, 465–472 (2009).
[CrossRef]

Braunstein, S. L.

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]

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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]

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]

Shapiro, J. H.

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[CrossRef]

Shih, Y.

J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007).
[CrossRef]

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]

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]

Shih, Y. H.

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]

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]

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]

Shimizu, R.

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]

Silberhorn, C.

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[CrossRef] [PubMed]

Sipe, J. E.

J. E. Sipe, “Photon wave functions,” Phys. Rev. A 52, 1875–1883 (1995).
[CrossRef] [PubMed]

Smith, B. J.

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]

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[CrossRef]

Solis-Prosser, M. A.

M. A. Solis-Prosser and L. Neves, “Remote state preparation of spatial qubits,” Phys. Rev. A 84, 012330 (2011).
[CrossRef]

Souto Ribeiro, P. H.

D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Continuous-variable quantum computation with spatial degrees of freedom of photons,” Phys. Rev. A 83, 052325 (2011).
[CrossRef]

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]

M. P. Almeida, S. P. Walborn, and P. H. Souto Ribeiro, “Experimental investigation of quantum key distribution with position and momentum of photon pairs,” Phys. Rev. A 72, 022313 (2005).
[CrossRef]

Strekalov, D. V.

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]

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]

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]

Taguchi, G.

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]

Takeuchi, S.

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]

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]

Tasca, D. S.

D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Continuous-variable quantum computation with spatial degrees of freedom of photons,” Phys. Rev. A 83, 052325 (2011).
[CrossRef]

Teich, M. C.

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]

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]

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]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Toscano, F.

D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Continuous-variable quantum computation with spatial degrees of freedom of photons,” Phys. Rev. A 83, 052325 (2011).
[CrossRef]

Vallone, G.

van Exter, M. P.

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]

Vargas, A.

Villoresi, P.

Waks, E.

E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4–8 (2006).
[CrossRef]

Walborn, S. P.

D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Continuous-variable quantum computation with spatial degrees of freedom of photons,” Phys. Rev. A 83, 052325 (2011).
[CrossRef]

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]

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]

M. P. Almeida, S. P. Walborn, and P. H. Souto Ribeiro, “Experimental investigation of quantum key distribution with position and momentum of photon pairs,” Phys. Rev. A 72, 022313 (2005).
[CrossRef]

Walmsley, I. A.

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[CrossRef] [PubMed]

Weinfurter, H.

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]

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).

Wen, J.

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]

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]

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]

J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007).
[CrossRef]

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]

Wilk, T.

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]

Williams, C. P.

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]

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]

Woerdman, J. P.

B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[CrossRef]

Wu, L.-A.

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]

Xiao, M.

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]

Xu, P.

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]

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]

Yamamoto, Y.

E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4–8 (2006).
[CrossRef]

Yang, T.

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]

Yao, X.-C.

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]

Yoshimoto, N.

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]

Yuan, Z.

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]

Zagury, N.

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]

Zeilinger, A.

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]

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).

Zhang, J.

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]

Zhang, L.

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[CrossRef] [PubMed]

Zhou, X.

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]

Zukowski, M.

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).

Acta Phys. Pol. A (1)

I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol. A 86, 97–116 (1994).

J. Opt. (1)

B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[CrossRef]

J. Opt. Soc. Am. B (3)

Nat. Phys. (3)

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]

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]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Nature (1)

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]

New J. Phys. (4)

E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4–8 (2006).
[CrossRef]

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]

P. L. Saldanha and C. H. Monken, “Interaction between light and matter: a photon wave function approach,” New J. Phys. 13, 073015 (2011).
[CrossRef]

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]

Opt. Commun. (1)

E. Brainis, C. Muldoon, L. Brandt, and A. Kuhn, “Coherent imaging of extended objects,” Opt. Commun. 282, 465–472 (2009).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (1)

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]

Phys. Rep. (1)

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]

Phys. Rev. A (23)

L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A 69, 042305 (2004).
[CrossRef]

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]

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]

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]

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]

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]

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]

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]

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[CrossRef]

J. E. Sipe, “Photon wave functions,” Phys. Rev. A 52, 1875–1883 (1995).
[CrossRef] [PubMed]

M. Hawton, “Photon position operator with commuting components,” Phys. Rev. A 59, 954–959 (1999).
[CrossRef]

M. Hawton and W. E. Baylis, “Photon position operators and localized bases,” Phys. Rev. A 64, 012101 (2001).
[CrossRef]

M. Hawton, “Photon wave functions in a localized coordinate space basis,” Phys. Rev. A 59, 3223–3227 (1999).
[CrossRef]

J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007).
[CrossRef]

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]

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]

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]

M. A. Solis-Prosser and L. Neves, “Remote state preparation of spatial qubits,” Phys. Rev. A 84, 012330 (2011).
[CrossRef]

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]

N. J. Cerf, M. Lévy, and G. V. Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[CrossRef]

M. P. Almeida, S. P. Walborn, and P. H. Souto Ribeiro, “Experimental investigation of quantum key distribution with position and momentum of photon pairs,” Phys. Rev. A 72, 022313 (2005).
[CrossRef]

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[CrossRef]

D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Continuous-variable quantum computation with spatial degrees of freedom of photons,” Phys. Rev. A 83, 052325 (2011).
[CrossRef]

Phys. Rev. Lett. (8)

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[CrossRef] [PubMed]

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]

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]

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989).
[CrossRef] [PubMed]

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]

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]

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]

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]

Science (1)

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]

Other (5)

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).

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.

Note that the quantum mechanical scalar product 〈Ψ(2)|Ψ(1)〉=∑h=±∫d3k[f±(2)(k)]*f±(1)(k)=∫d3r1∫d3r2[Ψ(2)(r2)]*⋅Ψ(1)(r1)𝒲(r1−r2) is evaluate using a double integral in the position representation with a non local kernel 𝒲 (ρ) = (h̄c2π2|ρ|2)−1.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005), 3rd ed.

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.

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Figures (2)

Fig. 1
Fig. 1

Generation of heralded linear superposition of localized two-photon states of light.

Fig. 2
Fig. 2

Scheme for quantum imaging with two independent photon pairs.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

ψ ± ( r , t ) = d 3 k h ¯ k c e ± ( k ) f ± ( k ) e i ( k r k c t ) ( 2 π ) 3 / 2 ,
Ψ ( r , t ) = ψ + ( r , t ) + ψ ( r , t ) .
Ψ ^ ( r , t ) = h = ± d 3 k h ¯ k c e h ( k ) a ^ h ( k ) e i ( k r k c t ) ( 2 π ) 3 / 2 = i 2 ɛ 0 E ^ ( + ) ( r , t ) .
| Ψ = h 1 , , h N d 3 k 1 d 3 k N f h 1 , , h N ( k 1 , , k N ) | 1 k 1 , h 1 , , 1 k N , h N
Ψ i 1 i N ( q 1 , , q N ) = ( i ) N ( 2 ɛ 0 ) N / 2 0 | E ^ i N ( + ) ( q N ) E ^ i 1 ( + ) ( q 1 ) | Ψ
Ψ i 1 i N * ( q 1 , , q N ) Ψ i 1 i N ( q 1 , , q N ) = ( 2 ɛ 0 ) N E ^ i 1 ( ) ( q 1 ) E ^ i N ( ) ( q N ) E ^ i N ( + ) ( q N ) E ^ i 1 ( + ) ( q 1 ) .
E ^ ( + ) ( r , t ) = 1 2 π c j d 2 ρ j d d t E ^ ( + ) ( ρ j , t | r ρ j | c ) | r ρ j | ,
Ψ ( r 1 , t 1 , , r N , t N ) = 1 ( 2 π c ) N 1 d 2 ρ 1 N d 2 ρ N d d t 1 d d t N Ψ ( ρ 1 , t 1 | r 1 ρ 1 | c , , ρ N , t N | r N ρ N | c ) | r 1 ρ 1 | | r N ρ N | .
a ( r 1 , t 1 , , r N , t N ) = ( i ) N λ 1 λ N 1 d 2 ρ 1 N d 2 ρ N e i 2 π λ 1 | r 1 ρ 1 | | r 1 ρ 1 | e i 2 π λ N | r N ρ N | | r N ρ N | × a ( ρ 1 , t 1 | r 1 ρ 1 | c , , ρ N , t N | r N ρ N | c ) .
h f s ( r i , ρ i ) = i λ i exp ( i 2 π λ i | r i ρ i | ) | r i ρ i |
a ( r 1 , t 1 , , r N , t N ) = 1 d 2 ρ 1 N d 2 ρ N h 1 ( r 1 , ρ 1 ) h N ( r N , ρ N ) a ( ρ 1 , t 1 l ( r 1 , ρ 1 ) c , , ρ N , t N l ( r N , ρ N ) c ) ,
a ( r 1 , t 1 , , r N , t N ) = 1 d 2 ρ 1 N d 2 ρ N k 1 , , k N a ( ρ 1 , t 1 l k 1 ( r 1 , ρ 1 ) c , , ρ N , t N l k N ( r N , ρ N ) c ) h 1 ( k 1 ) ( r 1 , ρ 1 ) h N ( k N ) ( r N , ρ N ) .
a ( ρ 1 , t 1 , ρ 2 , t 2 , ρ 3 , t 3 ) = E ( t 3 ) F ( ρ 3 ) δ ( ρ 1 ρ 3 ) δ ( ρ 2 ρ 3 ) δ ( t 1 t 3 ) δ ( t 2 t 3 ) ,
a ( r , t , r , t ) = δ ( t ( t * + τ ) ) δ ( t t ) δ ( r r ) F ( r / M ) exp ( i2 π | r | 2 ( 1 + 1 / M ) / ( λ 1 s 2 ) ) h 2 ( 0 , r / M ) .
a ( ρ 1 , t 1 , ρ 2 , t 2 , ρ 1 , t 1 , ρ 2 , t 2 ) = E ( t 2 ) E ( t 2 + Δ / c ) ( δ ( t 1 t 2 ) δ ( t 1 t 2 ) δ ( ρ 1 ρ 2 ) δ ( ρ 1 ρ 2 ) + δ ( t 1 t 2 ) δ ( t 1 t 2 ) δ ( ρ 1 ρ 2 ) δ ( ρ 1 ρ 2 ) )
a ( r a , t a , r b , t b , r c , t c , r d , t d ) = Γ 1 Γ 2 d 2 ρ 1 Γ 1 d 2 ρ 2 Γ 1 Γ 2 d ρ 1 Γ 2 d ρ 2 h a ( r a , ρ 1 ) h b ( r b , ρ 1 ) h c ( r c , ρ 2 ) h d ( r d , ρ 2 ) a ( ρ 1 , t a l ( r a , ρ 1 ) c , ρ 2 , t c l ( r c , ρ 2 ) c , ρ 1 , t b l ( r b , ρ 1 ) c , ρ 2 , t d l ( r d , ρ 2 ) c ) .
a ( r a , t a , r b , t b , r c , t c , r d , t d ) = E ( t c τ 1 ) E ( t d τ 1 ) [ δ ( t a t c τ 2 ) δ ( t b t d τ 2 ) Γ 1 d 2 ρ Γ 2 d 2 ρ h a ( r a , ρ ) h b ( r b , ρ ) h c ( r c , ρ ) h d ( r d , ρ ) ] + δ ( t b t c τ 2 ) δ ( t a t d τ 2 ) Γ 1 d 2 ρ Γ 2 d 2 ρ h a ( r a , ρ ) h b ( r b , ρ ) h c ( r c , ρ ) h d ( r d , ρ ) ] ,
a ( r a , t a , r b , t b ) = E 2 ( t * τ 1 ) δ ( t a t * τ 2 ) δ ( t a t b ) [ ϕ a c ( r a ) ϕ b d ( r b ) + ϕ a d ( r a ) ϕ b c ( r b ) ] ,
ϕ a c ( r a ) = Γ 1 h a ( r a , ρ ) h c ( r c , ρ ) d 2 ρ ,
ϕ a d ( r a ) = Γ 2 h a ( r a , ρ ) h d ( r d , ρ ) d 2 ρ ,
ϕ b d ( r b ) = Γ 2 h b ( r b , ρ ) h d ( r d , ρ ) d 2 ρ ,
ϕ b c ( r b ) = Γ 1 h b ( r b , ρ ) h c ( r c , ρ ) d 2 ρ .
a ( r a , t a , r b , t b ) = E 2 ( t * τ 1 ) δ ( t a t * τ 2 ) δ ( t a t b ) [ ϕ 1 ( r a ) ϕ 2 ( r b ) + ϕ 2 ( r a ) ϕ 1 ( r b ) ] .
1 + 1 p + q = 1 f ,
= ( 2 L l Δ p ) + λ 2 λ 1 ( L l Δ + s c ) ) = ( 2 L l p ) + λ 2 λ 1 ( L l + s d ) ) .
s c s d = λ 1 + λ 2 λ 2 Δ .
ϕ 1 ( r ) = 1 λ 1 2 ( p + q ) exp [ i 2 π λ 1 ( 2 L l Δ + q ) ] exp [ i 2 π λ 2 ( L l Δ + s c ) ] exp [ i π λ 1 x 2 + y 2 p + q ] M ˜ 1 ( x λ 1 ( p + q ) , y λ 1 ( p + q ) )
ϕ 2 ( r ) = 1 λ 1 2 ( p + q ) exp [ i 2 π λ 1 ( 2 L l + q ) ] exp [ i 2 π λ 2 ( L l + s d ) ] exp [ i π λ 1 x 2 + y 2 p + q ] M ˜ 2 ( x λ 1 ( p + q ) , y λ 1 ( p + q ) ) ,
M ˜ α ( ξ , η ) = d x d y M α ( x , y ) exp [ i2 π ( x ξ + y η ) ]
a ( r a , t a , r b , t b ) = E 2 ( t * τ 1 ) δ ( t a t * τ 2 ) δ ( t a t b ) [ M ˜ 1 ( r a ) M ˜ 2 ( r b ) + M ˜ 2 ( r a ) M ˜ 1 ( r b ) ]

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