Abstract

We analyze the quantum states obtained by optical parametric amplification of polarization NOON states. First we study, theoretically and experimentally, the amplification of a two-photon state by a collinear quantum injected optical parametric amplifier (QIOPA). We compare the stimulated emission regime with the spontaneous one, studied by Sciarrino et al. [Phys. Rev. A 77, 012324 (2008)] . As a second step, we show that the collinear amplifier cannot be successfully used for amplifying N-photon states with N>2, and we propose to adopt a different scheme, based on a noncollinear QIOPA. We show that the state obtained by the latter amplification process preserves the λN feature and exhibits a high resilience to losses. Furthermore, measurement of part of the output state can be adopted to increase the pattern visibility.

© 2009 Optical Society of America

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  1. 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]
  2. P. Kok, A. N. Boto, D. S. Abrams, C. P. William, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
    [CrossRef]
  3. M. J. Holland, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355-1358 (1993).
    [CrossRef] [PubMed]
  4. V. Giovannetti, S. LLoyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
    [CrossRef] [PubMed]
  5. J. P. Dowling, “Quantum optical metrology--the lowdown on high-NOON states,” Contemp. Phys. 49, 125-143 (2008).
    [CrossRef]
  6. M. D'Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
    [CrossRef] [PubMed]
  7. K. T. Kapale and J. P. Dowling, “A bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
    [CrossRef] [PubMed]
  8. H. Cable and J. P. Dowling, “Efficient generation of large number-path entanglement using only linear optics and feed-forward,” Phys. Rev. Lett. 99, 163604 (2007).
    [CrossRef] [PubMed]
  9. A. E. B. Nielsen and K. Molmer, “Conditional generation of path-entangled optical NOON states,” Phys. Rev. A 75, 063803 (2007).
    [CrossRef]
  10. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
    [CrossRef] [PubMed]
  11. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161-164 (2004).
    [CrossRef] [PubMed]
  12. P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161(2004).
    [CrossRef] [PubMed]
  13. 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]
  14. H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
    [CrossRef] [PubMed]
  15. H. F. Hofmann and T. Ono, “High-photon-number path entanglement in the interference of spontaneously down-converted photon pairs with coherent laser light,” Phys. Rev. A 76, 031806(R) (2007).
    [CrossRef]
  16. R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
    [CrossRef]
  17. U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
    [CrossRef] [PubMed]
  18. F. Sciarrino, C. Vitelli, F. De Martini, R. Glasser, H. Cable, and J. P. Dowling, “Experimental sub-Rayleigh resolution by an unseeded high-gain optical parametric amplifier for quantum lithography,” Phys. Rev. A 77, 012324 (2008).
    [CrossRef]
  19. A. Kolkiran and G. S. Agarwal, “Quantum interferometry using coherent beam stimulated parametric down-conversion,” Opt. Express 16, 6479-6485 (2008).
    [CrossRef] [PubMed]
  20. F. De Martini, V. Buzek, F. Sciarrino, and C. Sias, “Experimental realization of the quantum universal NOT gate,” Nature 419, 815-818 (2002).
    [CrossRef] [PubMed]
  21. F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement test on a microscopic-macroscopic system,” Phys. Rev. Lett. 100, 253601 (2008).
    [CrossRef] [PubMed]
  22. S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express 12, 5735-5740 (2004).
    [CrossRef] [PubMed]
  23. H. J. Chang, H. Shin, M. N. O'Sullivan-Hale, and R. W. Boyd, “Implementation of sub-Rayleigh lithography using an N-photon absorber,” J. Mod. Opt. 53, 2271-2277 (2006).
    [CrossRef]
  24. F. De Martini and F. Sciarrino, “Non-linear parametric processes in quantum information,” Prog. Quantum Electron. 29, 165-256 (2005).
    [CrossRef]
  25. G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable, and J. P. Dowling, “Quantum states of light produced by a high-gain optical parametric amplifier for use in quantum lithography,” J. Opt. Soc. Am. B 24, 270-274 (2007).
    [CrossRef]
  26. E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
    [CrossRef]
  27. H. S. Eisenberg, G. H. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
    [CrossRef] [PubMed]
  28. M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, “Entanglement, EPR correlations, and mesoscopic quantum superposition by the high-gain quantum injected parametric amplification,” Phys. Rev. A 73, 032312 (2006).
    [CrossRef]
  29. S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
    [CrossRef]
  30. N. Spagnolo, C. Vitelli, S. Giacomini, F. Sciarrino, and F. De Martini, “Polarization preserving ultra fast optical shutter for quantum information processing,” Opt. Express 16, 17609-17615 (2008).
    [CrossRef] [PubMed]

2009 (1)

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

2008 (6)

F. Sciarrino, C. Vitelli, F. De Martini, R. Glasser, H. Cable, and J. P. Dowling, “Experimental sub-Rayleigh resolution by an unseeded high-gain optical parametric amplifier for quantum lithography,” Phys. Rev. A 77, 012324 (2008).
[CrossRef]

F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement test on a microscopic-macroscopic system,” Phys. Rev. Lett. 100, 253601 (2008).
[CrossRef] [PubMed]

A. Kolkiran and G. S. Agarwal, “Quantum interferometry using coherent beam stimulated parametric down-conversion,” Opt. Express 16, 6479-6485 (2008).
[CrossRef] [PubMed]

N. Spagnolo, C. Vitelli, S. Giacomini, F. Sciarrino, and F. De Martini, “Polarization preserving ultra fast optical shutter for quantum information processing,” Opt. Express 16, 17609-17615 (2008).
[CrossRef] [PubMed]

J. P. Dowling, “Quantum optical metrology--the lowdown on high-NOON states,” Contemp. Phys. 49, 125-143 (2008).
[CrossRef]

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[CrossRef]

2007 (6)

H. F. Hofmann and T. Ono, “High-photon-number path entanglement in the interference of spontaneously down-converted photon pairs with coherent laser light,” Phys. Rev. A 76, 031806(R) (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]

K. T. Kapale and J. P. Dowling, “A bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
[CrossRef] [PubMed]

H. Cable and J. P. Dowling, “Efficient generation of large number-path entanglement using only linear optics and feed-forward,” Phys. Rev. Lett. 99, 163604 (2007).
[CrossRef] [PubMed]

A. E. B. Nielsen and K. Molmer, “Conditional generation of path-entangled optical NOON states,” Phys. Rev. A 75, 063803 (2007).
[CrossRef]

G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable, and J. P. Dowling, “Quantum states of light produced by a high-gain optical parametric amplifier for use in quantum lithography,” J. Opt. Soc. Am. B 24, 270-274 (2007).
[CrossRef]

2006 (3)

H. J. Chang, H. Shin, M. N. O'Sullivan-Hale, and R. W. Boyd, “Implementation of sub-Rayleigh lithography using an N-photon absorber,” J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, “Entanglement, EPR correlations, and mesoscopic quantum superposition by the high-gain quantum injected parametric amplification,” Phys. Rev. A 73, 032312 (2006).
[CrossRef]

V. Giovannetti, S. LLoyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

2005 (2)

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[CrossRef] [PubMed]

F. De Martini and F. Sciarrino, “Non-linear parametric processes in quantum information,” Prog. Quantum Electron. 29, 165-256 (2005).
[CrossRef]

2004 (5)

S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
[CrossRef]

S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express 12, 5735-5740 (2004).
[CrossRef] [PubMed]

H. S. Eisenberg, G. H. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161(2004).
[CrossRef] [PubMed]

2002 (2)

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

F. De Martini, V. Buzek, F. Sciarrino, and C. Sias, “Experimental realization of the quantum universal NOT gate,” Nature 419, 815-818 (2002).
[CrossRef] [PubMed]

2001 (3)

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

P. Kok, A. N. Boto, D. S. Abrams, C. P. William, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[CrossRef]

M. D'Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[CrossRef] [PubMed]

2000 (1)

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]

1993 (1)

M. J. Holland, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355-1358 (1993).
[CrossRef] [PubMed]

Abrams, D. S.

P. Kok, A. N. Boto, D. S. Abrams, C. P. William, 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]

Agarwal, G. S.

Aspelmeyer, M.

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161(2004).
[CrossRef] [PubMed]

Banaszek, K.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

Bentley, S. J.

S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express 12, 5735-5740 (2004).
[CrossRef] [PubMed]

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Boto, A. N.

P. Kok, A. N. Boto, D. S. Abrams, C. P. William, 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.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[CrossRef] [PubMed]

H. S. Eisenberg, G. H. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

Boyd, R. W.

G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable, and J. P. Dowling, “Quantum states of light produced by a high-gain optical parametric amplifier for use in quantum lithography,” J. Opt. Soc. Am. B 24, 270-274 (2007).
[CrossRef]

H. J. Chang, H. Shin, M. N. O'Sullivan-Hale, and R. W. Boyd, “Implementation of sub-Rayleigh lithography using an N-photon absorber,” J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express 12, 5735-5740 (2004).
[CrossRef] [PubMed]

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Braunstein, S. L.

P. Kok, A. N. Boto, D. S. Abrams, C. P. William, 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]

Buzek, V.

F. De Martini, V. Buzek, F. Sciarrino, and C. Sias, “Experimental realization of the quantum universal NOT gate,” Nature 419, 815-818 (2002).
[CrossRef] [PubMed]

Cable, H.

F. Sciarrino, C. Vitelli, F. De Martini, R. Glasser, H. Cable, and J. P. Dowling, “Experimental sub-Rayleigh resolution by an unseeded high-gain optical parametric amplifier for quantum lithography,” Phys. Rev. A 77, 012324 (2008).
[CrossRef]

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[CrossRef]

H. Cable and J. P. Dowling, “Efficient generation of large number-path entanglement using only linear optics and feed-forward,” Phys. Rev. Lett. 99, 163604 (2007).
[CrossRef] [PubMed]

G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable, and J. P. Dowling, “Quantum states of light produced by a high-gain optical parametric amplifier for use in quantum lithography,” J. Opt. Soc. Am. B 24, 270-274 (2007).
[CrossRef]

Caminati, M.

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, “Entanglement, EPR correlations, and mesoscopic quantum superposition by the high-gain quantum injected parametric amplification,” Phys. Rev. A 73, 032312 (2006).
[CrossRef]

Chan, K. W.

Chang, H. J.

H. J. Chang, H. Shin, M. N. O'Sullivan-Hale, and R. W. Boyd, “Implementation of sub-Rayleigh lithography using an N-photon absorber,” J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

Chekhova, M. V.

M. D'Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[CrossRef] [PubMed]

D'Angelo, M.

M. D'Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[CrossRef] [PubMed]

De Martini, F.

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[CrossRef]

F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement test on a microscopic-macroscopic system,” Phys. Rev. Lett. 100, 253601 (2008).
[CrossRef] [PubMed]

F. Sciarrino, C. Vitelli, F. De Martini, R. Glasser, H. Cable, and J. P. Dowling, “Experimental sub-Rayleigh resolution by an unseeded high-gain optical parametric amplifier for quantum lithography,” Phys. Rev. A 77, 012324 (2008).
[CrossRef]

N. Spagnolo, C. Vitelli, S. Giacomini, F. Sciarrino, and F. De Martini, “Polarization preserving ultra fast optical shutter for quantum information processing,” Opt. Express 16, 17609-17615 (2008).
[CrossRef] [PubMed]

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, “Entanglement, EPR correlations, and mesoscopic quantum superposition by the high-gain quantum injected parametric amplification,” Phys. Rev. A 73, 032312 (2006).
[CrossRef]

F. De Martini and F. Sciarrino, “Non-linear parametric processes in quantum information,” Prog. Quantum Electron. 29, 165-256 (2005).
[CrossRef]

F. De Martini, V. Buzek, F. Sciarrino, and C. Sias, “Experimental realization of the quantum universal NOT gate,” Nature 419, 815-818 (2002).
[CrossRef] [PubMed]

Demkowicz-Dobrzanski, R.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

Dorner, U.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

Dowling, J. P.

F. Sciarrino, C. Vitelli, F. De Martini, R. Glasser, H. Cable, and J. P. Dowling, “Experimental sub-Rayleigh resolution by an unseeded high-gain optical parametric amplifier for quantum lithography,” Phys. Rev. A 77, 012324 (2008).
[CrossRef]

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[CrossRef]

J. P. Dowling, “Quantum optical metrology--the lowdown on high-NOON states,” Contemp. Phys. 49, 125-143 (2008).
[CrossRef]

H. Cable and J. P. Dowling, “Efficient generation of large number-path entanglement using only linear optics and feed-forward,” Phys. Rev. Lett. 99, 163604 (2007).
[CrossRef] [PubMed]

K. T. Kapale and J. P. Dowling, “A bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
[CrossRef] [PubMed]

G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable, and J. P. Dowling, “Quantum states of light produced by a high-gain optical parametric amplifier for use in quantum lithography,” J. Opt. Soc. Am. B 24, 270-274 (2007).
[CrossRef]

P. Kok, A. N. Boto, D. S. Abrams, C. P. William, 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]

Durkin, G. A.

H. S. Eisenberg, G. H. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

Edamatsu, K.

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

Eisenberg, H. S.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[CrossRef] [PubMed]

H. S. Eisenberg, G. H. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

Gasparoni, S.

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161(2004).
[CrossRef] [PubMed]

Giacomini, S.

Giovannetti, V.

V. Giovannetti, S. LLoyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

Glasser, R.

F. Sciarrino, C. Vitelli, F. De Martini, R. Glasser, H. Cable, and J. P. Dowling, “Experimental sub-Rayleigh resolution by an unseeded high-gain optical parametric amplifier for quantum lithography,” Phys. Rev. A 77, 012324 (2008).
[CrossRef]

Glasser, R. T.

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[CrossRef]

Hodelin, J. F.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[CrossRef] [PubMed]

Hofmann, H. F.

H. F. Hofmann and T. Ono, “High-photon-number path entanglement in the interference of spontaneously down-converted photon pairs with coherent laser light,” Phys. Rev. A 76, 031806(R) (2007).
[CrossRef]

Holland, M. J.

M. J. Holland, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355-1358 (1993).
[CrossRef] [PubMed]

Itoh, T.

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

Kapale, K. T.

K. T. Kapale and J. P. Dowling, “A bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
[CrossRef] [PubMed]

Khoury, G.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[CrossRef] [PubMed]

Khoury, G. H.

H. S. Eisenberg, G. H. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

Kok, P.

P. Kok, A. N. Boto, D. S. Abrams, C. P. William, 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]

Kolkiran, A.

LLoyd, S.

V. Giovannetti, S. LLoyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

Lundeen, J. S.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

Maccone, L.

V. Giovannetti, S. LLoyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

Mitchell, M. W.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

Molmer, K.

A. E. B. Nielsen and K. Molmer, “Conditional generation of path-entangled optical NOON states,” Phys. Rev. A 75, 063803 (2007).
[CrossRef]

Nagasako, E. M.

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Nagata, T.

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]

Nielsen, A. E. B.

A. E. B. Nielsen and K. Molmer, “Conditional generation of path-entangled optical NOON states,” Phys. Rev. A 75, 063803 (2007).
[CrossRef]

O'Brien, J. L.

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]

Okamoto, R.

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]

Ono, T.

H. F. Hofmann and T. Ono, “High-photon-number path entanglement in the interference of spontaneously down-converted photon pairs with coherent laser light,” Phys. Rev. A 76, 031806(R) (2007).
[CrossRef]

O'Sullivan-Hale, M. N.

H. J. Chang, H. Shin, M. N. O'Sullivan-Hale, and R. W. Boyd, “Implementation of sub-Rayleigh lithography using an N-photon absorber,” J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

Pan, J. W.

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161(2004).
[CrossRef] [PubMed]

Perris, R.

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, “Entanglement, EPR correlations, and mesoscopic quantum superposition by the high-gain quantum injected parametric amplification,” Phys. Rev. A 73, 032312 (2006).
[CrossRef]

Rubin, M. H.

S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
[CrossRef]

Sasaki, K.

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]

Sciarrino, F.

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[CrossRef]

F. Sciarrino, C. Vitelli, F. De Martini, R. Glasser, H. Cable, and J. P. Dowling, “Experimental sub-Rayleigh resolution by an unseeded high-gain optical parametric amplifier for quantum lithography,” Phys. Rev. A 77, 012324 (2008).
[CrossRef]

F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement test on a microscopic-macroscopic system,” Phys. Rev. Lett. 100, 253601 (2008).
[CrossRef] [PubMed]

N. Spagnolo, C. Vitelli, S. Giacomini, F. Sciarrino, and F. De Martini, “Polarization preserving ultra fast optical shutter for quantum information processing,” Opt. Express 16, 17609-17615 (2008).
[CrossRef] [PubMed]

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, “Entanglement, EPR correlations, and mesoscopic quantum superposition by the high-gain quantum injected parametric amplification,” Phys. Rev. A 73, 032312 (2006).
[CrossRef]

F. De Martini and F. Sciarrino, “Non-linear parametric processes in quantum information,” Prog. Quantum Electron. 29, 165-256 (2005).
[CrossRef]

F. De Martini, V. Buzek, F. Sciarrino, and C. Sias, “Experimental realization of the quantum universal NOT gate,” Nature 419, 815-818 (2002).
[CrossRef] [PubMed]

Secondi, V.

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, “Entanglement, EPR correlations, and mesoscopic quantum superposition by the high-gain quantum injected parametric amplification,” Phys. Rev. A 73, 032312 (2006).
[CrossRef]

Shih, Y.

M. D'Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[CrossRef] [PubMed]

Shimizu, R.

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

Shin, H.

H. J. Chang, H. Shin, M. N. O'Sullivan-Hale, and R. W. Boyd, “Implementation of sub-Rayleigh lithography using an N-photon absorber,” J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

Sias, C.

F. De Martini, V. Buzek, F. Sciarrino, and C. Sias, “Experimental realization of the quantum universal NOT gate,” Nature 419, 815-818 (2002).
[CrossRef] [PubMed]

Simon, C.

H. S. Eisenberg, G. H. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

Smith, B. J.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

Spagnolo, N.

Steinberg, A. M.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

Takeuchi, S.

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]

Thanvanthri, S.

S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
[CrossRef]

Ursin, R.

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161(2004).
[CrossRef] [PubMed]

Vitelli, C.

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[CrossRef]

F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement test on a microscopic-macroscopic system,” Phys. Rev. Lett. 100, 253601 (2008).
[CrossRef] [PubMed]

F. Sciarrino, C. Vitelli, F. De Martini, R. Glasser, H. Cable, and J. P. Dowling, “Experimental sub-Rayleigh resolution by an unseeded high-gain optical parametric amplifier for quantum lithography,” Phys. Rev. A 77, 012324 (2008).
[CrossRef]

N. Spagnolo, C. Vitelli, S. Giacomini, F. Sciarrino, and F. De Martini, “Polarization preserving ultra fast optical shutter for quantum information processing,” Opt. Express 16, 17609-17615 (2008).
[CrossRef] [PubMed]

Walmsley, I. A.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

Walther, P.

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161(2004).
[CrossRef] [PubMed]

Wasilewski, W.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

William, C. P.

P. Kok, A. N. Boto, D. S. Abrams, C. P. William, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[CrossRef]

Williams, C. P.

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]

Zeilinger, A.

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161(2004).
[CrossRef] [PubMed]

Contemp. Phys. (1)

J. P. Dowling, “Quantum optical metrology--the lowdown on high-NOON states,” Contemp. Phys. 49, 125-143 (2008).
[CrossRef]

J. Mod. Opt. (1)

H. J. Chang, H. Shin, M. N. O'Sullivan-Hale, and R. W. Boyd, “Implementation of sub-Rayleigh lithography using an N-photon absorber,” J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (3)

F. De Martini, V. Buzek, F. Sciarrino, and C. Sias, “Experimental realization of the quantum universal NOT gate,” Nature 419, 815-818 (2002).
[CrossRef] [PubMed]

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161(2004).
[CrossRef] [PubMed]

Opt. Express (3)

Phys. Rev. A (8)

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802 (2001).
[CrossRef]

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, “Entanglement, EPR correlations, and mesoscopic quantum superposition by the high-gain quantum injected parametric amplification,” Phys. Rev. A 73, 032312 (2006).
[CrossRef]

S. Thanvanthri and M. H. Rubin, “Ghost interference with an optical parametric amplifier,” Phys. Rev. A 70, 063811 (2004).
[CrossRef]

F. Sciarrino, C. Vitelli, F. De Martini, R. Glasser, H. Cable, and J. P. Dowling, “Experimental sub-Rayleigh resolution by an unseeded high-gain optical parametric amplifier for quantum lithography,” Phys. Rev. A 77, 012324 (2008).
[CrossRef]

H. F. Hofmann and T. Ono, “High-photon-number path entanglement in the interference of spontaneously down-converted photon pairs with coherent laser light,” Phys. Rev. A 76, 031806(R) (2007).
[CrossRef]

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[CrossRef]

P. Kok, A. N. Boto, D. S. Abrams, C. P. William, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[CrossRef]

A. E. B. Nielsen and K. Molmer, “Conditional generation of path-entangled optical NOON states,” Phys. Rev. A 75, 063803 (2007).
[CrossRef]

Phys. Rev. Lett. (11)

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

M. J. Holland, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355-1358 (1993).
[CrossRef] [PubMed]

V. Giovannetti, S. LLoyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

M. D'Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[CrossRef] [PubMed]

K. T. Kapale and J. P. Dowling, “A bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
[CrossRef] [PubMed]

H. Cable and J. P. Dowling, “Efficient generation of large number-path entanglement using only linear optics and feed-forward,” Phys. Rev. Lett. 99, 163604 (2007).
[CrossRef] [PubMed]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[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]

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[CrossRef] [PubMed]

H. S. Eisenberg, G. H. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester, “Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93, 193901 (2004).
[CrossRef] [PubMed]

F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement test on a microscopic-macroscopic system,” Phys. Rev. Lett. 100, 253601 (2008).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

F. De Martini and F. Sciarrino, “Non-linear parametric processes in quantum information,” Prog. Quantum Electron. 29, 165-256 (2005).
[CrossRef]

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]

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Figures (12)

Fig. 1
Fig. 1

Quantum lithography based on (a) path entangled states and (b) polarization entangled states.

Fig. 2
Fig. 2

(a) Unseeded OPA. (b) Amplification of a polarization entangled NOON state.

Fig. 3
Fig. 3

Theoretical trend of the second-order visibility as a function of the number of photons generated by the amplification process in the two cases, spontaneous and stimulated.

Fig. 4
Fig. 4

Plot of the visibilities V N = 2 ( M ) with 2 M 6 for the collinear QIOPA in stimulated emission with the injection of a two-photon NOON state as a function of the nonlinear gain g. Solid curve, V N = 2 ( 2 ) ; dashed, V N = 2 ( 3 ) ; dotted, V N = 2 ( 4 ) ; dashed–dotted, V N = 2 ( 5 ) ; and short-dashed–dotted, V N = 2 ( 6 ) .

Fig. 5
Fig. 5

Experimental scheme adopted to amplify a two-photon state. By measuring coincidences between detector { D T , D T * } of spatial mode k T , the state of spatial mode k 1 is prepared in the two-photon NOON state ψ 2 1 . The rate of the trigger signal was around 10,000 Hz , and the rate of coincidences between ( D T , D T * ) was around 400 Hz .

Fig. 6
Fig. 6

(a) Fringe pattern of the twofold coincidences between detectors { D 1 B , D T } . (b) Fringe pattern of the fourfold coincidences between detectors { D 1 , D 1 B , D T , D T * } .

Fig. 7
Fig. 7

(a) Oscillation fringe patterns in the stimulated and spontaneous regimes. The unbalanced minima are due to different couplings of the π H and π V polarized signals with the single-mode fiber. This effect has been taken into account by fitting the experimental data points with a double-sinusoidal function. (b) Visibility value V N = 2 ( 2 ) as a function of nonlinear gain g in the spontaneous (rectangles) and stimulated case (circles). Experimental (points) and theoretical trend of visibility in the stimulated regime (curve) are shown. The theoretical curves used are V sp = 0.85 V N = 2 ( 2 ) and V stim = 0.85 V N = 2 ( 2 ) . The factor 0.85 has been inserted to consider experimental imperfections. The curves are parametric, plotted as a function of the respective number of generated photons, which are n ̂ sp = 2 sinh 2 g and n ̂ stim = 2 + 6 sinh 2 g . Data in the spontaneous regime refer to [18].

Fig. 8
Fig. 8

Experimental setup for the amplification of a NOON state by a noncollinear amplifier, implemented by a type II cut BBO crystal in the noncollinear configuration. The state ψ N 1 is injected into the input mode k 1 . The BS is inserted to simulate losses.

Fig. 9
Fig. 9

Plot of the asymptotic ( g and n ¯ ) Mth-order correlation function as a function of the order M in three cases. Squares, injection of a two-photon state; circles, three-photon state; triangles, four-photon state.

Fig. 10
Fig. 10

Scheme for the conditioned preparation of the Ψ k , q 1 state of mode k 1 . The discriminator identifies the detection of the state k + , q 2 of mode k 2 . Its output signal triggers the activation of the optical shutter of mode k 1 , which selects the desired state Ψ k , q 1 .

Fig. 11
Fig. 11

Decoherence model of the interferometric process. The phase shift in the NOON state ψ N is inserted by incorporating a Babinet compensator, while the BS with efficiency η models the decoherence process. The signal is then analyzed in polarization by the PBS and the N-photon absorbing device R ( N ) .

Fig. 12
Fig. 12

(a) Contour plot, distribution of fringe pattern visibility as a function of the detected photons k polarized π + and q polarized π of spatial mode k 2 , N = 5 , M = 10 . Clearer regions correspond to higher visibilities, which for this choice of N and M values are 15 % , and are obtained for equal numbers of revealed photons. (b) Comparison between the visibilities in the conditioned (red, top curve) and the unconditioned case (black, bottom curve) as a function of the order of correlation M, with k = q = 500 and N = 4 .

Equations (35)

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

G seed ( N ) = N ! 2 N [ 1 + ( 1 ) N + 1 cos ( N ϕ ) ] .
G seed ( M ) = N ! 2 M ( N M ) ! ,
| Φ 2 1 = 1 C n = 0 Γ n 1 ( n C 2 Γ 2 ) | n H ; n V 1 ,
G N = 2 ( 2 ) = 2 n ¯ ( 4 + 7 n ¯ ) + 1 2 ( 7 n ¯ 2 + 7 n ¯ + 1 ) [ 1 cos ( 2 θ ) ] .
V N ( M ) = G N ( M ) ( max ) G N ( M ) ( min ) G N ( M ) ( max ) + G N ( M ) ( min ) ,
V N = 2 ( 2 ) = 7 n ¯ 2 + 7 n ¯ + 1 35 n ¯ 2 + 23 n ¯ + 1 .
V N = 2 ( 2 ) = 1 + 7 n ¯ + 7 n ¯ 2 1 + 25 n ¯ + 35 n ¯ 2 ,
V N = 2 ( 3 ) = 12 + 48 n ¯ + 39 n ¯ 2 12 + 84 n ¯ + 91 n ¯ 2 ,
V N = 2 ( 4 ) = 12 + 291 n ¯ + 822 n ¯ 2 + 567 n ¯ 3 12 + 291 n ¯ + 1078 n ¯ 2 + 903 n ¯ 3 ,
V N = 2 ( 5 ) = 135 + 1315 n ¯ + 2845 n ¯ 2 + 1705 n ¯ 3 135 + 1315 n ¯ + 3245 n ¯ 2 + 2201 n ¯ 3 ,
V N = 2 ( 6 ) = 45 + 1745 n ¯ + 10080 n ¯ 2 + 1750 n ¯ 3 + 9245 n ¯ 4 45 + 1745 n ¯ + 10080 n ¯ 2 + 18657 n ¯ 3 + 10621 n ¯ 4 .
Φ 3 = 1 12 C 4 i , j = 0 ( Γ / 2 ) i ( Γ / 2 ) j i ! j ! { ( 2 i + 3 ) ! 2 j ! ( 2 i + 3 ) + , ( 2 j ) 2 i ! ( 2 j + 3 ) ! ( 2 i ) + , ( 2 j + 3 ) } Γ 3 2 C 2 i , j = 0 ( Γ / 2 ) i ( Γ / 2 ) j i ! j ! { ( 2 i + 1 ) ! 2 j ! ( 2 i + 1 ) + , ( 2 j ) + 2 i ! ( 2 j + 1 ) ! ( 2 i ) + , ( 2 j + 1 ) } .
G N = 3 ( 3 ) = a ( n ¯ ) + b ( n ¯ ) cos ( θ ) + c ( n ¯ ) cos ( 2 θ ) + d ( n ¯ ) cos ( 3 θ ) ,
H ̂ int = ι χ ( a ̂ 1 π a ̂ 2 π a ̂ 1 π a ̂ 2 π ) + h.c. ,
c ̂ 1 ( t ) = 1 2 [ a ̂ 1 + ( t ) e ι θ a ̂ 1 ( t ) ] .
a ̂ 1 ± ( t ) = a ̂ 1 ± cosh ( g ) + a ̂ 2 sinh ( g ) ,
U ̂ = e Γ ( a ̂ 1 + a ̂ 2 a ̂ 1 a ̂ 2 + ) e ln C ( 2 + n ̂ 1 + + n ̂ 1 + n ̂ 2 + + n ̂ 2 ) e Γ ( a ̂ 1 a ̂ 2 + a ̂ 1 + a ̂ 2 ) ,
Φ = 1 C 2 n = 0 Γ n m = 0 n ( 1 ) m ( n m ) + , m 1 m + , ( n m ) 2 .
G ̂ ( M ) = 1 2 M [ a ̂ 1 + C + a ̂ 2 S e ι θ a ̂ 1 C e ι θ a ̂ 2 + S ] M [ a ̂ 1 + C + a ̂ 2 S e ι θ a ̂ 1 C e ι θ a ̂ 2 + S ] M ,
G ̂ ( M ) = 1 2 M ( i , j , k g i j k ( a ̂ 1 + ) M i j k ( a ̂ 2 ) i ( a ̂ 1 ) j ( a ̂ 2 + ) k ) ( l , m , n g l m n * ( a ̂ 1 + ) M i j k ( a ̂ 2 ) i ( a ̂ 1 ) j ( a ̂ 2 + ) k ) ,
g i j k = ( 1 2 ) M ( M i , j , k ) ( 1 ) j + k ( e ι θ ) j + k C M i k S i + k ,
G 0 ( M ) = M ! S 2 M .
Φ N = 1 2 N ! C N + 1 n = 0 Γ n m = 0 n ( 1 ) m [ ( n m + N ) ! ( n m ) ! ( n m + N ) + , m 1 ( m + N ) ! m ! ( n m ) + , ( m + N ) 1 ] m + , ( n m ) 2 .
G N ( M ) = M ! 2 M { i = 0 M N j = 0 N C 2 j S 2 ( M j ) ( N j ) ( M i , j ) + i = M N + 1 M j = 0 M i C 2 j S 2 ( M j ) ( N j ) ( M i , j ) ( 1 ) M C 2 N S 2 ( M N ) [ i = 0 M N ( M i , N ) ] cos ( N θ ) } ,
G N ( M ) = M ! 2 M i = 0 M j = 0 M i C 2 j S 2 ( M j ) ( M i , j ) ( N j ) .
V ̃ N = 2 ( M ) ( n ¯ ) = M 2 M M 2 + 7 M + 8 ,
V ̃ N = 3 ( M ) ( n ¯ ) = M 3 3 M 2 + 2 M M 3 + 15 M 2 + 56 M + 48 ,
V ̃ N = 4 ( M ) ( n ¯ ) = M 4 6 M 3 + 11 M 2 6 M M 4 + 26 M 3 + 203 M 2 + 538 M + 384 .
ρ ̂ loss = η N ρ ̂ NOON + i = 0 N 1 ( N i ) η i ( 1 η ) N i ρ ̂ i ,
G loss ( N ) = η N G seed ( N ) .
d ̂ 1 ( t ) = η c ̂ 1 ( t ) + ι 1 η b ̂ i ( t ) ,
G ̃ N ( M ) = i , j = 0 M ( η ) i + j ( ι 1 η ) 2 M i j ( 1 ) M j 0 c 1 b 1 ψ N { [ c ̂ 1 ( t ) ] i [ b ̂ 1 ] M i [ c ̂ 1 ( t ) ] j [ b ̂ 1 ] M j } ψ N c 1 0 b 1 .
G ̃ N ( M ) = η M ψ N c 1 [ c ̂ 1 ( t ) ] M [ c ̂ 1 ( t ) ] M ψ N c 1 = η M G N ( M ) .
Ψ k , q 1 = 1 2 ( ( q + N ) ! q ! ( q + N ) + , k 1 ( k + N ) ! k ! q + , ( k + N ) 1 ) .
G N , k q ( M ) = 1 2 M + 1 { ( q + N ) ! q ! i = M k q + N ( M i ) 2 ( q + N ) ! ( q + N i ) ! k ! ( k M + i ) ! + ( k + N ) ! k ! i = M k N q ( M i ) 2 q ! ( q i ) ! ( k + N ) ! ( k + N M + i ) ! + i = M k q + N ( 1 ) 2 M N 2 i ( M i ) ( M N i ) ( q + N ) ! ( q + N i ) ! ( k + N ) ! ( k M + i ) ! cos ( N θ ) } .

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