Abstract

We present a method for using the Hong–Ou–Mandel (HOM) interference technique to quantify photon indistinguishability within an associated uncertainty. The method allows the relative importance of various experimental factors affecting the HOM visibility to be identified, and enables the actual indistinguishability, with an associated uncertainty, to be estimated from experimentally measured quantities. A measurement equation has been derived that accounts for the non-ideal performance of the interferometer. The origin of each term of the equation is explained, along with procedures for their experimental evaluation and uncertainty estimation. These uncertainties are combined to give an overall uncertainty for the derived photon indistinguishability. The analysis was applied to measurements from an inter ferometer sourced with photon pairs from a parametric downconversion process. The measured photon indistinguishably was found to be 0.954±0.036 by using the prescribed method.

© 2010 Optical Society of America

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  34. R. P. Feynman, R. B. Leighton, and M. Sands, Vol. 3 of The Feynman Lectures on Physics (Addison-Wesley, 1965).
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2009 (2)

P. J. Thomas, J. Y. Cheung, C. J. Chunnilall, and M. H. Dunn, “The Hong-Ou-Mandel interferometer: a new procedure for alignment,” Rev. Sci. Instrum. 80, 036101 (2009).
[CrossRef]

D. Calonico, F. Levi, L. Lorini, and G. Mana, “Bayesian inference of a negative quantity from positive measurement results,” Metrologia 46, 267-271 (2009).
[CrossRef]

2007 (5)

J. Cheung, J. L. Gardner, A. Migdall, S. Polyakov, and M. Ware, “High accuracy dual lens transmittance measurements,” Appl. Opt. 46, 5396-5403 (2007).
[CrossRef]

G. Fujii, N. Namekata, M. Motoya, S. Kurimura, and S. Inoue, “Bright narrowband source of photon pairs at optical telecommunication wavelengths using a type-II periodically poled lithium niobate waveguide,” Opt. Express 15, 12769-12776(2007).
[CrossRef]

T. B. Pittman, J. D. Franson, and B. C. Jacobs, “Investigation of a single-photon source based on quantum interference,” New J. Phys. 9, 195 (2007).
[CrossRef]

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature Phys. 3, 253-255 (2007).
[CrossRef]

S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, P. M. Petroff, and D. Bouwmeester, “High-frequency single-photon source with polarization control,” Nat. Photon. 1, 704-708(2007).
[CrossRef]

2006 (1)

C. Bödefeld, J. Ebbecke, J. Toivonen, M. Sopanen, H. Lipsanen, and A. Wixforth, “Experimental investigation towards a periodically pumped single-photon source,” Phys. Rev. B 74, 035407 (2006).
[CrossRef]

2005 (5)

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Informatics 1, 5-37 (2005).

P. P. Rohde and T. C. Ralph, “Frequency and temporal effects in linear optical quantum computing,” Phys. Rev. A 71, 032320(2005).
[CrossRef]

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N.C. Wong, and F. X. Kartner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94, 169903(2005).
[CrossRef]

E. Knill, “Quantum computing with realistically noisy devices,” Nature 434, 39-44 (2005).
[CrossRef]

E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, “Interference with correlated photons: five quantum mechanics experiments for undergraduates,” Am. J. Phys. 73, 127-140(2005).
[CrossRef]

2004 (3)

A. Kiraz, M. Atature, and A. Imamoglu, “Quantum-dot single-photon sources: prospects for applications in linear optics quantum-information processing,” Phys. Rev. A 69, 032305(2004).
[CrossRef]

J. Y. Cheung, C. J. Chunnilall, and J. Wang, “Radiometric applications of correlated photon metrology,” Proc. SPIE 5551, 220-230 (2004).
[CrossRef]

M. A. Albota and E. Dauler, “Single photon detection of degenerate photon pairs at 1.55 μm from a periodically poled lithium niobate parametric downconverter,” J. Mod. Opt. 51, 1417-1432 (2004).

2003 (3)

Y.-H. Kim, “Measurement of one-photon and two-photon wavepackets in spontaneous parametric downconversion,” J. Opt. Soc. Am. B 20, 1959-1966 (2003).
[CrossRef]

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, and N. Gisin, “Quantum interference with photon pairs created in spatially separated sources,” Phys. Rev. A 67, 022301 (2003).
[CrossRef]

Y. Shih, “Entangled biphoton source-property and preparation,” Rep. Prog. Phys. 66, 1009-1044 (2003).
[CrossRef]

2002 (4)

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A 65, 053817(2002).
[CrossRef]

P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
[CrossRef]

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594-597 (2002).
[CrossRef]

2001 (2)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef]

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P. Kulik, “Collinear two-photon state with spectral properties of type-I and polarization properties of type-II spontaneous parametric down-conversion: preparation and testing,” Phys. Rev. A 64, 041803 (2001).
[CrossRef]

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]

1999 (2)

E. Dauler, G. Jaeger, A. Muller, A. Migdall, and A. Sergienko, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1-10 (1999).

D. Branning, W. P. Grice, R. Erdmann, and I. A. Walmsley, “Engineering the indistinguishability and entanglement of two photons,” Phys. Rev. Lett. 83, 955-958 (1999).
[CrossRef]

1992 (1)

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a quantum eraser--a revival of coherence in a 2-photon interference experiment,” Phys. Rev. A 45, 7729-7739(1992).
[CrossRef]

1991 (2)

N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197-202 (1991).
[CrossRef]

L. Mandel, “Coherence and indistinguishability,” Opt. Lett. 16, 1882-1883 (1991).
[CrossRef]

1989 (1)

1988 (1)

Z. Y. Ou and L. Mandel, “Violation of Bell's inequality and classical probability in a 2-photon correlation experiment,” Phys. Rev. Lett. 61, 50-53 (1988).
[CrossRef]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between 2 photons by interference,” Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef]

1974 (1)

H. Mishina and T. Asakura, “2 Gaussian-beam interference,” Nouv. Rev. Opt. 5, 101-107 (1974).
[CrossRef]

1963 (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A 65, 053817(2002).
[CrossRef]

Abrams, D. S.

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]

Albota, M. A.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N.C. Wong, and F. X. Kartner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94, 169903(2005).
[CrossRef]

M. A. Albota and E. Dauler, “Single photon detection of degenerate photon pairs at 1.55 μm from a periodically poled lithium niobate parametric downconverter,” J. Mod. Opt. 51, 1417-1432 (2004).

Asakura, T.

H. Mishina and T. Asakura, “2 Gaussian-beam interference,” Nouv. Rev. Opt. 5, 101-107 (1974).
[CrossRef]

Atature, M.

A. Kiraz, M. Atature, and A. Imamoglu, “Quantum-dot single-photon sources: prospects for applications in linear optics quantum-information processing,” Phys. Rev. A 69, 032305(2004).
[CrossRef]

Beattie, N. S.

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Bödefeld, C.

C. Bödefeld, J. Ebbecke, J. Toivonen, M. Sopanen, H. Lipsanen, and A. Wixforth, “Experimental investigation towards a periodically pumped single-photon source,” Phys. Rev. B 74, 035407 (2006).
[CrossRef]

Boto, A. N.

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]

Bouwmeester, D.

S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, P. M. Petroff, and D. Bouwmeester, “High-frequency single-photon source with polarization control,” Nat. Photon. 1, 704-708(2007).
[CrossRef]

Branning, D.

D. Branning, W. P. Grice, R. Erdmann, and I. A. Walmsley, “Engineering the indistinguishability and entanglement of two photons,” Phys. Rev. Lett. 83, 955-958 (1999).
[CrossRef]

Braunstein, S. L.

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]

Burlakov, A. V.

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P. Kulik, “Collinear two-photon state with spectral properties of type-I and polarization properties of type-II spontaneous parametric down-conversion: preparation and testing,” Phys. Rev. A 64, 041803 (2001).
[CrossRef]

Calonico, D.

D. Calonico, F. Levi, L. Lorini, and G. Mana, “Bayesian inference of a negative quantity from positive measurement results,” Metrologia 46, 267-271 (2009).
[CrossRef]

Chekhova, M. V.

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P. Kulik, “Collinear two-photon state with spectral properties of type-I and polarization properties of type-II spontaneous parametric down-conversion: preparation and testing,” Phys. Rev. A 64, 041803 (2001).
[CrossRef]

Cheung, J.

Cheung, J. Y.

P. J. Thomas, J. Y. Cheung, C. J. Chunnilall, and M. H. Dunn, “The Hong-Ou-Mandel interferometer: a new procedure for alignment,” Rev. Sci. Instrum. 80, 036101 (2009).
[CrossRef]

J. Y. Cheung, C. J. Chunnilall, and J. Wang, “Radiometric applications of correlated photon metrology,” Proc. SPIE 5551, 220-230 (2004).
[CrossRef]

Chiao, R. Y.

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a quantum eraser--a revival of coherence in a 2-photon interference experiment,” Phys. Rev. A 45, 7729-7739(1992).
[CrossRef]

Chunnilall, C. J.

P. J. Thomas, J. Y. Cheung, C. J. Chunnilall, and M. H. Dunn, “The Hong-Ou-Mandel interferometer: a new procedure for alignment,” Rev. Sci. Instrum. 80, 036101 (2009).
[CrossRef]

J. Y. Cheung, C. J. Chunnilall, and J. Wang, “Radiometric applications of correlated photon metrology,” Proc. SPIE 5551, 220-230 (2004).
[CrossRef]

Coldren, L. A.

S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, P. M. Petroff, and D. Bouwmeester, “High-frequency single-photon source with polarization control,” Nat. Photon. 1, 704-708(2007).
[CrossRef]

Cooper, K.

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Courtemanche, N.

E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, “Interference with correlated photons: five quantum mechanics experiments for undergraduates,” Am. J. Phys. 73, 127-140(2005).
[CrossRef]

Dauler, E.

M. A. Albota and E. Dauler, “Single photon detection of degenerate photon pairs at 1.55 μm from a periodically poled lithium niobate parametric downconverter,” J. Mod. Opt. 51, 1417-1432 (2004).

E. Dauler, G. Jaeger, A. Muller, A. Migdall, and A. Sergienko, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1-10 (1999).

de Riedmatten, H.

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, and N. Gisin, “Quantum interference with photon pairs created in spatially separated sources,” Phys. Rev. A 67, 022301 (2003).
[CrossRef]

Diamanti, E.

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Informatics 1, 5-37 (2005).

Dowling, J. P.

P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
[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]

Dunn, M. H.

P. J. Thomas, J. Y. Cheung, C. J. Chunnilall, and M. H. Dunn, “The Hong-Ou-Mandel interferometer: a new procedure for alignment,” Rev. Sci. Instrum. 80, 036101 (2009).
[CrossRef]

Ebbecke, J.

C. Bödefeld, J. Ebbecke, J. Toivonen, M. Sopanen, H. Lipsanen, and A. Wixforth, “Experimental investigation towards a periodically pumped single-photon source,” Phys. Rev. B 74, 035407 (2006).
[CrossRef]

Erdmann, R.

D. Branning, W. P. Grice, R. Erdmann, and I. A. Walmsley, “Engineering the indistinguishability and entanglement of two photons,” Phys. Rev. Lett. 83, 955-958 (1999).
[CrossRef]

Fattal, D.

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Informatics 1, 5-37 (2005).

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594-597 (2002).
[CrossRef]

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, Vol. 3 of The Feynman Lectures on Physics (Addison-Wesley, 1965).

Fiorentino, M.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N.C. Wong, and F. X. Kartner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94, 169903(2005).
[CrossRef]

Fox, N. P.

N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197-202 (1991).
[CrossRef]

Franson, J. D.

T. B. Pittman, J. D. Franson, and B. C. Jacobs, “Investigation of a single-photon source based on quantum interference,” New J. Phys. 9, 195 (2007).
[CrossRef]

Fujii, G.

Galvez, E. J.

E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, “Interference with correlated photons: five quantum mechanics experiments for undergraduates,” Am. J. Phys. 73, 127-140(2005).
[CrossRef]

Gardner, J. L.

Gisin, N.

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, and N. Gisin, “Quantum interference with photon pairs created in spatially separated sources,” Phys. Rev. A 67, 022301 (2003).
[CrossRef]

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Grice, W. P.

D. Branning, W. P. Grice, R. Erdmann, and I. A. Walmsley, “Engineering the indistinguishability and entanglement of two photons,” Phys. Rev. Lett. 83, 955-958 (1999).
[CrossRef]

Hariharan, P.

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

Heilig, L.

E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, “Interference with correlated photons: five quantum mechanics experiments for undergraduates,” Am. J. Phys. 73, 127-140(2005).
[CrossRef]

Hijlkema, M.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature Phys. 3, 253-255 (2007).
[CrossRef]

Holbrow, C. H.

E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, “Interference with correlated photons: five quantum mechanics experiments for undergraduates,” Am. J. Phys. 73, 127-140(2005).
[CrossRef]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between 2 photons by interference,” Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef]

Imamoglu, A.

A. Kiraz, M. Atature, and A. Imamoglu, “Quantum-dot single-photon sources: prospects for applications in linear optics quantum-information processing,” Phys. Rev. A 69, 032305(2004).
[CrossRef]

Inoue, S.

Jacobs, B. C.

T. B. Pittman, J. D. Franson, and B. C. Jacobs, “Investigation of a single-photon source based on quantum interference,” New J. Phys. 9, 195 (2007).
[CrossRef]

Jaeger, G.

E. Dauler, G. Jaeger, A. Muller, A. Migdall, and A. Sergienko, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1-10 (1999).

Karabutova, O. A.

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P. Kulik, “Collinear two-photon state with spectral properties of type-I and polarization properties of type-II spontaneous parametric down-conversion: preparation and testing,” Phys. Rev. A 64, 041803 (2001).
[CrossRef]

Kardynal, B. E.

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Kartner, F. X.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N.C. Wong, and F. X. Kartner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94, 169903(2005).
[CrossRef]

Kim, Y.-H.

Kiraz, A.

A. Kiraz, M. Atature, and A. Imamoglu, “Quantum-dot single-photon sources: prospects for applications in linear optics quantum-information processing,” Phys. Rev. A 69, 032305(2004).
[CrossRef]

Knill, E.

E. Knill, “Quantum computing with realistically noisy devices,” Nature 434, 39-44 (2005).
[CrossRef]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef]

Kok, P.

P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
[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]

Kuhn, A.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature Phys. 3, 253-255 (2007).
[CrossRef]

Kulik, S. P.

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P. Kulik, “Collinear two-photon state with spectral properties of type-I and polarization properties of type-II spontaneous parametric down-conversion: preparation and testing,” Phys. Rev. A 64, 041803 (2001).
[CrossRef]

Kurimura, S.

Kuzucu, O.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N.C. Wong, and F. X. Kartner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94, 169903(2005).
[CrossRef]

Kwiat, P. G.

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a quantum eraser--a revival of coherence in a 2-photon interference experiment,” Phys. Rev. A 45, 7729-7739(1992).
[CrossRef]

Laflamme, R.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef]

Lee, H.

P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
[CrossRef]

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, Vol. 3 of The Feynman Lectures on Physics (Addison-Wesley, 1965).

Levi, F.

D. Calonico, F. Levi, L. Lorini, and G. Mana, “Bayesian inference of a negative quantity from positive measurement results,” Metrologia 46, 267-271 (2009).
[CrossRef]

Lipsanen, H.

C. Bödefeld, J. Ebbecke, J. Toivonen, M. Sopanen, H. Lipsanen, and A. Wixforth, “Experimental investigation towards a periodically pumped single-photon source,” Phys. Rev. B 74, 035407 (2006).
[CrossRef]

Lobo, C. J.

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Lorini, L.

D. Calonico, F. Levi, L. Lorini, and G. Mana, “Bayesian inference of a negative quantity from positive measurement results,” Metrologia 46, 267-271 (2009).
[CrossRef]

Mana, G.

D. Calonico, F. Levi, L. Lorini, and G. Mana, “Bayesian inference of a negative quantity from positive measurement results,” Metrologia 46, 267-271 (2009).
[CrossRef]

Mandel, L.

L. Mandel, “Coherence and indistinguishability,” Opt. Lett. 16, 1882-1883 (1991).
[CrossRef]

Z. Y. Ou and L. Mandel, “Violation of Bell's inequality and classical probability in a 2-photon correlation experiment,” Phys. Rev. Lett. 61, 50-53 (1988).
[CrossRef]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between 2 photons by interference,” Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef]

Marcikic, I.

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, and N. Gisin, “Quantum interference with photon pairs created in spatially separated sources,” Phys. Rev. A 67, 022301 (2003).
[CrossRef]

Martin, J. W.

E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, “Interference with correlated photons: five quantum mechanics experiments for undergraduates,” Am. J. Phys. 73, 127-140(2005).
[CrossRef]

Migdall, A.

J. Cheung, J. L. Gardner, A. Migdall, S. Polyakov, and M. Ware, “High accuracy dual lens transmittance measurements,” Appl. Opt. 46, 5396-5403 (2007).
[CrossRef]

E. Dauler, G. Jaeger, A. Muller, A. Migdall, and A. Sergienko, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1-10 (1999).

Milburn, G. J.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef]

Mishina, H.

H. Mishina and T. Asakura, “2 Gaussian-beam interference,” Nouv. Rev. Opt. 5, 101-107 (1974).
[CrossRef]

Motoya, M.

Muller, A.

E. Dauler, G. Jaeger, A. Muller, A. Migdall, and A. Sergienko, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1-10 (1999).

Namekata, N.

Nasr, M. B.

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A 65, 053817(2002).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou and L. Mandel, “Violation of Bell's inequality and classical probability in a 2-photon correlation experiment,” Phys. Rev. Lett. 61, 50-53 (1988).
[CrossRef]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between 2 photons by interference,” Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef]

Pepper, M.

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Petroff, P. M.

S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, P. M. Petroff, and D. Bouwmeester, “High-frequency single-photon source with polarization control,” Nat. Photon. 1, 704-708(2007).
[CrossRef]

Pittman, T. B.

T. B. Pittman, J. D. Franson, and B. C. Jacobs, “Investigation of a single-photon source based on quantum interference,” New J. Phys. 9, 195 (2007).
[CrossRef]

Polyakov, S.

Pysher, M. J.

E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, “Interference with correlated photons: five quantum mechanics experiments for undergraduates,” Am. J. Phys. 73, 127-140(2005).
[CrossRef]

Rakher, M. T.

S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, P. M. Petroff, and D. Bouwmeester, “High-frequency single-photon source with polarization control,” Nat. Photon. 1, 704-708(2007).
[CrossRef]

Ralph, T. C.

P. P. Rohde and T. C. Ralph, “Frequency and temporal effects in linear optical quantum computing,” Phys. Rev. A 71, 032320(2005).
[CrossRef]

Rarity, J. G.

Rempe, G.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature Phys. 3, 253-255 (2007).
[CrossRef]

Ritchie, D. A.

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Rohde, P. P.

P. P. Rohde and T. C. Ralph, “Frequency and temporal effects in linear optical quantum computing,” Phys. Rev. A 71, 032320(2005).
[CrossRef]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, Vol. 3 of The Feynman Lectures on Physics (Addison-Wesley, 1965).

Santori, C.

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Informatics 1, 5-37 (2005).

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594-597 (2002).
[CrossRef]

Sergienko, A.

E. Dauler, G. Jaeger, A. Muller, A. Migdall, and A. Sergienko, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1-10 (1999).

Sergienko, A. V.

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A 65, 053817(2002).
[CrossRef]

Shields, A. J.

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Shih, Y.

Y. Shih, “Entangled biphoton source-property and preparation,” Rep. Prog. Phys. 66, 1009-1044 (2003).
[CrossRef]

Solomon, G.

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Informatics 1, 5-37 (2005).

Solomon, G. S.

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594-597 (2002).
[CrossRef]

Sopanen, M.

C. Bödefeld, J. Ebbecke, J. Toivonen, M. Sopanen, H. Lipsanen, and A. Wixforth, “Experimental investigation towards a periodically pumped single-photon source,” Phys. Rev. B 74, 035407 (2006).
[CrossRef]

Specht, H. P.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature Phys. 3, 253-255 (2007).
[CrossRef]

Spencer, J.

E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, “Interference with correlated photons: five quantum mechanics experiments for undergraduates,” Am. J. Phys. 73, 127-140(2005).
[CrossRef]

Steinberg, A. M.

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a quantum eraser--a revival of coherence in a 2-photon interference experiment,” Phys. Rev. A 45, 7729-7739(1992).
[CrossRef]

Stevenson, R. M.

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Stoltz, N. G.

S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, P. M. Petroff, and D. Bouwmeester, “High-frequency single-photon source with polarization control,” Nat. Photon. 1, 704-708(2007).
[CrossRef]

Strauf, S.

S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, P. M. Petroff, and D. Bouwmeester, “High-frequency single-photon source with polarization control,” Nat. Photon. 1, 704-708(2007).
[CrossRef]

Tapster, P. R.

Teich, M. C.

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A 65, 053817(2002).
[CrossRef]

Thomas, P. J.

P. J. Thomas, J. Y. Cheung, C. J. Chunnilall, and M. H. Dunn, “The Hong-Ou-Mandel interferometer: a new procedure for alignment,” Rev. Sci. Instrum. 80, 036101 (2009).
[CrossRef]

Tittel, W.

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, and N. Gisin, “Quantum interference with photon pairs created in spatially separated sources,” Phys. Rev. A 67, 022301 (2003).
[CrossRef]

Toivonen, J.

C. Bödefeld, J. Ebbecke, J. Toivonen, M. Sopanen, H. Lipsanen, and A. Wixforth, “Experimental investigation towards a periodically pumped single-photon source,” Phys. Rev. B 74, 035407 (2006).
[CrossRef]

Vuckovic, J.

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Informatics 1, 5-37 (2005).

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594-597 (2002).
[CrossRef]

Waks, E.

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Informatics 1, 5-37 (2005).

Walmsley, I. A.

D. Branning, W. P. Grice, R. Erdmann, and I. A. Walmsley, “Engineering the indistinguishability and entanglement of two photons,” Phys. Rev. Lett. 83, 955-958 (1999).
[CrossRef]

Wang, J.

J. Y. Cheung, C. J. Chunnilall, and J. Wang, “Radiometric applications of correlated photon metrology,” Proc. SPIE 5551, 220-230 (2004).
[CrossRef]

Ware, M.

Weber, B.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature Phys. 3, 253-255 (2007).
[CrossRef]

Webster, S. C.

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature Phys. 3, 253-255 (2007).
[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]

Wixforth, A.

C. Bödefeld, J. Ebbecke, J. Toivonen, M. Sopanen, H. Lipsanen, and A. Wixforth, “Experimental investigation towards a periodically pumped single-photon source,” Phys. Rev. B 74, 035407 (2006).
[CrossRef]

Wong, F. N.C.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N.C. Wong, and F. X. Kartner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94, 169903(2005).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Informatics 1, 5-37 (2005).

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594-597 (2002).
[CrossRef]

Yuan, Z. L.

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Zbinden, H.

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, and N. Gisin, “Quantum interference with photon pairs created in spatially separated sources,” Phys. Rev. A 67, 022301 (2003).
[CrossRef]

Am. J. Phys. (1)

E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer, “Interference with correlated photons: five quantum mechanics experiments for undergraduates,” Am. J. Phys. 73, 127-140(2005).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

M. A. Albota and E. Dauler, “Single photon detection of degenerate photon pairs at 1.55 μm from a periodically poled lithium niobate parametric downconverter,” J. Mod. Opt. 51, 1417-1432 (2004).

J. Opt. Soc. Am. B (2)

J. Res. Natl. Inst. Stand. Technol. (1)

E. Dauler, G. Jaeger, A. Muller, A. Migdall, and A. Sergienko, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1-10 (1999).

Metrologia (2)

N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197-202 (1991).
[CrossRef]

D. Calonico, F. Levi, L. Lorini, and G. Mana, “Bayesian inference of a negative quantity from positive measurement results,” Metrologia 46, 267-271 (2009).
[CrossRef]

Nat. Photon. (1)

S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, P. M. Petroff, and D. Bouwmeester, “High-frequency single-photon source with polarization control,” Nat. Photon. 1, 704-708(2007).
[CrossRef]

Nature (3)

C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594-597 (2002).
[CrossRef]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef]

E. Knill, “Quantum computing with realistically noisy devices,” Nature 434, 39-44 (2005).
[CrossRef]

Nature Phys. (1)

M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature Phys. 3, 253-255 (2007).
[CrossRef]

New J. Phys. (1)

T. B. Pittman, J. D. Franson, and B. C. Jacobs, “Investigation of a single-photon source based on quantum interference,” New J. Phys. 9, 195 (2007).
[CrossRef]

Nouv. Rev. Opt. (1)

H. Mishina and T. Asakura, “2 Gaussian-beam interference,” Nouv. Rev. Opt. 5, 101-107 (1974).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Phys. Rev. A (7)

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a quantum eraser--a revival of coherence in a 2-photon interference experiment,” Phys. Rev. A 45, 7729-7739(1992).
[CrossRef]

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P. Kulik, “Collinear two-photon state with spectral properties of type-I and polarization properties of type-II spontaneous parametric down-conversion: preparation and testing,” Phys. Rev. A 64, 041803 (2001).
[CrossRef]

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, and N. Gisin, “Quantum interference with photon pairs created in spatially separated sources,” Phys. Rev. A 67, 022301 (2003).
[CrossRef]

A. Kiraz, M. Atature, and A. Imamoglu, “Quantum-dot single-photon sources: prospects for applications in linear optics quantum-information processing,” Phys. Rev. A 69, 032305(2004).
[CrossRef]

P. P. Rohde and T. C. Ralph, “Frequency and temporal effects in linear optical quantum computing,” Phys. Rev. A 71, 032320(2005).
[CrossRef]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum-optical coherence tomography with dispersion cancellation,” Phys. Rev. A 65, 053817(2002).
[CrossRef]

P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
[CrossRef]

Phys. Rev. B (1)

C. Bödefeld, J. Ebbecke, J. Toivonen, M. Sopanen, H. Lipsanen, and A. Wixforth, “Experimental investigation towards a periodically pumped single-photon source,” Phys. Rev. B 74, 035407 (2006).
[CrossRef]

Phys. Rev. Lett. (5)

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]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between 2 photons by interference,” Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef]

Z. Y. Ou and L. Mandel, “Violation of Bell's inequality and classical probability in a 2-photon correlation experiment,” Phys. Rev. Lett. 61, 50-53 (1988).
[CrossRef]

D. Branning, W. P. Grice, R. Erdmann, and I. A. Walmsley, “Engineering the indistinguishability and entanglement of two photons,” Phys. Rev. Lett. 83, 955-958 (1999).
[CrossRef]

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N.C. Wong, and F. X. Kartner, “Two-photon coincident-frequency entanglement via extended phase matching,” Phys. Rev. Lett. 94, 169903(2005).
[CrossRef]

Proc. SPIE (1)

J. Y. Cheung, C. J. Chunnilall, and J. Wang, “Radiometric applications of correlated photon metrology,” Proc. SPIE 5551, 220-230 (2004).
[CrossRef]

Prog. Informatics (1)

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Informatics 1, 5-37 (2005).

Rep. Prog. Phys. (1)

Y. Shih, “Entangled biphoton source-property and preparation,” Rep. Prog. Phys. 66, 1009-1044 (2003).
[CrossRef]

Rev. Sci. Instrum. (1)

P. J. Thomas, J. Y. Cheung, C. J. Chunnilall, and M. H. Dunn, “The Hong-Ou-Mandel interferometer: a new procedure for alignment,” Rev. Sci. Instrum. 80, 036101 (2009).
[CrossRef]

Science (1)

Z. L. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, and M. Pepper, “Electrically driven single-photon source,” Science 295, 102-105 (2002).
[CrossRef]

Other (3)

R. P. Feynman, R. B. Leighton, and M. Sands, Vol. 3 of The Feynman Lectures on Physics (Addison-Wesley, 1965).

“Guide to the expression of uncertainty in measurement” (GUM) (International Organization for Standardization, 1995).

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

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Figures (4)

Fig. 1
Fig. 1

Possible outcomes when two photons recombine from two separate input ports of a beam splitter. The defining feature of HOM interference is when (c) and (d) cease to be possibilities, and this occurs when the two photons are indistinguishable in all respects.

Fig. 2
Fig. 2

HOM interferometer from which measurements were made for this investigation. The wavelength dispersive prism (WDP) removed fluorescence from the Ar laser beam, while the half-wave plate (HWP) served as a switch for the downconversion process. A 1 m focal length lens positioned immediately before the BBO crystal served to improve the photon-pair collection efficiency.

Fig. 3
Fig. 3

Points are measured HOM dip data using the experimental setup in Fig. 2. The solid fit to the curve is based on the model described by Eq. (2).

Fig. 4
Fig. 4

(a) Black circles (white circles) are HOM dip (laser interference) visibility experimental data as a function of θ. The solid and dashed curves are model fits to the HOM and laser data, respectively, with corresponding maxima at θ HOM = 0 and θ laser = 0 . (b) Black circles (white circles) are HOM dip (laser interference) visibility experimental data as a function of ε. The solid and dashed curves are outputs of a fitted model to the HOM and laser interference data, respectively, with corresponding maxima at ε HOM = 0 and ε laser = 0 .

Tables (1)

Tables Icon

Table 1 Experimentally Determined Values of the Terms Defined by the Right-Hand Side of Eq. (3), together with their Associated Uncertainties, in Relation to the Setup in Fig. 2 a

Equations (19)

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V HOM = N c , Δ τ τ c N c , Δ τ = 0 N c , Δ τ τ c ,
N c ( τ ) = A ( 1 V HOM exp [ B ( τ C ) 2 ] × sinc [ D ( τ C ) ] ) .
P ind = V HOM f Δ φ int · f R , T · f ε · f θ · f surface .
u P ind 2 = i = 1 N ( P ind f i ) 2 · u f i 2 + 2 i = 1 N 1 j = i + 1 N 1 ( P ind f i ) ( P ind f j ) u ( f i , f j ) ,
| ψ Δ τ = 0 = 1 2 [ | 1 1 H 1 2 H + Δ φ int | 1 1 H + Δ φ int 1 2 H ] .
N c , Δ τ = 0 = ψ | a ^ 1 + a ^ 2 + a ^ 2 a ^ 1 | ψ = 1 2 sin 2 Δ φ int ,
N c , Δ τ τ c = 1 2 .
f Δ φ int = cos 2 Δ φ int .
N c , Δ τ τ c P 12 = | r r | 2 + | t t | 2 = R 2 + T 2 ,
N c , Δ τ = 0 P 12 = | r r + t t | 2 = R 2 + T 2 + 2 R T .
f R , T = 2 R T R 2 + T 2 .
f θ = V HOM ( θ opt ) V HOM ( θ HOM = 0 ) .
f ε = V HOM ( ε opt ) V HOM ( ε HOM = 0 ) .
P ind , laser = V laser f φ ind , laser · f I 1 I 2 · f ε laser · f θ laser · f surface .
f surface = V laser f φ int , laser · f I 1 I 2 · f ε laser · f θ laser .
V laser = I max I min I max + I min .
f I 1 , I 2 = 2 I 1 I 2 I 1 + I 2 ,
f θ laser = V laser ( θ opt ) V laser ( θ laser = 0 ) ,
f ε laser = V laser ( ε opt ) V laser ( ε laser = 0 ) .

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