Abstract

The classical Wiener–Khinchin theorem (WKT), which can extract spectral information by classical interferometers through Fourier transform, is a fundamental theorem used in many disciplines. However, there is still a need for a quantum version of WKT, which could connect correlated biphoton spectral information by quantum interferometers. Here, we extend the classical WKT to its quantum counterpart [i.e., extended WKT (e-WKT)], which is based on two-photon quantum interferometry. According to the e-WKT, the difference–frequency distribution of the biphoton wavefunctions can be extracted by applying a Fourier transform on the time-domain Hong–Ou–Mandel interference (HOMI) patterns, while the sum-frequency distribution can be extracted by applying a Fourier transform on the time-domain NOON state interference (NOONI) patterns. We also experimentally verified the WKT and e-WKT in a Mach–Zehnder interference (MZI), a HOMI, and a NOONI. This theorem can be directly applied to quantum spectroscopy, where the spectral correlation information of biphotons can be obtained from time-domain quantum interferences by Fourier transform. This may open a new path for the study of light–matter interaction at the single photon level.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2017 (1)

R.-B. Jin, G.-Q. Chen, H. Jing, C. Ren, P. Zhao, R. Shimizu, and P.-X. Lu, “Monotonic quantum-to-classical transition enabled by positively correlated biphotons,” Phys. Rev. A 95, 062341 (2017).
[Crossref]

2016 (4)

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, T. Yamamoto, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
[Crossref]

R.-B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol. 1, 015004 (2016).
[Crossref]

R.-B. Jin, M. Fujiwara, R. Shimizu, R. J. Collins, G. S. Buller, T. Yamashita, S. Miki, H. Terai, M. Takeoka, and M. Sasaki, “Detection-dependent six-photon Holland–Burnett state interference,” Sci. Rep. 6, 36914 (2016).
[Crossref]

M. Bergmann and P. van Loock, “Quantum error correction against photon loss using noon states,” Phys. Rev. A 94, 012311 (2016).
[Crossref]

2015 (2)

N. S. Bisht and R. Shimizu, “Spectral properties of broadband biphotons generated from PPMgSLT under a type-II phase-matching condition,” J. Opt. Soc. Am. B 32, 550–554 (2015).
[Crossref]

P. Chen, C. Shu, X. Guo, M. M. T. Loy, and S. Du, “Measuring the biphoton temporal wave function with polarization-dependent and time-resolved two-photon interference,” Phys. Rev. Lett. 114, 010401 (2015).
[Crossref]

2014 (2)

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112, 103604 (2014).
[Crossref]

N. Bruno, A. Martin, T. Guerreiro, B. Sanguinetti, and R. T. Thew, “Pulsed source of spectrally uncorrelated and indistinguishable photons at telecom wavelengths,” Opt. Express 22, 17246–17253 (2014).
[Crossref]

2013 (2)

2011 (3)

T. Gerrits, M. J. Stevens, B. Baek, B. Calkins, A. Lita, S. Glancy, E. Knill, S. W. Nam, R. P. Mirin, R. H. Hadfield, R. S. Bennink, W. P. Grice, S. Dorenbos, T. Zijlstra, T. Klapwijk, and V. Zwiller, “Generation of degenerate, factorizable, pulsed squeezed light at telecom wavelengths,” Opt. Express 19, 24434–24447 (2011).
[Crossref]

A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[Crossref]

R.-B. Jin, J. Zhang, R. Shimizu, N. Matsuda, Y. Mitsumori, H. Kosaka, and K. Edamatsu, “High-visibility nonclassical interference between intrinsically pure heralded single photons and photons from a weak coherent field,” Phys. Rev. A 83, 031805 (2011).
[Crossref]

2010 (1)

P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Bright source of spectrally uncorrelated polarization-entangled photons with nearly single-mode emission,” Phys. Rev. Lett. 105, 253601 (2010).
[Crossref]

2009 (1)

2008 (3)

O. Kuzucu and F. N. C. Wong, “Pulsed Sagnac source of narrow-band polarization-entangled photons,” Phys. Rev. A 77, 032314 (2008).
[Crossref]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[Crossref]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: a recipe for purity,” New J. Phys. 10, 093011 (2008).
[Crossref]

2006 (1)

J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darquie, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature 440, 779–782 (2006).
[Crossref]

2004 (3)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref]

K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004).
[Crossref]

F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84, 1644–1646 (2004).
[Crossref]

2003 (1)

M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of dispersion-canceled quantum-optical coherence tomography,” Phys. Rev. Lett. 91, 083601 (2003).
[Crossref]

2002 (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]

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]

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Generating entangled two-photon states with coincident frequencies,” Phys. Rev. Lett. 88, 183602 (2002).
[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]

1992 (2)

J. D. Franson, “Nonlocal cancellation of dispersion,” Phys. Rev. A 45, 3126–3132 (1992).
[Crossref]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer,” Phys. Rev. A 45, 6659–6665 (1992).
[Crossref]

1987 (1)

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

1934 (1)

A. Khintchine, “Korrelationstheorie der stationären stochastischen prozesse,” Math. Ann. 109, 604–615 (1934).

1930 (1)

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117–258 (1930).
[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]

Abrams, M. C.

S. P. Davis, M. C. Abrams, and J. W. Brault, Fourier Transform Spectrometry (Academic, 2001).

Ansari, V.

V. Ansari, B. Brecht, G. Harder, and C. Silberhorn, “Probing spectral-temporal correlations with a versatile integrated source of parametric down-conversion states,” arXiv:1404.7725 (2014).

Baek, B.

Benichi, H.

Bennink, R. S.

Bergmann, M.

M. Bergmann and P. van Loock, “Quantum error correction against photon loss using noon states,” Phys. Rev. A 94, 012311 (2016).
[Crossref]

Beugnon, J.

J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darquie, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature 440, 779–782 (2006).
[Crossref]

Bisht, N. S.

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]

Brault, J. W.

S. P. Davis, M. C. Abrams, and J. W. Brault, Fourier Transform Spectrometry (Academic, 2001).

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]

Brecht, B.

V. Ansari, B. Brecht, G. Harder, and C. Silberhorn, “Probing spectral-temporal correlations with a versatile integrated source of parametric down-conversion states,” arXiv:1404.7725 (2014).

Browaeys, A.

J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darquie, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature 440, 779–782 (2006).
[Crossref]

Bruno, N.

Buller, G. S.

R.-B. Jin, M. Fujiwara, R. Shimizu, R. J. Collins, G. S. Buller, T. Yamashita, S. Miki, H. Terai, M. Takeoka, and M. Sasaki, “Detection-dependent six-photon Holland–Burnett state interference,” Sci. Rep. 6, 36914 (2016).
[Crossref]

Calkins, B.

Chen, G.-Q.

R.-B. Jin, G.-Q. Chen, H. Jing, C. Ren, P. Zhao, R. Shimizu, and P.-X. Lu, “Monotonic quantum-to-classical transition enabled by positively correlated biphotons,” Phys. Rev. A 95, 062341 (2017).
[Crossref]

Chen, P.

P. Chen, C. Shu, X. Guo, M. M. T. Loy, and S. Du, “Measuring the biphoton temporal wave function with polarization-dependent and time-resolved two-photon interference,” Phys. Rev. Lett. 114, 010401 (2015).
[Crossref]

Chiao, R. Y.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer,” Phys. Rev. A 45, 6659–6665 (1992).
[Crossref]

Christ, A.

A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[Crossref]

Collins, R. J.

R.-B. Jin, M. Fujiwara, R. Shimizu, R. J. Collins, G. S. Buller, T. Yamashita, S. Miki, H. Terai, M. Takeoka, and M. Sasaki, “Detection-dependent six-photon Holland–Burnett state interference,” Sci. Rep. 6, 36914 (2016).
[Crossref]

Darquie, B.

J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darquie, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature 440, 779–782 (2006).
[Crossref]

Davis, S. P.

S. P. Davis, M. C. Abrams, and J. W. Brault, Fourier Transform Spectrometry (Academic, 2001).

De Haseth, J. A.

P. R. Griffiths and J. A. De Haseth, Fourier Transform Infrared Spectrometry, 2nd ed. (Wiley, 2007).

Dingjan, J.

J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darquie, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature 440, 779–782 (2006).
[Crossref]

Dorenbos, S.

Dowling, J. 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]

Du, S.

P. Chen, C. Shu, X. Guo, M. M. T. Loy, and S. Du, “Measuring the biphoton temporal wave function with polarization-dependent and time-resolved two-photon interference,” Phys. Rev. Lett. 114, 010401 (2015).
[Crossref]

Eckstein, A.

A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011).
[Crossref]

Edamatsu, K.

R.-B. Jin, J. Zhang, R. Shimizu, N. Matsuda, Y. Mitsumori, H. Kosaka, and K. Edamatsu, “High-visibility nonclassical interference between intrinsically pure heralded single photons and photons from a weak coherent field,” Phys. Rev. A 83, 031805 (2011).
[Crossref]

R. Shimizu and K. Edamatsu, “High-flux and broadband biphoton sources with controlled frequency entanglement,” Opt. Express 17, 16385–16393 (2009).
[Crossref]

K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004).
[Crossref]

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]

Evans, P. G.

P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Bright source of spectrally uncorrelated polarization-entangled photons with nearly single-mode emission,” Phys. Rev. Lett. 105, 253601 (2010).
[Crossref]

Fattal, D.

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]

Franson, J. D.

J. D. Franson, “Nonlocal cancellation of dispersion,” Phys. Rev. A 45, 3126–3132 (1992).
[Crossref]

Fujiwara, M.

R.-B. Jin, M. Fujiwara, R. Shimizu, R. J. Collins, G. S. Buller, T. Yamashita, S. Miki, H. Terai, M. Takeoka, and M. Sasaki, “Detection-dependent six-photon Holland–Burnett state interference,” Sci. Rep. 6, 36914 (2016).
[Crossref]

R.-B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol. 1, 015004 (2016).
[Crossref]

Gerrits, T.

R.-B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol. 1, 015004 (2016).
[Crossref]

T. Gerrits, M. J. Stevens, B. Baek, B. Calkins, A. Lita, S. Glancy, E. Knill, S. W. Nam, R. P. Mirin, R. H. Hadfield, R. S. Bennink, W. P. Grice, S. Dorenbos, T. Zijlstra, T. Klapwijk, and V. Zwiller, “Generation of degenerate, factorizable, pulsed squeezed light at telecom wavelengths,” Opt. Express 19, 24434–24447 (2011).
[Crossref]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref]

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Generating entangled two-photon states with coincident frequencies,” Phys. Rev. Lett. 88, 183602 (2002).
[Crossref]

Glancy, S.

Grangier, P.

J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darquie, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature 440, 779–782 (2006).
[Crossref]

Grice, W. P.

Griffiths, P. R.

P. R. Griffiths and J. A. De Haseth, Fourier Transform Infrared Spectrometry, 2nd ed. (Wiley, 2007).

Guerreiro, T.

Guo, X.

P. Chen, C. Shu, X. Guo, M. M. T. Loy, and S. Du, “Measuring the biphoton temporal wave function with polarization-dependent and time-resolved two-photon interference,” Phys. Rev. Lett. 114, 010401 (2015).
[Crossref]

Hadfield, R. H.

Harder, G.

V. Ansari, B. Brecht, G. Harder, and C. Silberhorn, “Probing spectral-temporal correlations with a versatile integrated source of parametric down-conversion states,” arXiv:1404.7725 (2014).

Hong, C. K.

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

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Acta Math. (1)

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Appl. Phys. Lett. (1)

F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84, 1644–1646 (2004).
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J. Opt. Soc. Am. B (1)

Math. Ann. (1)

A. Khintchine, “Korrelationstheorie der stationären stochastischen prozesse,” Math. Ann. 109, 604–615 (1934).

Nat. Commun. (1)

T. Ono, R. Okamoto, and S. Takeuchi, “An entanglement-enhanced microscope,” Nat. Commun. 4, 2426 (2013).
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Nat. Photonics (1)

T. Kobayashi, R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, T. Yamamoto, M. Koashi, and N. Imoto, “Frequency-domain Hong–Ou–Mandel interference,” Nat. Photonics 10, 441–444 (2016).
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Nature (3)

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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Model of the experimental setup. (a) Mach–Zehnder interferometer (MZI), (b) Hong–Ou–Mandel interferometer (HOMI), and (c) NOON-state interferometer (NOONI). s and i indicate the signal and idler photons. Both the signal and idler photons from SPDC are used for HOMI and NOONI, while only the signal photons are used for MZI.
Fig. 2.
Fig. 2. Experimental setup: M, mirror; QWP, quarter-wave plate; HWP, half-wave plate; PBS, polarization beam splitter; SMF, single-mode fiber; and APD, avalanche photodiode.
Fig. 3.
Fig. 3. Time domain interference patterns and their Fourier transformed frequency distribution. The first row shows the experimentally measured interference patterns: (a1) Mach–Zehnder interference pattern, (b1) Hong–Ou–Mandel interference pattern, and (c1) NOON state interference pattern. The visibility and temporal FWHM (Δτ) are shown in each figure. The figures in the second row [(a2), (b2), (c2)] show the corresponding frequency distribution, calculated from the interference patterns (a1), (b1), and (c1) by the Fourier transformation. The spectral FWHM (Δν) is shown in each figure.
Fig. 4.
Fig. 4. Experimental TSI and its projections onto three axes. (a) The experimentally measured two-photon spectral intensity (TSI) of the signal and idler photons from SPDC. The projections of the TSI onto the x-axis (b), antidiagonal axis (c), and diagonal axis (d) are shown. The corresponding FWHM values are 18.2 nm (2.18 THz), 1.9 nm (0.23 THz), and 24.6 nm (2.95 THz), respectively.

Tables (1)

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Table 1. Comparison of the Time Domain Data and Spectral Domain Dataa

Equations (4)

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P1(τ)=12[1+dω|f1(ω)|2cos(ωτ)],
F1(ω)|f1(ω)|2=12πdτG1(τ)eiωτ,
P2±(τ)=12[1±dωsdωi|f2(ωs,ωi)|2cos[(ωs±ωi)τ]],
F2±(ω±)=12πdτG2±(τ)eiω±τ,

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