Experimental demonstration of time-frequency duality of biphotons
Rui-Bo Jin Takuma Saito Ryosuke Shimizu
quant-ph







Laboratory of Optical Information Technology, Wuhan Institute of Technology, Wuhan 430205, China



The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, Japan


r-simizu@uec.ac.jp
The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, Japan





Time-frequency duality, which enables control of optical waveforms by manipulating amplitudes and phases of electromagnetic fields, plays a pivotal role in a wide range of modern optics.
The conventional one-dimensional (1D)  time-frequency duality has been successfully applied to characterize the behavior of classical light, such as ultrafast optical pulses from a laser.

However, the 1D treatment is not enough to characterize quantum mechanical correlations in the time-frequency behavior of multiple photons, such as the biphotons from parametric down conversion.  The two-dimensional treatment is essentially required, but has not been fully demonstrated yet due to the technical problem.

Here, we study the two-dimensional (2D)  time-frequency duality  duality of biphotons, by measuring two-photon distributions in both frequency and time domains.
It was found that generated biphotons satisfy the Fourier limited condition quantum mechanically, but not classically, by analyzing the time-bandwidth products in the 2D Fourier transform.

Our study provides an essential and deeper understanding of light beyond classical wave optics, and opens up new possibilities for optical synthesis in a high-dimensional frequency space in a quantum manner.















Experimental demonstration of time-frequency duality of biphotons
    Ryosuke Shimizu
    December 30, 2023
=================================================================




Introduction
Optical studies that utilize the relationship coupled by Fourier transform are called Fourier optics, and form a wide range of optical science and technology fields, including optical imaging <cit.>, spectroscopy <cit.>, optical measurement <cit.>, and optical signal processing <cit.>.
In particular, it is well-known that each electric field distribution in the time and frequency domains is connected by the one-dimensional Fourier transform (1DFT)

    A^(1)(t) = 1/√(2π)∫ dνÃ^(1)(ν )  e^ - 2π iν t

where A^(1)(t) is the electric field amplitude as a function of time t, and Ã^(1)(ν) is the corresponding distribution with a frequency ν, as shown in Fig. <ref>(a).
This relationship is a fundamental principle of cutting-edge technologies on  an ultrashort pulse laser, as seen in the recent developments in optical synthesis <cit.>.
On the other hand, Fourier optical phenomena can also be explained from the viewpoint of the particle nature of light, i.e., the photon.
Since the photons contained in a laser light pulse have no quantum correlation, the collective behavior of a many-photon system can be handled as a single-photon problem <cit.>.
As a result, the quantum mechanical treatment between the time and frequency domains of a laser light pulse can be explained by 1DFT, producing results in equivalent to an understanding with classical wave optics.
However, in principle, these phenomena should be treated  as a quantum many-body system, because a large number of photons are contained in an optical pulse output from an ultrashort pulse laser.




Recent progress in quantum optical technologies allows us to control not only a number of photons (i.e., photon statistics), but also the frequency quantum correlations in an optical pulse <cit.>.
Such a frequency quantum correlation would affect its temporal distribution directly through the time-frequency duality.
It is not reasonable to treat the behavior of a quantum-mechanically correlated photon  as a single-photon (or 1D) problem.
Therefore, an optical pulse containing the correlated photons requires a higher-order Fourier treatment that incorporates the quantum mechanics.
As the first step toward future photonics at the single-photon level in the time-frequency domain,  we focus here on the time-frequency behavior of biphotons, which requires the treatment of the two-dimensional Fourier transform (2DFT), as shown in Figs. <ref>b, c.

    A^(2)(t_1 ,t_2 ) = 1/2π∫∫  d  ν _1 dν _2 Ã^(2)(ν _1 ,ν _2 ) e^ - 2π i(ν _1  t_1  + ν _2  t_2)

where A^(2)(t_1 ,t_2 ) is the biphoton probability amplitude at time t_1 and t_2, and Ã^(2)(ν _1 ,ν _2 ) is the corresponding distribution with the frequency ν_1 and ν_2.











From the early stage of quantum optical experiments, time-frequency correlation of biphotons generated from spontaneous parametric down-conversion process have been extensively investigated <cit.>.

In these studies, they discussed the relationship between a classical frequency spectrum and time-domain quantum interference patterns <cit.>, the spectral properties <cit.>, or the temporal <cit.> properties of biphotons. These studies provide partial understanding of nonclassical behavior of biphotons.


However, it is necessary to discuss in the two-dimensional time-frequency space for the comprehensive understanding of biphoton behavior, but still has not been fully demonstrated yet due to the technical problem.

Here, we experimentally demonstrate the time-frequency duality of biphotons with the positive frequency correlations, by measuring both the two-photon spectral intensity distribution (TSI) and the two-photon temporal intensity distribution (TTI).
Furthermore, we show the variation of two-photon temporal distributions as a result of the two-photon spectral modulations, keeping quantum optical-Fourier-transform limited conditions.








Results
To generate biphotons  with  positive frequency correlation, we exploit the spontaneous parametric down-conversion process in a periodically poled  potassium titanyl phosphate (PPKTP) crystal pumped by a mode-locked titanium sapphire laser, operating at the center wavelength of 792 nm.
Thanks to the group velocity matching condition <cit.> with the femtosecond laser pulse pumping, we could generate biphotons with positive frequency correlations at the center wavelength of 1584 nm (see Supplementary Information for details).
Since biphotons generated via the type-II phase-matching condition have orthogonal polarizations, the polarization of the constituent photons were aligned along either the crystallographic y- or z- axis.
To manipulate a two-photon spectral distribution, we used  pump pulses with a bandwidth of either 8.1 or 2.8 nm.
In addition, we prepared two PPTKP crystals, one 30-mm and the other 10-mm long, and positioned one or the other in our experimental apparatus (see Methods).






We performed two experiments to characterize the biphoton distributions in the time-frequency domain.
A two-photon spectrometer consisting of two tunable bandpass filters, which had a  Gaussian-shaped filter function with a fixed FWHM of 0.56 nm, and a tunable central wavelength from 1560-1620 nm, followed by a two-photon detector enabled us to conduct the TSI measurements <cit.>.
For the TTI measurements, we utilized a time-resolved upconversion detection system with a spatial multiplexing technique <cit.>.
The TTI was measured by scanning the temporal delay between the constituent photons and the local oscillator pulse coming from the mode-locked laser with a step length of 0.13 ps, and we recorded coincidence events (see Methods).


Figure <ref> shows the experimentally measured TSIs (left) and TTIs (right) with a combination of the two different pump bandwidths (2.8 or 8.1 nm) and two different crystal lengths (10 or 30 mm).
We reproduced TSIs and TTIs deconvoluted from the original data, taking into account the resolutions of our two-photon detection systems (see Methods).











In Fig. <ref>, the horizontal (vertical) axis Δν _y (Δν _z) is the frequency-shift from the center frequency of each distribution in the y- (z-)direction-polarized photons.
The zero-shifted frequencies are 189.4 THz, corresponding to a center wavelength of 1584 nm.
The specific features of the time-frequency duality of the biphotons can be clearly observed in Fig. <ref>; the TSI of the biphotons from the PPKTP crystal has a positively correlated distribution along the diagonal (Δν_y = Δν _z) direction, while the corresponding TTI has a negatively correlated distribution along the anti-diagonal (Δν_y = -Δν _z) direction.





When the pump bandwidth was increased from 2.8 nm in Fig. <ref>a to 8.1 nm in Fig. <ref>b for the fixed crystal length of 30 mm, the TSI became slightly broader along the diagonal direction, while the width was unchanged along the anti-diagonal direction.
In the same manner, when the crystal length L was shortened from 30 mm in Fig. <ref>b to 10 mm in Fig. <ref>c, the TSI became broader along the anti-diagonal direction, while staying the same in the diagonal direction.

We can understand these phenomena from the following facts:
the biphoton spectral amplitude  Ã^(2)(Δν _y, Δν _z) is the product of a phase-matching function Φ (ν _-, L) and a pump spectral function α (ν _+); Ã^(2)(ν _+ ,ν _- ) = Φ (ν _-, L)α (ν _+), where ν_± = 1/√(2)(Δν_y - Δν_z) and L is the given PPKTP length.
Therefore the bandwidth of the TSI in the ν _+ direction is determined by the spectral function of the pump laser, while that in the ν _- direction is determined by the phase-matching function depending on the crystal length L (seeSupplementary Information).
Thus, in our case, the time-frequency duality of the biphoton can be expressed in the following form:

    A^(2)(Δτ_y ,Δτ_z)    =   1/√(2π)∫ dν_+  α (ν _+)  e^ - 2π i ν_+ τ_+
          ×1/√(2π)∫ dν_-  Φ (ν _-, L)  e^ - 2 π i ν_- τ_-
       =   A^(1)_1(τ_+)A^(1)_1(τ_-).

where τ_± = 1/√(2)(Δτ_y ±Δτ_z).
This indicates that the 2DFT can be decomposed into the product of two 1DFTs: A^(1)_1(τ_+) =   1/√(2π)∫ dν_+  α (ν _+)  e^ - 2π i ν_+ τ_+ and A^(1)_2(τ_-) = 1/√(2π)∫ dν_-  Φ (ν _-, L)  e^ - 2 π i ν_- τ_-.
Based on this, we confirmed the time-bandwidth product (TBP) between δτ_+ and δν_+ or δτ_- and δν_-, where δτ_± (δν_±) is the full width at half maximum (FWHM) values along the τ_± (ν_±) directions in the TTIs  (the TSIs).



Using the reproduced data in Fig. <ref>, we estimated δν_± and  δτ_± for all the cases in Fig. <ref> and calculated TBPs.

It is worth evaluating the TBP for the marginal distributions, as shown in Fig. <ref>, because they can be observed by classical spectroscopic measurements.
Here, δν_y (δτ_y) is the FWHM of the marginal distribution of the TSI (TTI) after the deconvolution.
All TBP values are summarized in Table  <ref>.










It is obvious that the TBP values between δτ_± and δν_± are less than 1, meaning that the generated biphoton wave packets satisfy the nearly Fourier transform limited condition.
The values of δν_+ ×δτ_+ are almost determined by the pump pulse spectra α (ν _+), and would have the value of approximately 0.44 when assuming a Gaussian.
On the other hand, δν_- ×δτ_- is associated with the phase-matching function Φ (ν _-, L), and the values would be approximately 0.87 due to the rectangular shape of the PPKTP crystals along the propagation direction of the pump pulse.
However, the value of δν_y ×δτ_y in Table  <ref> is clearly larger than 1.
This implies that the constituent photons of the biphoton wave packet no-longer satisfy the Fourier transform limited conditions (i.e., the coherence of  single-photon wave packets are degraded).
The disappearance of the single-photon coherence is the inherent characteristics in a quantum-mechanical biparticle system and is well-known as the degradation of purity in a qubit system <cit.>.



Discussion
We now consider the difference of time-frequency duality between classical and quantum regimes.
It is known that the Hamiltonian for a classical electromagnetic wave can be written as the sum of Hamiltonians for independent harmonic oscillators.
Thus, in terms of the photon picture, we can interpret classical optical phenomena as the collective behavior of uncorrelated photons, which results in a single photon (1D) treatment.
Under the 1D treatment, we understand the electric field amplitude, or one-photon probability amplitude, between time and frequency domains are conjugate physical quantities.
In contrast, the limitation of the 1D treatment is evident from the results presented in Figs. <ref> and <ref>.
In practice, the TBP of δν_y ×δτ_y implies the single-photon spectrum associated with the marginal distribution in the frequency domain is no longer conjugate with the single-photon temporal shape.
In our case, it is obvious that the marginal distributions in the frequency domain is mostly determined by the distribution along the diagonal direction in the TSIs, whose Fourier conjugate pair is the distribution along the diagonal direction in the time domain.
In other words, the single-photon spectral distribution strongly affects the temporal quantum correlation of the biphoton.
In the same manner, the frequency quantum correlation leads to the single-photon temporal distribution.
Through these considerations, we can understand classical optical theory only does not cause inconsistency for the collective behavior of uncorrelated photon such as laser pulses.
Higher-order treatments incorporates the quantum mechanics, such as a quantum many-body system <cit.>, is essentially required for general understanding the nature of light.






As a many-body quantum system, our study opens up possible new directions for optical science and technologies, taking into account the quantum correlation, which classical optics treats as the collective motion of uncorrelated photons.

Specifically, time-frequency duality with the higher-dimensional treatments has great potential to manage a multiple-photon wave packet in a higher dimensional time-frequency space.
Here, we only manipulated the real part of the two-photon spectral amplitude (i.e., the TSI). By modulating an imaginary two-photon spectral amplitude, i.e., the phase between two-photon spectral modes, in addition to the real part, however, we could easily extent our work to a new quantum optical synthesis technology based on a 2D Fourier optical treatment.
This may lead to a deeper understanding of the essential nature of light and to the development of future photonics at the single-photon level in the time-frequency domain.







Conclusion
In summary, we have demonstrated the correlation inversion between time and frequency domains for frequency-positive-correlated biphotons, by experimentally measuring two-photon distributions in both domains. It was discovered that the generated biphotons satisfy the Fourier limited condition quantum mechanically, but not classically. Our study opens up possible new directions for optical science and technologies a many-body and high-dimensional frequency space.



Acknowledgements
We thank Y. Guo for the helpful discussions. R.J. is supported by a fund from the Educational Department of Hubei Province, China (Grant No. D20161504), and by National Natural Science Foundations of China (Grant No.11704290). R.S. acknowledges support from the Research Foundation for Opto-Science and Technology, Hamamatsu, Japan, and support from Matsuo Foundation, Tokyo, Japan.






38
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Stark(1982)]Stark1982
  author author Henry Stark, @noop  title Applications of optical
  Fourier transforms (publisher Academic Press, year 1982)NoStop
[Goodman(2017)]Goodman2017
  author author Joseph W.  Goodman, @noop  title
  Introduction to Fourier Optics, edition 4th ed. (publisher Freeman, W. H., year 2017)NoStop
[Zhang et al.(2014)Zhang,
  Wei, Marhic, and Wong]Zhang2014
  author author Chi Zhang, author Xiaoming Wei,
  author Michel E. Marhic,  and  author Kenneth K. Y. Wong,  title title Ultrafast and versatile
  spectroscopy by temporal fourier transform, @noop  journal journal Sci. Rep. volume
  4, pages 5351 (year 2014)NoStop
[Smith(2011)]Smith2011
  author author Brian C.  Smith, @noop  title Fundamentals
  of Fourier Transform Infrared Spectroscopy, edition 2nd  ed. (publisher CRC Press, year 2011)NoStop
[Bell(1972)]Robert1972
  author author Robert Bell, @noop  title Introductory Fourier
  Transform Spectroscopy, edition 1st ed. (publisher Academic Press, year 1972)NoStop
[Sirohi(2017)]Sirohi2017
  author author Rajpal Sirohi, @noop  title Optical Methods of
  Measurement: Whole field Techniques, edition 2nd ed.  (publisher CRC Press, year 2017)NoStop
[Griffiths and De Haseth(2007)]Griffiths2007
  author author Peter R.  Griffiths and author James A.  De Haseth, @noop  title Fourier
  Transform Infrared Spectrometry, edition 2nd ed. (publisher Wiley-Interscience, year 2007)NoStop
[VanderLugt(2005)]VanderLugt2005
  author author Anthony VanderLugt, @noop  title Optical Signal
  Processing (publisher Wiley-Interscience, year
  2005)NoStop
[Rhodes(2009)]Rhodes2009
  author author W.T. Rhodes, @noop  title Fourier Optics and
  Optical Signal Processing (publisher Wiley-Blackwell,  year 2009)NoStop
[Chan et al.(2011)Chan,
  Hsieh, Liang, Kung,
  Lee, Lai, Pan, and  Peng]Chan2011
  author author Han-Sung  Chan, author Zhi-Ming  Hsieh, author Wei-Hong  Liang, author A. H. Kung, author Chao-Kuei Lee,
  author Chien-Jen Lai, author Ru-Pin Pan,  and author Lung-Han Peng, title title Synthesis and measurement of ultrafast
  waveforms from five discrete optical harmonics, @noop  journal journal Science volume
  331, pages 1165 (year 2011)NoStop
[Suhaimi et al.(2015)Suhaimi, Ohae, Gavara, Nakagawa, Hong, and Katsuragawa]Suhaimi2015
  author author Nurul Sheeda  Suhaimi, author Chiaki  Ohae, author Trivikramarao  Gavara, author Kenichi  Nakagawa, author Feng-Lei  Hong,  and author Masayuki  Katsuragawa, title title
  Generation of five phase-locked harmonics by implementing a divide-by-three
  optical frequency divider, @noop  journal journal Opt. Lett. volume 40, pages
  5802–5805 (year 2015)NoStop
[Kocsis et al.(2011)Kocsis,
  Braverman, Ravets, Stevens,
  Mirin, Shalm, and Steinberg]Kocsis2011
  author author Sacha Kocsis, author Boris Braverman, author Sylvain Ravets, author Martin J. Stevens, author Richard P. Mirin, author L. Krister Shalm,  and author Aephraim M. Steinberg, title title Observing the
  average trajectories of single photons in a two-slit interferometer,  http://science.sciencemag.org/content/332/6034/1170.abstract
  journal journal Science volume 332, pages 1170 (year
  2011)NoStop
[Aspden et al.(2016)Aspden,
  Padgett, and Spalding]Aspden2016
  author author Reuben S.  Aspden, author Miles J.  Padgett,  and author Gabriel C. Spalding, title title Video recording true single-photon double-slit
  interference, booktitle booktitle
  American Journal of Physics, https://doi.org/10.1119/1.4955173
  journal journal Am. J. Phys. volume 84, pages 671–677 (year
  2016)NoStop
[Eckstein et al.(2011)Eckstein, Christ, Mosley, and  Silberhorn]Eckstein2011
  author author Andreas Eckstein, author Andreas Christ, author Peter J. Mosley,  and author Christine Silberhorn, title title Highly
  efficient single-pass source of pulsed single-mode twin beams of light,  @noop  journal journal Phys. Rev. Lett.  volume 106, pages 013603 (year 2011)NoStop
[Jin et al.(2016)Jin,
  Shimizu, Fujiwara, Takeoka,
  Wakabayashi, Yamashita, Miki,
  Terai, Gerrits, and Sasaki]Jin2016QST
  author author Rui-Bo Jin, author Ryosuke Shimizu,
  author Mikio Fujiwara, author Masahiro Takeoka, author Ryota Wakabayashi, author Taro Yamashita, author
  Shigehito Miki, author
  Hirotaka Terai, author
  Thomas Gerrits,  and author
  Masahide Sasaki, title
  title Simple method of generating and distributing
  frequency-entangled qudits, @noop  journal journal Quantum Sci. Technol. volume 1,  pages 015004 (year 2016)NoStop
[Chen et al.(2017)Chen,
  Bo, Niu, Xu, Zhang, Shapiro, and Wong]Chen2017
  author author Changchen  Chen, author Cao Bo, author Murphy Yuezhen Niu, author Feihu Xu,
  author Zheshen Zhang, author Jeffrey H. Shapiro,  and  author Franco N. C. Wong,  title title Efficient generation and
  characterization of spectrally factorable biphotons, @noop 
  journal journal Opt. Express volume 25, pages 7300–7312 (year
  2017)NoStop
[Jin et al.(2017)Jin,
  Chen, Jing, Ren,
  Zhao, Shimizu, and Lu]Jin2017PRA
  author author Rui-Bo Jin, author Guo-Qun Chen,
  author Hui Jing, author Changliang Ren, author
  Pei Zhao, author Ryosuke Shimizu,  and author Pei-Xiang Lu, title title Monotonic quantum-to-classical transition enabled by positively
  correlated biphotons, @noop  journal journal Phys. Rev. A volume 95, pages 062341 (year 2017)NoStop
[Burnham and Weinberg(1970)]Burnham1970
  author author David C.  Burnham and author Donald L.  Weinberg, title title
  Observation of simultaneity in parametric production of optical photon
  pairs, 10.1103/PhysRevLett.25.84 journal
  journal Phys. Rev. Lett. volume 25,  pages 84–87 (year 1970)NoStop
[Hong et al.(1987)Hong,
  Ou, and Mandel]Hong1987
  author author C. K. Hong, author Z. Y. Ou,  and  author L. Mandel, title title Measurement of subpicosecond time
  intervals between two photons by interference, @noop  journal journal Phys. Rev. Lett. volume 59, pages 2044–2046 (year
  1987)NoStop
[Larchuk et al.(1993)Larchuk, Campos, Rarity, Tapster, Jakeman, Saleh, and  Teich]Larchuk1993
  author author T. S. Larchuk, author R. A. Campos,
  author J. G. Rarity, author P. R. Tapster, author
  E. Jakeman, author B. E. A. Saleh,  and author M. C. Teich, title title Interfering entangled photons of different colors, @noop 
  journal journal Phys. Rev. Lett. volume 70, pages 1603–1606 (year
  1993)NoStop
[Shih et al.(1994)Shih,
  Sergienko, Rubin, Kiess,  and Alley]Shih1994
  author author Y. H. Shih, author A. V. Sergienko,
  author M. H. Rubin, author T. E. Kiess,  and author C. O. Alley, title
  title Two-photon interference in a standard
  mach-zehnder interferometer, @noop  journal
  journal Phys. Rev. A volume 49,  pages 4243–4246 (year 1994)NoStop
[Giovannetti et al.(2002)Giovannetti, Maccone, Shapiro, and  Wong]Giovannetti2002
  author author Vittorio  Giovannetti, author Lorenzo  Maccone, author Jeffrey H.  Shapiro,  and author Franco
  N. C. Wong, title title Generating entangled two-photon states with coincident
  frequencies, @noop  journal journal
  Phys. Rev. Lett. volume 88, pages
  183602 (year 2002)NoStop
[Jin and Shimizu(2018)]Jin2018Optica
  author author Rui-Bo Jin and author Ryosuke Shimizu, title title Extended
  Wiener-Khinchin theorem for quantum spectral analysis, @noop 
  journal journal Optica volume 5, pages 93–98 (year
  2018)NoStop
[Kim and Grice(2005)]Kim2005
  author author Yoon-Ho Kim and author Warren P. Grice, title title easurement of
  the spectral properties of the two-photon state generated via type II
  spontaneous parametric downconversion, booktitle booktitle Optics Letters, http://ol.osa.org/abstract.cfm?URI=ol-30-8-908 journal
  journal Opt. Lett. volume 30,  pages 908–910 (year 2005)NoStop
[Shimizu and Edamatsu(2009)]Shimizu2009
  author author Ryosuke Shimizu and author Keiichi Edamatsu, title title High-flux and
  broadband biphoton sources with controlled frequency entanglement,  @noop  journal journal Opt. Express  volume 17, pages 16385–16393 (year 2009)NoStop
[Avenhaus et al.(2009)Avenhaus, Eckstein, Mosley, and  Silberhorn]Avenhaus2009
  author author Malte Avenhaus, author Andreas Eckstein, author Peter J. Mosley,  and author Christine Silberhorn, title title
  Fiber-assisted single-photon spectrograph, @noop  journal journal Opt. Lett. volume
  34, pages 2873–2875 (year 2009)NoStop
[Fang et al.(2014)Fang,
  Cohen, Liscidini, Sipe, and  Lorenz]Fang2014
  author author Bin Fang, author Offir Cohen,
  author Marco Liscidini, author John E. Sipe,  and author Virginia O. Lorenz, title title Fast and highly resolved capture of the
  joint spectral density of photon pairs, @noop  journal journal Optica volume 1,  pages 281 (year 2014)NoStop
[Allgaier et al.(2017)Allgaier, Vigh, Ansari, Eigner, Quiring, Ricken, Brecht, and Silberhorn]Allgaier2017
  author author Markus Allgaier, author Gesche Vigh,
  author Vahid Ansari, author Christof Eigner, author
  Viktor Quiring, author
  Raimund Ricken, author
  Benjamin Brecht,  and author
  Christine Silberhorn, title
  title Fast time-domain measurements on telecom single
  photons, @noop  journal journal
  arXiv:1702.03240  (year 2017)NoStop
[Kuzucu et al.(2008a)Kuzucu, Wong,
  Kurimura, and Tovstonog]Kuzucu2008PRL
  author author Onur Kuzucu, author Franco N. C. Wong, author Sunao Kurimura,
   and author Sergey Tovstonog,  title title Joint temporal density
  measurements for two-photon state characterization, @noop 
  journal journal Phys. Rev. Lett. volume 101, pages 153602 (year
  2008a)NoStop
[Cho et al.(2014)Cho,
  Park, Lee, and Kim]Cho2014
  author author Young-Wook  Cho, author Kwang-Kyoon  Park, author Jong-Chan  Lee,  and author Yoon-Ho  Kim, title title
  Engineering frequency-time quantum correlation of narrow-band biphotons from
  cold atoms, @noop  journal journal
  Phys. Rev. Lett. volume 113, pages
  063602 (year 2014)NoStop
[König and Wong(2004)]Konig2004
  author author Friedrich  König and author Franco
  N. C. Wong, title title Extended phase matching of second-harmonic generation in
  periodically poled KTiOPO_4 with zero group-velocity mismatch,  @noop  journal journal Appl. Phys.
  Lett. volume 84, pages 1644 (year 2004)NoStop
[Jin et al.(2013)Jin,
  Shimizu, Wakui, Benichi,  and Sasaki]Jin2013OE
  author author Rui-Bo Jin, author Ryosuke Shimizu,
  author Kentaro Wakui, author Hugo Benichi,  and author Masahide Sasaki, title title Widely tunable single photon source with
  high purity at telecom wavelength, @noop  journal
  journal Opt. Express volume 21,  pages 10659–10666 (year 2013)NoStop
[Kuzucu et al.(2008b)Kuzucu, Wong,
  Kurimura, and Tovstonog]Kuzucu2008OL
  author author Onur Kuzucu, author Franco N. C. Wong, author Sunao Kurimura,
   and author Sergey Tovstonog,  title title Time-resolved single-photon
  detection by femtosecond upconversion, @noop  journal journal Opt. Lett. volume
  33, pages 2257–2259 (year
  2008b)NoStop
[Nielsen and Chuang(2000)]Nielsen2000
  author author Michael A.  Nielsen and author Isaac L.  Chuang, @noop  title Quantum
  Computation and Quantum Information (publisher Cambridge
  University Press, year 2000)NoStop
[Schweigler et al.(2017)Schweigler, Kasper, Erne, Mazets, Rauer, Cataldini, Langen, Gasenzer, Berges, and  Schmiedmayer]Schweigler2017
  author author Thomas Schweigler, author Valentin Kasper, author Sebastian Erne, author Igor Mazets,
  author Bernhard Rauer, author Federica Cataldini, author Tim Langen, author
  Thomas Gasenzer, author
  Jürgen Berges,  and author Jörg Schmiedmayer,  title title Experimental
  characterization of a quantum many-body system via higher-order
  correlations, @noop  journal journal
  Nature volume 545, pages 323–326
  (year 2017)NoStop
[Bisht and Shimizu(2015)]Bisht2015
  author author Nandan S.  Bisht and author Ryosuke  Shimizu, title title
  Spectral properties of broadband biphotons generated from PPMgSLT under a
  type-II phase-matching condition, 10.1364/JOSAB.32.000550 journal journal J. Opt.
  Soc. Am. B volume 32, pages 550–554
  (year 2015)NoStop
[Evans et al.(2010)Evans,
  Bennink, Grice, Humble, and  Schaake]Evans2010
  author author P. G. Evans, author R. S. Bennink,
  author W. P. Grice, author T. S. Humble,  and author J. Schaake, title
  title Bright source of spectrally uncorrelated
  polarization-entangled photons with nearly single-mode emission,  @noop  journal journal Phys. Rev. Lett.  volume 105, pages 253601 (year 2010)NoStop
[Gerrits et al.(2011)Gerrits, Stevens, Baek, Calkins, Lita, Glancy, Knill, Nam, Mirin, Hadfield, Bennink, Grice, Dorenbos, Zijlstra, Klapwijk, and  Zwiller]Gerrits2011
  author author Thomas Gerrits, author Martin J. Stevens, author Burm Baek,
  author Brice Calkins, author Adriana Lita, author
  Scott Glancy, author
  Emanuel Knill, author
  Sae Woo Nam, author
  Richard P. Mirin, author
  Robert H. Hadfield, author
  Ryan S. Bennink, author
  Warren P. Grice, author
  Sander Dorenbos, author
  Tony Zijlstra, author
  Teun Klapwijk,  and author
  Val Zwiller, title title Generation of degenerate, factorizable, pulsed squeezed
  light at telecom wavelengths, @noop  journal
  journal Opt. Express volume 19,  pages 24434–24447 (year 2011)NoStop





















Methods

Schematic illustration of the whole  experimental setup
The experiment setup is shown in Fig. <ref>. We used femtosecond laser pulses with a repetition rate of 76 MHz from a mode-locked titanium sapphire laser, operating at the center wavelength of 792 nm with a bandwidth of 8.1 nm. These were divided into two paths by a polarization beam splitter (PBS) in order to generate and detect biphotons.
For the biphoton generation, pulses were sent to a periodically poled  potassium titanyl phosphate (PPKTP) crystal with a poling period of 46.1 μm for type-II group-velocity-matched spontaneous parametric down-conversion. 
To manipulate the two-photon spectral distribution, we also prepared the pump pulse with a bandwidth of 2.8 nm (by inserting a bandpass filter), and crystals of 30 and 10 mm in length.
Thanks to the group velocity matching condition with the femtosecond laser pulse pumping, we could generate biphotons with positive frequency correlations at the center wavelength of 1584 nm.
Since biphotons generated via the type-II phase-matching condition have orthogonal polarizations, the polarization of the constituent photons were aligned along either the crystallographic y- or z-axis.

The constituent photons were separated by a PBS and coupled into  two polarization-maintaining fibers (PMF).
The other path of the pump laser pulse was spatially filtered, and then used as a local oscillator (LO) for TTI measurement.

For TSI measurement <cit.>
, each constituent photon of the biphoton was sent to tunable bandpass filters (BPFs), which had a  Gaussian-shaped filter function  with a fixed FWHM of 0.56 nm and a tunable central wavelength from 1560-1620 nm, followed by a two-photon detector consisting of two fiber-coupled single-photon detectors and a time interval analyzer (TIA).

The TSI was measured by scanning the central wavelength of the two BPFs with a step length of 0.5 nm,  and recording the coincidence counts for each point.


For TTI measurements, we utilized a time-resolved upconversion detection system with spatial multiplexing <cit.>.
A periodically poled MgO-doped stoichiometric lithium tantalate (PPMgSLT) crystal (poling period of 8.5 μm; length 1 mm) under the type-0 phase-matching condition  was used for  noncollinear sum-frequency generation (1584 nm + 792 nm → 528 nm).
After filtering the backgrounds, the up-converted photons were detected by two Si avalanche photodiodes (Si-APD),  which were connected to a TIA for  coincidence counting.
The TTI was measured by scanning the temporal delay between the constituent photons and the LO pulse with a step length of 0.13 ps, and we recorded coincidence events.







The temporal width of the LO pulse was estimated to be approximately 0.12 ps from the bandwidth of the pump pulse. The group-velocity difference between the LO pulse and the photons passing through the PPMgSLT was calculated to be 0.22 ps. We therefore estimated the temporal resolution for a one-photon detection with Si-APD to be 0.12 + 0.22 = 0.34 ps, and for a two-photon detection, 0.34 ×√(2)≈ 0.48 ps.





Pump pulse spectra for 2.8-nm and 8.1-nm bandwidths
In our experiment, the femtosecond laser pulse had a Gaussian distribution with a center wavelength of 792 nm and bandwidth of 8.1 nm, as shown in Fig. <ref> (red curve). This spectrum was used for the TSIs and TTIs in Figs. <ref>b and c in the main text. After the insertion of a bandpass filter, the bandwidth became 2.8 nm, while the center wavelength was unchanged, as shown in Fig. <ref> (blue curve). This spectrum was used for the results in Fig. <ref>a in the main text.









Coordinates and parameters
We summarize the coordinates and parameters used in this paper, as shown in Figs. <ref>(a-f).
Δν_y,  Δν_z, ν_+ and ν_- are the horizontal, vertical, diagonal and anti-diagonal axes respectively in the frequency domain; Δτ_y,  Δτ_z, τ_+ and τ_- are the horizontal, vertical, diagonal and anti-diagonal axes,  respectively, in the time domain; δν_yc is the FWHM for the distribution long cross section of Δν_z = 0; meanwhile, δν_y, δν_+ and δν_-  are the FWHM for the marginal distribution by projecting the data onto the axes of Δν_y, ν_+ and ν_- respectively.
Finally, δτ_yc is the FWHM for the distribution along the cross section of Δτ_z = 0; and δτ_y, δτ_+ and δτ_-  are the FWHM for the marginal distribution by projecting the data onto the axes of Δτ_y, τ_+ and τ_- respectively.







Reproduction of the TSIs and TTIs
We take Fig. <ref>b in the main text as an example to show the details of the experimental data processing method.
The image on the left in Fig. <ref>b is the density plot of the TSI, which was scanned along the Δν_y direction at each Δν_z value.
Figure <ref>a is the waterfall plot of Fig. <ref>b.
We fitted the distributions along the Δν_z direction using Gaussian functions, and obtained an averaged FWHM value of δν_yc = 0.21 THz.
Next, we projected the TSI data onto the axis of Δν_y, and obtained the marginal distribution, shown in Fig. <ref>b.
The FWHM value of δν_y = 1.9 THz was estimated with a Gaussian function.





Similar results can also be obtained for the temporal data.
The density plot of the TTI in Fig. <ref>b was scanned by 76×76 steps.
Figure <ref>b is the waterfall plot of the TTI.
We fitted the distributions along the Δτ_z direction using Gaussian functions, and obtained a FWHM of δτ_yc = 0.26 ps.
From the projection of the TTI onto the  Δτ_y axis, we obtained the marginal distribution, as shown in Fig. <ref>b.
With a rectangular-like profile, we estimated a FWHM value of δτ_y = 4.3 ps.

The other four data measurements in Figs. <ref>a and c were processed similarly.
Using all the FWHM values, we reproduced the TSIs and TTIs in Fig. <ref>.

We summarize the parameters for TSI and TTI in Table  <ref>.








Measurement of TSI
The TSI was measured  using two center-wavelength-tunable BPFs, which had a Gaussian-shaped filter function  with an FWHM of 0.56 nm and a tunable central wavelength from 1560-1620 nm <cit.>.
The two single photon detectors used in this measurement were two InGaAs avalanche photodiode (APD) detectors (ID210, idQuantique), which had  a quantum efficiency of around 20%, with a dark count of around 2 kHz.
To measure the TSI of the photon pairs, we scanned the central wavelength of the two BPFs, and recorded the coincidence counts. The two BPFs were moved  0.1-nm per step and 60×60 steps in all. The coincidence counts were accumulated for 5 s for each point.














 §.§ Supplementary Information


S1:  Theoretical model of TSI and TTI
For a  deepper understanding of this experiment, we constructed a simple but effective mathematical model for the biphotons from the PPKTP crystal.
The two-photon spectral amplitude (TSA), f(ν_s, ν_i),  is the product of the pump-envelope function and phase-matching function.
For simplicity in the calculation, we define ν_s ≡Δν_y and ν_i ≡Δν_y as  the shifted-frequencies of the signal and idler photons in the main text.
Without loss of generality, we assume the pump-envelope function as α(ν_s + ν_i)= exp [ - a(ν _s  + ν _i )^2 ], where a is determined by the spectral width of the pump laser. α(ν_s + ν_i) has a distribution 135-degrees to the horizontal axis (i.e., it is distributed along the anti-diagonal direction, as shown in Fig. <ref>a).


The phase-matching function is Φ(ω_s, ω_i)= sinc[b(ν _s  - ν _i )], where b is determined by the length of the PPKTP crystal.
The PPKTP crystal satisfies the GVM condition at telecom wavelength <cit.>
, therefore, its phase-matching function has a distribution 45-degrees to the horizontal axis (i.e., distributed along the diagonal direction, as shown in Fig. <ref>b).


The TSA can be written as f(ν_s, ν_i)= exp [ - a(ν _s  + ν _i )^2 ]  sinc[b(ν _s  - ν _i )], which is the product of the pump-envelope function and phase-matching function. In this experiment, the  pump envelope is wider than the width of the phase-matching function; therefore,  their product also distributes along the diagonal direction. That is, the signal and idler are positively correlated, as shown in Fig. <ref>c.


The two-photon spectral intensity is |f(ν_s, ν_i)|^2, as shown in Fig. <ref>d.







After the long calculation  described in S2, the Fourier transform of f(ν_s, ν_i) can be analytically obtained as
F(t_s, t_i)= c_0 exp[ - (t_s  + t_i )^2/16a]  rect(t_s  - t_i/2b), where c_0=2π ^3 √(π)/b√(a).
Note that we define t_s ≡Δτ_y and t_i ≡Δτ_z as the shifted-time for the signal and idler photons in the main text.
 
Here, exp[-(t_s  + t_i )^2/16a] is the pump-envelope function in the time domain; rect(t_s  - t_i/2b) is the phase-matching function in the time domain; F(t_s, t_i) is the two-photon temporal amplitude (TSA) and   |F(t_s, t_i)|^2 is just the two-photon temporal intensity (TTI). The distribution of these functions is shown in Figs. <ref>e - h.


It is interesting to note that, ν _s and ν _i can not be written in a separated form in f(ν_s, ν_i). However, if we introduce new variables ν _±= 1/√(2)(ν_s ±ν_i), ν _+ and ν _- can be separated in f(ν_+, ν_-). The situation is similar for τ _s and τ _i, as explained in Eq. (3) in the main text. Another interesting feature is that the pump-envelope  and  phase-matching functions are independent from each other in both the spectral and temporal domains.

The spectral and temporal phase-matching functions are varied simultaneously by changing the parameter a, while phase-matching functions are not affected. Similar phenomena can be observed for the spectral and temporal pump-envelope functions by changing the parameter b:  varying  the sinc function in the spectral domain corresponds to varying the rect function in the time domain, while the Gaussian-shaped spectral and temporal  pump-envelope functions are not affected.




This model is used for the simulations of Fig. <ref> in the main text. For  Fig. <ref>a, we changed the pump-envelope function from a Gaussian function to  a convolution between a Gaussian function and rectangular function, because the pump laser for Fig. <ref>a was filtered by a near-rectangular shaped BPF.


S2: Theoretical calculation of TTA from TSA

For a pump-envelope function exp[ - a(ν _s  + ν _i )^2 ] and a phase-matching  function sinc[b(ν _s  - ν _i )],

the two-photon spectral amplitude (TSA) can be written as

    f(ν _s, ν _i)=exp[ - a(ν _s  + ν _i )^2 ]  sinc[b(ν _s  - ν _i )]

The two-photon temporal amplitude (TTA) is the Fourier transform (ℱ) of f(ν _s, ν _i), and can be calculated as follows.


    [                                                                                                                          ℱ{ f(ν _s ,ν _i )}                                                                                                                                           =                                                                                     ℱ{exp [ - a(ν _s  + ν _2 )^2 ] sinc [b(ν _s  - ν _2 )]};                                                                                                                                            ;                                                                                                                                                                                                                                                                                       =                                                                                 ℱ{exp [ - a(ν _s  + ν _i )^2 ]}⊗ℱ{ sinc [b(ν _s  - ν _i )]};                                                                                                                                            ;                                                                                                                                                                                                                                                                                       =                                                      [2π√(π/a)exp ( - t_s^2 /4a)δ (t_s  - t_i )] ⊗ [2π ^2 1/b rect (t_s /b)δ (t_s  + t_i )];                                                                                                                                            ;                                                                                                                                                                                                                                                                                       = ∫_ - ∞^∞∫_ - ∞^∞dτ _s dτ _i [ 2π√(π/a)exp ( - τ _s^2 /4a)δ (τ _s  - τ _i )][2π ^2 1/b rect (t_s  - τ _s /b)δ (t_s  + t_i  - τ _s  - τ _i )];                                                                                                                                            ;                                                                                                                                                                                                                                                                                       =                                ∫_ - ∞^∞dτ _s [ 2π√(π/a)exp ( - τ _s^2 /4a)][2π ^2 1/b rect (t_s  - τ _s /b)δ (t_s  + t_i  - τ _s  - τ _s )];                                                                                                                                            ;                                                                                                                                                                                                                                                                                       =                                   ∫_ - ∞^∞dτ _s [ 2π√(π/a)exp( - τ _s^2 /4a) ][2π ^2 1/b rect (t_s  - τ _s /b)1/2δ (τ _s  - t_s  + t_i /2)];                                                                                                                                            ;                                                                                                                                                                                                                                                                                       =                                                                          2π ^3 /b√(π/a)exp [ - (t_s  + t_i )^2 /16a]  rect (t_s  - t_i /2b);                                                                                                                                            ;                                                                                                                                                                                                                                                                                       =                                                                                   c_0 exp [ - (t_s  + t_i )^2 /16a]  rect (t_s  - t_i /2b),;                                                                                                                                            ;                                                                                                                                             ]

where ⊗ is the convolution symbol and c_0 =2π ^3 √(π)/b√(a). In conclusion, the TTA (F(t_s, t_i)) can be calculated from the Fourier transform of TSA (f(ν _s, ν _i)) with the following form:


    F(t_s, t_i)=ℱ{f(ν _s, ν _i)}=c_0 exp[ - (t_s  + t_i )^2 /16a] rect (t_s  - t_i/2b).









