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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 20401–20408
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Second order coherence of broadband down-converted light on ultrashort time scale determined by two photon absorption in semiconductor

Fabien Boitier, Antoine Godard, Aleksandr Ryasnyanskiy, Nicolas Dubreuil, Philippe Delaye, Claude Fabre, and Emmanuel Rosencher  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 20401-20408 (2010)
http://dx.doi.org/10.1364/OE.18.020401


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Abstract

We study the photon correlation properties of broadband parametric down-converted light. The measurement of the photon correlation is carried out thanks to a modified Hanbury Brown-Twiss interferometer based on two photon absorption in GaAs detector. Since this method is not affected by the phase matching conditions of the detecting apparatus (so called “final state post-selection”), the detection bandwidth can be extremely large. This is illustrated by studying, with the same apparatus, the degree of second order coherence of parametric light in both degenerate and non-degenerate cases. We show that our experiment is able to determine the coherent as well as the incoherent contributions to the degree of second order coherence of parametric light with a time resolution in the fs range scale.

© 2010 OSA

Photon correlation properties are now harnessed in numerous experiments and applications, at the classical level (photon correlation spectroscopy [1

1. H. Z. Cummins, and E. R. Pike, Photon correlation spectroscopy and light beating spectroscopy (Plenum Press, New York, 1974).

], high resolution profiling [2

2. Y. Tanaka, N. Sako, T. Kurokawa, H. Tsuda, and M. Takeda, “Profilometry based on two-photon absorption in a silicon avalanche photodiode,” Opt. Lett. 28(6), 402–404 (2003). [CrossRef] [PubMed]

]) as well as at the quantum level (quantum cryptography, quantum teleportation,…) [3

3. D. Bouwmeester, A. Ekert, and A. E. Zeilinger, The Physics of Quantum Information (Springer-Verlag, New York, 2000).

]. The determination of the correlation properties of photon fields is thus of paramount importance, especially these of two-photon pair states generated by parametric fluorescence. Several schemes have been proposed to analyze the quantum correlation of this kind of light. In Hanbury Brown–Twiss (HBT) interferometry [4

4. R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956). [CrossRef]

], the incident beam is split into two sub-beams by a beam splitter, sent on two separate photon counters the outputs of which are submitted to a correlation analysis. This technique has been used to characterize parametric fluorescence light sources for more than twenty years [5

5. S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985). [CrossRef] [PubMed]

] and has a time resolution limited by the detector response time, i.e. in the nanosecond range [6

6. M. Beck, “Comparing measurements of g(2)(0) performed with different coincidence detection techniques,” J. Opt. Soc. Am. B 24(12), 2972–2978 (2007). [CrossRef]

,7

7. D. B. Scarl, “Measurements of photon correlations in partially coherent light,” Phys. Rev. 175(5), 1661–1668 (1968). [CrossRef]

]. To circumvent this bandwidth issue, Abram et al. measured these time correlations by sum frequency generation of the two delayed sub-beams in a nonlinear crystal [8

8. I. Abram I, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986). [CrossRef] [PubMed]

]. More recently, following a pioneering experiment of Dayan et al. [9

9. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005). [CrossRef] [PubMed]

], several teams increased the performances of this latter experiment by compensating the dispersion and taking advantage of high non-linearity of current nonlinear crystals and detector improvement [10

10. F. Zäh, M. Halder, and T. Feurer, “Amplitude and phase modulation of time-energy entangled two-photon states,” Opt. Express 16(21), 16452–16458 (2008). [CrossRef] [PubMed]

,11

11. K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103(12), 123602 (2009). [CrossRef] [PubMed]

]. These experiments allow an excellent temporal resolution – down to the fs range – but are confined to narrow bandwidth final states imposed by phase-matching conditions which “postselect” only the contribution of photon pairs that are complementary in energy [12

12. B. Dayan, “Theory of two-photon interactions with broadband down-converted light and entangles photons,” Phys. Rev. A 76(4), 043813 (2007). [CrossRef]

]. The response time of the detector can also be circumvented by the use of time-gated detection by up-conversion scheme as investigated in Ref [13

13. O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008). [CrossRef] [PubMed]

]. However, the resolution time is then limited by the duration of the sampling pulse while the bandwidth is ultimately limited by the phase-matching acceptance of the up-conversion nonlinear crystal.

Recently, F. Boitier et al. [14

14. F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009). [CrossRef]

] have developed a new technique based on two photon conductivity in semiconductors that enables the characterization of optical sources with output power down to 0.1 µW, bandwidth in the 1.3 to 1.6 µm range and time resolution in the femtosecond range. Experimentally, the system is rather similar to a Hanbury Brown–Twiss (HBT) interferometer but, in our case, the two delayed sub-beams are recombined in a two-photon counting device [15

15. J. M. Roth, T. E. Murphy, and C. Xu, “Ultrasensitive and high-dynamic-range two-photon absorption in a GaAs photomultiplier tube,” Opt. Lett. 27(23), 2076–2078 (2002). [CrossRef]

]. We will refer our technique to Two-Photon Counting (TPC) interferometry. Since, as sketched on Fig. 1a
Fig. 1 (a) Two-photon absorption from valence band states to conduction band states in a direct gap semiconductor (e.g. GaAs). In a phototube, the electrons in the conduction band are emitted when reaching the extraction (or “vacuum”) level. Only photons arriving within time intervals shorter than the “virtual” state lifetime at midgap τH can induce TPA transitions. (b) The HBT apparatus is a Michelson interferometer with two arms: (Asph. L) is a 26 mm aspheric lens, (BS) beam splitter, (HPF) high pass filter, (M1) and (M2) mirrors and (PMT) is the GaAs photomultiplier tube. The source is based on a periodically poled lithium niobate (PPLN) crystal pumped at 780 nm by a mode-locked Ti:Sapphire laser delivering 10-ps pulses at a 80-MHz repetition rate. Estimated focal spot on the detector is 5 µm, far smaller than the detector size.
, two-photon absorption (TPA) in semiconductors occurs for photon energies larger than the semiconductor midgap and smaller than its gap, the detection bandwidth is very large, giving access to fs timescale correlation measurement and ultralarge bandwidth. Indeed, the virtual state lifetime can be estimated by the second Heisenberg uncertainty relation ΔE.Δτ/2, leading to Δτ0.5 fs for ΔE=Eg/2 i.e. 0.7 eV [16

16. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

]. It has been shown that the two photon counting rate measures the degree of second order coherence (DSOC) g(2)(τ) [17

17. R. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963). [CrossRef]

] given by:
g(2)(τ)=E^()(t+τ)E^()(t)E^(+)(t)E^(+)(t+τ)E^()(t)E^(+)(t)2
(1)
where τ is the delay between the two beams, E^(+)(t) and E^()(t) are the complex electric field operator and its hermitian conjugate respectively while stands for quantum expectation.

In the present TPC interferometry experiment, we used a H7421-50 Hamamatsu GaAs phototube as suggested in Ref [15

15. J. M. Roth, T. E. Murphy, and C. Xu, “Ultrasensitive and high-dynamic-range two-photon absorption in a GaAs photomultiplier tube,” Opt. Lett. 27(23), 2076–2078 (2002). [CrossRef]

]. so that optical fields with wavelengths between 900 nm and 1,800 nm can be studied. In such a device, the photocurrent is highly amplified by multiplication of photoelectrons emitted from the space charge region of the semiconductor into vacuum, allowing to detect TPC signal in the Geiger mode with experimentally determined 40 dark counts/s (substracted in the following experiments). Figure 1b shows a schematic diagram of the experiment. It is a standard Michelson interferometric apparatus where special attention is given to the complete filtering out of radiation wavelengths shorter that 900 nm in the incoming light, in order to eliminate any direct absorption in the GaAs detector. The investigated parametric downconverter is based on an undoped type 0 35-mm-long periodically poled lithium niobate (PPLN) crystal pumped at 780 nm by a mode-locked Ti:Sapphire laser delivering 10-ps pulses at a 80-MHz repetition rate. By temperature tuning of the quasi-phase matching condition in the PPLN crystal, the second-order coherence can be studied at the degeneracy point of the parametric downconversion as well as far from degeneracy, in our case at respectively 125°C and 128°C.

Figure 2a
Fig. 2 (a) DSOC spectrum, i.e., variation of the TPA photocounts as a function of the delay τ, of the TPA Michelson set-up of Fig. 1. The left inset shows the number of photoelectron counts as a function of the incident power. The quadratic behaviour clearly indicates a TPA process with an OPG in high gain regime. The right inset shows the spectrum of the degenerate OPG. (b) Zoom on small delay times τ which exhibits the DSOC g(2)(τ) features of the degenerate down-converted light. The red curve is TPALPF(τ) described in Eq. (2). (c) Theoretical modeling using the model described in the text.
shows a TPC interferogram carried out on the OPG light tuned at degeneracy, i.e. centred at 1.56 µm. The spectrum of the OPG emitted light is indicated in the inset of Fig. 2a. The broad peak in Fig. 2a reflects the pulse duration of the laser pump whereas the sharp peak gives access to the OPG correlation properties. This shows that, as far as the correlation properties of OPG photons are concerned, the experiment can be considered as cw. Figure 2b is a zoom of the Fig. 2a on the central peak where interference patterns are clearly observable. The red curve is the result of a low pass filter on the interferogram (TPALPF(τ)): it is shown in Ref [18

18. K. Mogi, K. Naganuma, and H. Yamada, “A novel real-time measurement method for ultrashort optical pulses,” Jpn. J. Appl. Phys. 27(Part 1, No. 11), 2078–2081 (1988). [CrossRef]

]. and [14

14. F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009). [CrossRef]

] that it is equal to:
TPALPF(τ)=1+2g(2)(τ)/g(2)(0)   ;
(2)
it thus measures the DSOC factor g(2)(τ) and points out the photon coincidences. The measured correlation time of about 70 fs is consistent with the 100 nm source bandwidth (see right inset of Fig. 2a) while, using g(2)(τ)1 for τ0.2 ps, one has g(2)(0)2, meaning that, in our experimental configuration, the properties of the DSOC are very close to what is obtained with chaotic light

We now show that the same TPC apparatus can be used to determine the correlation properties of the OPG beam away from degeneracy. The PPLN crystal temperature is changed in order to obtain two different radiations, the spectra of which is displayed in the inset Fig. 3a
Fig. 3 (a) Zoom on the non-degenerate OPG part of the DSOC spectrum. The inset shows the spectrum of the non-degenerate OPG. (b) Zoom on the non-degenerated OPG part of the DSOC spectrum in the case where the idler wavelengths were attenuated by a dichroic mirror. The inset shows the spectrum after attenuation. (c) Theoretical modeling of the mutual DSOC spectrum of Fig. 3a. (d) Theoretical modeling of the mutual DSOC spectrum of Fig. 3b. The red curve is TPALPF (τ) described in Eq. (2).
. The TPC interferogram of the two simultaneous radiations is shown in Fig. 3a while Fig. 3b shows the result with the signal only (the idler was attenuated by a dichroic mirror). Figure 3a shows that the influence of the two wavelengths is clearly observed. Using TPALPF(τ) (see Eq. (2), we can estimate a coherence time of about 140 fs which is again in compliance with the OPG bandwidth of 50 nm. Once again, we measure g (2)(0) ≈2 in both cases.

For a proper interpretation of our experimental results and to get some physical insights, we developed a quantum optics theoretical model of the experiment. We briefly present the main steps of the calculation while a detailed description of the model will be presented elsewhere. To derive the creation and annihilation operators, we use the continuous variables version [19

19. R. Loudon, The Quantum Theory of Light (Oxford Univ. Press., Oxford, 2000).

] of the formalism suggested by Huttner et al. [20

20. B. Huttner, S. Serulnik, and Y. Ben-Aryeh, “Quantum analysis of light propagation in a parametric amplifier,” Phys. Rev. A 42(9), 5594–5600 (1990). [CrossRef] [PubMed]

].

γ(ω)=g(ω)2Δk(ω)2/4.
(8)

Assuming negligible loss in the beam splitter, the annihilation operator at the detector position zd can be straightforwardly connected to the one at the crystal output according to the following expression:
a^(zd,ω)=12[iM(ω)(1+eiωτ)a^(zc,ω)ν^(ω)(1eiωτ)]
(9)
where M(ω)=eiφ(ω)accounts for dispersion experienced by the beam on its path from the crystal output to the TPC detector and ν^(ω)is the vacuum field operator for the angular frequency ω at the output port of the Michelson interferometer. The TPC signal S TPC is then proportional to [19

19. R. Loudon, The Quantum Theory of Light (Oxford Univ. Press., Oxford, 2000).

]:

STPCa^(zd,t)a^(zd,t)a^(zd,t)a^(zd,t)
(10)
with  a^(z,t)=12πdωa^(z,ω)eiωt.
(11)

We would like to particularly discuss the DSOC whose value at zero delay – g (2)(0) ≈2 in all our experimental cases – could seem surprising if one considers the expected extra-bunching for a twin-photon beam when compared to chaotic light. Expanding Eq. (10) and extracting only low frequency terms, it can be shown that the calculated DSOC at the detector position is given by:
g(2)(τ)=gI(2)(τ)+gC(2)(τ),
(12)
where gC(2)(τ) accounts for the so called “coherent” signal due to TPA induced by correlated twin-photon pairs while gI(2)(τ) accounts for the so called “incoherent” signal due to TPA from signal–signal, idler–idler and uncorrelated signal–idler photons (so called “accidental coincidences”).

On the other hand, the coherent term in Eq. (12), can be written

gC(2)(τ)=1Φ2|12π0ωpdωμ(zc,ω)ν(zc,ω)ei[φ(ω)+φ(ωpω)]eiωτ|2
(16)

In our case of high parametric gain, we can assume that

μ(zc,ω)ν(zc,ω)
(17)

Thus, we get:

gC(2)(τ)|12π0ωpdω   [12πdτg(1)(τ)eiωτ]ei[ϕ(ω)+ϕ(ωpω)]eiωτ|2
(18)

We have used this theory to account for our experimental results. As shown in Figs. 2 and 3, one can see that a very good agreement is obtained between experimental and calculated TPC interferograms when one accounts for dispersion of the optical elements. In our case, due to the dispersion of the optical components of our setup whose actual value is very close to ϕ4000fs2, twin-photon wave packets are stretched, leading to a very small coherent term observable in Fig. 3a. Indeed, this corresponds to the conditions when no signal can be measured in the experiment reported by O’Donnell et al [11

11. K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103(12), 123602 (2009). [CrossRef] [PubMed]

] where only the coherent part of the DSOC function can be measured. The observed interference pattern is thus dominated by by the incoherent term (accidental coincidences) whose properties are similar to the case of chaotic light

In summary, two-photon counting interferometry, which harnesses two photon absorption (TPA) in a semiconductor, has been used in order to measure the correlation properties of broadband down-converted light. Intensity fluctuation correlation times can be easily measured in the few femtosecond range. This technique displays an extremely large bandwidth since it is not affected by phase matching conditions. This is illustrated by studying the degree of second order coherence of the down converted light, and away from degeneracy. The TPC interferometer enables us to finely characterize the incoherent part of the down-converted light and to experimentally demonstrate that the behaviour of photon bunching is then rather similar to the one observed in chaotic sources [14

14. F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009). [CrossRef]

]. Work is in progress to enhance the overall quantum efficiency of TPC in order to investigate the low gain regime, as performed in Ref. [9

9. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005). [CrossRef] [PubMed]

].

References and links

1.

H. Z. Cummins, and E. R. Pike, Photon correlation spectroscopy and light beating spectroscopy (Plenum Press, New York, 1974).

2.

Y. Tanaka, N. Sako, T. Kurokawa, H. Tsuda, and M. Takeda, “Profilometry based on two-photon absorption in a silicon avalanche photodiode,” Opt. Lett. 28(6), 402–404 (2003). [CrossRef] [PubMed]

3.

D. Bouwmeester, A. Ekert, and A. E. Zeilinger, The Physics of Quantum Information (Springer-Verlag, New York, 2000).

4.

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956). [CrossRef]

5.

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985). [CrossRef] [PubMed]

6.

M. Beck, “Comparing measurements of g(2)(0) performed with different coincidence detection techniques,” J. Opt. Soc. Am. B 24(12), 2972–2978 (2007). [CrossRef]

7.

D. B. Scarl, “Measurements of photon correlations in partially coherent light,” Phys. Rev. 175(5), 1661–1668 (1968). [CrossRef]

8.

I. Abram I, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986). [CrossRef] [PubMed]

9.

B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005). [CrossRef] [PubMed]

10.

F. Zäh, M. Halder, and T. Feurer, “Amplitude and phase modulation of time-energy entangled two-photon states,” Opt. Express 16(21), 16452–16458 (2008). [CrossRef] [PubMed]

11.

K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103(12), 123602 (2009). [CrossRef] [PubMed]

12.

B. Dayan, “Theory of two-photon interactions with broadband down-converted light and entangles photons,” Phys. Rev. A 76(4), 043813 (2007). [CrossRef]

13.

O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008). [CrossRef] [PubMed]

14.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009). [CrossRef]

15.

J. M. Roth, T. E. Murphy, and C. Xu, “Ultrasensitive and high-dynamic-range two-photon absorption in a GaAs photomultiplier tube,” Opt. Lett. 27(23), 2076–2078 (2002). [CrossRef]

16.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

17.

R. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963). [CrossRef]

18.

K. Mogi, K. Naganuma, and H. Yamada, “A novel real-time measurement method for ultrashort optical pulses,” Jpn. J. Appl. Phys. 27(Part 1, No. 11), 2078–2081 (1988). [CrossRef]

19.

R. Loudon, The Quantum Theory of Light (Oxford Univ. Press., Oxford, 2000).

20.

B. Huttner, S. Serulnik, and Y. Ben-Aryeh, “Quantum analysis of light propagation in a parametric amplifier,” Phys. Rev. A 42(9), 5594–5600 (1990). [CrossRef] [PubMed]

21.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94(7), 073601 (2005). [CrossRef] [PubMed]

OCIS Codes
(270.4180) Quantum optics : Multiphoton processes
(270.5290) Quantum optics : Photon statistics

ToC Category:
Quantum Optics

History
Original Manuscript: April 7, 2010
Revised Manuscript: June 7, 2010
Manuscript Accepted: June 25, 2010
Published: September 10, 2010

Citation
Fabien Boitier, Antoine Godard, Aleksandr Ryasnyanskiy, Nicolas Dubreuil, Philippe Delaye, Claude Fabre, and Emmanuel Rosencher, "Second order coherence of broadband down-converted light on ultrashort time scale determined by two photon absorption in semiconductor," Opt. Express 18, 20401-20408 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20401


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References

  1. H. Z. Cummins, and E. R. Pike, Photon correlation spectroscopy and light beating spectroscopy (Plenum Press, New York, 1974).
  2. Y. Tanaka, N. Sako, T. Kurokawa, H. Tsuda, and M. Takeda, “Profilometry based on two-photon absorption in a silicon avalanche photodiode,” Opt. Lett. 28(6), 402–404 (2003). [CrossRef] [PubMed]
  3. D. Bouwmeester, A. Ekert, and A. E. Zeilinger, The Physics of Quantum Information (Springer-Verlag, New York, 2000).
  4. R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956). [CrossRef]
  5. S. Friberg, C. K. Hong, and L. Mandel, “Measurement of time delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985). [CrossRef] [PubMed]
  6. M. Beck, “Comparing measurements of g(2)(0) performed with different coincidence detection techniques,” J. Opt. Soc. Am. B 24(12), 2972–2978 (2007). [CrossRef]
  7. D. B. Scarl, “Measurements of photon correlations in partially coherent light,” Phys. Rev. 175(5), 1661–1668 (1968). [CrossRef]
  8. I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, “Direct observation of the second-order coherence of parametrically generated light,” Phys. Rev. Lett. 57(20), 2516–2519 (1986). [CrossRef] [PubMed]
  9. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94(4), 043602 (2005). [CrossRef] [PubMed]
  10. F. Zäh, M. Halder, and T. Feurer, “Amplitude and phase modulation of time-energy entangled two-photon states,” Opt. Express 16(21), 16452–16458 (2008). [CrossRef] [PubMed]
  11. K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103(12), 123602 (2009). [CrossRef] [PubMed]
  12. B. Dayan, “Theory of two-photon interactions with broadband down-converted light and entangles photons,” Phys. Rev. A 76(4), 043813 (2007). [CrossRef]
  13. O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. 101(15), 153602 (2008). [CrossRef] [PubMed]
  14. F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009). [CrossRef]
  15. J. M. Roth, T. E. Murphy, and C. Xu, “Ultrasensitive and high-dynamic-range two-photon absorption in a GaAs photomultiplier tube,” Opt. Lett. 27(23), 2076–2078 (2002). [CrossRef]
  16. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).
  17. R. Glauber, “Photon correlations,” Phys. Rev. Lett. 10(3), 84–86 (1963). [CrossRef]
  18. K. Mogi, K. Naganuma, and H. Yamada, “A novel real-time measurement method for ultrashort optical pulses,” Jpn. J. Appl. Phys. 27(Part 1, No. 11), 2078–2081 (1988). [CrossRef]
  19. R. Loudon, The Quantum Theory of Light (Oxford Univ. Press., Oxford, 2000).
  20. B. Huttner, S. Serulnik, and Y. Ben-Aryeh, “Quantum analysis of light propagation in a parametric amplifier,” Phys. Rev. A 42(9), 5594–5600 (1990). [CrossRef] [PubMed]
  21. A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94(7), 073601 (2005). [CrossRef] [PubMed]

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