Abstract

Cold atom based narrowband entangled photon sources are important for efficient atom–photon interaction, which is at the heart of long-distance quantum communication and quantum memory protocols. Complete characterization of the narrowband entangled photons requires acquiring the frequency-time two-photon wavefunction, involving both joint temporal intensity (JTI) and joint temporal phase (JTP) measurements. Here, we demonstrate stimulated emission tomography of the frequency-time two-photon wavefunction of narrowband entangled photons from cold atoms. We show more than six orders of magnitude (×106) improvement in the measurement time for obtaining JTI and JTP compared to the conventional direct photon counting method, thus paving the way toward ultrafast high-resolution quantum tomography of photonic quantum states.

© 2017 Optical Society of America

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    [Crossref]
  2. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
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  3. J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
    [Crossref]
  4. M. Mirhosseini, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, and R. W. Boyd, “Compressive direct measurement of the quantum wave function,” Phys. Rev. Lett. 113, 090402 (2014).
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  5. R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photonics 10, 576–579 (2016).
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  6. W. Wasilewski, P. Wasylczyk, P. Kolenderski, K. Banaszek, and C. Radzewicz, “Joint spectrum of photon pairs measured by coincidence Fourier spectroscopy,” Opt. Lett. 31, 1130–1132 (2006).
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    [Crossref]
  8. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
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  9. V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).
    [Crossref]
  10. S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
    [Crossref]
  11. S. Du, J. Wen, and M. H. Rubin, “Narrowband biphoton generation near atomic resonance,” J. Opt. Soc. Am. B 25, C98–C108 (2008).
    [Crossref]
  12. Y.-W. Cho, K.-K. Park, J.-C. Lee, and Y.-H. Kim, “Generation of nonclassical narrowband photon pairs from a cold rubidium cloud,” J. Korean Phys. Soc. 63, 943–950 (2013).
    [Crossref]
  13. X.-H. Bao, X.-F. Xu, C.-M. Li, Z.-S. Yuan, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation between remote atomic-ensemble quantum memories,” Proc. Natl. Acad. Sci. USA 109, 20347–20351 (2012).
    [Crossref]
  14. D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, B.-S. Shi, and G.-C. Guo, “Raman quantum memory of photonic polarized entanglement,” Nat. Photonics 9, 332–338 (2015).
    [Crossref]
  15. S.-J. Yang, X.-J. Wang, X.-H. Bao, and J.-W. Pan, “An efficient quantum light-matter interface with sub-second lifetime,” Nat. Photonics 10, 381–384 (2016).
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  16. H. Yan, S. Zhang, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Generation of narrow-band hyperentangled nondegenerate paired photons,” Phys. Rev. Lett. 106, 033601 (2011).
    [Crossref]
  17. Y.-W. Cho, K.-K. Park, J.-C. Lee, and Y.-H. Kim, “Engineering frequency-time quantum correlation of narrow-band biphotons from cold atoms,” Phys. Rev. Lett. 113, 063602 (2014).
    [Crossref]
  18. K. Liao, H. Yan, J. He, S. Du, Z.-M. Zhang, and S.-L. Zhu, “Subnatural-linewidth polarization-entangled photon pairs with controllable temporal length,” Phys. Rev. Lett. 112, 243602 (2014).
    [Crossref]
  19. D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
    [Crossref]
  20. J.-C. Lee, K.-K. Park, T.-M. Zhao, and Y.-H. Kim, “Einstein-Podolsky-Rosen entanglement of narrowband photons from cold atoms,” Phys. Rev. Lett. 117, 250501 (2016).
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  21. W. Zhang, D.-S. Ding, M.-X. Dong, S. Shi, K. Wang, S.-L. Liu, Y. Li, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Experimental realization of entanglement in multiple degrees of freedom between two quantum memories,” Nat. Commun. 7, 13514 (2016).
    [Crossref]
  22. F. A. Beduini, J. A. Zielińska, V. G. Lucivero, Y. A. de Icaza Astiz, and M. W. Mitchell, “Interferometric measurement of the biphoton wave function,” Phys. Rev. Lett. 113, 183602 (2014).
    [Crossref]
  23. C. Ren and H. F. Hofmann, “Analysis of the time-energy entanglement of down-converted photon pairs by correlated single-photon interference,” Phys. Rev. A 86, 043823 (2012).
    [Crossref]
  24. 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]
  25. M. Liscidini and J. E. Sipe, “Stimulated emission tomography,” Phys. Rev. Lett. 111, 193602 (2013).
    [Crossref]
  26. B. Fang, O. Cohen, M. Liscidini, J. E. Sipe, and V. O. Lorenz, “Fast and highly resolved capture of the joint spectral density of photon pairs,” Optica 1, 281–284 (2014).
    [Crossref]
  27. A. Eckstein, G. Boucher, A. Lemaître, P. Filloux, I. Favero, G. Leo, J. E. Sipe, M. Liscidini, and S. Ducci, “High-resolution spectral characterization of two photon states via classical measurements,” Laser Photon. Rev. 8, L76–L80 (2014).
    [Crossref]
  28. M. Avenhaus, B. Brecht, K. Laiho, and C. Silberhorn, “Time-frequency quantum process tomography of parametric down-conversion,” arXiv:1406.4252 (2014).
  29. L. A. Rozema, C. Wang, D. H. Mahler, A. Hayat, A. M. Steinberg, J. E. Sipe, and M. Liscidini, “Characterizing an entangled-photon source with classical detectors and measurements,” Optica 2, 430–433 (2015).
    [Crossref]
  30. I. Jizan, B. Bell, L. G. Helt, A. C. Bedoya, C. Xiong, and B. J. Eggleton, “Phase-sensitive tomography of the joint spectral amplitude of photon pair sources,” Opt. Lett. 41, 4803–4806 (2016).
    [Crossref]
  31. See Supplement 1 for the details of the derivation.
  32. M. Fleischhauer and M. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
    [Crossref]
  33. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
    [Crossref]
  34. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000).
    [Crossref]
  35. 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]

2016 (5)

R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photonics 10, 576–579 (2016).
[Crossref]

S.-J. Yang, X.-J. Wang, X.-H. Bao, and J.-W. Pan, “An efficient quantum light-matter interface with sub-second lifetime,” Nat. Photonics 10, 381–384 (2016).
[Crossref]

J.-C. Lee, K.-K. Park, T.-M. Zhao, and Y.-H. Kim, “Einstein-Podolsky-Rosen entanglement of narrowband photons from cold atoms,” Phys. Rev. Lett. 117, 250501 (2016).
[Crossref]

W. Zhang, D.-S. Ding, M.-X. Dong, S. Shi, K. Wang, S.-L. Liu, Y. Li, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Experimental realization of entanglement in multiple degrees of freedom between two quantum memories,” Nat. Commun. 7, 13514 (2016).
[Crossref]

I. Jizan, B. Bell, L. G. Helt, A. C. Bedoya, C. Xiong, and B. J. Eggleton, “Phase-sensitive tomography of the joint spectral amplitude of photon pair sources,” Opt. Lett. 41, 4803–4806 (2016).
[Crossref]

2015 (4)

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref]

L. A. Rozema, C. Wang, D. H. Mahler, A. Hayat, A. M. Steinberg, J. E. Sipe, and M. Liscidini, “Characterizing an entangled-photon source with classical detectors and measurements,” Optica 2, 430–433 (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]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, B.-S. Shi, and G.-C. Guo, “Raman quantum memory of photonic polarized entanglement,” Nat. Photonics 9, 332–338 (2015).
[Crossref]

2014 (6)

M. Mirhosseini, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, and R. W. Boyd, “Compressive direct measurement of the quantum wave function,” Phys. Rev. Lett. 113, 090402 (2014).
[Crossref]

Y.-W. Cho, K.-K. Park, J.-C. Lee, and Y.-H. Kim, “Engineering frequency-time quantum correlation of narrow-band biphotons from cold atoms,” Phys. Rev. Lett. 113, 063602 (2014).
[Crossref]

K. Liao, H. Yan, J. He, S. Du, Z.-M. Zhang, and S.-L. Zhu, “Subnatural-linewidth polarization-entangled photon pairs with controllable temporal length,” Phys. Rev. Lett. 112, 243602 (2014).
[Crossref]

F. A. Beduini, J. A. Zielińska, V. G. Lucivero, Y. A. de Icaza Astiz, and M. W. Mitchell, “Interferometric measurement of the biphoton wave function,” Phys. Rev. Lett. 113, 183602 (2014).
[Crossref]

B. Fang, O. Cohen, M. Liscidini, J. E. Sipe, and V. O. Lorenz, “Fast and highly resolved capture of the joint spectral density of photon pairs,” Optica 1, 281–284 (2014).
[Crossref]

A. Eckstein, G. Boucher, A. Lemaître, P. Filloux, I. Favero, G. Leo, J. E. Sipe, M. Liscidini, and S. Ducci, “High-resolution spectral characterization of two photon states via classical measurements,” Laser Photon. Rev. 8, L76–L80 (2014).
[Crossref]

2013 (2)

M. Liscidini and J. E. Sipe, “Stimulated emission tomography,” Phys. Rev. Lett. 111, 193602 (2013).
[Crossref]

Y.-W. Cho, K.-K. Park, J.-C. Lee, and Y.-H. Kim, “Generation of nonclassical narrowband photon pairs from a cold rubidium cloud,” J. Korean Phys. Soc. 63, 943–950 (2013).
[Crossref]

2012 (2)

X.-H. Bao, X.-F. Xu, C.-M. Li, Z.-S. Yuan, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation between remote atomic-ensemble quantum memories,” Proc. Natl. Acad. Sci. USA 109, 20347–20351 (2012).
[Crossref]

C. Ren and H. F. Hofmann, “Analysis of the time-energy entanglement of down-converted photon pairs by correlated single-photon interference,” Phys. Rev. A 86, 043823 (2012).
[Crossref]

2011 (2)

H. Yan, S. Zhang, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Generation of narrow-band hyperentangled nondegenerate paired photons,” Phys. Rev. Lett. 106, 033601 (2011).
[Crossref]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref]

2008 (3)

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref]

S. Du, J. Wen, and M. H. Rubin, “Narrowband biphoton generation near atomic resonance,” J. Opt. Soc. Am. B 25, C98–C108 (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]

2007 (1)

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99, 123601 (2007).
[Crossref]

2006 (1)

2005 (2)

V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

2003 (1)

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[Crossref]

2001 (1)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

2000 (2)

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000).
[Crossref]

M. Fleischhauer and M. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref]

1999 (1)

A. G. White, D. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally entangled states: production, characterization, and utilization,” Phys. Rev. Lett. 83, 3103–3107 (1999).
[Crossref]

Avenhaus, M.

M. Avenhaus, B. Brecht, K. Laiho, and C. Silberhorn, “Time-frequency quantum process tomography of parametric down-conversion,” arXiv:1406.4252 (2014).

Balic, V.

V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).
[Crossref]

Bamber, C.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref]

Banaszek, K.

Bao, X.-H.

S.-J. Yang, X.-J. Wang, X.-H. Bao, and J.-W. Pan, “An efficient quantum light-matter interface with sub-second lifetime,” Nat. Photonics 10, 381–384 (2016).
[Crossref]

X.-H. Bao, X.-F. Xu, C.-M. Li, Z.-S. Yuan, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation between remote atomic-ensemble quantum memories,” Proc. Natl. Acad. Sci. USA 109, 20347–20351 (2012).
[Crossref]

Bedoya, A. C.

Beduini, F. A.

F. A. Beduini, J. A. Zielińska, V. G. Lucivero, Y. A. de Icaza Astiz, and M. W. Mitchell, “Interferometric measurement of the biphoton wave function,” Phys. Rev. Lett. 113, 183602 (2014).
[Crossref]

Bell, B.

Belthangady, C.

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref]

Boca, A.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[Crossref]

Boozer, A. D.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[Crossref]

Boucher, G.

A. Eckstein, G. Boucher, A. Lemaître, P. Filloux, I. Favero, G. Leo, J. E. Sipe, M. Liscidini, and S. Ducci, “High-resolution spectral characterization of two photon states via classical measurements,” Laser Photon. Rev. 8, L76–L80 (2014).
[Crossref]

Bowen, W. P.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[Crossref]

Boyd, R. W.

M. Mirhosseini, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, and R. W. Boyd, “Compressive direct measurement of the quantum wave function,” Phys. Rev. Lett. 113, 090402 (2014).
[Crossref]

Braje, D. A.

V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).
[Crossref]

Brecht, B.

M. Avenhaus, B. Brecht, K. Laiho, and C. Silberhorn, “Time-frequency quantum process tomography of parametric down-conversion,” arXiv:1406.4252 (2014).

Chen, J. F.

H. Yan, S. Zhang, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Generation of narrow-band hyperentangled nondegenerate paired photons,” Phys. Rev. Lett. 106, 033601 (2011).
[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]

Cho, Y.-W.

Y.-W. Cho, K.-K. Park, J.-C. Lee, and Y.-H. Kim, “Engineering frequency-time quantum correlation of narrow-band biphotons from cold atoms,” Phys. Rev. Lett. 113, 063602 (2014).
[Crossref]

Y.-W. Cho, K.-K. Park, J.-C. Lee, and Y.-H. Kim, “Generation of nonclassical narrowband photon pairs from a cold rubidium cloud,” J. Korean Phys. Soc. 63, 943–950 (2013).
[Crossref]

Chou, C. W.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[Crossref]

Chrapkiewicz, R.

R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photonics 10, 576–579 (2016).
[Crossref]

Cohen, O.

de Icaza Astiz, Y. A.

F. A. Beduini, J. A. Zielińska, V. G. Lucivero, Y. A. de Icaza Astiz, and M. W. Mitchell, “Interferometric measurement of the biphoton wave function,” Phys. Rev. Lett. 113, 183602 (2014).
[Crossref]

Ding, D.-S.

W. Zhang, D.-S. Ding, M.-X. Dong, S. Shi, K. Wang, S.-L. Liu, Y. Li, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Experimental realization of entanglement in multiple degrees of freedom between two quantum memories,” Nat. Commun. 7, 13514 (2016).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, B.-S. Shi, and G.-C. Guo, “Raman quantum memory of photonic polarized entanglement,” Nat. Photonics 9, 332–338 (2015).
[Crossref]

Dong, M.-X.

W. Zhang, D.-S. Ding, M.-X. Dong, S. Shi, K. Wang, S.-L. Liu, Y. Li, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Experimental realization of entanglement in multiple degrees of freedom between two quantum memories,” Nat. Commun. 7, 13514 (2016).
[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]

K. Liao, H. Yan, J. He, S. Du, Z.-M. Zhang, and S.-L. Zhu, “Subnatural-linewidth polarization-entangled photon pairs with controllable temporal length,” Phys. Rev. Lett. 112, 243602 (2014).
[Crossref]

H. Yan, S. Zhang, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Generation of narrow-band hyperentangled nondegenerate paired photons,” Phys. Rev. Lett. 106, 033601 (2011).
[Crossref]

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref]

S. Du, J. Wen, and M. H. Rubin, “Narrowband biphoton generation near atomic resonance,” J. Opt. Soc. Am. B 25, C98–C108 (2008).
[Crossref]

Duan, L.-M.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[Crossref]

Ducci, S.

A. Eckstein, G. Boucher, A. Lemaître, P. Filloux, I. Favero, G. Leo, J. E. Sipe, M. Liscidini, and S. Ducci, “High-resolution spectral characterization of two photon states via classical measurements,” Laser Photon. Rev. 8, L76–L80 (2014).
[Crossref]

Eberhard, P. H.

A. G. White, D. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally entangled states: production, characterization, and utilization,” Phys. Rev. Lett. 83, 3103–3107 (1999).
[Crossref]

Eberly, J. H.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000).
[Crossref]

Eckstein, A.

A. Eckstein, G. Boucher, A. Lemaître, P. Filloux, I. Favero, G. Leo, J. E. Sipe, M. Liscidini, and S. Ducci, “High-resolution spectral characterization of two photon states via classical measurements,” Laser Photon. Rev. 8, L76–L80 (2014).
[Crossref]

Eggleton, B. J.

Fang, B.

Favero, I.

A. Eckstein, G. Boucher, A. Lemaître, P. Filloux, I. Favero, G. Leo, J. E. Sipe, M. Liscidini, and S. Ducci, “High-resolution spectral characterization of two photon states via classical measurements,” Laser Photon. Rev. 8, L76–L80 (2014).
[Crossref]

Filloux, P.

A. Eckstein, G. Boucher, A. Lemaître, P. Filloux, I. Favero, G. Leo, J. E. Sipe, M. Liscidini, and S. Ducci, “High-resolution spectral characterization of two photon states via classical measurements,” Laser Photon. Rev. 8, L76–L80 (2014).
[Crossref]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

M. Fleischhauer and M. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref]

Frankowski, R.

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See Supplement 1 for the details of the derivation.

Supplementary Material (1)

NameDescription
» Supplement 1       the details of the derivation

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Figures (4)

Fig. 1.
Fig. 1. Experimental setup. (a) JTI measurement scheme using time-resolved two-photon coincidence detection. (b) JTI and JTP measurement scheme with stimulated four-wave mixing. Note that unlike the spontaneous case, the etalon is unnecessary for blocking pumping lasers due to the high intensity of the stimulated field. PBS, polarizing beamsplitter; QWP, quarter wave plate.

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Fig. 2.
Fig. 2. Two-photon waveform measurement. (a) Time-resolved two-photon detection. The red dots represent the coincidence counts accumulated for 60 s (excluding the MOT preparation time). (b) Stimulated four-wave mixing. The intensity of anti-Stokes emission is measured when the Stokes seed pulse of 20 ns is applied at ts. The blue dots represent the averaged intensity of 20 oscilloscope traces during the 34 μs accumulation time. The black solid line in (a) and (b) is the theoretical curve for the two-photon waveform calculated with the experimental parameters OD=53, γgs/2π=16.8  kHz, and Ωc/2π=7.2  MHz. Note that, since the TCSPC measures the coincidence histogram in the tas−ts axis via the start–stop measurement, the two-photon measurement data in (a) correspond to the red cross section of the JTI shown in (c). For the stimulated emission, the anti-Stokes pulse is measured at a specific Stokes pulse time ts, corresponding to the blue cross section shown in (d), which is not corresponding to the tas−ts axis. For the same comparison, the time axis in (b) thus has been scaled by 1/2 to project onto the tas−ts axis.

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Fig. 3.
Fig. 3. Frequency-time two-photon wavefunction for time-correlated narrowband photon pair. (a) Experimental JTI measured via stimulated four-wave mixing. The color bar represents the intensity of the stimulated anti-Stokes pulse. (b) Theoretical JTI. The unit of the color bar is 10−5  μs−2. (c) Corresponding experimental JTP from interference. The color bar corresponds to the cosine of JTP. The black areas represent phase undefined regions due to vanishing JTI. (d) Theoretical JTP. The black contour lines in (c) and (d) represent the isoheight JTI lines.

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Fig. 4.
Fig. 4. Frequency-time two-photon wavefunction for time-uncorrelated narrowband photon pair. (a) Experimental JTI measured via stimulated four-wave mixing. The color bar represents the intensity of the stimulated anti-Stokes pulse. (b) Theoretical JTI. The unit of the color bar is 10−4  μs−2. (c) Corresponding experimental JTP from interference. The color bar corresponds to the cosine of JTP. The black areas represent phase undefined regions due to vanishing JTI. (d) Theoretical JTP. The black contour lines in (c) and (d) represent the isoheight JTI lines.

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Equations (4)

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|ψ⟩=∫dtsdtasΨ(ts,tas)a^s†(ts)a^as†(tas)|0⟩,
⟨ψ|a^s†(ts)a^as†(tas)a^as(tas)a^s(ts)|ψ⟩∝|Ψ(ts,tas)|2.
⟨ψ|Ds†[α(ts)]a^s†(ts)a^as†(tas)a^as(tas)a^s(ts)Ds[α(ts)]|ψ⟩∝|α(ts)|2|Ψ(ts,tas)|2,
ΔI∝|Ψ(ts,tas)α(ts)β|cos[ϕ(ts,tas)−φseed],

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