Abstract

Multiphoton processes in dense atomic vapors such as four-wave mixing or coherent blue light generation are typically viewed from single-atom perspective. Here we study the surprisingly important effect of phase matching near two-photon resonances that arises due to spatial extent of the atomic medium within which the multiphoton process occurs. The non-unit refractive index of the atomic vapor may inhibit generation of light in nonlinear processes, significantly shift the efficiency maxima in frequencies and redirect emitted beam. We present these effects on an example of four-wave mixing in dense rubidium vapors in a double-ladder configuration. By deriving a simple theory that takes into account essential spatial properties of the process, we give precise predictions and confirm their validity in the experiment. The model allows us to improve on the geometry of the experiment and engineer more efficient four-wave mixing.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (4)

M. Parniak, A. Leszczyński, and W. Wasilewski, “Coupling of four-wave mixing and Raman scattering by ground-state atomic coherence,” Phys. Rev. A 93, 053821 (2016).
[Crossref]

M. Parniak, A. Leszczyński, and W. Wasilewski, “Magneto-optical polarization rotation in a ladder-type atomic system for tunable offset locking,” Appl. Phys. Lett. 108, 161103 (2016).
[Crossref]

R. F. Offer, J. W. C. Conway, E. Riis, S. Franke-Arnold, and A. S. Arnold, “Cavity-enhanced frequency up-conversion in rubidium vapor,” Opt. Lett. 41, 2177–2180 (2016).
[Crossref] [PubMed]

M. Mazelanik, M. Dąbrowski, and W. Wasilewski, “Correlation steering in the angularly multimode Raman atomic memory,” Opt. Express 24, 21995–22003 (2016).
[Crossref] [PubMed]

2015 (5)

E. Brekke and E. Herman, “Frequency characteristics of far-detuned parametric four-wave mixing in Rb,” Opt. Lett. 40, 5674–5677 (2015).
[Crossref] [PubMed]

J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner, and W. Li, “Lensing effect of electromagnetically induced transparency involving a Rydberg state,” Phys. Rev. A 92, 063824 (2015).
[Crossref]

A. B. Mirza and S. Singh, “Electromagnetically induced transparency and steady-state propagation characteristics in Doppler broadened diamond systems,” J. Mod. Opt. 62, 16–26 (2015).
[Crossref]

N. R. de Melo and S. S. Vianna, “Frequency shift in three-photon resonant four-wave mixing by internal atom-field interaction,” Phys. Rev. A 92, 053830 (2015).
[Crossref]

M. Parniak and W. Wasilewski, “Interference and nonlinear properties of four-wave-mixing resonances in thermal vapor: Analytical results and experimental verification,” Phys. Rev. A 91, 023418 (2015).
[Crossref]

2014 (5)

2013 (3)

M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, “Role of the phase-matching condition in nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light,” Phys. Rev. A 88, 033845 (2013).
[Crossref]

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble,” Phys. Rev. Lett. 111, 123602 (2013).
[Crossref] [PubMed]

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

2012 (2)

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

A. B. Mirza and S. Singh, “Wave-vector mismatch effects in electromagnetically induced transparency in Y-type systems,” Phys. Rev. A 85, 053837 (2012).
[Crossref]

2011 (2)

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Photon statistics and polarization correlations at telecommunications wavelengths from a warm atomic ensemble,” Opt. Express 19, 14632–14641 (2011).
[Crossref] [PubMed]

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

2010 (2)

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Phys. 6, 894–899 (2010).
[Crossref]

A. Vernier, S. Franke-Arnold, E. Riis, and A. S. Arnold, “Enhanced frequency up-conversion in Rb vapor,” Opt. Express 18, 17020–17026 (2010).
[Crossref] [PubMed]

2009 (2)

A. A. M. Akulshin, R. J. R. McLean, A. I. Sidorov, and P. Hannaford, “Coherent and collimated blue light generated by four-wave mixing in Rb vapour,” Opt. Express 17, 22861–22870 (2009).
[Crossref]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[Crossref]

2008 (2)

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: The diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[Crossref]

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[Crossref]

2006 (2)

T. Chanelière, D. Matsukevich, S. Jenkins, T. Kennedy, M. Chapman, and A. Kuzmich, “Quantum Telecommunication Based on Atomic Cascade Transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[Crossref] [PubMed]

T. Meijer, J. D. White, B. Smeets, M. Jeppesen, and R. E. Scholten, “Blue five-level frequency-upconversion system in rubidium,” Opt. Lett. 31, 1002–1004 (2006).
[Crossref] [PubMed]

2002 (1)

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[Crossref]

2001 (1)

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes,” Phys. Rev. A 63, 063406 (2001).
[Crossref]

Akulshin, A. A. M.

Altin, P. A.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Arnold, A. S.

Becerra, F. E.

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Photon statistics and polarization correlations at telecommunications wavelengths from a warm atomic ensemble,” Opt. Express 19, 14632–14641 (2011).
[Crossref] [PubMed]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[Crossref]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: The diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[Crossref]

Bernu, J.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Bharti, V.

V. Bharti and A. Wasan, “Complete wavelength mismatching effect in a Doppler broadened Y-type six-level EIT atomic medium,” Opt. Commun. 324, 238–244 (2014).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008, 3 Edition).

Boyer, V.

M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, “Role of the phase-matching condition in nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light,” Phys. Rev. A 88, 033845 (2013).
[Crossref]

Brekke, E.

E. Brekke and E. Herman, “Frequency characteristics of far-detuned parametric four-wave mixing in Rb,” Opt. Lett. 40, 5674–5677 (2015).
[Crossref] [PubMed]

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[Crossref]

Buchler, B. C.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Chanelière, T.

T. Chanelière, D. Matsukevich, S. Jenkins, T. Kennedy, M. Chapman, and A. Kuzmich, “Quantum Telecommunication Based on Atomic Cascade Transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[Crossref] [PubMed]

Chapman, M.

T. Chanelière, D. Matsukevich, S. Jenkins, T. Kennedy, M. Chapman, and A. Kuzmich, “Quantum Telecommunication Based on Atomic Cascade Transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[Crossref] [PubMed]

Chng, B.

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble,” Phys. Rev. Lett. 111, 123602 (2013).
[Crossref] [PubMed]

Clemmen, S.

Conway, J. W. C.

Dabrowski, M.

Day, J. O.

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[Crossref]

de Melo, N. R.

N. R. de Melo and S. S. Vianna, “Frequency shift in three-photon resonant four-wave mixing by internal atom-field interaction,” Phys. Rev. A 92, 053830 (2015).
[Crossref]

N. R. de Melo and S. S. Vianna, “Two-photon resonant forward four-wave mixing in rubidium vapor involving Rydberg states,” J. Opt. Soc. Am. B 31, 1735–1740 (2014).
[Crossref]

DePaola, B. D.

Donvalkar, P. S.

Dudin, Y. O.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Phys. 6, 894–899 (2010).
[Crossref]

Embrey, C. S.

M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, “Role of the phase-matching condition in nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light,” Phys. Rev. A 88, 033845 (2013).
[Crossref]

Firstenberg, O.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Franke-Arnold, S.

Gaeta, A. L.

Gearba, M. A.

Geng, J.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Glorieux, Q.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Gorshkov, A. V.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Gulati, G. K.

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble,” Phys. Rev. Lett. 111, 123602 (2013).
[Crossref] [PubMed]

Guo, R.

J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner, and W. Li, “Lensing effect of electromagnetically induced transparency involving a Rydberg state,” Phys. Rev. A 92, 063824 (2015).
[Crossref]

Han, J.

J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner, and W. Li, “Lensing effect of electromagnetically induced transparency involving a Rydberg state,” Phys. Rev. A 92, 063824 (2015).
[Crossref]

Hannaford, P.

Herman, E.

Hofferberth, S.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Hollberg, L.

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[Crossref]

Hosseini, M.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Huber, B.

B. Huber, A. Kölle, and T. Pfau, “Motion-induced signal revival in pulsed Rydberg four-wave mixing beyond the frozen gas limit,” Phys. Rev. A 90, 053806 (2014).
[Crossref]

Jen, H. H.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Phys. 6, 894–899 (2010).
[Crossref]

Jenkins, S.

T. Chanelière, D. Matsukevich, S. Jenkins, T. Kennedy, M. Chapman, and A. Kuzmich, “Quantum Telecommunication Based on Atomic Cascade Transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[Crossref] [PubMed]

Jenkins, S. D.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Phys. 6, 894–899 (2010).
[Crossref]

Jeppesen, M.

Kennedy, T.

T. Chanelière, D. Matsukevich, S. Jenkins, T. Kennedy, M. Chapman, and A. Kuzmich, “Quantum Telecommunication Based on Atomic Cascade Transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[Crossref] [PubMed]

Kennedy, T. A. B.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Phys. 6, 894–899 (2010).
[Crossref]

Kiffner, M.

J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner, and W. Li, “Lensing effect of electromagnetically induced transparency involving a Rydberg state,” Phys. Rev. A 92, 063824 (2015).
[Crossref]

Knize, R. J.

Kölle, A.

B. Huber, A. Kölle, and T. Pfau, “Motion-induced signal revival in pulsed Rydberg four-wave mixing beyond the frozen gas limit,” Phys. Rev. A 90, 053806 (2014).
[Crossref]

Kurtsiefer, C.

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble,” Phys. Rev. Lett. 111, 123602 (2013).
[Crossref] [PubMed]

Kuzmich, A.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Phys. 6, 894–899 (2010).
[Crossref]

T. Chanelière, D. Matsukevich, S. Jenkins, T. Kennedy, M. Chapman, and A. Kuzmich, “Quantum Telecommunication Based on Atomic Cascade Transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[Crossref] [PubMed]

Lam, P. K.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Leszczynski, A.

M. Parniak, A. Leszczyński, and W. Wasilewski, “Coupling of four-wave mixing and Raman scattering by ground-state atomic coherence,” Phys. Rev. A 93, 053821 (2016).
[Crossref]

M. Parniak, A. Leszczyński, and W. Wasilewski, “Magneto-optical polarization rotation in a ladder-type atomic system for tunable offset locking,” Appl. Phys. Lett. 108, 161103 (2016).
[Crossref]

Li, C.

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

Li, P.

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

Li, W.

J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner, and W. Li, “Lensing effect of electromagnetically induced transparency involving a Rydberg state,” Phys. Rev. A 92, 063824 (2015).
[Crossref]

Liang, Q.-Y.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Lukin, M. D.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[Crossref]

Magno, W. C.

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes,” Phys. Rev. A 63, 063406 (2001).
[Crossref]

Manjappa, M.

J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner, and W. Li, “Lensing effect of electromagnetically induced transparency involving a Rydberg state,” Phys. Rev. A 92, 063824 (2015).
[Crossref]

Marino, A. M.

M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, “Role of the phase-matching condition in nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light,” Phys. Rev. A 88, 033845 (2013).
[Crossref]

Maslennikov, G.

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble,” Phys. Rev. Lett. 111, 123602 (2013).
[Crossref] [PubMed]

Matsukevich, D.

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble,” Phys. Rev. Lett. 111, 123602 (2013).
[Crossref] [PubMed]

T. Chanelière, D. Matsukevich, S. Jenkins, T. Kennedy, M. Chapman, and A. Kuzmich, “Quantum Telecommunication Based on Atomic Cascade Transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[Crossref] [PubMed]

Mazelanik, M.

McLean, R. J. R.

Meijer, T.

Mirza, A. B.

A. B. Mirza and S. Singh, “Electromagnetically induced transparency and steady-state propagation characteristics in Doppler broadened diamond systems,” J. Mod. Opt. 62, 16–26 (2015).
[Crossref]

A. B. Mirza and S. Singh, “Wave-vector mismatch effects in electromagnetically induced transparency in Y-type systems,” Phys. Rev. A 85, 053837 (2012).
[Crossref]

Nussenzveig, P.

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes,” Phys. Rev. A 63, 063406 (2001).
[Crossref]

Offer, R. F.

Orozco, L. A.

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Photon statistics and polarization correlations at telecommunications wavelengths from a warm atomic ensemble,” Opt. Express 19, 14632–14641 (2011).
[Crossref] [PubMed]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[Crossref]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: The diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[Crossref]

Parniak, M.

M. Parniak, A. Leszczyński, and W. Wasilewski, “Magneto-optical polarization rotation in a ladder-type atomic system for tunable offset locking,” Appl. Phys. Lett. 108, 161103 (2016).
[Crossref]

M. Parniak, A. Leszczyński, and W. Wasilewski, “Coupling of four-wave mixing and Raman scattering by ground-state atomic coherence,” Phys. Rev. A 93, 053821 (2016).
[Crossref]

M. Parniak and W. Wasilewski, “Interference and nonlinear properties of four-wave-mixing resonances in thermal vapor: Analytical results and experimental verification,” Phys. Rev. A 91, 023418 (2015).
[Crossref]

Petrov, P. G.

M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, “Role of the phase-matching condition in nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light,” Phys. Rev. A 88, 033845 (2013).
[Crossref]

Peyronel, T.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Pfau, T.

B. Huber, A. Kölle, and T. Pfau, “Motion-induced signal revival in pulsed Rydberg four-wave mixing beyond the frozen gas limit,” Phys. Rev. A 90, 053806 (2014).
[Crossref]

Pohl, T.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Prandini, R. B.

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes,” Phys. Rev. A 63, 063406 (2001).
[Crossref]

Radnaev, A. G.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Phys. 6, 894–899 (2010).
[Crossref]

Riis, E.

Robins, N. P.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Rolston, S. L.

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Photon statistics and polarization correlations at telecommunications wavelengths from a warm atomic ensemble,” Opt. Express 19, 14632–14641 (2011).
[Crossref] [PubMed]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[Crossref]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: The diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[Crossref]

Saha, K.

Sang, S.

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

Scholten, R. E.

Scully, M. O.

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[Crossref]

Sell, J. F.

Sidorov, A. I.

Singh, S.

A. B. Mirza and S. Singh, “Electromagnetically induced transparency and steady-state propagation characteristics in Doppler broadened diamond systems,” J. Mod. Opt. 62, 16–26 (2015).
[Crossref]

A. B. Mirza and S. Singh, “Wave-vector mismatch effects in electromagnetically induced transparency in Y-type systems,” Phys. Rev. A 85, 053837 (2012).
[Crossref]

Smeets, B.

Sparkes, B. M.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Srivathsan, B.

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble,” Phys. Rev. Lett. 111, 123602 (2013).
[Crossref] [PubMed]

Turnbull, M. T.

M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, “Role of the phase-matching condition in nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light,” Phys. Rev. A 88, 033845 (2013).
[Crossref]

Venkataraman, V.

Vernier, A.

Vianna, S. S.

N. R. de Melo and S. S. Vianna, “Frequency shift in three-photon resonant four-wave mixing by internal atom-field interaction,” Phys. Rev. A 92, 053830 (2015).
[Crossref]

N. R. de Melo and S. S. Vianna, “Two-photon resonant forward four-wave mixing in rubidium vapor involving Rydberg states,” J. Opt. Soc. Am. B 31, 1735–1740 (2014).
[Crossref]

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes,” Phys. Rev. A 63, 063406 (2001).
[Crossref]

Vogt, T.

J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner, and W. Li, “Lensing effect of electromagnetically induced transparency involving a Rydberg state,” Phys. Rev. A 92, 063824 (2015).
[Crossref]

Vuletic, V.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Walker, T. G.

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[Crossref]

Wang, Z.

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

Wasan, A.

V. Bharti and A. Wasan, “Complete wavelength mismatching effect in a Doppler broadened Y-type six-level EIT atomic medium,” Opt. Commun. 324, 238–244 (2014).
[Crossref]

Wasilewski, W.

M. Parniak, A. Leszczyński, and W. Wasilewski, “Magneto-optical polarization rotation in a ladder-type atomic system for tunable offset locking,” Appl. Phys. Lett. 108, 161103 (2016).
[Crossref]

M. Parniak, A. Leszczyński, and W. Wasilewski, “Coupling of four-wave mixing and Raman scattering by ground-state atomic coherence,” Phys. Rev. A 93, 053821 (2016).
[Crossref]

M. Mazelanik, M. Dąbrowski, and W. Wasilewski, “Correlation steering in the angularly multimode Raman atomic memory,” Opt. Express 24, 21995–22003 (2016).
[Crossref] [PubMed]

M. Parniak and W. Wasilewski, “Interference and nonlinear properties of four-wave-mixing resonances in thermal vapor: Analytical results and experimental verification,” Phys. Rev. A 91, 023418 (2015).
[Crossref]

White, J. D.

Willis, R. T.

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Photon statistics and polarization correlations at telecommunications wavelengths from a warm atomic ensemble,” Opt. Express 19, 14632–14641 (2011).
[Crossref] [PubMed]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[Crossref]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: The diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[Crossref]

Xiao, M.

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

Yuan, C.

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

Zhang, Y.

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

Zhao, R.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Phys. 6, 894–899 (2010).
[Crossref]

Zheng, H.

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

Zibrov, A. S.

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[Crossref]

Appl. Phys. B (1)

Z. Wang, Y. Zhang, P. Li, S. Sang, C. Yuan, H. Zheng, C. Li, and M. Xiao, “Observation of polarization-controlled spatial splitting of four-wave mixing in a three-level atomic system,” Appl. Phys. B 104, 633–638 (2011).
[Crossref]

Appl. Phys. Lett. (1)

M. Parniak, A. Leszczyński, and W. Wasilewski, “Magneto-optical polarization rotation in a ladder-type atomic system for tunable offset locking,” Appl. Phys. Lett. 108, 161103 (2016).
[Crossref]

J. Mod. Opt. (1)

A. B. Mirza and S. Singh, “Electromagnetically induced transparency and steady-state propagation characteristics in Doppler broadened diamond systems,” J. Mod. Opt. 62, 16–26 (2015).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (1)

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Nature Phys. (1)

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Phys. 6, 894–899 (2010).
[Crossref]

New J. Phys. (1)

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Opt. Commun. (1)

V. Bharti and A. Wasan, “Complete wavelength mismatching effect in a Doppler broadened Y-type six-level EIT atomic medium,” Opt. Commun. 324, 238–244 (2014).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. A (12)

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes,” Phys. Rev. A 63, 063406 (2001).
[Crossref]

A. B. Mirza and S. Singh, “Wave-vector mismatch effects in electromagnetically induced transparency in Y-type systems,” Phys. Rev. A 85, 053837 (2012).
[Crossref]

M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, “Role of the phase-matching condition in nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light,” Phys. Rev. A 88, 033845 (2013).
[Crossref]

N. R. de Melo and S. S. Vianna, “Frequency shift in three-photon resonant four-wave mixing by internal atom-field interaction,” Phys. Rev. A 92, 053830 (2015).
[Crossref]

J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner, and W. Li, “Lensing effect of electromagnetically induced transparency involving a Rydberg state,” Phys. Rev. A 92, 063824 (2015).
[Crossref]

M. Parniak, A. Leszczyński, and W. Wasilewski, “Coupling of four-wave mixing and Raman scattering by ground-state atomic coherence,” Phys. Rev. A 93, 053821 (2016).
[Crossref]

B. Huber, A. Kölle, and T. Pfau, “Motion-induced signal revival in pulsed Rydberg four-wave mixing beyond the frozen gas limit,” Phys. Rev. A 90, 053806 (2014).
[Crossref]

M. Parniak and W. Wasilewski, “Interference and nonlinear properties of four-wave-mixing resonances in thermal vapor: Analytical results and experimental verification,” Phys. Rev. A 91, 023418 (2015).
[Crossref]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[Crossref]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: The diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[Crossref]

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[Crossref]

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[Crossref]

Phys. Rev. Lett. (2)

T. Chanelière, D. Matsukevich, S. Jenkins, T. Kennedy, M. Chapman, and A. Kuzmich, “Quantum Telecommunication Based on Atomic Cascade Transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[Crossref] [PubMed]

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble,” Phys. Rev. Lett. 111, 123602 (2013).
[Crossref] [PubMed]

Other (1)

R. W. Boyd, Nonlinear Optics (Academic, 2008, 3 Edition).

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Figures (6)

Fig. 1
Fig. 1

(a) Atomic level double-ladder configuration used in the experiment. For the ground state 52S1/2 we take one of its hyperfine components with F = 2 and for the highest excited state 52D5/2 we take its hyperfine component with F′ = 4. For the intermediate state 52P3/2 we consider all its hyperfine components, that lead to multiple wave-mixing paths. (b) Beam directions: three-dimensional sketch of the laser beams intersecting inside a cell filled with rubidium vapor. Laser beams marked with green and blue colors have wavelength of 780 nm. Laser beam marked with the red arrow (776 nm) propagates in the opposite direction. The four-wave mixing signal is marked with the orange dotted line. The signal is emitted roughly in direction determined by transverse phase matching, where the sum of transverse components of wavevectors k1 + k2 is equal to the respective k3 + k4 sum. Due to this condition the four points in the transverse k-plane, corresponding to the four beams, form a parallelogram. (c) Central part of the experimental setup. Emitted four-wave mixing signal is filtered using a half-wave plate (λ/2), a polarizer and an interference filter and then registered by with an avalanche photodiode or a CCD camera situated in the far field with respect to the rubidium vapor cell.

Fig. 2
Fig. 2

Theoretical amplitudes of emitted four-wave mixing signal intensity assuming that 1/e2 diameters of incoming beams equal to 400 μm. Panel (a) portrays the Gaussian intensity distribution without taking into account the longitudinal phase-matching condition. Panel (b) shows the value of the phase-matching factor sinc 2 ( k 3 L 2 δ ) . The black circle corresponds to a region where the longitudinal phase matching is perfect. Variations of the refractive index induced by changing laser detuning cause the radius of this circle to change as well, which in turn impacts the direction of emission. Panel (c) presents product of (a) and (b), reflecting small change in shape and radial translation.

Fig. 3
Fig. 3

Dependence of the four-wave mixing signal intensity and emission angle on detuning Δ4 for Δ1/2π = −3500 MHz and Δ2 = 0 at T = 145°C. Dots correspond to the experimental result, while solid lines correspond to theoretical prediction.

Fig. 4
Fig. 4

Transmission profile of laser 1 light (blue curve and points) and the intensity of four-wave mixing signal (red curve and points) as a function of two-photon detuning Δ2 in the vicinity of the two-photon resonance. Dots correspond to experimental result for Δ1/2π = −3000 MHz and Δ4/2π = −2760 MHz, while solid lines correspond to the theoretical prediction. Vertical dotted lines mark maxima of the four-wave mixing signal intensity and two-photon absorption. The frequency shift between the two maxima is marked as ΔS, while the maximum intensity of the four-wave mixing signal attained at the resonance is marked as Imax .

Fig. 5
Fig. 5

Maximal four-wave mixing signal intensities Imax (blue curves, left axis) and corresponding frequency shifts ΔS (red curves, right axis) for a set of Δ1 and Δ4 detuning. Panels (a) and (b) correspond to experimental and theoretical results, respectively. Subsequent curves were measured/calculated for different Δ1 detunings in 500 2π×MHz steps starting from Δ1/2π = −5000 MHz on the left.

Fig. 6
Fig. 6

(a) Tilts of crossing beams to the z-axis. Dots and crosses inside circles correspond to direction to the positive and negative values on the z-axis. (b) Dependence of the four-wave mixing frequency shift ΔS in relation to the two-photon absorption resonance and (c) maximum intensity Imax on Δ4 detuning. Subsequent experimental points correspond to measurements at various detuning Δ4 and angles as shown in panel (a) with constant Δ1/2π = −2000 MHz. Dots correspond to experimental result, while solid lines correspond to theoretical prediction.

Equations (17)

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z A ( x , y , z ) = i ω 2 2 k 0 c 2 ε 0 P NL ( x , y , z ) + i 2 k 0 Δ A ( x , y , z ) ,
z A ˜ ( k , z ) = i ω 2 2 k 0 c 2 ε 0 P ˜ NL ( k , z ) exp ( i k 2 2 k 0 z ) ,
A ˜ ( k , L ) = i ω 2 2 k 0 c 2 ε 0 L P ˜ NL ( k , z ) exp ( i k 2 2 k 0 ( z L ) ) d z .
A j ( x , y , z ) = a j exp ( x 2 + y 2 2 σ 2 ) exp ( i k j r ) .
P NL ( x , y , z ) = χ 3 ( 3 ) A 1 A 2 A 4 * = a 3 exp ( 3 ( x 2 + y 2 ) 2 σ 2 ) exp ( i K r ) ,
K k 3 Φ = k 1 φ 1 + k 2 φ 2 k 4 φ 4
A ˜ 3 ( φ 3 , L ) ~ a 3 exp ( k 3 2 σ 2 | φ 3 Φ | 2 6 ) sinc ( k 3 L 2 δ ) ,
δ = k 1 k 3 ( 1 | φ 1 | 2 2 ) + k 2 k 3 ( 1 | φ 2 | 2 2 ) k 4 k 3 ( 1 | φ 4 | 2 2 ) 1 θ 3 2 / 2 + | φ 3 | 2 2 .
Δ θ 3 = π / ( 2 k 3 L θ 3 ) .
θ 3 k j ± 1 k j θ 3 .
θ 3 φ j x y = ± k j φ j x y k 3 θ 30
χ 3 ( 3 ) = N d 12 d 23 d 23 * d 14 * 4 3 Δ ˜ 1 Δ ˜ 2 Δ ˜ 4 * ,
χ 1 = N ε 0 ( d 12 2 Δ ˜ 1 d 12 2 Ω 2 2 4 Δ ˜ 1 2 Δ ˜ 2 ) ,
χ 2 = N d 23 2 Ω 1 2 4 ε 0 Δ ˜ 1 Δ 1 * Δ ˜ 2 ,
χ j ( ω 1 , , ω 4 ) χ j ( ω 1 k 1 v , , ω 4 k 4 v ) g ( v ) d v ,
g ( v ) = m 2 π k b T exp ( m v 2 2 k B T ) ,
α 3 = d φ 3 | φ 3 | | A ˜ 3 ( φ 3 , L ) | 2 d φ 3 | A ˜ 3 ( φ 3 , L ) | 2 .

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