Abstract

Eight-wave mixing (EWM) is a seven-order nonlinear process that can reflect nonclassical features within multiple optical fields, thus imparting certain advantages. In this study, we directly observed the EWM spectrum and spatial images that show Rydberg atoms under a circularly polarized probe field in a five-level coherently prepared atomic system. Such circular polarization dressing fields can obtain high-contrast Rydberg EWM overcome the difficulties of several multi-wave mixing (MWM) signals always coexist, and the multi-parameter controlling Rydberg EWM mechanism is established by changing the power and detuning and polarization of the dressing fields. These controllable high-order MWM processes present a contrast ratio of 96% and a narrow linewidth of <30 MHz compared with low-order mixing processes under identical conditions (e.g., six-wave mixing). The corresponding MWM spatial images are presented, and they can partly reflect the underlying nonlinear phase variation, whereas the given theory can predict the experimental results.

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

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References

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    [Crossref]
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    [Crossref]
  24. S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
    [Crossref] [PubMed]
  25. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
    [Crossref] [PubMed]
  26. O. Firstenberg, C. S. Adams, and S. Hofferberth, “Nonlieanr quantum optics mediated by Rydberg interactions,” J. Phys. At. Mol. Opt. Phys. 49(15), 152003 (2016).
    [Crossref]
  27. Y. P. Zhang, Z. Q. Nie, H. B. Zheng, C. B. Li, J. P. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013825 (2009).
    [Crossref]
  28. W. G. Yang, A. Joshi, and M. Xiao, “Controlling dynamic instability of three-level atoms inside an optical ring cavity,” Phys. Rev. A 70(3), 033807 (2004).
    [Crossref]
  29. M. M. Müller, A. Kölle, R. Löw, T. Pfau, T. Calarco, and S. Montangero, “Room-temperature Rydberg single-photon source,” Phys. Rev. A 87(5), 053412 (2013).
    [Crossref]
  30. L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
    [Crossref] [PubMed]

2016 (4)

D. A. Anderson, S. A. Miller, G. Raithel, J. A. Gordon, M. L. Butler, and C. L. Holloway, “Optical Measurements of Strong Microwave Fields with Rydberg Atoms in a Vapor Cell,” Phys. Rev. Appl. 5(3), 034003 (2016).
[Crossref]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

M. Kiffner, A. Feizpour, T. K. Kaczmarek, D. Jaksch, and J. Nunn, “Two-way interconversion of millimeter-wave and optical fields in Rydberg gases,” New J. Phys. 18(9), 093030 (2016).
[Crossref]

O. Firstenberg, C. S. Adams, and S. Hofferberth, “Nonlieanr quantum optics mediated by Rydberg interactions,” J. Phys. At. Mol. Opt. Phys. 49(15), 152003 (2016).
[Crossref]

2015 (2)

Z. Zhang, J. Che, D. Zhang, Z. Liu, X. Wang, and Y. Zhang, “Eight-wave mixing process in a Rydberg-dressing atomic ensemble,” Opt. Express 23(11), 13814–13822 (2015).
[Crossref] [PubMed]

J. L. Che, J. Q. Ma, H. B. Zheng, Z. Y. Zhang, X. Yao, Y. Q. Zhang, and Y. P. Zhang, “Rydberg six-wave mixing process,” Europhys. Lett. 109(3), 33001 (2015).
[Crossref]

2014 (3)

J. A. Gordon, C. L. Holloway, A. Schwarzkopf, D. A. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Millimeter wave detection via Autler-Townes splitting in rubidium Rydberg atoms,” Appl. Phys. Lett. 105(2), 024104 (2014).
[Crossref]

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24(4), 045404 (2014).
[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(5), 053806 (2014).
[Crossref]

2013 (2)

M. M. Müller, A. Kölle, R. Löw, T. Pfau, T. Calarco, and S. Montangero, “Room-temperature Rydberg single-photon source,” Phys. Rev. A 87(5), 053412 (2013).
[Crossref]

T. Baluktsian, B. Huber, R. Löw, and T. Pfau, “Evidence for strong van der Waals Type Rydberg-Rydberg Interaction in a Thermal Vapor,” Phys. Rev. Lett. 110(12), 123001 (2013).
[Crossref] [PubMed]

2012 (1)

A. Kölle, G. Epple, H. Kübler, R. Löw, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85(6), 063821 (2012).
[Crossref]

2011 (3)

B. Huber, T. Baluktsian, M. Schlagmüller, A. Kölle, H. Kübler, R. Löw, and T. Pfau, “GHz Rabi flopping to Rydberg states in hot atomic vapor cells,” Phys. Rev. Lett. 107(24), 243001 (2011).
[Crossref] [PubMed]

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
[Crossref] [PubMed]

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(15), 14632–14641 (2011).
[Crossref] [PubMed]

2009 (3)

M. Scheid, D. Kolbe, F. Markert, T. W. Hänsch, and J. Walz, “Continuous-wave Lyman-α generation with solid-state lasers,” Opt. Express 17(14), 11274–11280 (2009).
[Crossref] [PubMed]

Y. P. Zhang, Z. Q. Nie, H. B. Zheng, C. B. Li, J. P. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

Y. G. Du, Y. P. Zhang, C. C. Zuo, C. B. Li, Z. Q. Nie, H. B. Zheng, M. Z. Shi, R. M. Wang, J. P. Song, K. Q. Lu, and M. Xiao, “Controlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparency,” Phys. Rev. A 79(6), 063839 (2009).
[Crossref]

2008 (1)

X. M. Liu, “Theory and experiments for multiple four-wave-mixing processes with multi-frequency pumps in optical fibers,” Phys. Rev. A 77(4), 043818 (2008).
[Crossref]

2007 (2)

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98(11), 113003 (2007).
[Crossref] [PubMed]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening Four-Wave Mixing And Six-Wave Mixing Channels via Dual Electromagnetically Induced Transparency Windows,” Phys. Rev. Lett. 99(12), 123603 (2007).
[Crossref] [PubMed]

2005 (2)

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atom pairs correlated to the ns–ns, np–np and nd–nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
[Crossref]

X. Liu, X. Yang, F. Lu, J. Ng, X. Zhou, and C. Lu, “Stable and uniform dual-wavelength erbium-doped fiber laser based on fiber Bragg gratings and photonic crystal fiber,” Opt. Express 13(1), 142–147 (2005).
[Crossref] [PubMed]

2004 (4)

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurásek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432(7016), 482–486 (2004).
[Crossref] [PubMed]

W. G. Yang, A. Joshi, and M. Xiao, “Controlling dynamic instability of three-level atoms inside an optical ring cavity,” Phys. Rev. A 70(3), 033807 (2004).
[Crossref]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

2001 (1)

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref] [PubMed]

1998 (1)

I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nosbaum, V. M. Akulin, and P. Pillet, “Many-Body Effects in a Frozen Rydberg Gas,” Phys. Rev. Lett. 80(2), 253–256 (1998).
[Crossref]

Adams, C. S.

O. Firstenberg, C. S. Adams, and S. Hofferberth, “Nonlieanr quantum optics mediated by Rydberg interactions,” J. Phys. At. Mol. Opt. Phys. 49(15), 152003 (2016).
[Crossref]

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98(11), 113003 (2007).
[Crossref] [PubMed]

Akulin, V. M.

I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nosbaum, V. M. Akulin, and P. Pillet, “Many-Body Effects in a Frozen Rydberg Gas,” Phys. Rev. Lett. 80(2), 253–256 (1998).
[Crossref]

Anderson, D. A.

D. A. Anderson, S. A. Miller, G. Raithel, J. A. Gordon, M. L. Butler, and C. L. Holloway, “Optical Measurements of Strong Microwave Fields with Rydberg Atoms in a Vapor Cell,” Phys. Rev. Appl. 5(3), 034003 (2016).
[Crossref]

J. A. Gordon, C. L. Holloway, A. Schwarzkopf, D. A. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Millimeter wave detection via Autler-Townes splitting in rubidium Rydberg atoms,” Appl. Phys. Lett. 105(2), 024104 (2014).
[Crossref]

Ates, C.

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
[Crossref] [PubMed]

Baluktsian, T.

T. Baluktsian, B. Huber, R. Löw, and T. Pfau, “Evidence for strong van der Waals Type Rydberg-Rydberg Interaction in a Thermal Vapor,” Phys. Rev. Lett. 110(12), 123001 (2013).
[Crossref] [PubMed]

B. Huber, T. Baluktsian, M. Schlagmüller, A. Kölle, H. Kübler, R. Löw, and T. Pfau, “GHz Rabi flopping to Rydberg states in hot atomic vapor cells,” Phys. Rev. Lett. 107(24), 243001 (2011).
[Crossref] [PubMed]

Becerra, F. E.

Brown, A. W.

Y. Zhang, A. W. Brown, and M. Xiao, “Opening Four-Wave Mixing And Six-Wave Mixing Channels via Dual Electromagnetically Induced Transparency Windows,” Phys. Rev. Lett. 99(12), 123603 (2007).
[Crossref] [PubMed]

Butler, M. L.

D. A. Anderson, S. A. Miller, G. Raithel, J. A. Gordon, M. L. Butler, and C. L. Holloway, “Optical Measurements of Strong Microwave Fields with Rydberg Atoms in a Vapor Cell,” Phys. Rev. Appl. 5(3), 034003 (2016).
[Crossref]

Calarco, T.

M. M. Müller, A. Kölle, R. Löw, T. Pfau, T. Calarco, and S. Montangero, “Room-temperature Rydberg single-photon source,” Phys. Rev. A 87(5), 053412 (2013).
[Crossref]

Che, J.

Che, J. L.

J. L. Che, J. Q. Ma, H. B. Zheng, Z. Y. Zhang, X. Yao, Y. Q. Zhang, and Y. P. Zhang, “Rydberg six-wave mixing process,” Europhys. Lett. 109(3), 33001 (2015).
[Crossref]

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24(4), 045404 (2014).
[Crossref]

Christodoulides, D. N.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Cirac, J. I.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurásek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432(7016), 482–486 (2004).
[Crossref] [PubMed]

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref] [PubMed]

Comparat, D.

I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nosbaum, V. M. Akulin, and P. Pillet, “Many-Body Effects in a Frozen Rydberg Gas,” Phys. Rev. Lett. 80(2), 253–256 (1998).
[Crossref]

Côté, R.

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atom pairs correlated to the ns–ns, np–np and nd–nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
[Crossref]

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

de Tomasi, F.

I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nosbaum, V. M. Akulin, and P. Pillet, “Many-Body Effects in a Frozen Rydberg Gas,” Phys. Rev. Lett. 80(2), 253–256 (1998).
[Crossref]

Dennis, M. R.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

Du, Y. G.

Y. G. Du, Y. P. Zhang, C. C. Zuo, C. B. Li, Z. Q. Nie, H. B. Zheng, M. Z. Shi, R. M. Wang, J. P. Song, K. Q. Lu, and M. Xiao, “Controlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparency,” Phys. Rev. A 79(6), 063839 (2009).
[Crossref]

Duan, L. M.

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref] [PubMed]

Epple, G.

A. Kölle, G. Epple, H. Kübler, R. Löw, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85(6), 063821 (2012).
[Crossref]

Eyler, E. E.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Farooqi, S. M.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Feizpour, A.

M. Kiffner, A. Feizpour, T. K. Kaczmarek, D. Jaksch, and J. Nunn, “Two-way interconversion of millimeter-wave and optical fields in Rydberg gases,” New J. Phys. 18(9), 093030 (2016).
[Crossref]

Fioretti, A.

I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nosbaum, V. M. Akulin, and P. Pillet, “Many-Body Effects in a Frozen Rydberg Gas,” Phys. Rev. Lett. 80(2), 253–256 (1998).
[Crossref]

Firstenberg, O.

O. Firstenberg, C. S. Adams, and S. Hofferberth, “Nonlieanr quantum optics mediated by Rydberg interactions,” J. Phys. At. Mol. Opt. Phys. 49(15), 152003 (2016).
[Crossref]

Fiurásek, J.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurásek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432(7016), 482–486 (2004).
[Crossref] [PubMed]

Gordon, J. A.

D. A. Anderson, S. A. Miller, G. Raithel, J. A. Gordon, M. L. Butler, and C. L. Holloway, “Optical Measurements of Strong Microwave Fields with Rydberg Atoms in a Vapor Cell,” Phys. Rev. Appl. 5(3), 034003 (2016).
[Crossref]

J. A. Gordon, C. L. Holloway, A. Schwarzkopf, D. A. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Millimeter wave detection via Autler-Townes splitting in rubidium Rydberg atoms,” Appl. Phys. Lett. 105(2), 024104 (2014).
[Crossref]

Gould, P. L.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Hänsch, T. W.

He, B.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

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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(5), 053806 (2014).
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T. Baluktsian, B. Huber, R. Löw, and T. Pfau, “Evidence for strong van der Waals Type Rydberg-Rydberg Interaction in a Thermal Vapor,” Phys. Rev. Lett. 110(12), 123001 (2013).
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B. Huber, T. Baluktsian, M. Schlagmüller, A. Kölle, H. Kübler, R. Löw, and T. Pfau, “GHz Rabi flopping to Rydberg states in hot atomic vapor cells,” Phys. Rev. Lett. 107(24), 243001 (2011).
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B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurásek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432(7016), 482–486 (2004).
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B. Huber, T. Baluktsian, M. Schlagmüller, A. Kölle, H. Kübler, R. Löw, and T. Pfau, “GHz Rabi flopping to Rydberg states in hot atomic vapor cells,” Phys. Rev. Lett. 107(24), 243001 (2011).
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B. Huber, T. Baluktsian, M. Schlagmüller, A. Kölle, H. Kübler, R. Löw, and T. Pfau, “GHz Rabi flopping to Rydberg states in hot atomic vapor cells,” Phys. Rev. Lett. 107(24), 243001 (2011).
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M. M. Müller, A. Kölle, R. Löw, T. Pfau, T. Calarco, and S. Montangero, “Room-temperature Rydberg single-photon source,” Phys. Rev. A 87(5), 053412 (2013).
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T. Baluktsian, B. Huber, R. Löw, and T. Pfau, “Evidence for strong van der Waals Type Rydberg-Rydberg Interaction in a Thermal Vapor,” Phys. Rev. Lett. 110(12), 123001 (2013).
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A. Kölle, G. Epple, H. Kübler, R. Löw, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85(6), 063821 (2012).
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B. Huber, T. Baluktsian, M. Schlagmüller, A. Kölle, H. Kübler, R. Löw, and T. Pfau, “GHz Rabi flopping to Rydberg states in hot atomic vapor cells,” Phys. Rev. Lett. 107(24), 243001 (2011).
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J. A. Gordon, C. L. Holloway, A. Schwarzkopf, D. A. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Millimeter wave detection via Autler-Townes splitting in rubidium Rydberg atoms,” Appl. Phys. Lett. 105(2), 024104 (2014).
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D. A. Anderson, S. A. Miller, G. Raithel, J. A. Gordon, M. L. Butler, and C. L. Holloway, “Optical Measurements of Strong Microwave Fields with Rydberg Atoms in a Vapor Cell,” Phys. Rev. Appl. 5(3), 034003 (2016).
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Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

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A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98(11), 113003 (2007).
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M. M. Müller, A. Kölle, R. Löw, T. Pfau, T. Calarco, and S. Montangero, “Room-temperature Rydberg single-photon source,” Phys. Rev. A 87(5), 053412 (2013).
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M. M. Müller, A. Kölle, R. Löw, T. Pfau, T. Calarco, and S. Montangero, “Room-temperature Rydberg single-photon source,” Phys. Rev. A 87(5), 053412 (2013).
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Y. P. Zhang, Z. Q. Nie, H. B. Zheng, C. B. Li, J. P. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013825 (2009).
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M. Kiffner, A. Feizpour, T. K. Kaczmarek, D. Jaksch, and J. Nunn, “Two-way interconversion of millimeter-wave and optical fields in Rydberg gases,” New J. Phys. 18(9), 093030 (2016).
[Crossref]

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Padgett, M. J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[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(5), 053806 (2014).
[Crossref]

T. Baluktsian, B. Huber, R. Löw, and T. Pfau, “Evidence for strong van der Waals Type Rydberg-Rydberg Interaction in a Thermal Vapor,” Phys. Rev. Lett. 110(12), 123001 (2013).
[Crossref] [PubMed]

M. M. Müller, A. Kölle, R. Löw, T. Pfau, T. Calarco, and S. Montangero, “Room-temperature Rydberg single-photon source,” Phys. Rev. A 87(5), 053412 (2013).
[Crossref]

A. Kölle, G. Epple, H. Kübler, R. Löw, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85(6), 063821 (2012).
[Crossref]

B. Huber, T. Baluktsian, M. Schlagmüller, A. Kölle, H. Kübler, R. Löw, and T. Pfau, “GHz Rabi flopping to Rydberg states in hot atomic vapor cells,” Phys. Rev. Lett. 107(24), 243001 (2011).
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I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nosbaum, V. M. Akulin, and P. Pillet, “Many-Body Effects in a Frozen Rydberg Gas,” Phys. Rev. Lett. 80(2), 253–256 (1998).
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S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
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B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurásek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432(7016), 482–486 (2004).
[Crossref] [PubMed]

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D. A. Anderson, S. A. Miller, G. Raithel, J. A. Gordon, M. L. Butler, and C. L. Holloway, “Optical Measurements of Strong Microwave Fields with Rydberg Atoms in a Vapor Cell,” Phys. Rev. Appl. 5(3), 034003 (2016).
[Crossref]

J. A. Gordon, C. L. Holloway, A. Schwarzkopf, D. A. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Millimeter wave detection via Autler-Townes splitting in rubidium Rydberg atoms,” Appl. Phys. Lett. 105(2), 024104 (2014).
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Scheid, M.

Schlagmüller, M.

B. Huber, T. Baluktsian, M. Schlagmüller, A. Kölle, H. Kübler, R. Löw, and T. Pfau, “GHz Rabi flopping to Rydberg states in hot atomic vapor cells,” Phys. Rev. Lett. 107(24), 243001 (2011).
[Crossref] [PubMed]

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J. A. Gordon, C. L. Holloway, A. Schwarzkopf, D. A. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Millimeter wave detection via Autler-Townes splitting in rubidium Rydberg atoms,” Appl. Phys. Lett. 105(2), 024104 (2014).
[Crossref]

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S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
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Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

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B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurásek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432(7016), 482–486 (2004).
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Y. G. Du, Y. P. Zhang, C. C. Zuo, C. B. Li, Z. Q. Nie, H. B. Zheng, M. Z. Shi, R. M. Wang, J. P. Song, K. Q. Lu, and M. Xiao, “Controlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparency,” Phys. Rev. A 79(6), 063839 (2009).
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K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atom pairs correlated to the ns–ns, np–np and nd–nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
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Y. G. Du, Y. P. Zhang, C. C. Zuo, C. B. Li, Z. Q. Nie, H. B. Zheng, M. Z. Shi, R. M. Wang, J. P. Song, K. Q. Lu, and M. Xiao, “Controlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparency,” Phys. Rev. A 79(6), 063839 (2009).
[Crossref]

Y. P. Zhang, Z. Q. Nie, H. B. Zheng, C. B. Li, J. P. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013825 (2009).
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K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atom pairs correlated to the ns–ns, np–np and nd–nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
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D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Thaicharoen, N.

J. A. Gordon, C. L. Holloway, A. Schwarzkopf, D. A. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Millimeter wave detection via Autler-Townes splitting in rubidium Rydberg atoms,” Appl. Phys. Lett. 105(2), 024104 (2014).
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Y. G. Du, Y. P. Zhang, C. C. Zuo, C. B. Li, Z. Q. Nie, H. B. Zheng, M. Z. Shi, R. M. Wang, J. P. Song, K. Q. Lu, and M. Xiao, “Controlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparency,” Phys. Rev. A 79(6), 063839 (2009).
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Weidemüller, M.

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atom pairs correlated to the ns–ns, np–np and nd–nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
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Xiao, M.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24(4), 045404 (2014).
[Crossref]

Y. G. Du, Y. P. Zhang, C. C. Zuo, C. B. Li, Z. Q. Nie, H. B. Zheng, M. Z. Shi, R. M. Wang, J. P. Song, K. Q. Lu, and M. Xiao, “Controlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparency,” Phys. Rev. A 79(6), 063839 (2009).
[Crossref]

Y. P. Zhang, Z. Q. Nie, H. B. Zheng, C. B. Li, J. P. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening Four-Wave Mixing And Six-Wave Mixing Channels via Dual Electromagnetically Induced Transparency Windows,” Phys. Rev. Lett. 99(12), 123603 (2007).
[Crossref] [PubMed]

W. G. Yang, A. Joshi, and M. Xiao, “Controlling dynamic instability of three-level atoms inside an optical ring cavity,” Phys. Rev. A 70(3), 033807 (2004).
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Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
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W. G. Yang, A. Joshi, and M. Xiao, “Controlling dynamic instability of three-level atoms inside an optical ring cavity,” Phys. Rev. A 70(3), 033807 (2004).
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Yao, X.

J. L. Che, J. Q. Ma, H. B. Zheng, Z. Y. Zhang, X. Yao, Y. Q. Zhang, and Y. P. Zhang, “Rydberg six-wave mixing process,” Europhys. Lett. 109(3), 33001 (2015).
[Crossref]

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24(4), 045404 (2014).
[Crossref]

Zhang, D.

Zhang, Y.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Z. Zhang, J. Che, D. Zhang, Z. Liu, X. Wang, and Y. Zhang, “Eight-wave mixing process in a Rydberg-dressing atomic ensemble,” Opt. Express 23(11), 13814–13822 (2015).
[Crossref] [PubMed]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening Four-Wave Mixing And Six-Wave Mixing Channels via Dual Electromagnetically Induced Transparency Windows,” Phys. Rev. Lett. 99(12), 123603 (2007).
[Crossref] [PubMed]

Zhang, Y. P.

J. L. Che, J. Q. Ma, H. B. Zheng, Z. Y. Zhang, X. Yao, Y. Q. Zhang, and Y. P. Zhang, “Rydberg six-wave mixing process,” Europhys. Lett. 109(3), 33001 (2015).
[Crossref]

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24(4), 045404 (2014).
[Crossref]

Y. G. Du, Y. P. Zhang, C. C. Zuo, C. B. Li, Z. Q. Nie, H. B. Zheng, M. Z. Shi, R. M. Wang, J. P. Song, K. Q. Lu, and M. Xiao, “Controlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparency,” Phys. Rev. A 79(6), 063839 (2009).
[Crossref]

Y. P. Zhang, Z. Q. Nie, H. B. Zheng, C. B. Li, J. P. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004).
[Crossref] [PubMed]

Zhang, Y. Q.

J. L. Che, J. Q. Ma, H. B. Zheng, Z. Y. Zhang, X. Yao, Y. Q. Zhang, and Y. P. Zhang, “Rydberg six-wave mixing process,” Europhys. Lett. 109(3), 33001 (2015).
[Crossref]

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24(4), 045404 (2014).
[Crossref]

Zhang, Z.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M. A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Z. Zhang, J. Che, D. Zhang, Z. Liu, X. Wang, and Y. Zhang, “Eight-wave mixing process in a Rydberg-dressing atomic ensemble,” Opt. Express 23(11), 13814–13822 (2015).
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Zhang, Z. Y.

J. L. Che, J. Q. Ma, H. B. Zheng, Z. Y. Zhang, X. Yao, Y. Q. Zhang, and Y. P. Zhang, “Rydberg six-wave mixing process,” Europhys. Lett. 109(3), 33001 (2015).
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H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24(4), 045404 (2014).
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Zheng, H. B.

J. L. Che, J. Q. Ma, H. B. Zheng, Z. Y. Zhang, X. Yao, Y. Q. Zhang, and Y. P. Zhang, “Rydberg six-wave mixing process,” Europhys. Lett. 109(3), 33001 (2015).
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H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24(4), 045404 (2014).
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Y. G. Du, Y. P. Zhang, C. C. Zuo, C. B. Li, Z. Q. Nie, H. B. Zheng, M. Z. Shi, R. M. Wang, J. P. Song, K. Q. Lu, and M. Xiao, “Controlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparency,” Phys. Rev. A 79(6), 063839 (2009).
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Y. P. Zhang, Z. Q. Nie, H. B. Zheng, C. B. Li, J. P. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013825 (2009).
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Zoller, P.

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
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Y. G. Du, Y. P. Zhang, C. C. Zuo, C. B. Li, Z. Q. Nie, H. B. Zheng, M. Z. Shi, R. M. Wang, J. P. Song, K. Q. Lu, and M. Xiao, “Controlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparency,” Phys. Rev. A 79(6), 063839 (2009).
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Appl. Phys. Lett. (1)

J. A. Gordon, C. L. Holloway, A. Schwarzkopf, D. A. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Millimeter wave detection via Autler-Townes splitting in rubidium Rydberg atoms,” Appl. Phys. Lett. 105(2), 024104 (2014).
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Europhys. Lett. (1)

J. L. Che, J. Q. Ma, H. B. Zheng, Z. Y. Zhang, X. Yao, Y. Q. Zhang, and Y. P. Zhang, “Rydberg six-wave mixing process,” Europhys. Lett. 109(3), 33001 (2015).
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J. Phys. At. Mol. Opt. Phys. (2)

O. Firstenberg, C. S. Adams, and S. Hofferberth, “Nonlieanr quantum optics mediated by Rydberg interactions,” J. Phys. At. Mol. Opt. Phys. 49(15), 152003 (2016).
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K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atom pairs correlated to the ns–ns, np–np and nd–nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
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Laser Phys. (1)

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24(4), 045404 (2014).
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Nature (3)

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M. Kiffner, A. Feizpour, T. K. Kaczmarek, D. Jaksch, and J. Nunn, “Two-way interconversion of millimeter-wave and optical fields in Rydberg gases,” New J. Phys. 18(9), 093030 (2016).
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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(5), 053806 (2014).
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Y. P. Zhang, Z. Q. Nie, H. B. Zheng, C. B. Li, J. P. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013825 (2009).
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W. G. Yang, A. Joshi, and M. Xiao, “Controlling dynamic instability of three-level atoms inside an optical ring cavity,” Phys. Rev. A 70(3), 033807 (2004).
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Other (1)

T. F. Gallagher, Rydberg Atoms (Cambridge University, 1994).

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Figures (5)

Fig. 1
Fig. 1 (colour online) (a) K-type five-level system of 85Rb atoms, where F = 3 (|0〉) and F = 2 (|3〉) which are two hyperfine state of the ground state 5S1/2, and a first excited state 5P3/2, F = 3(|1〉), a low-lying excited state 5D5/2, F = 2 (|4〉), and a highly excited Rydberg state 37D5/2 (|2〉) are involved. (b) Spatial arrangement of laser beams in the experiment and EWM phase matching condition diagram, where E i on behalf of different laser field, E M is the MWM field. (c) Zeeman sublevels with various transition pathways.
Fig. 2
Fig. 2 EIT and MWM spectra with changes in the polarization of the probe field versus Δ2. (a) EIT (a1) and MWM (a2) intensity spectra, which vary with the θ of QWP. (b) Blocking of E 4, EIT (b1) and SWM (b2) intensity spectra, which vary with the θ of QWP. (c) EIT (c1) and SWM (c2) intensity spectra changes with the power of E 1 when θ = 45°. (d) Identical to (c) except the power of E 2 is changed. (e) Images of the probe transmission at θ = 0° (e1), 35° (e2), 45° (e3), 55° (e4) and 65° (e5). (f) SWM images that correspond to (e). Corresponding Rabi frequency excluding (c) and (d), with Ω1 = 2π × 54 MHz, Ω2 = 2π × 7.6 MHz, Ω3 = 2π × 142 MHz, and Ω3′ = 2π × 224 MHz, and the ground atom density is 1.0 × 1012 cm−3. For the color coding each distribution has been normalized by the actual maximum intensity.
Fig. 3
Fig. 3 MWM signals with different powers of E 4 by scanning Δ2 when Δ1 = −400 MHz. Signal evolution with fixed E 4 power by setting the E 1 power to 100 μW (Ω1 = 2π × 10 MHz) (a1), 500 μW (Ω1 = 2π × 54 MHz) (a2), and 1 mW (Ω1 = 2π × 108 MHz) (a3). Corresponding dependence curves are shown in (b). Corresponding images of the probe (c) and SWM (d) at different powers of E 12 = 2π × 7.6 MHz; Ω3 = 2π × 142 MHz; Ω3′ = 2π × 224 MHz; and maximum Ω4 = 2π × 140 MHz). The color coding is same with Fig. 2.
Fig. 4
Fig. 4 (a) Variation of MWM signals with Δ4 when Δ2 is scanned and Δ1 is fixed at −400 MHz. (b) Theoretical simulation that corresponds to (a). (c) Switch between SWM and EWM by controlling the power of E 1 by scanning Δ2. Corresponding Rabi frequency at Ω1 = 2π × 54 MHz; Ω2 = 2π × 7.6 MHz; Ω3 = 2π × 142 MHz; Ω3′ = 2π × 224 MHz; and Ω4 = 2π × 100 MHz.
Fig. 5
Fig. 5 (a1) EWM signals versus Δ2 at different polarizations of the dressing field E 4 when the probe field is circularly polarized and Δ1 is fixed at −400 MHz. (a2) Identical to (a1) but with Δ1 = −700 MHz. (b) Dependence curves with respect to the angle of QWP added on E 4. (c) Corresponding Zeeman-level diagram between 5P3/2 and 5D5/2. The Rabi-frequencies of each field are identical to those in Fig. 4.

Equations (8)

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ρ 1 M + q 0 M ( 1 ) = i Ω 1 M + q 0 , ± /( d 1 M + q + | Ω 2 M + q | 0.4 / n 11 d 2 M + q + | Ω 3 M + q | 2 / d 3 M + q + | Ω 4 M + q | 2 / d 4 M + q ),
ρ 1 M + q 0 M ( 3 ) = i Ω 1 M + q 0 , ± | Ω 3 M + q | 2 e i φ / ( d 1 M + q + Ω 1 M + q 0 , ± / Γ 0 M + q 0 M ) 2 d 3 M + q .
ρ 1 M + q 0 M ( 5 ) = i Ω 1 M + q 0 , ± | Ω 3 M + q | 2 | Ω 2 M + q | 0.4 e i ( Δ Φ R + φ ) / n 4.4 m 1 3 d 3 M + q d 2 M + q
Re [ χ ] ω 2 = 3 π N 0 λ 1 3 Γ 10 4 Ω 2 M + q 2 16 Γ 10 2 Γ 20 2 + 8 Γ 10 Γ 20 Ω 2 M + q 2 + Ω 2 M + q 4 ,
Δ ω 2 = N 2 V U ( r r ) d 3 r ,
Δ n r = 12 π N 0 λ 1 3 Γ 10 N 2 Ω 2 M + q 2 16 Γ 4 + 8 Γ 2 Ω 2 M + q 2 + Ω 2 M + q 4 V U ( r r ) d 3 r .
Δ Φ R ( U ) = L ω 2 Δ n r / c = 4 π 2 N 0 μ 10 4 L ω 2 N 2 ε 0 n 0 2 c 2 4 Re ( | Ω 2 M + q | 2 d 1 M + q d 2 M + q D 1 2 d 2 M + q 2 ) V U ( r r ) d 3 r ,
ρ 1 M + q 0 M ( 7 ) = i Ω 1 M + q 0 , ± ( | Ω 2 M + q | / n 11 ) 0.4 | Ω 3 M + q | 2 | Ω 4 M + q | 2 e i ( Δ Φ R + φ ) / m 1 4 d 3 M + q d 2 M + q d 4 M + q .

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