Abstract

We investigate the way to control multi-wave mixing (MWM) process in Rydberg atoms via the interaction between Rydberg blockade and light field dressing effect. Considering both of the primary and secondary blockades, we theoretically study the MWM process in both diatomic and quadratomic systems, in which the enhancement, suppression and avoided crossing can be affected by the atomic internuclear distance or external electric field intensity. In the diatomic system, we also can eliminate the primary blockade by the dressing effect. Such investigations have potential applications in quantum computing with Rydberg atom as the carrier of qubit.

© 2013 OSA

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  1. H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. Büchler, “A Rydberg quantum simulator,” Nat. Phys.6(5), 382–388 (2010).
    [CrossRef]
  2. M. Saffman, T. G. Walker, and K. Mölmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys.82(3), 2313–2363 (2010).
    [CrossRef]
  3. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett.85(10), 2208–2211 (2000).
    [CrossRef] [PubMed]
  4. I. E. Protsenko, G. Reymond, N. Schlosser, and P. Grangier, “Operation of a quantum phase gate using neutral atoms in microscopic dipole traps,” Phys. Rev. A65(5), 052301 (2002).
    [CrossRef]
  5. 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]
  6. J. Stanojevic, R. Côté, D. Tong, E. E. Eyler, and P. L. Gould, “Long-range potentials and (n−1)d+ns molecular resonances in an ultracold Rydberg gas,” Phys. Rev. A78(5), 052709 (2008).
    [CrossRef]
  7. 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]
  8. A. Reinhard, T. C. Liebisch, B. Knuffman, and G. Raithel, “Level shifts of rubidium Rydberg states due to binary interactions,” Phys. Rev. A75(3), 032712 (2007).
    [CrossRef]
  9. M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A52(1), 311–328 (1995).
    [CrossRef] [PubMed]
  10. 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]
  11. J. Han and T. F. Gallagher, “Millimeter-wave rubidium Rydberg van der Waals spectroscopy,” Phys. Rev. A79(5), 053409 (2009).
    [CrossRef]
  12. E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys.5(2), 110–114 (2009).
    [CrossRef]
  13. D. Comparat and P. Pillet, “Dipole blockade in a cold Rydberg atomic sample,” J. Opt. Soc. Am. B27(6), A208–A232 (2010).
    [CrossRef]
  14. A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of twoindividual atoms in the Rydberg blockade regime,” Nat. Phys.5(2), 115–118 (2009).
    [CrossRef]
  15. T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, “Dipole blockade at Förster resonances in high resolution laser excitation of Rydberg states of cesium atoms,” Phys. Rev. Lett.97(8), 083003 (2006).
    [CrossRef] [PubMed]
  16. M. D. Lukin, M. Fleischhauer, R. Côté, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett.87(3), 037901 (2001).
    [CrossRef] [PubMed]
  17. S. E. Harris, “Electromagnetically Induced Transparency,” Phys. Today50(7), 36–42 (1997).
    [CrossRef]
  18. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
    [CrossRef]
  19. K. J. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
    [CrossRef] [PubMed]
  20. H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett.87(7), 073601 (2001).
    [CrossRef] [PubMed]
  21. J. Kou, R. G. Wan, Z. H. Kang, H. H. Wang, L. Jiang, X. J. Zhang, Y. Jiang, and J. Y. Gao, “EIT-assisted large cross-Kerr nonlinearity in a four-level inverted-Y atomic system,” J. Opt. Soc. Am. B27(10), 2035–2039 (2010).
    [CrossRef]
  22. D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett.93(18), 183601 (2004).
    [CrossRef] [PubMed]
  23. Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A67(1), 013811 (2003).
    [CrossRef]
  24. Y. P. 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]
  25. Y. P. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system,” Appl. Phys. Lett.90(11), 111104 (2007).
    [CrossRef]
  26. Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97(19), 193904 (2006).
    [CrossRef] [PubMed]
  27. H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
    [CrossRef] [PubMed]
  28. D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett.100(17), 170504 (2008).
    [CrossRef] [PubMed]
  29. 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]
  30. C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Antiblockade in Rydberg excitation of an ultracold lattice gas,” Phys. Rev. Lett.98(2), 023002 (2007).
    [CrossRef] [PubMed]
  31. T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, “Evidence of antiblockade in an ultracold Rydberg gas,” Phys. Rev. Lett.104(1), 013001 (2010).
    [CrossRef] [PubMed]
  32. E. Brekke, J. O. Day, and T. G. Walker, Phys. “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A78(6), 063830 (2008).
    [CrossRef]
  33. A. Kölle, G. Epple, H. Kübler, R. Löw, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A85(6), 063821 (2012).
    [CrossRef]

2012 (1)

A. Kölle, G. Epple, H. Kübler, R. Löw, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A85(6), 063821 (2012).
[CrossRef]

2011 (1)

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]

2010 (6)

H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. Büchler, “A Rydberg quantum simulator,” Nat. Phys.6(5), 382–388 (2010).
[CrossRef]

M. Saffman, T. G. Walker, and K. Mölmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys.82(3), 2313–2363 (2010).
[CrossRef]

D. Comparat and P. Pillet, “Dipole blockade in a cold Rydberg atomic sample,” J. Opt. Soc. Am. B27(6), A208–A232 (2010).
[CrossRef]

J. Kou, R. G. Wan, Z. H. Kang, H. H. Wang, L. Jiang, X. J. Zhang, Y. Jiang, and J. Y. Gao, “EIT-assisted large cross-Kerr nonlinearity in a four-level inverted-Y atomic system,” J. Opt. Soc. Am. B27(10), 2035–2039 (2010).
[CrossRef]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
[CrossRef] [PubMed]

T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, “Evidence of antiblockade in an ultracold Rydberg gas,” Phys. Rev. Lett.104(1), 013001 (2010).
[CrossRef] [PubMed]

2009 (3)

J. Han and T. F. Gallagher, “Millimeter-wave rubidium Rydberg van der Waals spectroscopy,” Phys. Rev. A79(5), 053409 (2009).
[CrossRef]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys.5(2), 110–114 (2009).
[CrossRef]

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of twoindividual atoms in the Rydberg blockade regime,” Nat. Phys.5(2), 115–118 (2009).
[CrossRef]

2008 (3)

J. Stanojevic, R. Côté, D. Tong, E. E. Eyler, and P. L. Gould, “Long-range potentials and (n−1)d+ns molecular resonances in an ultracold Rydberg gas,” Phys. Rev. A78(5), 052709 (2008).
[CrossRef]

E. Brekke, J. O. Day, and T. G. Walker, Phys. “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A78(6), 063830 (2008).
[CrossRef]

D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett.100(17), 170504 (2008).
[CrossRef] [PubMed]

2007 (5)

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]

C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Antiblockade in Rydberg excitation of an ultracold lattice gas,” Phys. Rev. Lett.98(2), 023002 (2007).
[CrossRef] [PubMed]

Y. P. 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]

Y. P. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system,” Appl. Phys. Lett.90(11), 111104 (2007).
[CrossRef]

A. Reinhard, T. C. Liebisch, B. Knuffman, and G. Raithel, “Level shifts of rubidium Rydberg states due to binary interactions,” Phys. Rev. A75(3), 032712 (2007).
[CrossRef]

2006 (2)

T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, “Dipole blockade at Förster resonances in high resolution laser excitation of Rydberg states of cesium atoms,” Phys. Rev. Lett.97(8), 083003 (2006).
[CrossRef] [PubMed]

Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97(19), 193904 (2006).
[CrossRef] [PubMed]

2005 (2)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

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]

2004 (2)

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]

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett.93(18), 183601 (2004).
[CrossRef] [PubMed]

2003 (1)

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A67(1), 013811 (2003).
[CrossRef]

2002 (1)

I. E. Protsenko, G. Reymond, N. Schlosser, and P. Grangier, “Operation of a quantum phase gate using neutral atoms in microscopic dipole traps,” Phys. Rev. A65(5), 052301 (2002).
[CrossRef]

2001 (2)

M. D. Lukin, M. Fleischhauer, R. Côté, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett.87(3), 037901 (2001).
[CrossRef] [PubMed]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett.87(7), 073601 (2001).
[CrossRef] [PubMed]

2000 (1)

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett.85(10), 2208–2211 (2000).
[CrossRef] [PubMed]

1997 (1)

S. E. Harris, “Electromagnetically Induced Transparency,” Phys. Today50(7), 36–42 (1997).
[CrossRef]

1995 (1)

M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A52(1), 311–328 (1995).
[CrossRef] [PubMed]

1991 (1)

K. J. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

Adams, C. S.

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]

Amthor, T.

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
[CrossRef] [PubMed]

T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, “Evidence of antiblockade in an ultracold Rydberg gas,” Phys. Rev. Lett.104(1), 013001 (2010).
[CrossRef] [PubMed]

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]

C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Antiblockade in Rydberg excitation of an ultracold lattice gas,” Phys. Rev. Lett.98(2), 023002 (2007).
[CrossRef] [PubMed]

Balic, V.

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett.93(18), 183601 (2004).
[CrossRef] [PubMed]

Boller, K. J.

K. J. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66(20), 2593–2596 (1991).
[CrossRef] [PubMed]

Braje, D. A.

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett.93(18), 183601 (2004).
[CrossRef] [PubMed]

Brekke, E.

E. Brekke, J. O. Day, and T. G. Walker, Phys. “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A78(6), 063830 (2008).
[CrossRef]

Browaeys, A.

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of twoindividual atoms in the Rydberg blockade regime,” Nat. Phys.5(2), 115–118 (2009).
[CrossRef]

Brown, A. W.

Y. P. 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]

Büchler, H. P.

H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. Büchler, “A Rydberg quantum simulator,” Nat. Phys.6(5), 382–388 (2010).
[CrossRef]

Chotia, A.

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of twoindividual atoms in the Rydberg blockade regime,” Nat. Phys.5(2), 115–118 (2009).
[CrossRef]

T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, “Dipole blockade at Förster resonances in high resolution laser excitation of Rydberg states of cesium atoms,” Phys. Rev. Lett.97(8), 083003 (2006).
[CrossRef] [PubMed]

Cirac, J. I.

M. D. Lukin, M. Fleischhauer, R. Côté, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett.87(3), 037901 (2001).
[CrossRef] [PubMed]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett.85(10), 2208–2211 (2000).
[CrossRef] [PubMed]

Comparat, D.

D. Comparat and P. Pillet, “Dipole blockade in a cold Rydberg atomic sample,” J. Opt. Soc. Am. B27(6), A208–A232 (2010).
[CrossRef]

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of twoindividual atoms in the Rydberg blockade regime,” Nat. Phys.5(2), 115–118 (2009).
[CrossRef]

T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, “Dipole blockade at Förster resonances in high resolution laser excitation of Rydberg states of cesium atoms,” Phys. Rev. Lett.97(8), 083003 (2006).
[CrossRef] [PubMed]

Côté, R.

J. Stanojevic, R. Côté, D. Tong, E. E. Eyler, and P. L. Gould, “Long-range potentials and (n−1)d+ns molecular resonances in an ultracold Rydberg gas,” Phys. Rev. A78(5), 052709 (2008).
[CrossRef]

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]

M. D. Lukin, M. Fleischhauer, R. Côté, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett.87(3), 037901 (2001).
[CrossRef] [PubMed]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett.85(10), 2208–2211 (2000).
[CrossRef] [PubMed]

Dalgarno, A.

M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A52(1), 311–328 (1995).
[CrossRef] [PubMed]

Day, J. O.

E. Brekke, J. O. Day, and T. G. Walker, Phys. “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A78(6), 063830 (2008).
[CrossRef]

Depaola, B. D.

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
[CrossRef] [PubMed]

Duan, L. M.

M. D. Lukin, M. Fleischhauer, R. Côté, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett.87(3), 037901 (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. A85(6), 063821 (2012).
[CrossRef]

Eyler, E. E.

J. Stanojevic, R. Côté, D. Tong, E. E. Eyler, and P. L. Gould, “Long-range potentials and (n−1)d+ns molecular resonances in an ultracold Rydberg gas,” Phys. Rev. A78(5), 052709 (2008).
[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]

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]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

M. D. Lukin, M. Fleischhauer, R. Côté, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett.87(3), 037901 (2001).
[CrossRef] [PubMed]

Fu, G.

Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97(19), 193904 (2006).
[CrossRef] [PubMed]

Fu, P.

Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97(19), 193904 (2006).
[CrossRef] [PubMed]

Gaëtan, A.

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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).
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H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
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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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H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
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E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys.5(2), 110–114 (2009).
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A. Reinhard, T. C. Liebisch, B. Knuffman, and G. Raithel, “Level shifts of rubidium Rydberg states due to binary interactions,” Phys. Rev. A75(3), 032712 (2007).
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Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97(19), 193904 (2006).
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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. A85(6), 063821 (2012).
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M. D. Lukin, M. Fleischhauer, R. Côté, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett.87(3), 037901 (2001).
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D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett.85(10), 2208–2211 (2000).
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M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
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[CrossRef]

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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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D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett.100(17), 170504 (2008).
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M. Saffman, T. G. Walker, and K. Mölmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys.82(3), 2313–2363 (2010).
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D. Møller, L. B. Madsen, and K. Mølmer, “Quantum gates and multiparticle entanglement by Rydberg excitation blockade and adiabatic passage,” Phys. Rev. Lett.100(17), 170504 (2008).
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H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. Büchler, “A Rydberg quantum simulator,” Nat. Phys.6(5), 382–388 (2010).
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C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Antiblockade in Rydberg excitation of an ultracold lattice gas,” Phys. Rev. Lett.98(2), 023002 (2007).
[CrossRef] [PubMed]

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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. A85(6), 063821 (2012).
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D. Comparat and P. Pillet, “Dipole blockade in a cold Rydberg atomic sample,” J. Opt. Soc. Am. B27(6), A208–A232 (2010).
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A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of twoindividual atoms in the Rydberg blockade regime,” Nat. Phys.5(2), 115–118 (2009).
[CrossRef]

T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, “Dipole blockade at Förster resonances in high resolution laser excitation of Rydberg states of cesium atoms,” Phys. Rev. Lett.97(8), 083003 (2006).
[CrossRef] [PubMed]

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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).
[CrossRef] [PubMed]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
[CrossRef] [PubMed]

C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Antiblockade in Rydberg excitation of an ultracold lattice gas,” Phys. Rev. Lett.98(2), 023002 (2007).
[CrossRef] [PubMed]

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I. E. Protsenko, G. Reymond, N. Schlosser, and P. Grangier, “Operation of a quantum phase gate using neutral atoms in microscopic dipole traps,” Phys. Rev. A65(5), 052301 (2002).
[CrossRef]

Raithel, G.

A. Reinhard, T. C. Liebisch, B. Knuffman, and G. Raithel, “Level shifts of rubidium Rydberg states due to binary interactions,” Phys. Rev. A75(3), 032712 (2007).
[CrossRef]

Reinhard, A.

A. Reinhard, T. C. Liebisch, B. Knuffman, and G. Raithel, “Level shifts of rubidium Rydberg states due to binary interactions,” Phys. Rev. A75(3), 032712 (2007).
[CrossRef]

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I. E. Protsenko, G. Reymond, N. Schlosser, and P. Grangier, “Operation of a quantum phase gate using neutral atoms in microscopic dipole traps,” Phys. Rev. A65(5), 052301 (2002).
[CrossRef]

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D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett.85(10), 2208–2211 (2000).
[CrossRef] [PubMed]

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C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Antiblockade in Rydberg excitation of an ultracold lattice gas,” Phys. Rev. Lett.98(2), 023002 (2007).
[CrossRef] [PubMed]

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M. Saffman, T. G. Walker, and K. Mölmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys.82(3), 2313–2363 (2010).
[CrossRef]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys.5(2), 110–114 (2009).
[CrossRef]

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Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A67(1), 013811 (2003).
[CrossRef]

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H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
[CrossRef] [PubMed]

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H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
[CrossRef] [PubMed]

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I. E. Protsenko, G. Reymond, N. Schlosser, and P. Grangier, “Operation of a quantum phase gate using neutral atoms in microscopic dipole traps,” Phys. Rev. A65(5), 052301 (2002).
[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).
[CrossRef] [PubMed]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
[CrossRef] [PubMed]

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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).
[CrossRef]

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J. Stanojevic, R. Côté, D. Tong, E. E. Eyler, and P. L. Gould, “Long-range potentials and (n−1)d+ns molecular resonances in an ultracold Rydberg gas,” Phys. Rev. A78(5), 052709 (2008).
[CrossRef]

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]

Sun, J.

Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97(19), 193904 (2006).
[CrossRef] [PubMed]

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J. Stanojevic, R. Côté, D. Tong, E. E. Eyler, and P. L. Gould, “Long-range potentials and (n−1)d+ns molecular resonances in an ultracold Rydberg gas,” Phys. Rev. A78(5), 052709 (2008).
[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]

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E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys.5(2), 110–114 (2009).
[CrossRef]

Viteau, M.

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of twoindividual atoms in the Rydberg blockade regime,” Nat. Phys.5(2), 115–118 (2009).
[CrossRef]

T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, “Dipole blockade at Förster resonances in high resolution laser excitation of Rydberg states of cesium atoms,” Phys. Rev. Lett.97(8), 083003 (2006).
[CrossRef] [PubMed]

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T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, “Dipole blockade at Förster resonances in high resolution laser excitation of Rydberg states of cesium atoms,” Phys. Rev. Lett.97(8), 083003 (2006).
[CrossRef] [PubMed]

Walker, T. G.

M. Saffman, T. G. Walker, and K. Mölmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys.82(3), 2313–2363 (2010).
[CrossRef]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys.5(2), 110–114 (2009).
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E. Brekke, J. O. Day, and T. G. Walker, Phys. “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A78(6), 063830 (2008).
[CrossRef]

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Wang, H.

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett.87(7), 073601 (2001).
[CrossRef] [PubMed]

Wang, H. H.

Weidemüller, M.

T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, “Evidence of antiblockade in an ultracold Rydberg gas,” Phys. Rev. Lett.104(1), 013001 (2010).
[CrossRef] [PubMed]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
[CrossRef] [PubMed]

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]

Weimer, H.

H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. Büchler, “A Rydberg quantum simulator,” Nat. Phys.6(5), 382–388 (2010).
[CrossRef]

Wilk, T.

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of twoindividual atoms in the Rydberg blockade regime,” Nat. Phys.5(2), 115–118 (2009).
[CrossRef]

Wu, L. A.

Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97(19), 193904 (2006).
[CrossRef] [PubMed]

Wu, Y.

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A67(1), 013811 (2003).
[CrossRef]

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Y. P. 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).
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Y. P. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system,” Appl. Phys. Lett.90(11), 111104 (2007).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett.87(7), 073601 (2001).
[CrossRef] [PubMed]

Yavuz, D. D.

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys.5(2), 110–114 (2009).
[CrossRef]

Yin, G. Y.

D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett.93(18), 183601 (2004).
[CrossRef] [PubMed]

Zhang, X. J.

Zhang, Y. P.

Y. P. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system,” Appl. Phys. Lett.90(11), 111104 (2007).
[CrossRef]

Y. P. 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]

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]

Zhao, J.

T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, “Dipole blockade at Förster resonances in high resolution laser excitation of Rydberg states of cesium atoms,” Phys. Rev. Lett.97(8), 083003 (2006).
[CrossRef] [PubMed]

Zhu, Y. F.

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A67(1), 013811 (2003).
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Zoller, P.

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Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97(19), 193904 (2006).
[CrossRef] [PubMed]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. Depaola, T. Amthor, M. Weidemüller, S. Sevinçli, and T. Pohl, “Coherent population trapping with controlled interparticle interactions,” Phys. Rev. Lett.104(17), 173602 (2010).
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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) The diagram of 85Rb energy levels with different coupling schemes in the five-level system. (b) The beam geometry diagram, in which the elliptic sample is designed so that atom density is uniform and the statistical variation of internuclear distance R can be neglected. (c) and (d) The diatomic systems consisting of 3 × 3 and 4 × 4 levels, respectively. The lines correspond to different light beams. Solid lines: E1; long dashed lines: E2 ( E 2 ); short dashed lines: E4; dash-dotted lines: E3 ( E 3 ). (e) The energy level shift curves for different symmetries of 70s-70s (dashed lines), 70p-70p (dotted lines) and 69d-69d (solid lines) of rubidium. (f) and (g) The diagrams of dressed FWM process in a Y-type four-level diatomic system and dressed SWM process in a reverse-Y type four-level diatomic system, with Rydberg blockade, respectively.

Fig. 2
Fig. 2

(a) The theoretical calculations of the intensity of (a1) the probe transmission and (a2) the doubly-dressed FWM signal E F DD versus Δ1 at discrete internuclear distances R = 77000, 78800, 80200, 82000 and 84000 a.u. (Δ4 is fixed at Δ4 = −70 MHz). (b) The theoretical calculations of the intensity of (b1) the probe transmission and (b2) E F DD versus R at discrete probe detunings Δ1 = −50, −25, −11, −5, 0, 5, 11, 25, and 50 MHz (Δ4 is fixed at Δ4 = −60 MHz). (c) The theoretical calculations of the intensity of E F DD versus (c1) probe detuning Δ1 (horizontal axis) and dressing detuning Δ4 (vertical axis), (c2) Δ1 and R, and (c3) Δ4 and R. (d) The energy levels corresponding to (b). (d1) to (d9) correspond to the nine curves from left to right in (b1) and (b2). The beam connecting |10>−|20> split |10> into |G2 ± 0>, and |G2+0> will be split into |G2+G4 ± 0> by the beam connecting |10>−|40>.

Fig. 3
Fig. 3

(a) The theoretical calculations of the intensity of (a1) the probe transmission and (a2) the singly-dressed SWM signal E S D versus Δ 1 at discrete internuclear distances R = 58000, 59500, 62000, 65000 and 68300 a.u. ( Δ 4 is fixed at Δ 4 = 100 MHz). (b) The theoretical calculations of the intensity of (b1) the probe transmission, (b2) the enhancement and suppression of E S D without considering d 40 , and (b3) the complete E S D versus R at discrete probe detunings Δ 1 = −80, −20, 0, 20, and 80 MHz ( Δ 4 is fixed at Δ 4 = 90 MHz). (c) The theoretical calculations of the intensity of E S D versus (c1) Δ 1 (horizontal axis) and dressing detuning Δ 4 (vertical axis), (c2) Δ 1 and R, and (c3) Δ 4 and R. (d) The energy levels corresponding to (b). (d1) to (d5) correspond to the curves from left to right in (b).

Fig. 4
Fig. 4

(a) The left figure presents the energy levels of a quadratomic system in atom states; μ1 is the dipole matrix element between |40s> and |40p>. The right figure illustrates the corresponding energy levels in pair sates, in which Δ is the energy gap between |40s40s> and |40p40p>. (b) The energy level shift curves of |40s40s> and |40p40p> when only the primary blockade is considered, with the external electric field intensity ε = 0. Solid line: |40s40s>; dashed lines: |40p40p>. The only line ascending with increasing R of |40p40p> is selected in calculating Δ. (c) The eigen-frequencies of the two levels E+(R) (the upper curve) and E(R) (the lower curve) with both the primary and secondary Rydberg blockades considered, versus R with ε fixed, when the eigen-frequency of |40s40s> without any blockade is set as zero. The neighborhoods of points A, B and C correspond to the regions where only resonant dipole-dipole interaction works, both resonant dipole-dipole and van der Waals interactions work, and only van der Waals interaction works, respectively. (d) The same quantity in (c) but versus ε with R fixed.

Fig. 5
Fig. 5

(a) The intensity of the doubly-dressed FWM signal E F DD in the quadratomic system, with energy level shift Δ|ss>+ versus R at discrete probe detunings Δ1 = −100, −50, −20, −10, 0, 10, 20, 50, and 100 MHz with the conditions (a1) |ΔΕ(ε, R)/2|>> | μ 1 2 / R 3 | , (a2) |ΔΕ(ε, R)/2| at intermediate value compared with | μ 1 2 / R 3 | , and (a3) |ΔΕ(ε, R)/2|<< | μ 1 2 / R 3 | . (b) The intensity of the singly-dressed SWM signal E S D in the quadratomic system, with the same condition as that in (a), except that Δ1 = −100, −20, 0, 20, and 100 MHz from the left curve to right one in each row.

Fig. 6
Fig. 6

(a) The intensity of the doubly-dressed FWM signal E F DD in the quadratomic system, versus ε at discrete probe detunings Δ1 = −100, −50, −20, −10, 0, 10, 20, 50, and 100 MHz, considering both primary and secondary Rydberg interactions for (a1) the upper energy level shift Δ|ss>+(ε), and (a2) the lower energy level shift Δ|ss>−(ε); R is fixed at R = 9000 a.u. (b) The intensity of the singly-dressed SWM signal E S D in the quadratomic system, with the same condition as that in (a), except that Δ1 = −100, −20, 0, 20, and 100 MHz, from the left curve to right one in each row.

Fig. 7
Fig. 7

(a1) The intensity of the undressed FWM signal E F in a ladder-type system, versus the probe detuning Δ1 at discrete atomic internuclear distances R = 1000000, 125000, 115000, 106000 and 98000 a.u. from top to bottom. For each curve in (a2) and (a3), R is the same as that of the corresponding curve in (a1). (a2) The intensity of the singly-dressed FWM signal, versus Δ1 at discrete dressing detunings Δ1 = 0, −13, −22, and −44 MHz, corresponding to the second to the fifth curves from top to bottom, respectively. The top curve is the same as the top one in (a1). (a3) The intensity of the singly-dressed FWM signal versus Δ1 at discrete Rabi frequencies of dressing field G5 = 15, 23, 32 and 41 MHz, corresponding to the second to fifth curves from top to bottom. The top curve is the same as the top one in (a1). Δ4 = 0. (b) The energy levels corresponding to (a).

Fig. 8
Fig. 8

The theoretical calculations of (a) singly-dressed FWM signal E F D and (b) the probe transmission versus R at discrete probe detunings Δ1 = 5, 10, 15, 20, 25, 30, 35 and 40 MHz from top to bottom. (c) The energy diagrams corresponding to (a).

Equations (11)

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ρ F (3) = G F d 20 ( d 10 +| G 4 | 2 / d 40 +| G 2 | 2 / d 20 ) 2 .
ρ S (5) = G S d 10 d 30 d 40 ( d 10 +| G 4 | 2 / d 40 ) 2 .
ρ F (3) = G F d 20 ( d 10 +| G 4 | 2 / d 40 +| G 2 | 2 / d 20 ) 2 .
ρ S (5) = G S d 10 d 30 d 40 ( d 10 +| G 4 | 2 / d 40 ) 2 ,
H=[ 0 μ 1 2 R 3 μ 1 2 R 3 ΔE(ε,R) ],
E ± =Δ E | ss ± (ε,R)= ΔE(ε,R) 2 ± ( ΔE(ε,R) 2 ) 2 + ( μ 1 2 R 3 ) 2 ,
E ± =Δ E | ss ± ΔE 2 ±[ ΔE 2 + 1 ΔE ( μ 1 2 R 3 ) 2 + ],
E ± =Δ E | ss ± =± μ 1 2 R 3 =± C 3 R 3 ,
ρ 1000 ( 3 ) = i G 1 G 2 ( G 2 ) * exp( i k F r ) ( Γ 10 +i Δ 1 + | G 2 | 2 Γ 20 +i( Δ 1 + Δ 2 ) + | G 4 | 2 Γ 40 +i( Δ 1 + Δ 2 +Δ E | ss ± /ћ) ) 2 [ Γ 20 +i( Δ 1 + Δ 2 ) ] ,
ρ 1000 ( 5 ) = i G 1 G 2 ( G 2 ) * G 4 ( G 4 ) * exp( i k S r ) ( Γ 10 +i Δ 1 + | G 4 | 2 Γ 40 +i( Δ 1 + Δ 2 +Δ E | ss ± /ћ) ) 2 [ Γ 30 +i( Δ 1 + Δ 3 ) ] × 1 Γ 40 +i( Δ 1 + Δ 2 +Δ E | ss ± /ћ) ,
ρ 1000 (3) = i G 1 G 2 ( G 2 ) * ( d 10 +| G 4 | / 2 d 40 ) 2 d 20 ,

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