Abstract

The four-wave mixing susceptibility χ(3)(ω1, ω1, -ω2) of a system of two-level atoms driven by weak external fields of frequencies ω1 and ω2 in a high-Q cavity is evaluated in the presence of atomic collisions. That susceptibility exhibits resonances corresponding to dipole-allowed transitions between the first two excited manifolds of the dressed states and the ground state and also the collision-induced Bloembergen resonances. The resonances corresponding to transitions between the first two manifolds of excited states that characterize the quantum and nonlinear nature of the atom–field interaction are, however, flattened in the absence of collisions. The collisions thus induce the quantum effects in four-wave mixing in a high-Q cavity.

© 1997 Optical Society of America

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  1. E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
    [CrossRef]
  2. G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
    [CrossRef] [PubMed]
  3. G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
    [CrossRef]
  4. G. S. Agarwal, J. Opt. Soc. Am. B 2, 480 (1985).
    [CrossRef]
  5. H. J. Carmichael, Phys. Rev. A 32, 3262 (1986).
    [CrossRef]
  6. M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, Phys. Rev. Lett. 63, 240 (1989).
    [CrossRef] [PubMed]
  7. R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. 68, 1132 (1992).
    [CrossRef] [PubMed]
  8. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
    [CrossRef] [PubMed]
  9. J. J. Sánchez-Mondragón, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
    [CrossRef]
  10. T. Yajima, Opt. Commun. 14, 378 (1975).
    [CrossRef]
  11. A. R. Bogdan, Y. Prior, and N. Bloembergen, Opt. Lett. 6, 82 (1981).
    [CrossRef] [PubMed]
  12. N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy V, A. R. W. Keller, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 157.
  13. N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 340.
  14. R. R. Puri and G. S. Agarwal, Phys. Rev. A 33, 3610 (1986); 35, 3433 (1987).
    [CrossRef] [PubMed]
  15. G. Grynberg, in Laser Spectroscopy V, A. R. W. Keller, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 174.
  16. H. J. Carmichael, Phys. Rev. Lett. 43, 1106 (1979).
    [CrossRef]
  17. G. V. Varada, M. S. Kumar, and G. S. Agarwal, Opt. Commun. 62, 328 (1987).
    [CrossRef]
  18. G. Scharf, Helv. Phys. Acta. 43, 806 (1976).
  19. G. S. Agarwal, R. K. Bullough, and N. Nayak, Opt. Commun. 85, 202 (1991).
    [CrossRef]
  20. J. I. Cirac, H. Ritsch, and P. Zoller, Phys. Rev. A 44, 4541 (1991).
    [CrossRef] [PubMed]

1992 (1)

R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. 68, 1132 (1992).
[CrossRef] [PubMed]

1991 (2)

G. S. Agarwal, R. K. Bullough, and N. Nayak, Opt. Commun. 85, 202 (1991).
[CrossRef]

J. I. Cirac, H. Ritsch, and P. Zoller, Phys. Rev. A 44, 4541 (1991).
[CrossRef] [PubMed]

1990 (1)

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

1989 (1)

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, Phys. Rev. Lett. 63, 240 (1989).
[CrossRef] [PubMed]

1987 (2)

G. V. Varada, M. S. Kumar, and G. S. Agarwal, Opt. Commun. 62, 328 (1987).
[CrossRef]

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

1986 (1)

H. J. Carmichael, Phys. Rev. A 32, 3262 (1986).
[CrossRef]

1985 (1)

1984 (1)

G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
[CrossRef]

1983 (1)

J. J. Sánchez-Mondragón, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

1981 (1)

1979 (1)

H. J. Carmichael, Phys. Rev. Lett. 43, 1106 (1979).
[CrossRef]

1976 (1)

G. Scharf, Helv. Phys. Acta. 43, 806 (1976).

1975 (1)

T. Yajima, Opt. Commun. 14, 378 (1975).
[CrossRef]

1963 (1)

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, R. K. Bullough, and N. Nayak, Opt. Commun. 85, 202 (1991).
[CrossRef]

G. V. Varada, M. S. Kumar, and G. S. Agarwal, Opt. Commun. 62, 328 (1987).
[CrossRef]

G. S. Agarwal, J. Opt. Soc. Am. B 2, 480 (1985).
[CrossRef]

G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
[CrossRef]

Bloembergen, N.

Bogdan, A. R.

Brecha, R. J.

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, Phys. Rev. Lett. 63, 240 (1989).
[CrossRef] [PubMed]

Bullough, R. K.

G. S. Agarwal, R. K. Bullough, and N. Nayak, Opt. Commun. 85, 202 (1991).
[CrossRef]

Carmichael, H. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, Phys. Rev. Lett. 63, 240 (1989).
[CrossRef] [PubMed]

H. J. Carmichael, Phys. Rev. A 32, 3262 (1986).
[CrossRef]

H. J. Carmichael, Phys. Rev. Lett. 43, 1106 (1979).
[CrossRef]

Cirac, J. I.

J. I. Cirac, H. Ritsch, and P. Zoller, Phys. Rev. A 44, 4541 (1991).
[CrossRef] [PubMed]

Cummings, F. W.

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Eberly, J. H.

J. J. Sánchez-Mondragón, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Gauthier, D. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Jaynes, E. T.

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Kimble, H. J.

R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. 68, 1132 (1992).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, Phys. Rev. Lett. 63, 240 (1989).
[CrossRef] [PubMed]

Klein, N.

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

Kumar, M. S.

G. V. Varada, M. S. Kumar, and G. S. Agarwal, Opt. Commun. 62, 328 (1987).
[CrossRef]

Morin, S. E.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Mossberg, T. W.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Narozhny, N. B.

J. J. Sánchez-Mondragón, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Nayak, N.

G. S. Agarwal, R. K. Bullough, and N. Nayak, Opt. Commun. 85, 202 (1991).
[CrossRef]

Prior, Y.

Raizen, M. G.

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, Phys. Rev. Lett. 63, 240 (1989).
[CrossRef] [PubMed]

Rempe, G.

R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. 68, 1132 (1992).
[CrossRef] [PubMed]

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

Ritsch, H.

J. I. Cirac, H. Ritsch, and P. Zoller, Phys. Rev. A 44, 4541 (1991).
[CrossRef] [PubMed]

Sánchez-Mondragón, J. J.

J. J. Sánchez-Mondragón, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Scharf, G.

G. Scharf, Helv. Phys. Acta. 43, 806 (1976).

Thompson, R. J.

R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. 68, 1132 (1992).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, Phys. Rev. Lett. 63, 240 (1989).
[CrossRef] [PubMed]

Varada, G. V.

G. V. Varada, M. S. Kumar, and G. S. Agarwal, Opt. Commun. 62, 328 (1987).
[CrossRef]

Walther, H.

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

Wu, Q.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Yajima, T.

T. Yajima, Opt. Commun. 14, 378 (1975).
[CrossRef]

Zhu, Y.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Zoller, P.

J. I. Cirac, H. Ritsch, and P. Zoller, Phys. Rev. A 44, 4541 (1991).
[CrossRef] [PubMed]

Helv. Phys. Acta. (1)

G. Scharf, Helv. Phys. Acta. 43, 806 (1976).

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

Opt. Commun. (3)

G. S. Agarwal, R. K. Bullough, and N. Nayak, Opt. Commun. 85, 202 (1991).
[CrossRef]

T. Yajima, Opt. Commun. 14, 378 (1975).
[CrossRef]

G. V. Varada, M. S. Kumar, and G. S. Agarwal, Opt. Commun. 62, 328 (1987).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

J. I. Cirac, H. Ritsch, and P. Zoller, Phys. Rev. A 44, 4541 (1991).
[CrossRef] [PubMed]

H. J. Carmichael, Phys. Rev. A 32, 3262 (1986).
[CrossRef]

Phys. Rev. Lett. (7)

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, Phys. Rev. Lett. 63, 240 (1989).
[CrossRef] [PubMed]

R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. 68, 1132 (1992).
[CrossRef] [PubMed]

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

J. J. Sánchez-Mondragón, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
[CrossRef]

H. J. Carmichael, Phys. Rev. Lett. 43, 1106 (1979).
[CrossRef]

Proc. IEEE (1)

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[CrossRef]

Other (4)

N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy V, A. R. W. Keller, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 157.

N. Bloembergen, A. R. Bogdan, and M. W. Downer, in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, Berlin, 1979), p. 340.

R. R. Puri and G. S. Agarwal, Phys. Rev. A 33, 3610 (1986); 35, 3433 (1987).
[CrossRef] [PubMed]

G. Grynberg, in Laser Spectroscopy V, A. R. W. Keller, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p. 174.

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Figures (5)

Fig. 1
Fig. 1

S=4g3|χ(3)|/N|d|4 as a function of (ω2-ω0)/g for ω1 =ω0; γ=0.001g, κ=0.05g, and γc=0.

Fig. 2
Fig. 2

Same as Fig. 1 but for γc=0.04g.

Fig. 3
Fig. 3

Same as Fig. 1 but for γc=0.05g.

Fig. 4
Fig. 4

Same as Fig. 1 but for γc=0.08g.

Fig. 5
Fig. 5

Same as Fig. 1 but for γc=0.2g.

Equations (35)

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H0=ω0Sz+ω0aa+g(aS++aS-),
He(t)=d[1 exp(-iω1t)+2 exp(-iω2t)]S++h.c.,
dρdt=Le(t)ρ+Lρ
Le(t)ρ=-i[He(t), ρ],Lρ=-i[H0, ρ]+(Λ+Λc)ρ,
Λρ=κ(2aρa-aaρ-ρaa)+γ(2S-ρS+-S+S-ρ-ρS+S-)
Λcρ=γc(2SzρSz-Sz2ρ-ρSz2).
ρ(3)(Ω)=d2d*1L+iΩ×S+, 1L+i(ω1-ω2)×S+, 1L-iω2[S-, ρ(0)]+S+, 1L+i(ω1-ω2)×S-,1L+iω1[S+, ρ(0)]+S-,1L+2iω1×S+, 1L+iω1[S+, ρ(0)],
ρ(0)=|0, -1/20, -1/2|,
χ(3)(ω1, ω1, -ω2)=N|d|4[(ω2-ω1)-2i(γ+γc)][(ω0-Ω)-i(γ+γc)][(ω0-ω1)-i(γ+γc)]×1[(ω2-ω0)-i(γ+γc)][(ω2-ω1)-2iγ]
|ψm±=12(|m, ½±|m+1, -½),
m=0, 1, 2, 
λm±=ω0(m+½)±gm+1.
S+|ψm±=±½(|ψm+1++|ψm+1-),
S-|ψm±=½(|ψm-1+-|ψm-1-),
a|ψm±=±½(Γ±,m|ψm-1+-Γ,m|ψm-1-),
a|ψm±=±½(Γ±,m+1|ψm+1+-Γ,m+1|ψm+1-),
Sz|ψm±=½|ψm,
Γ±,m=m+1±m.
Λρ=(κ+γ)(|0, -1/2ψ0+|ρ|ψ0+0, -1/2|+|0, -1/2×ψ0-|ρ|ψ0-0, -1/2|)+12m=0((κΓ+,m+12+γ)×(|ψm+ψm+1+|ρ|ψm+1+ψm+|+|ψm-×ψm+1-|ρ|ψm+1-ψm-|)+(κΓ-,m+12+γ)×(|ψm+ψm+1-|ρ|ψm+1-ψm+|+|ψm-×ψm+1+|ρ|ψm+1+ψm-|)-{[κ(2m+1)+γ]}×[(|ψm+ψm+|+|ψm-ψm-|)ρ+ρ(|ψm+ψm+|+|ψm-ψm-|)]),
Λcρ=γc2(|ψm-ψm+|ρ|ψm+ψm-|+|ψm+ψm-|ρ|ψm-×ψm+|+|0, -1/20, -1/2|ρ|0, -1/2×0, -1/2|-ρ).
χ(3)(ω1, ω1, -ω2)=N|d|441D1+(Ω)2p1+p2-p4+igD1+(ω1)D1-(ω1)1D2+(2ω1)+1D2-(2ω1)+1D1-(Ω)p1+2p2-p3-igD1+(ω1)D1-(ω1)×1D2+(2ω1)+1D2-(2ω1)-1D3+(Ω)×p1+p32+igD1+(ω1)D1-(ω1)D2+(2ω1)-1D4+(Ω)p2+p42-igD1+(ω1)D1-(ω1)D2+(2ω1)-1D3-(Ω)×p2+p42-igD1+(ω1)D1-(ω1)D2-(2ω1)-1D4-(Ω)p1+p32+igD1+(ω1)D1-(ω1)D2-(2ω1),
p1=1i(ω1-ω2)-(κ+γ)×i(ω1-ω2)-(κ+γ+γc/2)D1+*(ω2)D1+(ω1)-γc/2D1-*(ω2)D1-(ω1),
p2=1i(ω1-ω2)-(κ+γ)×i(ω1-ω2)-(κ+γ+γc/2)D1-*(ω2)D1-(ω1)-γc/2D1+*(ω2)D1+(ω1),
p3=-1D1+*(ω2)D1-(ω1)×2ig+i(ω1-ω2)-(κ+γ+γc)2ig+i(ω1-ω2)-(κ+γ+γc/2),
p4=-1D1-*(ω2)D1+(ω1)×-2ig+i(ω1-ω2)-(κ+γ+γc)-2ig+i(ω1-ω2)-(κ+γ+γc/2);
D1±(α)=-i(ω0-α±g)-κ+γ+γc2,
D2±(α)=-i(2ω0-α±g2)-3κ+γ+γc2,
D3±(α)=-i[ω0-α±g(2-1)]-2κ+γ+γc2,
D4±(α)=-i[ω0-α±g(2+1)]-2κ+γ+γc2.
(1)ω2=ω0±g,
(2)ω2=2ω1-ω0±g,
(3)ω2=2ω1-ω0±g(2-1),
(4)ω2=2ω1-ω0±g(2+1),
(5)ω2=ω1,
(6)ω2=ω1±2g.

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