Abstract

Quantum corrections to the Lorentz-Lorenz formula are given for a dense ensemble of atoms interacting with the quantized radiation field. The influence of these corrections on local-field effects in two-level systems is discussed in the non-cooperative limit. For initially inverted atoms we find superluminescence and radiation trapping. Furthermore it is shown that the quantum corrections set strong limitations to intrinsic optical bistability.

© 1997 Optical Society of America

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References

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  1. H. A. Lorentz, Wiedem. Ann. 9, 641 (1880).
  2. L. Lorenz, Wiedem. Ann. 11, 70 (1881).
  3. M. Born and E. Wolf, Principles of Optics, (Wiley, New York, 1975).
  4. N. Bloembergen, Nonlinear Optics, (Benjamin, New York, 1965).
  5. C. M. Bowden and J. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A 47, 1247 (1993).
    [Crossref] [PubMed]
  6. J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and Nonlinear Optical Measurements of the Lorentz Local Field,” Phys. Rev. Lett. 67, 972 (1991).
    [Crossref] [PubMed]
  7. V. A. Sautenkov, H. van Kampen, E. R. Eliel, and J. P. Woerdman, “Dipole-Dipole Broadenend Lineshape in a Partially Excited Dense Atomic Gas,” Phys. Rev. Lett. 77, 3327 (1996).
    [Crossref] [PubMed]
  8. R. W. Boyd, Nonlinear Optics, (Academic Press, Boston, 1992).
  9. C. M. Bowden and C. C. Sung, “First and second order phase transitions in the Dicke model: Relation to optical bistability,” Phys. Rev. A 19, 2392 (1979).
    [Crossref]
  10. F. A. Hopf, C. M. Bowden, and W. H. Louisell, “Mirrorless optical bistability with the use of the local-field correction,” Phys. Rev. A 29, 2591 (1984).
    [Crossref]
  11. A.S. Manka, J.P. Dowling, C.M. Bowden, and M. Fleischhauer, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms,” Phys. Rev. Lett. 73, 1789 (1994).
    [Crossref] [PubMed]
  12. R. H. Dicke, “Coherence in Spontaneous Radiation Processes,”Phys. Rev. 93, 99 (1954).
    [Crossref]
  13. See for example the review article: M. Gross and S. Haroche, “Supperradiance: An essay on the theory of collective spontaneous emission” Phys. Rep. 93, 302–396 (1982).
    [Crossref]
  14. See also the recent textbook: A. V. Andreev, V.I. Emelyanov, and Yu. A. Ilinskii, Cooperative Effects in Optics, (Malvern Physics Series, IOP Publishing, London1993).
  15. M. Fleischhauer and S. F Yelin, “Quantum corrections to the Lorentz-Lorenz relation: Generalized Maxwell-Bloch equations for radiative interactions in dense atomic media”, (unpublished).
  16. see for example: W. H. Louisell, Quantum Statistical Properties of Radiation, (John Wiley & Sons, New York, 1973).
  17. M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. 73, 1103 (1994).
    [Crossref] [PubMed]
  18. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quasi-dark resonances in phase-coherent media,” (unpublished)

1996 (1)

V. A. Sautenkov, H. van Kampen, E. R. Eliel, and J. P. Woerdman, “Dipole-Dipole Broadenend Lineshape in a Partially Excited Dense Atomic Gas,” Phys. Rev. Lett. 77, 3327 (1996).
[Crossref] [PubMed]

1994 (2)

A.S. Manka, J.P. Dowling, C.M. Bowden, and M. Fleischhauer, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms,” Phys. Rev. Lett. 73, 1789 (1994).
[Crossref] [PubMed]

M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. 73, 1103 (1994).
[Crossref] [PubMed]

1993 (1)

C. M. Bowden and J. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A 47, 1247 (1993).
[Crossref] [PubMed]

1991 (1)

J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and Nonlinear Optical Measurements of the Lorentz Local Field,” Phys. Rev. Lett. 67, 972 (1991).
[Crossref] [PubMed]

1984 (1)

F. A. Hopf, C. M. Bowden, and W. H. Louisell, “Mirrorless optical bistability with the use of the local-field correction,” Phys. Rev. A 29, 2591 (1984).
[Crossref]

1982 (1)

See for example the review article: M. Gross and S. Haroche, “Supperradiance: An essay on the theory of collective spontaneous emission” Phys. Rep. 93, 302–396 (1982).
[Crossref]

1979 (1)

C. M. Bowden and C. C. Sung, “First and second order phase transitions in the Dicke model: Relation to optical bistability,” Phys. Rev. A 19, 2392 (1979).
[Crossref]

1954 (1)

R. H. Dicke, “Coherence in Spontaneous Radiation Processes,”Phys. Rev. 93, 99 (1954).
[Crossref]

Andreev, A. V.

See also the recent textbook: A. V. Andreev, V.I. Emelyanov, and Yu. A. Ilinskii, Cooperative Effects in Optics, (Malvern Physics Series, IOP Publishing, London1993).

Bloembergen, N.

N. Bloembergen, Nonlinear Optics, (Benjamin, New York, 1965).

Born, M.

M. Born and E. Wolf, Principles of Optics, (Wiley, New York, 1975).

Bowden, C. M.

C. M. Bowden and J. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A 47, 1247 (1993).
[Crossref] [PubMed]

F. A. Hopf, C. M. Bowden, and W. H. Louisell, “Mirrorless optical bistability with the use of the local-field correction,” Phys. Rev. A 29, 2591 (1984).
[Crossref]

C. M. Bowden and C. C. Sung, “First and second order phase transitions in the Dicke model: Relation to optical bistability,” Phys. Rev. A 19, 2392 (1979).
[Crossref]

Bowden, C.M.

A.S. Manka, J.P. Dowling, C.M. Bowden, and M. Fleischhauer, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms,” Phys. Rev. Lett. 73, 1789 (1994).
[Crossref] [PubMed]

Boyd, R. W.

J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and Nonlinear Optical Measurements of the Lorentz Local Field,” Phys. Rev. Lett. 67, 972 (1991).
[Crossref] [PubMed]

R. W. Boyd, Nonlinear Optics, (Academic Press, Boston, 1992).

Dicke, R. H.

R. H. Dicke, “Coherence in Spontaneous Radiation Processes,”Phys. Rev. 93, 99 (1954).
[Crossref]

Dowling, J.

C. M. Bowden and J. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A 47, 1247 (1993).
[Crossref] [PubMed]

Dowling, J.P.

A.S. Manka, J.P. Dowling, C.M. Bowden, and M. Fleischhauer, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms,” Phys. Rev. Lett. 73, 1789 (1994).
[Crossref] [PubMed]

Eliel, E. R.

V. A. Sautenkov, H. van Kampen, E. R. Eliel, and J. P. Woerdman, “Dipole-Dipole Broadenend Lineshape in a Partially Excited Dense Atomic Gas,” Phys. Rev. Lett. 77, 3327 (1996).
[Crossref] [PubMed]

Emelyanov, V.I.

See also the recent textbook: A. V. Andreev, V.I. Emelyanov, and Yu. A. Ilinskii, Cooperative Effects in Optics, (Malvern Physics Series, IOP Publishing, London1993).

Fleischhauer, M.

A.S. Manka, J.P. Dowling, C.M. Bowden, and M. Fleischhauer, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms,” Phys. Rev. Lett. 73, 1789 (1994).
[Crossref] [PubMed]

M. Fleischhauer and S. F Yelin, “Quantum corrections to the Lorentz-Lorenz relation: Generalized Maxwell-Bloch equations for radiative interactions in dense atomic media”, (unpublished).

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quasi-dark resonances in phase-coherent media,” (unpublished)

Gross, M.

See for example the review article: M. Gross and S. Haroche, “Supperradiance: An essay on the theory of collective spontaneous emission” Phys. Rep. 93, 302–396 (1982).
[Crossref]

Güdel, H. U.

M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. 73, 1103 (1994).
[Crossref] [PubMed]

Haroche, S.

See for example the review article: M. Gross and S. Haroche, “Supperradiance: An essay on the theory of collective spontaneous emission” Phys. Rep. 93, 302–396 (1982).
[Crossref]

Hehlen, M. P.

M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. 73, 1103 (1994).
[Crossref] [PubMed]

Hopf, F. A.

F. A. Hopf, C. M. Bowden, and W. H. Louisell, “Mirrorless optical bistability with the use of the local-field correction,” Phys. Rev. A 29, 2591 (1984).
[Crossref]

Ilinskii, Yu. A.

See also the recent textbook: A. V. Andreev, V.I. Emelyanov, and Yu. A. Ilinskii, Cooperative Effects in Optics, (Malvern Physics Series, IOP Publishing, London1993).

Lorentz, H. A.

H. A. Lorentz, Wiedem. Ann. 9, 641 (1880).

Lorenz, L.

L. Lorenz, Wiedem. Ann. 11, 70 (1881).

Louisell, W. H.

F. A. Hopf, C. M. Bowden, and W. H. Louisell, “Mirrorless optical bistability with the use of the local-field correction,” Phys. Rev. A 29, 2591 (1984).
[Crossref]

see for example: W. H. Louisell, Quantum Statistical Properties of Radiation, (John Wiley & Sons, New York, 1973).

Lukin, M. D.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quasi-dark resonances in phase-coherent media,” (unpublished)

Maki, J. J.

J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and Nonlinear Optical Measurements of the Lorentz Local Field,” Phys. Rev. Lett. 67, 972 (1991).
[Crossref] [PubMed]

Malcuit, M. S.

J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and Nonlinear Optical Measurements of the Lorentz Local Field,” Phys. Rev. Lett. 67, 972 (1991).
[Crossref] [PubMed]

Manka, A.S.

A.S. Manka, J.P. Dowling, C.M. Bowden, and M. Fleischhauer, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms,” Phys. Rev. Lett. 73, 1789 (1994).
[Crossref] [PubMed]

Rai, J.

M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. 73, 1103 (1994).
[Crossref] [PubMed]

Rai, S.

M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. 73, 1103 (1994).
[Crossref] [PubMed]

Rand, S. C.

M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. 73, 1103 (1994).
[Crossref] [PubMed]

Sautenkov, V. A.

V. A. Sautenkov, H. van Kampen, E. R. Eliel, and J. P. Woerdman, “Dipole-Dipole Broadenend Lineshape in a Partially Excited Dense Atomic Gas,” Phys. Rev. Lett. 77, 3327 (1996).
[Crossref] [PubMed]

Scully, M. O.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quasi-dark resonances in phase-coherent media,” (unpublished)

Shu, Q.

M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. 73, 1103 (1994).
[Crossref] [PubMed]

Sipe, J. E.

J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and Nonlinear Optical Measurements of the Lorentz Local Field,” Phys. Rev. Lett. 67, 972 (1991).
[Crossref] [PubMed]

Sung, C. C.

C. M. Bowden and C. C. Sung, “First and second order phase transitions in the Dicke model: Relation to optical bistability,” Phys. Rev. A 19, 2392 (1979).
[Crossref]

van Kampen, H.

V. A. Sautenkov, H. van Kampen, E. R. Eliel, and J. P. Woerdman, “Dipole-Dipole Broadenend Lineshape in a Partially Excited Dense Atomic Gas,” Phys. Rev. Lett. 77, 3327 (1996).
[Crossref] [PubMed]

Woerdman, J. P.

V. A. Sautenkov, H. van Kampen, E. R. Eliel, and J. P. Woerdman, “Dipole-Dipole Broadenend Lineshape in a Partially Excited Dense Atomic Gas,” Phys. Rev. Lett. 77, 3327 (1996).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, (Wiley, New York, 1975).

Yelin, S. F

M. Fleischhauer and S. F Yelin, “Quantum corrections to the Lorentz-Lorenz relation: Generalized Maxwell-Bloch equations for radiative interactions in dense atomic media”, (unpublished).

Yelin, S. F.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quasi-dark resonances in phase-coherent media,” (unpublished)

Phys. Rep. (1)

See for example the review article: M. Gross and S. Haroche, “Supperradiance: An essay on the theory of collective spontaneous emission” Phys. Rep. 93, 302–396 (1982).
[Crossref]

Phys. Rev. (1)

R. H. Dicke, “Coherence in Spontaneous Radiation Processes,”Phys. Rev. 93, 99 (1954).
[Crossref]

Phys. Rev. A (3)

C. M. Bowden and C. C. Sung, “First and second order phase transitions in the Dicke model: Relation to optical bistability,” Phys. Rev. A 19, 2392 (1979).
[Crossref]

F. A. Hopf, C. M. Bowden, and W. H. Louisell, “Mirrorless optical bistability with the use of the local-field correction,” Phys. Rev. A 29, 2591 (1984).
[Crossref]

C. M. Bowden and J. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A 47, 1247 (1993).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and Nonlinear Optical Measurements of the Lorentz Local Field,” Phys. Rev. Lett. 67, 972 (1991).
[Crossref] [PubMed]

V. A. Sautenkov, H. van Kampen, E. R. Eliel, and J. P. Woerdman, “Dipole-Dipole Broadenend Lineshape in a Partially Excited Dense Atomic Gas,” Phys. Rev. Lett. 77, 3327 (1996).
[Crossref] [PubMed]

A.S. Manka, J.P. Dowling, C.M. Bowden, and M. Fleischhauer, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms,” Phys. Rev. Lett. 73, 1789 (1994).
[Crossref] [PubMed]

M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. 73, 1103 (1994).
[Crossref] [PubMed]

Other (9)

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quasi-dark resonances in phase-coherent media,” (unpublished)

See also the recent textbook: A. V. Andreev, V.I. Emelyanov, and Yu. A. Ilinskii, Cooperative Effects in Optics, (Malvern Physics Series, IOP Publishing, London1993).

M. Fleischhauer and S. F Yelin, “Quantum corrections to the Lorentz-Lorenz relation: Generalized Maxwell-Bloch equations for radiative interactions in dense atomic media”, (unpublished).

see for example: W. H. Louisell, Quantum Statistical Properties of Radiation, (John Wiley & Sons, New York, 1973).

R. W. Boyd, Nonlinear Optics, (Academic Press, Boston, 1992).

H. A. Lorentz, Wiedem. Ann. 9, 641 (1880).

L. Lorenz, Wiedem. Ann. 11, 70 (1881).

M. Born and E. Wolf, Principles of Optics, (Wiley, New York, 1975).

N. Bloembergen, Nonlinear Optics, (Benjamin, New York, 1965).

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Figures (4)

Fig.1:
Fig.1:

Two level system with decay γ and cw coherent field Ω.

Fig.2:
Fig.2:

The decay of the the upper-level population ρaa for a ratio of the radiative to the inhomogeneous linewidth γ D = 0.1 including quantum corrections for different values of η.

Fig.3:
Fig.3:

The upper-level population ρaa plotted as function of the driving-field Rabi frequency Ω. Quantum corrections are not taken into account. Different curves correspond to different cooperativity ��. The transition between normal and bistable regime occurs at �� ~ 2.6.

Fig.4:
Fig.4:

The curves of the previous figure, but including quantum corrections for �� = 2.6 (a) and �� = 4 (b). The parameter ρ assumes the values 0, 0.002, 0.004.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

T R 1 N e γμ
ω ˙ ( l ) = ( γ + 2 gl ) ( ω ( l ) ω 0 ( l ) ) 2 ( ρ ab ( l ) c . c . )
ρ ˙ ab ( l ) = [ γ 2 + gl + i ( Δ + h l ) ] ρ ab ( l ) i Ω lL ω ( l )
Ω lL = Ω + C γ ρ ab ( l ) .
C = λ 3 N 4 π 2 ,
w 0 ( l ) = γ γ + 2 gl
g l = 2 Re [ s l ] ,
h l = 2 Im [ s l ] ,
s l ( t 1 ) = 4 ħ 4 j t 1 d t 2 t 1 d t 3 t 1 d t 4 e i ω 0 ( ( t 1 t 2 ) ( t 3 t 4 ) ) ×
× D ret ( r l , t 1 ; r j , t 3 ) D ret ( r l , t 2 ; r j , t 4 ) σ j ( t 4 ) σ j + ( t 3 ) .
D ret ( r 1 , t 1 ; r 2 , t 2 ) = i ħ 4 π 0 c Θ ( τ ) [ 2 τ 2 c 2 cos 2 ϑ 2 τ 2 ] δ ( r ) r ,
ρ ˙ aa = ( γ + 2 g ) ρ aa + g .
g γ = η 1 + 2 g γ 1 + ( γ Δ D ) 2 ( 1 2 + g γ ) 2 ρ aa ,
η = ( 3 4 ) 3 N λ 2 d ( γ Δ D ) 2 = 27 π 2 16 C d λ ( γ Δ D ) 2
ρ aa ss ( 2 ) 1 2 γ 4 Δ D η 1 .
g = 24 π 5 γ 2 [ γ + 2 g Γ 2 ( ρ aa ρ ab 2 ) Ω L 2 ( ( γ / 2 + g ) 2 ( Δ + h ) 2 ) Γ 4 ( α + γ / 2 + g ) ρ aa
+ i α + γ / 2 + g ( Ω L ρ ba Γ Ω L * ρ ab Γ * ) ( ρ aa i ( Ω L * ρ ab Γ Ω L ρ ba Γ * ) ) ] ,
h = 24 π 5 C ϱ γ 2 [ 2 Δ + h Γ 2 ( ρ aa ρ ab 2 ) + 2 Ω L 2 ( γ / 2 + g ) ( Δ + h ) Γ 4 ( α + γ / 2 + g ) ρ aa
+ 1 α + γ / 2 + g ( Ω L ρ ba Γ + Ω L * ρ ab Γ * ) ( ρ aa i ( Ω L * ρ ab Γ Ω L ρ ba Γ * ) ) ] ,
Γ = ( γ 2 + g ) + i ( Δ + h ) ,
α = ( γ + 2 g ) Ω L 2 Γ 2 .
P inc P coh 8 π C ϱ ( γ Ω ) 2 ,

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