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.

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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-eld 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 Eects 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, London 1993).
  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. G udel, 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)

Other (18)

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).

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]

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]

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

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-eld correction," Phys. Rev. A 29, 2591 (1984).
[CrossRef]

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

R. H. Dicke, "Coherence in Spontaneous Radiation Processes,"Phys. Rev. 93, 99 (1954).
[CrossRef]

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]

See also the recent textbook: A. V. Andreev, V.I. Emelyanov, and Yu. A. Ilinskii, Cooperative Effects in Optics, (Malvern Physics Series, IOP Publishing, London 1993).

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).

M.P.Hehlen,H. U. G udel, 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]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, "Quasi-dark resonances in phase-coherent media," (unpublished)

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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)

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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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