Abstract

Using a master equation with cooperative interaction of radiative nature included, we demonstrate the generation and relaxation characteristics of the coherent population trapping state. We also show how the microscopic master equation in the mean field approximation leads to density matrix equations obtained from local field considerations.

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References

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  1. For a recent review on coherent population trapping see E. Arimondo in Progress in Optics, Vol. XXXV, ed. E. Wolf (North-Holland, Amsterdam, 1996) p. 257 and references therein.
  2. I.V. Jyotsna and G.S. Agarwal, Phys. Rev. A 53, 1690 (1996).
  3. C.M. Bowden, A.S. Manka, J.P. Dowling and M. Fleischhauer, in Coherence and Quantum Optics, eds. J.H. Eberly, L. Mandel and E. Wolf (Plenum, NewYork, 1996) p. 271.
  4. G.S. Agarwal, Quantum Optics (Springer-Verlag, Berlin, 1974) Sec. 6.
  5. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980) Chap. 2.
  6. R. Friedberg, S.R. Hartmann, and Jamal T. Manassah, Phys. Rev. A 40, 2446 (1989); Phys. Rev. A 42, 494 (1990).
  7. J.J. Maki, M.S. Malcuit, J.E. Sipe, and R.W. Boyd, Phys. Rev. Lett. 67, 972 (1991).
    [CrossRef] [PubMed]
  8. Nonlinear density matrix equations have been used earlier in quantum optics, see e.g. [2,3] and G.S. Agarwal, Phys. Rev. A 4, 1791 (1971); K. Molmer and Y. Castin, Ref. 3, p. 193.

Other (8)

For a recent review on coherent population trapping see E. Arimondo in Progress in Optics, Vol. XXXV, ed. E. Wolf (North-Holland, Amsterdam, 1996) p. 257 and references therein.

I.V. Jyotsna and G.S. Agarwal, Phys. Rev. A 53, 1690 (1996).

C.M. Bowden, A.S. Manka, J.P. Dowling and M. Fleischhauer, in Coherence and Quantum Optics, eds. J.H. Eberly, L. Mandel and E. Wolf (Plenum, NewYork, 1996) p. 271.

G.S. Agarwal, Quantum Optics (Springer-Verlag, Berlin, 1974) Sec. 6.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980) Chap. 2.

R. Friedberg, S.R. Hartmann, and Jamal T. Manassah, Phys. Rev. A 40, 2446 (1989); Phys. Rev. A 42, 494 (1990).

J.J. Maki, M.S. Malcuit, J.E. Sipe, and R.W. Boyd, Phys. Rev. Lett. 67, 972 (1991).
[CrossRef] [PubMed]

Nonlinear density matrix equations have been used earlier in quantum optics, see e.g. [2,3] and G.S. Agarwal, Phys. Rev. A 4, 1791 (1971); K. Molmer and Y. Castin, Ref. 3, p. 193.

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram (a) of the system and its energy level and (b) of the interaction with the dipole field.

Fig. 2
Fig. 2

Relaxation of various elements of the density matrix. We plot deviation ρ̃ from steady state (ρ̃ ≡ ρ - ρ s ) for G 1 = G 2 = γ 1 = γ 2 = 1, Δ1 = Δ2 =0, α 1 = α 2=0 (solid), 0.9 (dot) and 1.5 (dashed).

Equations (25)

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ρ t = i [ H , ρ ] + Λ ρ ,
H = j H j ,
H j = ħ Δ 1 A 11 j + ħ ( Δ 1 Δ 2 ) A 22 j
ħ ( G 1 j A 13 j + G 2 j A 12 j + H . C . ) ,
Λ ρ jl γ jl ( 1 ) ( A 13 j A 31 2 A 31 A 13 j + ρA 13 j A 31 l )
jl γ jl ( 2 ) ( A 12 j A 21 2 A 21 A 12 j + ρA 12 j A 21 l )
i j l Ω jl ( 1 ) [ A 13 j A 31 l , ρ ] i j l Ω jl ( 2 ) [ A 12 j A 21 l , ρ ] .
Ω jl ( 1 ) + jl ( 1 ) = 1 ħ κμ ( d 13 ) κ ( d 13 * ) μ χ κμ r j r l ω 13 ,
χ κμ r 1 r 2 ω = ( ω 2 c 2 δ κμ + 2 r 1 κ r 2 μ ) e i ω c r 1 r 2 r 1 r 2 .
ρ s = j ψ jj ψ ,
ψ j = G 2 j 3 j G 1 j 2 j ( G ij 2 + G 2 j 2 ) 1 / 2 .
ρ j ρ ( j ) ,
ρ ( j ) t = i [ H ˜ ( j ) , ρ ( j ) ]
γ jj ( 1 ) ( A 13 j A 31 A 31 A 13 j + H . C . )
γ jj ( 2 ) ( A 12 j A 21 A 21 A 12 j + H . C . ) ,
G 1 j G 1 j + l j [ Ω jl ( 1 ) + jl ( 1 ) ] A 31 l ,
G 2 j G 2 j + l j [ Ω jl ( 2 ) + jl ( 2 ) ] A 21 l ,
ρ ˙ 11 = 2 ( γ 1 + γ 2 ) ρ 11 + i G 1 ρ 31 + i G 2 ρ 21 + c . c . ,
ρ ˙ 12 = { γ 1 + γ 2 i [ Δ 2 + α 2 ( ρ 22 ρ 11 ) ] } ρ 12 + iG 1 ρ 32
+ iG 2 ( ρ 22 ρ 11 ) + 1 ρ 13 ρ 32 ,
ρ ˙ 13 = { γ 1 + γ 2 i [ Δ 1 + α 1 ( 1 2 ρ 11 ρ 22 ) ] } ρ 13 + iG 2 ρ 23
+ iG 1 ( 1 2 ρ 11 ρ 22 ) + 2 ρ 12 ρ 23 ,
ρ ˙ 22 = 2 γ 2 ρ 11 iG 2 ρ 21 + c . c . ,
ρ ˙ 23 = i ( Δ 1 + Δ 2 ) ρ 23 i G 1 ρ 21 + iG 2 * + i ( α 2 α 1 ) ρ 13 ρ 21 .
α 1 = 4 πn d 13 2 3 ħ , α 2 = 4 πn d 12 2 3 ħ .

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