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.

© 1997 Optical Society of America

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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 .

1996 (1)

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

1991 (1)

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

1989 (1)

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

1971 (1)

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 .

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 .

Agarwal, G.S.

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

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 .

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

Arimondo, E.

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.

Born, M.

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

Bowden, C.M.

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 .

Boyd, R.W.

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

Castin, Y.

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 .

Dowling, J.P.

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 .

Fleischhauer, M.

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 .

Friedberg, R.

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

Hartmann, S.R.

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

Jyotsna, I.V.

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

Maki, J.J.

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

Malcuit, M.S.

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

Manassah, Jamal T.

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

Manka, A.S.

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 .

Molmer, K.

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 .

Sipe, J.E.

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

Wolf, E.

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

Phys. Rev. (3)

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

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

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 .

Phys. Rev. Lett. (1)

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

Other (4)

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.

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.

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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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