Abstract

One- and two-photon renormalization is applied to three-level optical double resonance. An exact steady-state solution to the optical Bloch equations is given for arbitrarily strong, near-resonant, monochromatic fields connecting any two of the three possible level pairings. The results are discussed under various conditions and limits.

© 1994 Optical Society of America

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References

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  1. A. Javan, Phys. Rev. 107, 1579 (1957).
    [CrossRef]
  2. K. Shimoda and T. Shimizu, in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, New York, 1972), Vol. 2, Part 2.
  3. R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
    [CrossRef]
  4. R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
    [CrossRef]
  5. O. Blum, I. Kim, T. K. Gustafson, P. L. Kelley, and J.-P. E. Taran in Technical Digest of the Topical Conference on Nonlinear Optics: Materials, Phenomena, and Devices (IEEE, Piscataway, N.J., 1990), paper WP9.
  6. O. Blum, P. J. Harshman, T. K. Gustafson, and P. L. Kelley, Phys. Rev. A 47, 5165 (1993).
    [CrossRef] [PubMed]
  7. For a preliminary account of this work see P. L. Kelley, P. J. Harshman, O. Blum, and T. K. Gustafson in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper TuD.
  8. We have reversed the sign of the Γii’s for i≠ i′ compared with the definition given in Ref. 6. This simplifies the further analysis.

1993 (1)

O. Blum, P. J. Harshman, T. K. Gustafson, and P. L. Kelley, Phys. Rev. A 47, 5165 (1993).
[CrossRef] [PubMed]

1976 (1)

R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

1975 (1)

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[CrossRef]

1957 (1)

A. Javan, Phys. Rev. 107, 1579 (1957).
[CrossRef]

Blum, O.

O. Blum, P. J. Harshman, T. K. Gustafson, and P. L. Kelley, Phys. Rev. A 47, 5165 (1993).
[CrossRef] [PubMed]

For a preliminary account of this work see P. L. Kelley, P. J. Harshman, O. Blum, and T. K. Gustafson in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper TuD.

O. Blum, I. Kim, T. K. Gustafson, P. L. Kelley, and J.-P. E. Taran in Technical Digest of the Topical Conference on Nonlinear Optics: Materials, Phenomena, and Devices (IEEE, Piscataway, N.J., 1990), paper WP9.

Brewer, R. G.

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[CrossRef]

Gustafson, T. K.

O. Blum, P. J. Harshman, T. K. Gustafson, and P. L. Kelley, Phys. Rev. A 47, 5165 (1993).
[CrossRef] [PubMed]

For a preliminary account of this work see P. L. Kelley, P. J. Harshman, O. Blum, and T. K. Gustafson in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper TuD.

O. Blum, I. Kim, T. K. Gustafson, P. L. Kelley, and J.-P. E. Taran in Technical Digest of the Topical Conference on Nonlinear Optics: Materials, Phenomena, and Devices (IEEE, Piscataway, N.J., 1990), paper WP9.

Hahn, E. L.

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[CrossRef]

Harshman, P. J.

O. Blum, P. J. Harshman, T. K. Gustafson, and P. L. Kelley, Phys. Rev. A 47, 5165 (1993).
[CrossRef] [PubMed]

For a preliminary account of this work see P. L. Kelley, P. J. Harshman, O. Blum, and T. K. Gustafson in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper TuD.

Javan, A.

A. Javan, Phys. Rev. 107, 1579 (1957).
[CrossRef]

Kelley, P. L.

O. Blum, P. J. Harshman, T. K. Gustafson, and P. L. Kelley, Phys. Rev. A 47, 5165 (1993).
[CrossRef] [PubMed]

For a preliminary account of this work see P. L. Kelley, P. J. Harshman, O. Blum, and T. K. Gustafson in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper TuD.

O. Blum, I. Kim, T. K. Gustafson, P. L. Kelley, and J.-P. E. Taran in Technical Digest of the Topical Conference on Nonlinear Optics: Materials, Phenomena, and Devices (IEEE, Piscataway, N.J., 1990), paper WP9.

Kim, I.

O. Blum, I. Kim, T. K. Gustafson, P. L. Kelley, and J.-P. E. Taran in Technical Digest of the Topical Conference on Nonlinear Optics: Materials, Phenomena, and Devices (IEEE, Piscataway, N.J., 1990), paper WP9.

Shimizu, T.

K. Shimoda and T. Shimizu, in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, New York, 1972), Vol. 2, Part 2.

Shimoda, K.

K. Shimoda and T. Shimizu, in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, New York, 1972), Vol. 2, Part 2.

Stroud, C. R.

R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

Taran, J.-P. E.

O. Blum, I. Kim, T. K. Gustafson, P. L. Kelley, and J.-P. E. Taran in Technical Digest of the Topical Conference on Nonlinear Optics: Materials, Phenomena, and Devices (IEEE, Piscataway, N.J., 1990), paper WP9.

Whitley, R. M.

R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

Phys. Rev. (1)

A. Javan, Phys. Rev. 107, 1579 (1957).
[CrossRef]

Phys. Rev. A (3)

R. G. Brewer and E. L. Hahn, Phys. Rev. A 11, 1641 (1975).
[CrossRef]

R. M. Whitley and C. R. Stroud, Phys. Rev. A 14, 1498 (1976).
[CrossRef]

O. Blum, P. J. Harshman, T. K. Gustafson, and P. L. Kelley, Phys. Rev. A 47, 5165 (1993).
[CrossRef] [PubMed]

Other (4)

For a preliminary account of this work see P. L. Kelley, P. J. Harshman, O. Blum, and T. K. Gustafson in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper TuD.

We have reversed the sign of the Γii’s for i≠ i′ compared with the definition given in Ref. 6. This simplifies the further analysis.

O. Blum, I. Kim, T. K. Gustafson, P. L. Kelley, and J.-P. E. Taran in Technical Digest of the Topical Conference on Nonlinear Optics: Materials, Phenomena, and Devices (IEEE, Piscataway, N.J., 1990), paper WP9.

K. Shimoda and T. Shimizu, in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, New York, 1972), Vol. 2, Part 2.

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Figures (1)

Fig. 1
Fig. 1

Three possible energy diagrams. Levels 1 and 3 are connected by dipole transitions only to level 2. (a) Successive excitation double resonance. Here E1E2E3 and ω21, ω32, ωa, ωb ≥ 0. (b) Raman double resonance. Here E1E3E2, ω21, ωa, ≥ 0, and ω32, ωb ≤ 0. (c) Ground-state coupled double resonance. Here E2E1, E3 and ω21, ω32, ωa, ωb ≤ 0.

Equations (48)

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i d ρ ( t ) d t = 1 [ H ( t ) , ρ ( t ) ] + R ρ ( t ) ,
i d ρ ( t ) d t = [ L ( t ) + R ] ρ ( t ) ,
i d ρ ( t ) d t = [ L 0 + R + Ω ( t ) ] ρ ( t ) .
{ A ρ ( t ) } i j = i j A i j i j ρ i j ( t ) .
R i j i j = - i ( 1 - δ i j ) δ i i δ j j Γ i j - i δ i j δ i j Γ i i .
Ω i j i j ( ω ) = μ i j i j E ( ω ) / ,
μ i j i j = μ i i δ j j - δ i i μ j j .
L 0 = L 0 + R ,
i d ρ ( t ) d t = [ L 0 + Ω ( t ) ] ρ ( t ) .
ω ρ ( ω ) = L 0 ρ ( ω ) + f 1 Ω f 1 ρ ( ω - ω f 1 ) ;
ρ ( ω ) = 1 ω - L 0 f 1 Ω f 1 ρ ( ω - ω f 1 ) .
( ω - L 0 ) ρ ( ω ) = f 1 , f 2 Ω f 2 1 ω - ω f 2 - L 0 Ω f 1 ρ ( ω - ω f 1 - ω f 2 ) ,
Δ ρ a = γ a 0 ( S b + T b b - γ b 2 ) + γ b 0 ( T a b + γ a 2 ) ( S a + T a a - γ a 1 ) ( S b + T b b - γ b 2 ) - ( T a b + γ b 1 ) ( T a b + γ a 2 ) ,
Δ ρ b = γ b 0 ( S a + T a a - γ a 1 ) + γ a 0 ( T a b + γ b 1 ) ( S a + T a a - γ a 1 ) ( S b + T b b - γ b 2 ) - ( T a b + γ b 1 ) ( T a b + γ a 2 ) .
S a = Ω a 2 ( 1 D a + 1 D a * ) ,
S b = Ω b 2 ( 1 D b + 1 D b * ) ,
T a a = T D a 2 + c . c . ,
T a b = T D a D b + c . c . , T b b = T D b 2 + c . c .
D a = - i Δ ω a + Γ 21 ,
D b = - i Δ ω b + Γ 32 ,
D a b = - i ( Δ ω a + Δ ω b ) + Γ 31 ,
T = - Ω a 2 Ω b 2 ( 1 + S ) D a b ,
S = 1 D a b ( Ω a 2 D b + Ω b 2 D a ) .
γ i 0 = 1 3 ( Γ i 1 + Γ i 2 + Γ i 3 ) ,
γ i 1 = 1 3 ( 2 Γ i 1 - Γ i 2 - Γ i 3 ) ,
γ i 2 = 1 3 ( Γ i 1 + Γ i 2 - 2 Γ i 3 ) .
ρ a b = - Ω a Ω b ( 1 + S ) D a b ( Δ ρ a D a - Δ ρ b D b ) ,
ρ a = - i D a ( Ω a Δ ρ a + Ω b * ρ a b ) ,
ρ b = - i D b ( Ω b Δ ρ b - Ω a * ρ a b ) .
Δ ρ a = 3 Γ 0 × Δ ρ a ( 0 ) ( 2 S b + 2 T b b + T a b + Γ 0 ) + Δ ρ b ( 0 ) ( S b + T b b + T a b ) ( 3 S a + 3 T a a + 2 Γ 0 ) ( 3 S b + 3 T b b + 2 Γ 0 ) - ( 3 T a b - Γ 0 ) 2 ,
Δ ρ b = 3 Γ 0 × Δ ρ b ( 0 ) ( 2 S a + 2 T a a + T a b + Γ 0 ) + Δ ρ a ( 0 ) ( S a + T a a + T a b ) ( 3 S a + 3 T a a + 2 Γ 0 ) ( 3 S b + 3 T b b + 2 Γ 0 ) - ( 3 T a b - Γ 0 ) 2 .
Δ ρ a = - γ a 0 ( S b - γ b 2 ) + γ b 0 γ a 2 γ a 1 ( S b - γ b 2 ) + γ b 1 γ a 2 ,
Δ ρ b = γ b 0 γ a 1 - γ a 0 γ b 1 γ a 1 ( S b - γ b 2 ) + γ b 1 γ a 2 .
ρ a b = - Ω a Ω b D a D a b + Ω b 2 ,
ρ a = - i Ω a D a b D a D a b + Ω b 2 ,
ρ b = 0.
ρ a = - i Ω a D c + Ω c Ω b * D a D c + Ω b 2 ,
ρ b = 0 ,
ρ c = - Ω a Ω b + i Ω c D a D a D c + Ω b 2 ,
Δ ρ a = γ a 0 ( T - γ b 2 ) - γ b 0 ( T - γ a 2 ) ( T - γ a 1 ) ( T - γ b 2 ) - ( T - γ b 1 ) ( T - γ a 2 ) ,
Δ ρ b = γ b 0 ( T - γ a 1 ) - γ a 0 ( T - γ b 1 ) ( T - γ a 1 ) ( T - γ b 2 ) - ( T - γ b 1 ) ( T - γ a 2 ) .
Δ ρ a = g + T g + 2 T ,
Δ ρ b = - T g + 2 T .
ρ a b = - Ω a Ω b * ( 1 + S ) D a b D a ( 1 g + 2 T ) ( g + T D a + D b D b ) ,
ρ a = - i Ω a D a ( 1 g + 2 T ) × [ g + T - Ω b 2 ( 1 + S ) D a b D a ( g + T D a + D b D b ) ] ,
ρ b = i Ω b * D a ( 1 g + 2 T ) × [ T - Ω a 2 ( 1 + S ) D a b D a ( g + T D a + D b D b ) ] ,
ρ b = - i Ω b * Ω a 2 ( Δ ω a ) 2 ( D a b + Ω a 2 i Δ ω a ) ;
ω b = ω a - ω 31 + Ω a 2 ω a - ω 21 .

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