Abstract

We show theoretically that velocity-selective coherent population trapping in one dimension may be realized in atomic transitions other than Jg = 1 ↔ Je = 1. The atomic momentum distribution resulting from irradiation by counterpropagating σ+σ waves on the Jg = 3/2 ↔ Je = 1/2 and the Jg = 2 ↔ Je = 1 atomic transitions is investigated through solution of the optical Bloch equations and determination of the effective loss rates for atomic eigenstates. An inverted-W atomic level configuration is also used to investigate the features of velocity-selective coherent population trapping. The momentum distribution exhibits peaks at the ±ħk or the ±ħ2k and the 0 momenta, depending on the atomic transitions and the laser intensity. These structures, generated by atomic states that do not interact with the laser radiation, are stable when associated with eigenstates of the kinetic energy, or metastables; i.e., the structures last several hundred spontaneous lifetimes when generated by nonexact kinetic-energy eigenstates.

© 1992 Optical Society of America

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  1. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
    [CrossRef] [PubMed]
  2. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
    [CrossRef]
  3. V. S. Smirnov, A. M. Tumaikin, V. I. Yudin, Sov. Phys. JETP 69, 913 (1989).
  4. Y. Castin, H. Wallis, J. Dalibard, J. Opt. Soc. Am. B 6, 2046 (1989).
    [CrossRef]
  5. R. J. Cook, Phys. Rev. A 22, 1078 (1980); J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980); J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
    [CrossRef]
  6. C. Cohen-Tannoudji, Metrologia 13, 161 (1977); C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Processus d’Interaction entre Photons and Atoms (CNRS, Paris, 1988), p. 197.
    [CrossRef]
  7. A. Messiah, Mécanique Quantique (Dunod, Paris, 1960), Vol. 2.
  8. J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
    [CrossRef]
  9. If decay out of the cycling transition and repumping by an incoherent light source are considered, VSCPT is again realized, but on a longer time scale.

1989

1988

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

1980

R. J. Cook, Phys. Rev. A 22, 1078 (1980); J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980); J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

1977

C. Cohen-Tannoudji, Metrologia 13, 161 (1977); C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Processus d’Interaction entre Photons and Atoms (CNRS, Paris, 1988), p. 197.
[CrossRef]

Arimondo, E.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Aspect, A.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Castin, Y.

Cohen-Tannoudji, C.

J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

C. Cohen-Tannoudji, Metrologia 13, 161 (1977); C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Processus d’Interaction entre Photons and Atoms (CNRS, Paris, 1988), p. 197.
[CrossRef]

Cook, R. J.

R. J. Cook, Phys. Rev. A 22, 1078 (1980); J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980); J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

Dalibard, J.

Kaiser, R.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Messiah, A.

A. Messiah, Mécanique Quantique (Dunod, Paris, 1960), Vol. 2.

Smirnov, V. S.

V. S. Smirnov, A. M. Tumaikin, V. I. Yudin, Sov. Phys. JETP 69, 913 (1989).

Tumaikin, A. M.

V. S. Smirnov, A. M. Tumaikin, V. I. Yudin, Sov. Phys. JETP 69, 913 (1989).

Vansteenkiste, N.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2112 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Wallis, H.

Yudin, V. I.

V. S. Smirnov, A. M. Tumaikin, V. I. Yudin, Sov. Phys. JETP 69, 913 (1989).

J. Opt. Soc. Am. B

Metrologia

C. Cohen-Tannoudji, Metrologia 13, 161 (1977); C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Processus d’Interaction entre Photons and Atoms (CNRS, Paris, 1988), p. 197.
[CrossRef]

Phys. Rev. A

R. J. Cook, Phys. Rev. A 22, 1078 (1980); J. Javanainen, S. Stenholm, Appl. Phys. 21, 35 (1980); J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

Phys. Rev. Lett.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).
[CrossRef] [PubMed]

Sov. Phys. JETP

V. S. Smirnov, A. M. Tumaikin, V. I. Yudin, Sov. Phys. JETP 69, 913 (1989).

Other

A. Messiah, Mécanique Quantique (Dunod, Paris, 1960), Vol. 2.

If decay out of the cycling transition and repumping by an incoherent light source are considered, VSCPT is again realized, but on a longer time scale.

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Figures (9)

Fig. 1
Fig. 1

Representations of atomic transitions, interacting with σ+σ. light beams, investigated in this paper. a, Λ configuration; b, Jg = 3/2 ↔ Je = 1/2 transition; c, Jg = 2 ↔ Je = 1 transition; d, inverted-W case. In d we also assume that all spontaneous-emission decays are equal to Γ/2. The states defining the closed families of states coupled by the interaction with the two laser waves are reported. The vertical dashed lines represent spontaneous-emission coupling between different families.

Fig. 2
Fig. 2

Coupling constants and effective loss rates in the basis of coupled and uncoupled states for, a, the Λ configuration at K+ = K and, b, the inverted-W configuration for α ≫ 1.

Fig. 3
Fig. 3

Atomic level schemes and Clebsch–Gordan coefficients for the Jg = 3/2 ↔ Je = 1/2 and Jg = 2 ↔ Je = 1 transitions.

Fig. 4
Fig. 4

Numerical results for VSCPT on the Jg = 3/2 ↔ Je = 1/2 transition. 6Li parameters introduced in the numerical simulation with Δ = 0, K + 1 = K - 2 = 3 K - 1 = 3 K + 2 = 0.45 Γ, and an initial Gaussian atomic momentum distribution with a 30ħk width. a, Atomic momentum P(pz) distribution at interaction time Θ = 1600/Γ. b, Fractional areas of the peaks, i.e., the fraction of the total number of coherently trapped atoms as a function of the interaction time Θ.

Fig. 5
Fig. 5

Damping rates in the inverted-W configuration, ΓSC′(q), ΓWC′(q), and Γ″(q), from top to bottom, as function of the q label of the family, at Δ = 0. a, Results for α = 1.48; b, results for α = 0.18. All damping rates are measured in units of Γ′.

Fig. 6
Fig. 6

Numerical results for the atomic momentum P(pz) distribution on an inverted-W configuration. 4He parameters are introduced into the numerical simulation, with KIW = 0.77Γ Δ = 0, and interaction time Θ = 200/Γ The dashed line represents the initial Gaussian atomic distribution with a 30ħk width. The three-peak structure corresponding to the large α = 1.48 value appears in the atomic momentum distributions.

Fig. 7
Fig. 7

Numerical results for the atomic momentum P(pz) distribution on an inverted-W configuration. 4He parameters are introduced into the numerical simulation with Δ = 0, KIW = 0.14Γ, and interaction time Θ = 200/Γ The dashed line represents the initial Gaussian atomic momentum distribution with a 30ħk width. The two-peak structure, corresponding to the small α = 0.18 value, appears in the atomic momentum distribution.

Fig. 8
Fig. 8

Numerical results for the Jg = 2 ↔ Je = 1 atomic transition. The total atomic momentum P(pz) distributions are shown in a and b and the partial momentum distributions, PΛ(pz) and PIW(pz), in c and d. 4He parameters are introduced into the numerical simulation with Δ = −0.5Γ, KeIW = 0.77Γ, other coupling constants as given by the Clebsch–Gordan coefficients, and an initial Gaussian atomic momentum distribution with a 30ħk width. The dashed lines represent the initial atomic distributions. a, At Δ = 50/Γ the contributions of the inverted-W and the Λ configurations lead to a three-peak structure in the total momentum distribution. b, At Θ = 200/Γ the dominant Λ configuration contribution leads to a two-peak structure in the total momentum distribution. In c and d, at Θ = 200/Γ the contributions to the momentum distribution by the various inverted-W and Λ configurations are shown.

Fig. 9
Fig. 9

Numerical results for the atomic momentum P(pz) distribution in VSCPT on the Jg = 2 ↔ Je = 1 atomic transitions. 7Li parameters are introduced into the numerical simulation with Θ = 0, KeIW = 0.46Γ, an initial Gaussian atomic momentum distribution with a 30ħk width, and interaction time Θ = 1600/Γ. At such a long interaction time only the stable two-peak structure of the Λ configuration remains in the P(pz) atomic momentum distribution. The dashed line represents the initial atomic distribution.

Equations (53)

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H A = H kin + H 0 = p z 2 / ( 2 M ) + ω 0 P e ,
P e = e 0 e 0 .
E ( z , t ) = ½ { ^ + E + exp [ i ( k z - ω L t ) ] + h . c . } + ½ { ^ - E - exp [ i ( - k z - ω L t ) ] + h . c . } ,
K ± = - ( d ± E ± ) / ,
V AL = q ( K + 2 e 0 , q g - , q - k + K - 2 e 0 , q × g + , q + k ) exp ( - i ω L t ) + h . c .
e 0 , q ,             g - , q - k ,             g + , q + k .
q - 2 k q q + 2 k .
d d t ρ = 1 i [ H A + V AL , ρ ] - Γ 2 ( P e ρ + ρ P e ) + 3 Γ 8 π d 2 Ω u ^ u ( Δ + · ^ ) + P e × exp ( - i k u · R ) ρ exp ( i k u · R ) P e ( Δ + · ^ ) ,
ψ NC ( q ) = ( 1 / K ) ( K - g - , q - k ) - K + g + , q + k ) ,
ψ C ( q ) = ( 1 / K ) ( K + g - , q - k + K - g + , q + k ) ,
e 0 , q V AL ψ NC ( q ) = 0 ,
e 0 , q V AL ψ c ( q ) = K 2 exp ( - i ω L t ) .
ψ C ( q ) H kin ψ NC ( q ) = - 2 K - K + K 2 q k M .
d d t ρ ˜ = 1 i ( O ρ ˜ - ρ ˜ O + ) ,
ρ ˜ = exp ( i P e ω L t ) ρ exp ( - i P e ω L t ) ,
V ˜ AL = exp ( i P e ω L t ) V AL exp ( - i P e ω L t )
O = H A + V ˜ AL - i ( Γ / 2 ) P e - ω P e .
O o j = [ Δ E j - i ( Γ j / 2 ) ] o j .
d d t o j ρ ˜ o j = - Γ j o j ρ ˜ o j
H eff = V ˜ AL P e 1 - [ i ( Γ / 2 ) - Δ ] P e V ˜ AL ,
Γ ( q ) = K 2 / Γ ,
Γ ( q ) = 4 ψ C ( q ) H kin / ψ NC ( q ) 2 Γ ( q ) = 16 ( K - K + K 2 ) 2 ( q k M ) 2 Γ K 2 .
F ( p z ) = - d V AL d z = i k K + [ ρ e 0 , g - 1 ( p z ) - ρ g - 1 , e 0 ( p z ) ] - i k K - [ ρ e 0 , g 1 ( p z ) - ρ g 1 , e 0 ( p z ) ] ,
P e = e + 1 / 2 e + 1 / 2 + e - 1 / 2 e - 1 / 2 .
V AL = q ( K + 1 2 e - 1 / 2 , q g - 3 / 2 , q - k + K + 2 2 e + 1 / 2 , q × g - 1 / 2 , q - k + K - 1 2 e - 1 / 2 , q g + 1 / 2 , q - k + K - 2 2 e + 1 / 2 , q g + 3 / 2 , q + k ) exp ( - i ω L t ) + h . c . ,
F 1 ( q ) = { g - 3 / 2 , q - k , e - 1 / 2 , q , g + 1 / 2 , q + k } , F 2 ( q ) = { g - 1 / 2 , q - k , e + 1 / 2 , q , g + 3 / 2 , q + k } .
P e = e - 1 e - 1 + e 0 e 0 + e + 1 e + 1 .
V AL IW = q 2 ( K e IW e - 1 , q g - 2 , q - k + K i IW e - 1 , q × g 0 , q + k + K i IW e 1 , q g 0 , q - k + K e IW e + 1 , q g + 2 , q + k ) exp ( - i ω L t ) + h . c . ,
V AL Λ = q 2 ( K Λ e 0 , q g - 1 , q - k + K Λ e 0 , q × g + 1 , q + k ) exp ( - i ω L t ) + h . c . ,
V AL = V AL IW + V AL Λ .
F IW ( q ) = { g - 2 , q - 2 k , e - 1 , q - k , g 0 , q × e + 1 , q + k , g + 2 , q + 2 k } ,
F Λ ( q ) = { g - 1 , q - k , e 0 , q , g + 1 , q + k } .
α = ( K IW ) 2 4 Γ E R = Γ IW 8 E R ,
Γ IW = 2 ( K IW ) 2 / Γ .
Γ SC ( q = 0 ) = 3 ( K IW ) 2 / Γ ,
Γ WC ( q = 0 ) = ( K IW ) 2 / Γ .
ψ SC ( q = 0 ) = ( 1 / 6 ) ( g - 2 , 2 k + 2 g 0 , 0 + g + 2 , + 2 k ) ,
ψ WC ( q = 0 ) = ( 1 / 2 ) ( g - 2 , - 2 k - g + 2 , + 2 k ) ,
ψ NC ( q = 0 ) = ( 1 / 3 ) ( g - 2 , - 2 k - g 0 + g + 2 , + 2 k ) .
Γ ( q ) = Γ WC ( q ) + Γ SC ( q ) = 32 3 ( 2 q k 3 M ) 2 1 Γ WC + 32 9 ( k 2 M ) 2 1 Γ SC .
F ( p z ) = - d V AL IW d z = i k [ K i IW ρ ˜ e 1 , g 0 ( p z ) + K e IW ρ ˜ e - 1 , g - 2 ( p z ) ] - i k [ K i IW ρ ˜ e - 1 , g 0 ( p z ) + K e IW ρ ˜ e - 1 , g 0 ( p z ) ] + h . c .
ψ NC ( q = k ) = { [ ( 1 / 2 ) + ( 1 / 20 ) 1 / 2 ] 1 / 2 g - 2 , - k - [ ( 1 / 2 ) - ( 1 / 20 ) 1 / 2 ] 1 / 2 g 0 , + k } ,
ψ NC ( q = - k ) = { [ ( 1 / 2 ) - ( 1 / 20 ) 1 / 2 ] 1 / 2 g 0 , - k - [ ( 1 / 2 ) + ( 1 / 20 ) 1 / 2 ] 1 / 2 g 2 , + k } ,
Γ ( q = ± k ) = 3 - 5 2 ( K IW ) 2 Γ .
ψ NC ( q ) = 3 2 ( 1 6 g - 2 , q - 2 k - g 0 , q + 1 6 g + 2 , q + 2 k ) .
d ρ ˜ g - 1 , g - 1 ( p z ) d t = Γ 2 - k k d u H e 0 , g - 1 ( u ) ρ ˜ e 0 , e 0 ( p z + k + u ) - i K + 2 [ ρ ˜ e 0 , g - 1 ( p z ) - ρ ˜ g - 1 , e 0 ( p z ) ] ,
d ρ ˜ g - 1 , g 1 ( p z ) d t = Γ 2 - k k d u H e 0 , g 1 ( u ) ρ ˜ e 0 , e 0 ( p z - k + u ) - i K - 2 [ ρ ˜ e 0 , g 1 ( p z ) - ρ ˜ g 1 , e 0 ( p z ) ] ,
F ( p z ) = k Γ 2 - k k d u H e 0 , g 1 ( u ) ρ ˜ e 0 , e 0 ( p z + u ) - k Γ 2 - k k d u H e 0 , g - 1 ( u ) ρ ˜ e 0 , e 0 ( p z + u ) .
d ρ ˜ g - 2 , g - 2 ( p z ) d t = Γ e ρ ˜ e - 1 , e - 1 ( p z ) - i K e IW 2 [ ρ ˜ e - 1 , g - 2 ( p z ) - ρ ˜ g - 2 , e - 1 ( p z ) ] ,
d ρ ˜ g 2 , g 2 ( p z ) d t = Γ e ρ ˜ e 1 , e 1 ( p z ) - i K e IW 2 [ ρ ˜ e 1 , g 2 ( p z ) - ρ ˜ g 2 , e 1 ( p z ) ] ,
d ρ ˜ e - 1 , e - 1 ( p z ) d t = - Γ e ρ ˜ e - 1 , e - 1 ( p z ) + i K e IW 2 [ ρ ˜ e - 1 , g - 2 ( p z ) - ρ ˜ g - 2 , e - 1 ( p z ) ] - Γ i ρ ˜ e - 1 , e - 1 ( p z ) + i K i IW 2 × [ ρ ˜ e - 1 , g 0 ( p z ) - ρ ˜ g 0 , e - 1 ( p z ) ] ,
d ρ ˜ e 1 , e 1 ( p z ) d t = - Γ e ρ ˜ e 1 , e 1 ( p z ) + i K e IW 2 [ ρ ˜ e 1 , g - 2 ( p z ) - ρ ˜ g 2 , e 1 ( p z ) ] - Γ i ρ ˜ e 1 , e 1 ( p z ) + i K i IW 2 × [ ρ ˜ e 1 , g 0 ( p z ) - ρ ˜ g 0 , e 1 ( p z ) ] .
F ( p z ) = k ( Γ e - Γ i ) ρ ˜ e - 1 , e - 1 ( p z ) - k ( Γ e - Γ i ) ρ ˜ e + 1 , e + 1 ( p z ) .

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