Abstract

We show that the dressed-atom approach provides a quantitative understanding of the main features of radiative dipole forces (mean value, fluctuations, velocity dependence) in the high-intensity limit where perturbative treatments are no longer valid. In an inhomogeneous laser beam, the energies of the dressed states vary in space, and this gives rise to dressed-state-dependent forces. Spontaneous transitions between dressed states lead to a multivalued instantaneous force fluctuating around a mean value. The velocity dependence of the mean force is related to the modification, induced by the atomic motion, of the population balance between the different dressed states. The corresponding modification of the atomic energy is associated with a change of the fluorescence spectrum emitted by the atom. The particular case of atomic motion in a standing wave is investigated, and two regimes are identified in which the mean dipole force averaged over a wavelength exhibits a simple velocity dependence. The large values of this force achievable with reasonable laser powers are pointed out with view to slowing down atoms with dipole

© 1985 Optical Society of America

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  1. J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
    [CrossRef]
  2. R. J. Cook, Comments At. Mol. Phys. 10, 267 (1981).
  3. V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 1 (1981).
    [CrossRef]
  4. A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, J. Phys. (Paris) 42, 1231 (1981).
    [CrossRef]
  5. D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
    [CrossRef]
  6. S. Stenholm, Phys. Rev. A 27, 2513 (1983).
    [CrossRef]
  7. E. Kyrölä, S. Stenholm, Opt. Commun. 22, 123 (1977).
    [CrossRef]
  8. A. F. Bernhardt, B. W. Shore, Phys. Rev. A 23, 1290 (1981).
    [CrossRef]
  9. R. J. Cook, Phys. Rev. A 20, 224 (1979).
    [CrossRef]
  10. V. G. Minogin, O. T. Serimaa, Opt. Commun. 30, 373 (1979).
    [CrossRef]
  11. V. G. Minogin, Opt. Commun. 37, 442 (1981).
    [CrossRef]
  12. J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
    [CrossRef]
  13. R. J. Cook, Phys. Rev. A 22, 1078 (1980).
    [CrossRef]
  14. C. Cohen Tannoudji, S. Reynaud, in Multiphoton Processes, J. H. Eberly, P. Lambropoulos, eds. (Wiley, New York, 1978), p. 103.
  15. C. Cohen-Tannoudji, S. Reynaud, J. Phys. B 10, 345 (1977).
    [CrossRef]
  16. S. Reynaud, Ann. Phys. (Paris) 8, 315 (1983).
  17. B. R. Mollow, Phys. Rev. 188, 1969 (1969).
    [CrossRef]
  18. A. Messiah, Mécanique Quantique II (Dunod, Paris, 1964), p. 642.
  19. The phase of the standing wave is constant so that radiation pressure does not play any role here.
  20. The counterpart of this advantage is that the diffusion coefficient is bigger in a strong standing wave, so that the equilibrium velocities after cooling are larger than for usual radiation-pressure cooling (See Ref. 5).
  21. E. Fiordilino, M. H. Mittleman, Phys. Rev. A 30, 177 (1984).
    [CrossRef]
  22. I. S. Gradshteyn, L. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).
  23. A. P. Kazantsev, Sov. Phys. Usp. 21, 58 (1978).
    [CrossRef]

1985

J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

1984

E. Fiordilino, M. H. Mittleman, Phys. Rev. A 30, 177 (1984).
[CrossRef]

1983

S. Reynaud, Ann. Phys. (Paris) 8, 315 (1983).

S. Stenholm, Phys. Rev. A 27, 2513 (1983).
[CrossRef]

1981

A. F. Bernhardt, B. W. Shore, Phys. Rev. A 23, 1290 (1981).
[CrossRef]

R. J. Cook, Comments At. Mol. Phys. 10, 267 (1981).

V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 1 (1981).
[CrossRef]

A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, J. Phys. (Paris) 42, 1231 (1981).
[CrossRef]

V. G. Minogin, Opt. Commun. 37, 442 (1981).
[CrossRef]

1980

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

R. J. Cook, Phys. Rev. A 22, 1078 (1980).
[CrossRef]

1979

D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

R. J. Cook, Phys. Rev. A 20, 224 (1979).
[CrossRef]

V. G. Minogin, O. T. Serimaa, Opt. Commun. 30, 373 (1979).
[CrossRef]

1978

A. P. Kazantsev, Sov. Phys. Usp. 21, 58 (1978).
[CrossRef]

1977

C. Cohen-Tannoudji, S. Reynaud, J. Phys. B 10, 345 (1977).
[CrossRef]

E. Kyrölä, S. Stenholm, Opt. Commun. 22, 123 (1977).
[CrossRef]

1969

B. R. Mollow, Phys. Rev. 188, 1969 (1969).
[CrossRef]

Ashkin, A.

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

Bernhardt, A. F.

A. F. Bernhardt, B. W. Shore, Phys. Rev. A 23, 1290 (1981).
[CrossRef]

Cohen Tannoudji, C.

C. Cohen Tannoudji, S. Reynaud, in Multiphoton Processes, J. H. Eberly, P. Lambropoulos, eds. (Wiley, New York, 1978), p. 103.

Cohen-Tannoudji, C.

J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

C. Cohen-Tannoudji, S. Reynaud, J. Phys. B 10, 345 (1977).
[CrossRef]

Cook, R. J.

R. J. Cook, Comments At. Mol. Phys. 10, 267 (1981).

R. J. Cook, Phys. Rev. A 22, 1078 (1980).
[CrossRef]

R. J. Cook, Phys. Rev. A 20, 224 (1979).
[CrossRef]

Dalibard, J.

J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

Fiordilino, E.

E. Fiordilino, M. H. Mittleman, Phys. Rev. A 30, 177 (1984).
[CrossRef]

Gordon, J. P.

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, L. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

Itano, W. M.

D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

Kazantsev, A. P.

A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, J. Phys. (Paris) 42, 1231 (1981).
[CrossRef]

A. P. Kazantsev, Sov. Phys. Usp. 21, 58 (1978).
[CrossRef]

Kyrölä, E.

E. Kyrölä, S. Stenholm, Opt. Commun. 22, 123 (1977).
[CrossRef]

Letokhov, V. S.

V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 1 (1981).
[CrossRef]

Messiah, A.

A. Messiah, Mécanique Quantique II (Dunod, Paris, 1964), p. 642.

Minogin, V. G.

V. G. Minogin, Opt. Commun. 37, 442 (1981).
[CrossRef]

V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 1 (1981).
[CrossRef]

V. G. Minogin, O. T. Serimaa, Opt. Commun. 30, 373 (1979).
[CrossRef]

Mittleman, M. H.

E. Fiordilino, M. H. Mittleman, Phys. Rev. A 30, 177 (1984).
[CrossRef]

Mollow, B. R.

B. R. Mollow, Phys. Rev. 188, 1969 (1969).
[CrossRef]

Reynaud, S.

S. Reynaud, Ann. Phys. (Paris) 8, 315 (1983).

C. Cohen-Tannoudji, S. Reynaud, J. Phys. B 10, 345 (1977).
[CrossRef]

C. Cohen Tannoudji, S. Reynaud, in Multiphoton Processes, J. H. Eberly, P. Lambropoulos, eds. (Wiley, New York, 1978), p. 103.

Ryzhik, L. M.

I. S. Gradshteyn, L. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

Serimaa, O. T.

V. G. Minogin, O. T. Serimaa, Opt. Commun. 30, 373 (1979).
[CrossRef]

Shore, B. W.

A. F. Bernhardt, B. W. Shore, Phys. Rev. A 23, 1290 (1981).
[CrossRef]

Stenholm, S.

S. Stenholm, Phys. Rev. A 27, 2513 (1983).
[CrossRef]

E. Kyrölä, S. Stenholm, Opt. Commun. 22, 123 (1977).
[CrossRef]

Surdutovich, G. I.

A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, J. Phys. (Paris) 42, 1231 (1981).
[CrossRef]

Wineland, D. J.

D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

Yakovlev, V. P.

A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, J. Phys. (Paris) 42, 1231 (1981).
[CrossRef]

Ann. Phys. (Paris)

S. Reynaud, Ann. Phys. (Paris) 8, 315 (1983).

Comments At. Mol. Phys.

R. J. Cook, Comments At. Mol. Phys. 10, 267 (1981).

J. Phys. (Paris)

A. P. Kazantsev, G. I. Surdutovich, V. P. Yakovlev, J. Phys. (Paris) 42, 1231 (1981).
[CrossRef]

J. Phys. B

J. Dalibard, C. Cohen-Tannoudji, J. Phys. B 18, 1661 (1985).
[CrossRef]

C. Cohen-Tannoudji, S. Reynaud, J. Phys. B 10, 345 (1977).
[CrossRef]

Opt. Commun.

E. Kyrölä, S. Stenholm, Opt. Commun. 22, 123 (1977).
[CrossRef]

V. G. Minogin, O. T. Serimaa, Opt. Commun. 30, 373 (1979).
[CrossRef]

V. G. Minogin, Opt. Commun. 37, 442 (1981).
[CrossRef]

Phys. Rep.

V. S. Letokhov, V. G. Minogin, Phys. Rep. 73, 1 (1981).
[CrossRef]

Phys. Rev.

B. R. Mollow, Phys. Rev. 188, 1969 (1969).
[CrossRef]

Phys. Rev. A

J. P. Gordon, A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[CrossRef]

R. J. Cook, Phys. Rev. A 22, 1078 (1980).
[CrossRef]

D. J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979).
[CrossRef]

S. Stenholm, Phys. Rev. A 27, 2513 (1983).
[CrossRef]

A. F. Bernhardt, B. W. Shore, Phys. Rev. A 23, 1290 (1981).
[CrossRef]

R. J. Cook, Phys. Rev. A 20, 224 (1979).
[CrossRef]

E. Fiordilino, M. H. Mittleman, Phys. Rev. A 30, 177 (1984).
[CrossRef]

Sov. Phys. Usp.

A. P. Kazantsev, Sov. Phys. Usp. 21, 58 (1978).
[CrossRef]

Other

I. S. Gradshteyn, L. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

C. Cohen Tannoudji, S. Reynaud, in Multiphoton Processes, J. H. Eberly, P. Lambropoulos, eds. (Wiley, New York, 1978), p. 103.

A. Messiah, Mécanique Quantique II (Dunod, Paris, 1964), p. 642.

The phase of the standing wave is constant so that radiation pressure does not play any role here.

The counterpart of this advantage is that the diffusion coefficient is bigger in a strong standing wave, so that the equilibrium velocities after cooling are larger than for usual radiation-pressure cooling (See Ref. 5).

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Figures (7)

Fig. 1
Fig. 1

Dressed-atom energy diagram. a, States of the combined atom–laser mode system without coupling, bunched in well-separated two-dimensional manifolds. b, Dressed states in a given point r. The laser–atom coupling produces a r-dependent splitting Ω(r) between the two dressed states of a given manifold. c, Variation across the laser beam of the dressed-atom energy levels. The energy splitting and the wave functions both depend on r. Out of the laser beam, the energy levels connect with the uncoupled states of a.

Fig. 2
Fig. 2

Spontaneous radiative transitions between two adjacent manifolds, giving rise to the three components of the fluorescence spectrum with frequencies ωL + Ω, ωL, ωL − Ω. Γji is the transition rate from a level i to a level j.

Fig. 3
Fig. 3

Interpretation of the sign of the mean dipole force, which is the average of the two dressed-state-dependent forces weighted by the steady-state populations of these states represented by the filled circles. a, When δ = ωLω0 > 0, the state |1, n; r〉, which connects to |g, n + 1〉 out of the laser beam, is less contaminated by |e, n〉 than |2, n; r〉 and therefore more populated. The mean dipole force then expels the atom out of the laser beam. b, When δ = ωLω0 < 0, the state |2, n; r〉 is more populated than |1, n; r〉, and the mean dipole force attracts the atom toward the high-intensity region.

Fig. 4
Fig. 4

Interpretation of the sign of the velocity-dependent dipole force (δ > 0) for a slowly moving atom. a, Level |1, n; r〉 connects with |g, n + 1〉 out of the laser beam, so that the steady-state population Π1st decreases as the atom is put in higher-intensity regions: Π1st(r − dr) (hatched circle) is larger than Π1st(r) (filled circle), and consequently, Π2st(r − dr) < Π2st(r). b, For an atom entering with a velocity v in the laser beam, because of the time lag τpop of radiative relaxation the instantaneous population in r, Π1(r), is the steady population at point rvτpop, which is larger than Π1st(r). The mean dipole force then expels the moving atom out of the laser beam more than if it were at rest. The component of the force linear in v is therefore a damping term.

Fig. 5
Fig. 5

The instantaneous dipole force switches back and forth between the two dressed-state-dependent forces −∇E1 and −∇E2. The intervals of time τ1 and τ2 spent in each dressed state between two successive jumps are random variables.

Fig. 6
Fig. 6

Interpretation of the slowing down for δ > 0 of an atom moving in a standing wave with intermediate velocities. The emission of the upper sideband of the fluorescence spectrum (transitions 1 → 2) occurs preferentially at the antinodes of the standing wave where the contaminations of |1, n + 1; r〉 by |e, n + 1〉 and |2, n, r〉 by |g, n + 1〉 are the largest. By contrast, for the lower sideband, the transitions 2 → 1 occur preferentially at the nodes where |2, n; r〉 and |1, n − 1; r〉 coincide, respectively, with |e, n〉 and |g, n〉. Consequently, between two transitions 1 → 2 or 2 → 1, the moving atom “sees” on the average more “uphill” parts that “downhill” ones in the dressed-atom energy diagram and is therefore slowed down.

Fig. 7
Fig. 7

Mean dipole force f ¯ (in units of ħkΓ/2) versus velocity (in units of Γ/k) for an atom moving in a standing wave ( ω ˜ 1= 1000Γ, δ = 200Γ). The solid curve is an exact numerical solution obtained by the method of continued fractions. The two dashed lines represent the analytical predictions of the dressed-atom approach for very low velocities (kv ≪ Γ) (see also the insert) and for intermediate velocities (Γ ≪ kvkvcr). The structures appearing in the exact solution in the high-velocity domain are a signature of the breakdown of the adiabatic approximation (“Doppleron” resonances).

Equations (105)

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H = H A + H R + V ,
H A = P 2 2 m + ω 0 b + b .
b = g e ,             b + = e g
H R = λ ω λ a λ + a λ ,
V = - d · [ b + E + ( R ) + b E - ( R ) ] ,
E + ( R ) = λ E λ ( R ) a λ , E - ( R ) = λ E λ * ( R ) a λ + .
Δ r λ .
k Δ v Γ .
m Δ r Δ v ,
2 k 2 2 m Γ .
H D A ( r ) = ( ω L - δ ) b + b + ω L a L + a L - [ d · E L ( r ) b + a L + d · E L * ( r ) b a L + ] ,
δ = ω L - ω 0 ω L , ω 0 .
2 e , n V g , n + 1 = - 2 n + 1 d · E L ( r ) = ω 1 ( r ) exp [ i φ ( r ) ] .
E 1 n ( r ) = ( n + 1 ) ω L - δ 2 + Ω ( r ) 2 , E 2 n ( r ) = ( n + 1 ) ω L - δ 2 - Ω ( r ) 2 ,
Ω ( r ) = [ ω 1 2 ( r ) + δ 2 ] 1 / 2 ,
1 , n ; r = exp [ i φ ( r ) / 2 ] cos θ ( r ) e , n + exp [ - i φ ( r ) / 2 ] sin θ ( r ) g , n + 1 , 2 , n ; r = - exp [ i φ ( r ) / 2 ] sin θ ( r ) e , n + exp [ - i φ ( r ) / 2 ] cos θ ( r ) g , n + 1 ,
cos 2 θ ( r ) = - δ / Ω ( r ) ,             sin 2 θ ( r ) = ω 1 ( r ) / Ω ( r ) .
d i j ( r ) = i , n - 1 ; r d ( b + b + ) j , n ; r , d 11 = - d 22 = d cos θ sin θ e i φ , d 12 = - d e i φ sin 2 θ ,             d 21 = d e i φ cos 2 θ .
Π i ( r ) = n i , n ; r ρ i , n ; r ,
ρ i j ( r ) = n i , n ; r ρ j , n ; r .
Ω ( r ) Γ ,
Π ˙ 1 ( r ) = - Γ 21 ( r ) Π 1 ( r ) + Γ 12 ( r ) Π 2 ( r ) Π ˙ 2 ( r ) = Γ 21 ( r ) Π 1 ( r ) - Γ 12 ( r ) Π 2 ( r ) ,
Γ 12 ( r ) = Γ sin 4 Θ ( r ) ,             Γ 21 ( r ) = Γ cos 4 Θ ( r ) .
ρ ˙ 12 ( r ) = [ - i Ω ( r ) - Γ coh ( r ) ] ρ 12 ( r ) , ρ ˙ 12 ( r ) = [ i Ω ( r ) - Γ coh ( r ) ] ρ 21 ( r ) ,
Γ coh ( r ) = Γ [ 1 2 + cos 2 Θ ( r ) · sin 2 Θ ( r ) ] .
Π 1 st ( r ) = Γ 12 ( r ) Γ 12 ( r ) + Γ 21 ( r ) = sin 4 Θ ( r ) sin 4 Θ ( r ) + cos 4 Θ ( r ) , Π 2 st ( r ) = Γ 21 ( r ) Γ 12 ( r ) + Γ 21 ( r ) = cos 4 Θ ( r ) sin 4 Θ ( r ) + cos 4 Θ ( r ) , ρ 12 st ( r ) = ρ 21 st ( r ) = 0.
Γ pop ( r ) = Γ 12 ( r ) + Γ 21 ( r ) = Γ [ cos 4 Θ ( r ) + sin 4 Θ ( r ) ]
Π ˙ i ( r ) = - Γ pop ( r ) [ Π i ( r ) - Π i st ( r ) ] .
Π ˙ i ( r ) = n i , n ; r ρ ˙ i , n ; r + i , n ; r ¯ . ρ i , n ; r + i , n ; r ρ i , n ; r ¯ .
i , n ; r ¯ . = v · i , n ; r
Π ˙ 1 = - Γ pop ( Π 1 - Π 1 st ) + v · Θ ( ρ 12 + ρ 21 ) + i v · φ sin Θ cos Θ ( ρ 21 - ρ 12 ) ,
ρ ˙ 12 = - [ i ( Ω + v · φ cos 2 θ ) + Γ coh ] ρ 12 + [ v · Θ + i v · φ sin Θ cos Θ ] ( Π 2 - Π 1 ) .
d Π i = ( d Π i ) Rad + ( d Π i ) NA .
( d Π i ) Rad = - Γ pop ( Π i - Π i st ) .
( d Π 1 ) NA = - v d t · [ Θ ( ρ 12 + ρ 21 ) + i sin Θ cos Θ φ ( ρ 21 - ρ 12 ) ] = - ( d Π 2 ) NA .
p i j sup { j , n ; r ( t ) i , n ; r ( t ) ¯ . 2 ω i j ( t ) 2 } .
i , n ; r ( t ) ¯ . = ± v · Θ j , n ; r ( t ) ,
Θ = - δ ω 1 2 ˙ ( δ 2 + ω 1 2 ) .
Θ = k 2 | δ ω ˜ 1 sin k · r δ 2 + ω ˜ 1 2 cos 2 k · r | k 2 | ω ˜ 1 δ |
p i j ( k v 2 ) 2 δ ω ˜ 1 sin kr 2 δ 2 + ω ˜ 1 2 cos 2 kr 2 | k v 2 · ω ˜ 1 δ 2 | 2 .
k v k v c r = ( 2 Π Γ δ 4 / ω ˜ 1 2 ) 1 / 3 .
F = d P d t = i [ H , P ] = - R H = - R V .
f ( r ) = F ( R ) .
f ( r ) = b + a L [ d · E L ( r ) ] + b a L + [ d · E L * ( r ) ] .
f ( r ) = ω 1 2 i φ ( ρ e g e - i φ - ρ g e e i φ ) - ω 1 2 ( ρ e g e - i φ + ρ g e e i φ ) ,
ρ e g = n e , n ρ g , n + 1 , ρ g e = n g , n + 1 ρ e , n .
f dip = Ω 2 ( Π 2 - Π 1 ) - Ω Θ ( ρ 12 + ρ 21 ) .
d W = - f dip · d r = - Ω 2 · d r ( Π 2 - Π 1 ) + Ω Θ · d r ( ρ 12 + ρ 21 ) .
- d Ω 2 ( Π 2 - Π 1 ) = i = 1 , 2 Π i d E i ,
E 1 ( r ) = 1 2 Ω ( r ) , E 2 ( r ) = - 1 2 Ω ( r ) = - E 1 ( r ) .
Ω Θ · v d t ( ρ 12 + ρ 21 ) = Ω ( d Π 1 ) NA = - Ω ( d Π 2 ) NA = Ω 2 [ ( d Π 1 ) NA - ( d Π 2 ) NA ] = i E i ( d Π i ) NA .
d W = i = 1 , 2 [ Π i d E i + E i ( d Π i ) NA ] .
d W = i Π i d E i             if             v v c r .
U A = i E i Π i ,
d W d U A .
d W = d U A - i E i ( d Π i ) Rad .
d W = d U A + ( Γ 21 Π 1 Ω - Γ 12 Π 2 Ω ) d t .
d U F = ( Γ 12 Π 1 Ω - Γ 12 Π 2 Ω ) d t .
d W = - f dip · d r = d U A + d U F ,
f dip st = - i Π i st E i = - Π 1 st E 1 - Π 2 st E 2 .
f dip st = - δ ω 1 2 ω 1 2 + 2 δ 2 α = - [ δ 2 log ( 1 + ω 1 2 2 δ 2 ) ] ,
α = ω 1 ω 1 = Ω ω 1 2 Ω .
f dip = - Π 1 E 1 - Π 2 E 2 .
k v Γ 1                         v Γ - 1 λ .
v · Π i ( r ) = - Γ pop ( r ) [ Π i ( r ) - Π i st ( r ) ] .
Π i ( r ) Π i st ( r ) - v τ pop · Π i st ( r ) ,
τ pop ( r ) = 1 / Γ pop ( r ) .
Π i ( r ) Π i st ( r - v τ pop ) ,
Π 1 ( r ) = Π 1 st ( r - v τ pop ) > Π 1 st ( r ) .
Π 2 ( r ) = Π 2 st ( r - v τ pop ) < Π 2 st ( r ) .
f dip = - Ω 2 [ Π 1 ( r ) - Π 2 ( r ) ] ,
f dip ( r , v ) = f dip st ( r ) - 2 δ Γ ( ω 1 2 ( r ) ω 1 2 ( r ) + 2 δ 2 ) 3 ( α · v ) α ,
ρ 12 , ρ 21 Max { k v Γ × Γ Ω , k v Γ × Γ ω 1 Ω δ } ,
D dip = 0 d t [ F ( t ) · F ( t + τ ) - f dip 2 ] ,
F ( t ) · F ( t + τ ) = i = 1 , 2 j = 1 , 2 ( - E i ) ( - E j ) P ( i , t ; j , t + τ ) .
P ( i , t ; j , t + τ ) = P i P ( j , τ / i , 0 ) ,
P i = Π i st ,
P ( i , τ / i , 0 ) = Π i st exp ( - Γ pop τ ) ,
P ( j , τ / i , 0 ) = 1 - P ( i , τ / i , 0 ) .
D dip = 2 2 Γ ( ω 1 2 ω 1 2 + 2 δ 2 ) 3 ( ω 1 ) 2 .
ω 1 ( z ) = ω ˜ 1 cos k z ,
Δ W = z z + λ f dip d z = λ f ¯ .
f ¯ = - 1 λ z z + λ d U A - 1 λ z z + λ d U F .
- 1 λ z z + λ d U A = - 1 λ [ U A ( z + λ ) - U A ( z ) ] = 0.
f ¯ = - 1 λ z z + λ d U F .
f ¯ = 1 λ - λ / 2 λ / 2 f dip ( z , v ) d z = 1 λ - λ / 2 λ / 2 f dip st ( z ) d z - 1 λ - λ / 2 λ / 2 2 δ Γ ( ω 1 2 ω 1 2 + 2 δ 2 ) 3 α 2 v d z .
f ¯ = - m τ v ,
1 τ = 1 τ 0 δ Γ { 3 4 1 + s + 3 2 1 + s - 1 4 ( 1 + s ) 3 / 2 - 2 } ,
τ 0 = m / k 2 ,             s = ω ˜ 1 2 / 2 δ 2 .
v = d Π i d z = - Γ pop ( Π i - Π i st )
Π i ( z ) = e ( z 0 , z ) Π i ( z 0 ) + z 0 z d z Γ pop Π i st v e ( z , z ) , e ( z 1 , z 2 ) = exp { - z 1 z 2 d z Γ pop v } .
Π i ( z ) = z - λ z d z Γ pop Π i st v e ( z , z ) 1 - e ( z - λ , z ) .
e ( z 1 , z 2 ) 1 - z 1 z 2 d z Γ pop v .
Π i ( z ) Γ pop Π i ¯ Γ ¯ pop ,
Π 1 Γ ¯ 12 Γ ¯ 12 + Γ ¯ 21 ,             Π 2 Γ ¯ 21 Γ ¯ 12 + Γ ¯ 21 .
d U F = ( Ω Γ 21 Γ ¯ 12 Γ ¯ 12 + Γ ¯ 21 - Ω Γ 12 Γ ¯ 21 Γ ¯ 12 + Γ ¯ 21 ) d t
f ¯ = - v Ω Γ 21 ¯ × Γ ¯ 12 - Ω Γ 12 ¯ × Γ ¯ 21 Γ ¯ 12 + Γ ¯ 21 .
- f ¯ v = d U F d t = ( ω L + Ω ) Γ 21 ¯ Γ ¯ 21 + ( ω L - Ω ) Γ 12 ¯ Γ ¯ 12 - 2 ω L 1 Γ ¯ 12 + 1 Γ ¯ 21 .
( ω L + Ω ) Γ 21 ¯ Γ ¯ 21 [ ω L + ( ω ˜ 1 2 + δ 2 ) 1 / 2 ] .
( ω L - Ω ) Γ 12 ¯ Γ ¯ 12 ( ω L - δ ) .
- f ¯ v ( ω 1 2 + δ 2 ) ¯ 1 / 2 - δ 1 Γ 12 + 1 Γ 21 ,
Γ ¯ 12 21 = Γ 4 { 1 + 1 4 ( 1 + 2 s ) 1 / 2 [ 1 ± 4 π K ( 2 s 1 + 2 s ) 1 / 2 ] } , Ω Γ 12 21 ¯ = Γ δ 2 π { ( 1 + 2 s ) 1 / 2 E ( 2 s 1 + 2 s ) 1 / 2 } + 1 ( 1 + 2 s ) 1 / 2 K ( 2 s 1 + 2 s ) 1 / 2 ± π } ,
K ( k ) = 0 π / 2 d α ( 1 - k 2 sin 2 α ) 1 / 2 , E ( k ) = 0 π / 2 d α ( 1 - k 2 sin 2 α ) 1 / 2 ,
k v Γ ,             f ¯ = - 460 ( k Γ 2 ) ( k v Γ ) ,
Γ k v k v c r 20 Γ ,             f ¯ = - 40 ( k Γ 2 ) ( Γ k v ) .

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