Abstract

We present a master equation that describes the dynamics of laser cooling of a trapped ion. It is valid in the Lamb–Dicke limit and rests on an adiabatic elimination combined with a degenerate perturbation treatment. It describes relaxation of probabilities and coherences in the harmonic-trap degrees of freedom. The eigenvalue spectrum the of the time-evolution operator is derived, and it follows that only one zero eigenvalue exists, giving the unique steady-state probability distribution. The coherences all decay to zero with time. The ultimate steady state is distribution, which can be characterized by a temperature. We also report a numerical calculation that a Planck supports our analytical work. The final energy of the cooling is given and discussed. Finally there is a comparison between the present results and our earlier, approximate, treatments.

© 1984 Optical Society of America

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  1. W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 233–236 (1978).
    [CrossRef]
  2. D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639–1642 (1978).
    [CrossRef]
  3. D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521–1540 (1979); Phys. Rev. A 25, 35–54 (1982).
    [CrossRef]
  4. J. Javanainen and S. Stenholm, Appl. Phys. 21, 283–291 (1980).
    [CrossRef]
  5. J. Javanainen and S. Stenholm, Appl. Phys. 24, 71–84 (1981).
    [CrossRef]
  6. J. Javanainen and S. Stenholm, Appl. Phys. 24, 151–162 (1981).
    [CrossRef]
  7. J. Javanainen, J. Phys. B. 14, 2519–2534 (1981); J. Phys. B. 14, 4191–4205 (1981).
    [CrossRef]
  8. J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606–1616 (1980).
    [CrossRef]
  9. R. J. Cook, Phys. Rev. A 22, 1078–1098 (1980).
    [CrossRef]
  10. C. Bloch, Nucl. Phys. 6, 329–347 (1958).
    [CrossRef]
  11. N. N. Bogoliubov, Lectures on Quantum Statistics (Gordon and Breach, New York, 1967), Vol. 1, pp. 152–157.
  12. V. Dohm, Phys. Rev. A 14, 393–407 (1976).
    [CrossRef]
  13. H. Haken, Rev. Mod. Phys. 47, 67–121 (1975).
    [CrossRef]
  14. U. M. Titulaer, Physica (The Hague) 100A, 234–250 (1980).
  15. V. G. Minogin, Sov. Phys. JETP 52, 1032–1038 (1980); Sov. Phys. JETP 53, 1164–1170 (1981).
  16. S. Stenholm, Lectures on Quantum Statistics (Plenum, New York, to be published), Vol 1, pp. 152–157.
  17. For the ordinary case of a dipole interaction α= 2/5, it is often replaced by the simple isotropic value α= 1/3. For a detailed treatment, see J. Javanainen and S. Stenholm, Appl. Phys. 21, 35–45 (1980) or L. Mandel, J. Opt. (Paris) 10, 51–64 (1979).
    [CrossRef]

1981 (3)

J. Javanainen and S. Stenholm, Appl. Phys. 24, 71–84 (1981).
[CrossRef]

J. Javanainen and S. Stenholm, Appl. Phys. 24, 151–162 (1981).
[CrossRef]

J. Javanainen, J. Phys. B. 14, 2519–2534 (1981); J. Phys. B. 14, 4191–4205 (1981).
[CrossRef]

1980 (6)

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606–1616 (1980).
[CrossRef]

R. J. Cook, Phys. Rev. A 22, 1078–1098 (1980).
[CrossRef]

J. Javanainen and S. Stenholm, Appl. Phys. 21, 283–291 (1980).
[CrossRef]

U. M. Titulaer, Physica (The Hague) 100A, 234–250 (1980).

V. G. Minogin, Sov. Phys. JETP 52, 1032–1038 (1980); Sov. Phys. JETP 53, 1164–1170 (1981).

For the ordinary case of a dipole interaction α= 2/5, it is often replaced by the simple isotropic value α= 1/3. For a detailed treatment, see J. Javanainen and S. Stenholm, Appl. Phys. 21, 35–45 (1980) or L. Mandel, J. Opt. (Paris) 10, 51–64 (1979).
[CrossRef]

1979 (1)

D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521–1540 (1979); Phys. Rev. A 25, 35–54 (1982).
[CrossRef]

1978 (2)

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 233–236 (1978).
[CrossRef]

D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639–1642 (1978).
[CrossRef]

1976 (1)

V. Dohm, Phys. Rev. A 14, 393–407 (1976).
[CrossRef]

1975 (1)

H. Haken, Rev. Mod. Phys. 47, 67–121 (1975).
[CrossRef]

1958 (1)

C. Bloch, Nucl. Phys. 6, 329–347 (1958).
[CrossRef]

Ashkin, A.

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606–1616 (1980).
[CrossRef]

Bloch, C.

C. Bloch, Nucl. Phys. 6, 329–347 (1958).
[CrossRef]

Bogoliubov, N. N.

N. N. Bogoliubov, Lectures on Quantum Statistics (Gordon and Breach, New York, 1967), Vol. 1, pp. 152–157.

Cook, R. J.

R. J. Cook, Phys. Rev. A 22, 1078–1098 (1980).
[CrossRef]

Dehmelt, H. G.

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 233–236 (1978).
[CrossRef]

Dohm, V.

V. Dohm, Phys. Rev. A 14, 393–407 (1976).
[CrossRef]

Drullinger, R. E.

D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639–1642 (1978).
[CrossRef]

Gordon, J. P.

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606–1616 (1980).
[CrossRef]

Haken, H.

H. Haken, Rev. Mod. Phys. 47, 67–121 (1975).
[CrossRef]

Hohenstatt, M.

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 233–236 (1978).
[CrossRef]

Itano, W. M.

D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521–1540 (1979); Phys. Rev. A 25, 35–54 (1982).
[CrossRef]

Javanainen, J.

J. Javanainen, J. Phys. B. 14, 2519–2534 (1981); J. Phys. B. 14, 4191–4205 (1981).
[CrossRef]

J. Javanainen and S. Stenholm, Appl. Phys. 24, 71–84 (1981).
[CrossRef]

J. Javanainen and S. Stenholm, Appl. Phys. 24, 151–162 (1981).
[CrossRef]

J. Javanainen and S. Stenholm, Appl. Phys. 21, 283–291 (1980).
[CrossRef]

For the ordinary case of a dipole interaction α= 2/5, it is often replaced by the simple isotropic value α= 1/3. For a detailed treatment, see J. Javanainen and S. Stenholm, Appl. Phys. 21, 35–45 (1980) or L. Mandel, J. Opt. (Paris) 10, 51–64 (1979).
[CrossRef]

Minogin, V. G.

V. G. Minogin, Sov. Phys. JETP 52, 1032–1038 (1980); Sov. Phys. JETP 53, 1164–1170 (1981).

Neuhauser, W.

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 233–236 (1978).
[CrossRef]

Stenholm, S.

J. Javanainen and S. Stenholm, Appl. Phys. 24, 151–162 (1981).
[CrossRef]

J. Javanainen and S. Stenholm, Appl. Phys. 24, 71–84 (1981).
[CrossRef]

J. Javanainen and S. Stenholm, Appl. Phys. 21, 283–291 (1980).
[CrossRef]

For the ordinary case of a dipole interaction α= 2/5, it is often replaced by the simple isotropic value α= 1/3. For a detailed treatment, see J. Javanainen and S. Stenholm, Appl. Phys. 21, 35–45 (1980) or L. Mandel, J. Opt. (Paris) 10, 51–64 (1979).
[CrossRef]

S. Stenholm, Lectures on Quantum Statistics (Plenum, New York, to be published), Vol 1, pp. 152–157.

Titulaer, U. M.

U. M. Titulaer, Physica (The Hague) 100A, 234–250 (1980).

Toschek, P.

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 233–236 (1978).
[CrossRef]

Walls, F. L.

D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639–1642 (1978).
[CrossRef]

Wineland, D. J.

D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521–1540 (1979); Phys. Rev. A 25, 35–54 (1982).
[CrossRef]

D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639–1642 (1978).
[CrossRef]

Appl. Phys. (4)

J. Javanainen and S. Stenholm, Appl. Phys. 21, 283–291 (1980).
[CrossRef]

J. Javanainen and S. Stenholm, Appl. Phys. 24, 71–84 (1981).
[CrossRef]

J. Javanainen and S. Stenholm, Appl. Phys. 24, 151–162 (1981).
[CrossRef]

For the ordinary case of a dipole interaction α= 2/5, it is often replaced by the simple isotropic value α= 1/3. For a detailed treatment, see J. Javanainen and S. Stenholm, Appl. Phys. 21, 35–45 (1980) or L. Mandel, J. Opt. (Paris) 10, 51–64 (1979).
[CrossRef]

J. Phys. B. (1)

J. Javanainen, J. Phys. B. 14, 2519–2534 (1981); J. Phys. B. 14, 4191–4205 (1981).
[CrossRef]

Nucl. Phys. (1)

C. Bloch, Nucl. Phys. 6, 329–347 (1958).
[CrossRef]

Phys. Rev. A (4)

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606–1616 (1980).
[CrossRef]

R. J. Cook, Phys. Rev. A 22, 1078–1098 (1980).
[CrossRef]

D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521–1540 (1979); Phys. Rev. A 25, 35–54 (1982).
[CrossRef]

V. Dohm, Phys. Rev. A 14, 393–407 (1976).
[CrossRef]

Phys. Rev. Lett. (2)

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 233–236 (1978).
[CrossRef]

D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639–1642 (1978).
[CrossRef]

Physica (The Hague) (1)

U. M. Titulaer, Physica (The Hague) 100A, 234–250 (1980).

Rev. Mod. Phys. (1)

H. Haken, Rev. Mod. Phys. 47, 67–121 (1975).
[CrossRef]

Sov. Phys. JETP (1)

V. G. Minogin, Sov. Phys. JETP 52, 1032–1038 (1980); Sov. Phys. JETP 53, 1164–1170 (1981).

Other (2)

S. Stenholm, Lectures on Quantum Statistics (Plenum, New York, to be published), Vol 1, pp. 152–157.

N. N. Bogoliubov, Lectures on Quantum Statistics (Gordon and Breach, New York, 1967), Vol. 1, pp. 152–157.

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Figures (1)

Fig. 1
Fig. 1

We compare the ultimate energy of the cooling in three of our calculations as a function of detuning Δ. The optimum is near the resonance point Δ ≈ ν. We also have Γ = 2γ2 = 0.5ν and κ = 0.1ν. Curve a is the present, exact result. Curve b is the result from Ref. 7, which is acceptable near the minimum but not toward zero detuning. Curve c is the earlier, rate result of Ref. 6, which shows the correct global behavior but is numerically not accurate. For convenience the zero-point energy ½ħν is subtracted in these curves.

Tables (1)

Tables Icon

Table 1 Calculated Value of 〈n〉 = (E/ħν) − 1/2a

Equations (33)

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U n n = n exp [ - i η ( a + a ) ] n ,
η 2 = q 2 / 2 M ν .
f ˙ a a ( n , n ) = - i ν ( n - n ) f a a ( n , n ) - Γ f a a ( n , n ) + i k n [ U n n * f b a ( n , n ) - U n n f a b ( n , n ) ] ,
f ˙ b b ( n , n ) = - i ν ( n - n ) f b b ( n , n ) + Γ 2 η - η + η n n f a a ( n , n ) U n n ( η ) U n n * ( η ) d η + i κ n [ U n n f a b ( n , n ) - U n n * f b a ( n , n ) ] ,
f ˙ a b ( n , n ) = - i [ Δ + ν ( n - n ) - i γ 2 ] f a b ( n , n ) + i κ n [ U n n * f b b ( n , n ) - U n n * f a a ( n , n ) ] ,
F ( n , n ) = f a a ( n , n ) + f b b ( n , n ) ,
R 3 ( n , n ) = f a a ( n , n ) - f b b ( n , n ) ,
R + ( n , n ) = f a b ( n , n ) = R - * ( n , n ) .
k = n - n ,
( d / d t ) F ( n , n + k ) = - i ν k F ( n , n + k ) + O ( η ) ,
( d / d t ) F ( n , n ) = - i ( n - n ) ( ν + η 2 A 0 ) F ( n , n ) + η 2 { [ ( n + 1 ) ( n + 1 ) ] 1 / 2 A - F ( n + 1 , n + 1 ) - ½ ( n + n + 2 ) A + F ( n , n ) - ½ ( n + n ) A - F ( n , n ) + ( n n ) 1 / 2 A + F ( n - 1 , n - 1 ) } .
A - > A + ,
A - ( ν ) = A + ( - ν ) = Γ P ( Δ ) Γ + 2 P ( Δ ) ( α - 1 ) + 2 Γ + 2 P ( Δ ) Re { ( ν + i Γ ) [ ν + i Γ + 2 κ 2 × ( 1 Δ - ν - i γ 2 - 1 Δ + ν + i γ 2 ) ] - 1 × i κ 2 [ Γ ( 1 Δ + i γ 2 - 1 Δ - ν - i γ 2 ) - ( 1 + i Γ ν ) × P ( Δ ) ( 1 Δ - ν - i γ 2 - 1 Δ + ν + i γ 2 ) ] } ,
P ( Δ ) = 2 κ 2 γ 2 Δ 2 + γ 2 2
F ( n , n + k ) exp [ - ( μ η 2 A - - i k ν ) t ] .
G k ( z ) = exp [ ( μ η 2 A - - i ν k ) t ] × n = 0 z n [ ( n + k ) ! n ! ] 1 / 2 F ( n , n + k ) .
M n = 1 n ! ( n G k z n ) k = 0 z = 1 ,
F ( n , n + k ) 2 F ( n , n ) F ( n + k , n + k ) .
( 1 - q z ) ( 1 - z ) d G k d z = [ k 2 ( 1 - q ) + q ( k + 1 ) ( 1 - z ) - μ ] G k ,
q = A + / A - ,
G k ( z ) = C ( 1 - q z ) - ( 1 + k + σ ) ( 1 - z ) σ ,
σ = - ½ k + μ / ( 1 - q ) .
μ = ( 1 - q ) ( ½ k + N ) ,
( d / d t ) F ( n ) = η 2 [ A - ( n + 1 ) F ( n + 1 ) - ( n + 1 ) A + F ( n ) - n A - F ( n ) + n A + F ( n - 1 ) ] ,
F ( n ) = ( 1 - q ) q n .
E = ν ( n + ½ ) = ν ( A + A - - A + + ½ ) K 1 4 Δ K 2 ,
K 1 = { α [ ν 2 ( Δ 2 + 5 γ 2 2 + 4 κ 2 - ν 2 ) 2 + 4 γ 2 2 ( Δ 2 + γ 2 2 + 2 κ 2 - 2 ν 2 ) 2 ] + ν 2 ( Δ 2 + γ 2 2 + 2 κ 2 ) ( Δ 2 + 5 γ 2 2 + 8 κ 2 + ν 2 ) + 4 γ 2 2 [ ( Δ 2 + γ 2 2 + 2 κ 2 ) 2 + 2 κ 2 ( Δ 2 - 3 γ 2 2 - 3 ν 2 ) ] } ,
K 2 = [ ν 2 ( Δ 2 + γ 2 2 + 6 κ 2 ) + 4 γ 2 2 ( Δ 2 + γ 2 2 + 2 κ 2 ) ] .
ν γ 2 .
E = ν [ 1 2 + ( α + 1 4 ) ( γ 2 ν ) 2 + ( κ ν ) 2 ( 2 γ 2 Γ - 1 ) ] ,
n [ f a a ( n , n ) + f b b ( n , n ) ] = 1 .
Γ f a a ( n , n ) ½ ( Γ n + Γ n ) f a a ( n , n )
Γ n = Γ 2 η n = 0 n max - η + η d η U n n ( η ) U n n * ( η ) ;

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