Abstract

A model is given for the continual measurement of the inversion in a two-level system. The system is assumed to undergo oscillations between the two states. The effect of the measurement is determined by a Markovian master equation and one parameter Γ, which is a function of the measurement response bandwidth and the measuring device noise level. For rapid, accurate measurements, Γ is large. The two-level oscillation frequency provides a dynamical threshold. When Γ exceeds this threshold a two-level system prepared initially in the excited state remains so on a time scale of Γ. Thus, in the limit of Γ infinitely large, the dynamics is frozen. This is the Zeno effect. The relationship of this behavior to motional narrowing is discussed.

© 1988 Optical Society of America

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References

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  1. B. Misra, E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
    [CrossRef]
  2. A. Peres, Am. J. Phys. 48, 931 (1980).
    [CrossRef]
  3. W. H. Zurek, presented at International Symposium on Foundations of Quantum Mechanics, Tokyo, 1983.
  4. E. Joos, Phys. Rev. D 29, 1626 (1984).
    [CrossRef]
  5. A. Barchielli, L. Lanz, G. M. Prosperi, Nuovo Cimento B 72, 79 (1982).
    [CrossRef]
  6. A. Barchielli, Phys. Rev. A 34, 1642 (1982).
    [CrossRef]
  7. C. M. Caves, Phys. Rev. D 33, 1643 (1986);Phys. Rev. D 35, 1815 (1987).
    [CrossRef]
  8. C. M. Caves, G. J. Milburn, Phys. Rev. A 36, 5543 (1987).
    [CrossRef] [PubMed]
  9. G. J. Milburn, Phys. Rev. A 36, 5271 (1987).
    [CrossRef] [PubMed]
  10. L. Allen, J. H. Eberly, Optical Resonance and Two Level Atoms (Wiley, New York, 1975).
  11. A. J. Leggett, S. Charkravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
    [CrossRef]
  12. J. M. Eberly, K. Wodkiewicz, B. W. Shore, Phys. Rev. A 30, 2381 (1984).
    [CrossRef]
  13. E. B. Davies, Quantum Theory of Open Systems (Academic, New York, 1979).
  14. G. Ludwig, Foundations of Quantum Mechanics, I and II (Springer-Verlag, Berlin, 1983).
    [CrossRef]
  15. K. Kraus, States, Effects and Operations; Fundamental Notions of Quantum Theory (Springer-Verlag, Berlin, 1983).
    [CrossRef]
  16. G. J. Milburn, Phys. Rev. A 15, 744 (1987).
    [CrossRef]
  17. P. D. Drummond, D. F. Walls, Phys. Rev. A 23, 2563 (1981).
    [CrossRef]

1987 (4)

C. M. Caves, G. J. Milburn, Phys. Rev. A 36, 5543 (1987).
[CrossRef] [PubMed]

G. J. Milburn, Phys. Rev. A 36, 5271 (1987).
[CrossRef] [PubMed]

A. J. Leggett, S. Charkravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
[CrossRef]

G. J. Milburn, Phys. Rev. A 15, 744 (1987).
[CrossRef]

1986 (1)

C. M. Caves, Phys. Rev. D 33, 1643 (1986);Phys. Rev. D 35, 1815 (1987).
[CrossRef]

1984 (2)

J. M. Eberly, K. Wodkiewicz, B. W. Shore, Phys. Rev. A 30, 2381 (1984).
[CrossRef]

E. Joos, Phys. Rev. D 29, 1626 (1984).
[CrossRef]

1982 (2)

A. Barchielli, L. Lanz, G. M. Prosperi, Nuovo Cimento B 72, 79 (1982).
[CrossRef]

A. Barchielli, Phys. Rev. A 34, 1642 (1982).
[CrossRef]

1981 (1)

P. D. Drummond, D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

1980 (1)

A. Peres, Am. J. Phys. 48, 931 (1980).
[CrossRef]

1977 (1)

B. Misra, E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[CrossRef]

Allen, L.

L. Allen, J. H. Eberly, Optical Resonance and Two Level Atoms (Wiley, New York, 1975).

Barchielli, A.

A. Barchielli, Phys. Rev. A 34, 1642 (1982).
[CrossRef]

A. Barchielli, L. Lanz, G. M. Prosperi, Nuovo Cimento B 72, 79 (1982).
[CrossRef]

Caves, C. M.

C. M. Caves, G. J. Milburn, Phys. Rev. A 36, 5543 (1987).
[CrossRef] [PubMed]

C. M. Caves, Phys. Rev. D 33, 1643 (1986);Phys. Rev. D 35, 1815 (1987).
[CrossRef]

Charkravarty, S.

A. J. Leggett, S. Charkravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
[CrossRef]

Davies, E. B.

E. B. Davies, Quantum Theory of Open Systems (Academic, New York, 1979).

Dorsey, A. T.

A. J. Leggett, S. Charkravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
[CrossRef]

Drummond, P. D.

P. D. Drummond, D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

Eberly, J. H.

L. Allen, J. H. Eberly, Optical Resonance and Two Level Atoms (Wiley, New York, 1975).

Eberly, J. M.

J. M. Eberly, K. Wodkiewicz, B. W. Shore, Phys. Rev. A 30, 2381 (1984).
[CrossRef]

Fisher, M. P. A.

A. J. Leggett, S. Charkravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
[CrossRef]

Garg, A.

A. J. Leggett, S. Charkravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
[CrossRef]

Joos, E.

E. Joos, Phys. Rev. D 29, 1626 (1984).
[CrossRef]

Kraus, K.

K. Kraus, States, Effects and Operations; Fundamental Notions of Quantum Theory (Springer-Verlag, Berlin, 1983).
[CrossRef]

Lanz, L.

A. Barchielli, L. Lanz, G. M. Prosperi, Nuovo Cimento B 72, 79 (1982).
[CrossRef]

Leggett, A. J.

A. J. Leggett, S. Charkravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
[CrossRef]

Ludwig, G.

G. Ludwig, Foundations of Quantum Mechanics, I and II (Springer-Verlag, Berlin, 1983).
[CrossRef]

Milburn, G. J.

G. J. Milburn, Phys. Rev. A 36, 5271 (1987).
[CrossRef] [PubMed]

G. J. Milburn, Phys. Rev. A 15, 744 (1987).
[CrossRef]

C. M. Caves, G. J. Milburn, Phys. Rev. A 36, 5543 (1987).
[CrossRef] [PubMed]

Misra, B.

B. Misra, E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[CrossRef]

Peres, A.

A. Peres, Am. J. Phys. 48, 931 (1980).
[CrossRef]

Prosperi, G. M.

A. Barchielli, L. Lanz, G. M. Prosperi, Nuovo Cimento B 72, 79 (1982).
[CrossRef]

Shore, B. W.

J. M. Eberly, K. Wodkiewicz, B. W. Shore, Phys. Rev. A 30, 2381 (1984).
[CrossRef]

Sudarshan, E. C. G.

B. Misra, E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[CrossRef]

Walls, D. F.

P. D. Drummond, D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

Wodkiewicz, K.

J. M. Eberly, K. Wodkiewicz, B. W. Shore, Phys. Rev. A 30, 2381 (1984).
[CrossRef]

Zurek, W. H.

W. H. Zurek, presented at International Symposium on Foundations of Quantum Mechanics, Tokyo, 1983.

Zweger, W.

A. J. Leggett, S. Charkravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
[CrossRef]

Am. J. Phys. (1)

A. Peres, Am. J. Phys. 48, 931 (1980).
[CrossRef]

J. Math. Phys. (1)

B. Misra, E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[CrossRef]

Nuovo Cimento B (1)

A. Barchielli, L. Lanz, G. M. Prosperi, Nuovo Cimento B 72, 79 (1982).
[CrossRef]

Phys. Rev. A (6)

A. Barchielli, Phys. Rev. A 34, 1642 (1982).
[CrossRef]

C. M. Caves, G. J. Milburn, Phys. Rev. A 36, 5543 (1987).
[CrossRef] [PubMed]

G. J. Milburn, Phys. Rev. A 36, 5271 (1987).
[CrossRef] [PubMed]

J. M. Eberly, K. Wodkiewicz, B. W. Shore, Phys. Rev. A 30, 2381 (1984).
[CrossRef]

G. J. Milburn, Phys. Rev. A 15, 744 (1987).
[CrossRef]

P. D. Drummond, D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

Phys. Rev. D (2)

C. M. Caves, Phys. Rev. D 33, 1643 (1986);Phys. Rev. D 35, 1815 (1987).
[CrossRef]

E. Joos, Phys. Rev. D 29, 1626 (1984).
[CrossRef]

Rev. Mod. Phys. (1)

A. J. Leggett, S. Charkravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
[CrossRef]

Other (5)

E. B. Davies, Quantum Theory of Open Systems (Academic, New York, 1979).

G. Ludwig, Foundations of Quantum Mechanics, I and II (Springer-Verlag, Berlin, 1983).
[CrossRef]

K. Kraus, States, Effects and Operations; Fundamental Notions of Quantum Theory (Springer-Verlag, Berlin, 1983).
[CrossRef]

W. H. Zurek, presented at International Symposium on Foundations of Quantum Mechanics, Tokyo, 1983.

L. Allen, J. H. Eberly, Optical Resonance and Two Level Atoms (Wiley, New York, 1975).

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Equations (45)

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P ( a ) = tr ( ρ ̂ | a a | ) .
ρ ̂ ( s ) | a a | ρ ̂ | a a | tr ( ρ ̂ | a a | ) = | a a | .
ρ ̂ ( NS ) a | a a | ρ ̂ | a a | = a P ( a ) | a a | ,
P Δ ( a ) = tr [ ρ ̂ F ̂ Δ ( a ) ] ,
ρ ̂ Δ ( s ) = ϕ Δ ( a ) ρ ̂ tr [ ϕ Δ ( a ) ρ ̂ ] ,
P Δ ( a ) = tr [ ϕ Δ ( a ) ρ ̂ ] = tr [ ρ ̂ F ̂ Δ ( a ) ] .
ρ ̂ ( NS ) = a ϕ Δ ( a ) ρ ̂ ϕ Δ ρ ̂ ,
F ̂ Δ ( x ) = ( 2 π Δ ) 1 / 2 exp [ ( x σ ̂ z ) 2 / 2 Δ ] ,
lim Δ 0 F ̂ Δ ( x ) = { | 1 1 | if x = ½ | 2 2 | if x = + ½ } ,
d ρ ̂ d t = i [ H ̂ , ρ ̂ ] + γ ( ϕ Δ ρ ̂ ρ ̂ ) ,
H ̂ = ω 0 σ ̂ x ,
σ ̂ z ½ ( | 2 2 | | 1 1 | ) .
σ ̂ y 1 2 i ( | 2 1 | | 1 2 | ) .
P 2 ( t ) = 2 | ρ ( t ) | 2 = ½ ( 1 + cos ω 0 t ) .
H I = κ σ ̂ z y ̂ r .
ρ ̂ ( s ) = tr r [ S ̂ ( ρ ̂ ρ ̂ r ) S ̂ | x r x r | ] ϕ ( x r ) ρ ̂ ,
S ̂ exp ( i κ σ ̂ z y ̂ r )
x r | Ψ r = ( 2 π Δ ) 1 / 4 exp ( x r 2 / 4 Δ ) .
ϕ ( x r ) ρ ̂ = Φ ̂ ( x r ) ρ ̂ Φ ̂ ( x r ) ,
Φ ̂ ( x r ) = ( 2 π Δ ) 1 / 4 exp [ ( x r κ σ ̂ z ) 2 / 4 Δ ] .
F ̂ ( x r ) = Φ ̂ ( x r ) Φ ̂ ( x r ) .
P ( x r ) = tr [ ρ ̂ F ̂ ( x r ) ] ,
| Ψ ( t ) = a ( t ) | 1 + b ( t ) | 2 ,
P ( x r ) = ( 2 π Δ ) 1 / 2 { | a ( t ) | 2 exp [ ( x r κ / 2 ) 2 / 2 Δ ] + | b ( t ) | 2 exp [ ( x r κ / 2 ) 2 / 2 Δ ] } .
E ( x r ) = κ σ ̂ z ( t ) ,
E ( x r 2 ) E ( x r ) 2 = Δ ,
a r + 1 = a r [ | a r | 2 + | b r | 2 exp ( κ x r / Δ ) ] 1 / 2 ,
b r + 1 = b r [ | b r | 2 + | a r | 2 exp ( κ x r / Δ ) ] 1 / 2 ,
ϕ d x r ϕ ( x r ) .
Φ ̂ ( x 1 ) A ̂ Φ ̂ ( x r ) = ( 2 π Δ ) 1 / 2 exp [ ( κ 2 / 4 x 2 ) / 2 Δ ] × { μ ( x r ) A ̂ + 2 ν ( x r ) ( σ ̂ z A ̂ + A ̂ σ ̂ z ) + 4 [ μ ( x r ) 1 ] σ ̂ z A ̂ σ ̂ z } ,
μ ( x r ) = 1 2 [ 1 + cosh ( κ x r 2 Δ ) ] , ν ( x r ) = 1 2 sinh ( κ x r 2 Δ ) .
ϕ ρ = 2 [ 1 exp ( κ 2 / 8 Δ ) ] σ z ρ σ z + ½ [ 1 + exp ( κ 2 / 8 Δ ) ] ρ .
d ρ ̂ d t = i [ H ̂ , ρ ̂ ] Γ [ σ ̂ z , [ σ ̂ z , ρ ̂ ] ] ,
Γ γ [ 1 exp ( κ 2 / 8 Δ ) ] .
d d t n | ρ ̂ | m = ( free motion ) 2 Γ ( n m ) 2 n | ρ ̂ | m .
d y d t = 2 Γ y ω 0 z ,
d z d t = ω 0 y ,
P 2 ( t ) = e Γ t 2 Ω [ sin ( Ω Γ t ) + Ω cos ( Ω Γ t ) ] + 1 2 ,
Ω = ( ω 0 2 Γ 2 1 ) 1 / 2 .
P 2 ( t ) = e Γ t 2 Ω [ sinh ( Ω Γ t ) + Ω sinh ( Ω Γ t ) ] + 1 2 ,
Ω = ( 1 ω 0 2 Γ 2 ) 1 / 2 .
lim Γ P 2 ( t ) = 1 ,
P 2 ( t ) ½ [ 1 + exp ( ω 0 2 t / 2 Γ ) ] ,
z + 2 T z ¨ + ω 0 2 z ˙ + 2 T ω 0 2 cos 2 ( a ) z = 0 ,
z + 2 Γ z ¨ + ω 0 2 z ˙ = 0 .

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