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 and 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, and 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 and 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 and 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, and W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
    [Crossref]
  12. J. M. Eberly, K. Wodkiewicz, and 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 and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
    [Crossref]

1987 (4)

C. M. Caves and 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, and 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, and 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, and G. M. Prosperi, Nuovo Cimento B 72, 79 (1982).
[Crossref]

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

1981 (1)

P. D. Drummond and 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 and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[Crossref]

Allen, L.

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

Barchielli, A.

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

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

Caves, C. M.

C. M. Caves and 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, and 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, and W. Zweger, Rev. Mod. Phys. 59, 1 (1987).
[Crossref]

Drummond, P. D.

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

Eberly, J. H.

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

Eberly, J. M.

J. M. Eberly, K. Wodkiewicz, and 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, and 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, and 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, and 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, and 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 15, 744 (1987).
[Crossref]

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

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

Misra, B.

B. Misra and 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, and G. M. Prosperi, Nuovo Cimento B 72, 79 (1982).
[Crossref]

Shore, B. W.

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

Sudarshan, E. C. G.

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

Walls, D. F.

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

Wodkiewicz, K.

J. M. Eberly, K. Wodkiewicz, and 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, and 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 and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[Crossref]

Nuovo Cimento B (1)

A. Barchielli, L. Lanz, and 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 and 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, and B. W. Shore, Phys. Rev. A 30, 2381 (1984).
[Crossref]

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

P. D. Drummond and 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, and 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 and 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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