Abstract

Novel methods are discussed for the state control of atoms coupled to multi-mode reservoirs with non-Markovian spectra:

The rich arsenal of control methods described above can improve the performance of single-atom devices. It can also advance the state-of-the-art of quantum information encoding and processing.

©1998 Optical Society of America

Full Article  |  PDF Article
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References

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  1. A.G. Kofman and G. Kurizki, “Control of decay into Non-Markov reservoirs by the quantum Zeno effect,” (preprint).
  2. A. Kozhekin, G. Kurizki, and V. Yudson, “Resonant population transfer in three-level atom by a single photon: spontaneous emission control,” (preprint).
  3. G. Harel, A. Kozhekin, and G. Kurizki, “State control by interfering interaction histories,” (preprint).
  4. B. Misra and E. C. G. Sudarshan, “The Zeno’s paradox in quantum theory,” J. Math. Phys. 18, 756 (1977).
    [Crossref]
  5. J. Maddox, “Fuzzy-sets make fuzzy-logic,” Nature (London) 306, 111 (1983).
  6. A. Peres, “Quantum limited detectors for weak classical signals,” Phys. Rev. D 39, 2943 (1989).
    [Crossref]
  7. W. M. Itano et al., “Quantum Zeno effect,” Phys. Rev. A 41, 2295 (1990).
    [Crossref] [PubMed]
  8. P. L. Knight, “The quantum Zeno effect,” Nature (London) 344, 493 (1990).
    [Crossref]
  9. T. Petrosky, S. Tasaki, and I. Prigogine, “Quantum Zeno effect,” Phys. Lett. A 151, 109 (1990).
    [Crossref]
  10. E. Block and P. R. Berman, “Quantum Zeno effect and quantum Zeno paradox in atomic physics,” Phys. Rev. A 44, 1466 (1991).
    [Crossref] [PubMed]
  11. L. E. Ballentine, “Quantum Zeno effect - comment,” Phys. Rev. A 43, 5165 (1991).
    [Crossref] [PubMed]
  12. V. Frerichs and A. Schenzle, “Quantum Zeno effect without collapse of the wave packet,” Phys. Rev. A 44, 1962 (1991).
    [Crossref] [PubMed]
  13. M. B. Plenio, P. L. Knight, and R. C. Thompson, “Inhibition of spontaneous decay by continuous measurements - proposal for realizable experiment,” Opt. Commun. 123, 278 (1996).
    [Crossref]
  14. A. Luis and J. Per̆ina, “Zeno effect in parametric down-conversion,” Phys. Rev. Lett. 76, 4340 (1996).
    [Crossref] [PubMed]
  15. A. G. Kofman and G. Kurizki, “Quantum Zeno effect on atomic excitation decay in resonators,” Phys. Rev. A 54, R3750 (1996).
    [Crossref] [PubMed]
  16. A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneousand induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353 (1994).
    [Crossref]
  17. See O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175 (1992) and references therein.
    [Crossref]
  18. M. O. Scully, “From lasers and masers to phaseonium and phasers,” Phys. Rep. 219, 191 (1992).
    [Crossref]
  19. A. Imamoglu, J. E. Feld, and S. E. Harris, “Nonlinear optical processes using electrically induced transparency,” Phys. Rev. Lett. 661154 (1991).
  20. I. A. Walmsley, M. Mitsunaga, and C. L. Tang, “Theory of quantum beats in optical transmission-correlation and pump-probe experiments for a general Raman configuration,” Phys. Rev. A 38, 4681 (1988).
    [Crossref] [PubMed]
  21. H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978).
    [Crossref] [PubMed]
  22. P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a 3-level system,” J. Phys. B 15, 561 (1982).
    [Crossref]
  23. J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in multilevel systems,” Phys. Rev. A 29, 690 (1984).
    [Crossref]
  24. J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72, 56 (1994).
    [Crossref] [PubMed]
  25. S. E. Harris, “Normal modes for electromagnetically induced transparency,” Phys. Rev. Lett. 72, 52 (1994).
    [Crossref] [PubMed]

1996 (3)

M. B. Plenio, P. L. Knight, and R. C. Thompson, “Inhibition of spontaneous decay by continuous measurements - proposal for realizable experiment,” Opt. Commun. 123, 278 (1996).
[Crossref]

A. Luis and J. Per̆ina, “Zeno effect in parametric down-conversion,” Phys. Rev. Lett. 76, 4340 (1996).
[Crossref] [PubMed]

A. G. Kofman and G. Kurizki, “Quantum Zeno effect on atomic excitation decay in resonators,” Phys. Rev. A 54, R3750 (1996).
[Crossref] [PubMed]

1994 (3)

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneousand induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353 (1994).
[Crossref]

J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72, 56 (1994).
[Crossref] [PubMed]

S. E. Harris, “Normal modes for electromagnetically induced transparency,” Phys. Rev. Lett. 72, 52 (1994).
[Crossref] [PubMed]

1992 (2)

See O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175 (1992) and references therein.
[Crossref]

M. O. Scully, “From lasers and masers to phaseonium and phasers,” Phys. Rep. 219, 191 (1992).
[Crossref]

1991 (4)

A. Imamoglu, J. E. Feld, and S. E. Harris, “Nonlinear optical processes using electrically induced transparency,” Phys. Rev. Lett. 661154 (1991).

E. Block and P. R. Berman, “Quantum Zeno effect and quantum Zeno paradox in atomic physics,” Phys. Rev. A 44, 1466 (1991).
[Crossref] [PubMed]

L. E. Ballentine, “Quantum Zeno effect - comment,” Phys. Rev. A 43, 5165 (1991).
[Crossref] [PubMed]

V. Frerichs and A. Schenzle, “Quantum Zeno effect without collapse of the wave packet,” Phys. Rev. A 44, 1962 (1991).
[Crossref] [PubMed]

1990 (3)

W. M. Itano et al., “Quantum Zeno effect,” Phys. Rev. A 41, 2295 (1990).
[Crossref] [PubMed]

P. L. Knight, “The quantum Zeno effect,” Nature (London) 344, 493 (1990).
[Crossref]

T. Petrosky, S. Tasaki, and I. Prigogine, “Quantum Zeno effect,” Phys. Lett. A 151, 109 (1990).
[Crossref]

1989 (1)

A. Peres, “Quantum limited detectors for weak classical signals,” Phys. Rev. D 39, 2943 (1989).
[Crossref]

1988 (1)

I. A. Walmsley, M. Mitsunaga, and C. L. Tang, “Theory of quantum beats in optical transmission-correlation and pump-probe experiments for a general Raman configuration,” Phys. Rev. A 38, 4681 (1988).
[Crossref] [PubMed]

1984 (1)

J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in multilevel systems,” Phys. Rev. A 29, 690 (1984).
[Crossref]

1983 (1)

J. Maddox, “Fuzzy-sets make fuzzy-logic,” Nature (London) 306, 111 (1983).

1982 (1)

P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a 3-level system,” J. Phys. B 15, 561 (1982).
[Crossref]

1978 (1)

1977 (1)

B. Misra and E. C. G. Sudarshan, “The Zeno’s paradox in quantum theory,” J. Math. Phys. 18, 756 (1977).
[Crossref]

Ballentine, L. E.

L. E. Ballentine, “Quantum Zeno effect - comment,” Phys. Rev. A 43, 5165 (1991).
[Crossref] [PubMed]

Berman, P. R.

E. Block and P. R. Berman, “Quantum Zeno effect and quantum Zeno paradox in atomic physics,” Phys. Rev. A 44, 1466 (1991).
[Crossref] [PubMed]

Block, E.

E. Block and P. R. Berman, “Quantum Zeno effect and quantum Zeno paradox in atomic physics,” Phys. Rev. A 44, 1466 (1991).
[Crossref] [PubMed]

Eberly, J. H.

J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72, 56 (1994).
[Crossref] [PubMed]

J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in multilevel systems,” Phys. Rev. A 29, 690 (1984).
[Crossref]

Feld, J. E.

A. Imamoglu, J. E. Feld, and S. E. Harris, “Nonlinear optical processes using electrically induced transparency,” Phys. Rev. Lett. 661154 (1991).

Frerichs, V.

V. Frerichs and A. Schenzle, “Quantum Zeno effect without collapse of the wave packet,” Phys. Rev. A 44, 1962 (1991).
[Crossref] [PubMed]

Gray, H. R.

Haq, H. R.

J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72, 56 (1994).
[Crossref] [PubMed]

Harel, G.

G. Harel, A. Kozhekin, and G. Kurizki, “State control by interfering interaction histories,” (preprint).

Harris, S. E.

S. E. Harris, “Normal modes for electromagnetically induced transparency,” Phys. Rev. Lett. 72, 52 (1994).
[Crossref] [PubMed]

A. Imamoglu, J. E. Feld, and S. E. Harris, “Nonlinear optical processes using electrically induced transparency,” Phys. Rev. Lett. 661154 (1991).

Hioe, F. T.

J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in multilevel systems,” Phys. Rev. A 29, 690 (1984).
[Crossref]

Imamoglu, A.

A. Imamoglu, J. E. Feld, and S. E. Harris, “Nonlinear optical processes using electrically induced transparency,” Phys. Rev. Lett. 661154 (1991).

Itano, W. M.

W. M. Itano et al., “Quantum Zeno effect,” Phys. Rev. A 41, 2295 (1990).
[Crossref] [PubMed]

Knight, P. L.

M. B. Plenio, P. L. Knight, and R. C. Thompson, “Inhibition of spontaneous decay by continuous measurements - proposal for realizable experiment,” Opt. Commun. 123, 278 (1996).
[Crossref]

P. L. Knight, “The quantum Zeno effect,” Nature (London) 344, 493 (1990).
[Crossref]

P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a 3-level system,” J. Phys. B 15, 561 (1982).
[Crossref]

Kocharovskaya, O.

See O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175 (1992) and references therein.
[Crossref]

Kofman, A. G.

A. G. Kofman and G. Kurizki, “Quantum Zeno effect on atomic excitation decay in resonators,” Phys. Rev. A 54, R3750 (1996).
[Crossref] [PubMed]

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneousand induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353 (1994).
[Crossref]

Kofman, A.G.

A.G. Kofman and G. Kurizki, “Control of decay into Non-Markov reservoirs by the quantum Zeno effect,” (preprint).

Kozhekin, A.

A. Kozhekin, G. Kurizki, and V. Yudson, “Resonant population transfer in three-level atom by a single photon: spontaneous emission control,” (preprint).

G. Harel, A. Kozhekin, and G. Kurizki, “State control by interfering interaction histories,” (preprint).

Kurizki, G.

A. G. Kofman and G. Kurizki, “Quantum Zeno effect on atomic excitation decay in resonators,” Phys. Rev. A 54, R3750 (1996).
[Crossref] [PubMed]

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneousand induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353 (1994).
[Crossref]

G. Harel, A. Kozhekin, and G. Kurizki, “State control by interfering interaction histories,” (preprint).

A. Kozhekin, G. Kurizki, and V. Yudson, “Resonant population transfer in three-level atom by a single photon: spontaneous emission control,” (preprint).

A.G. Kofman and G. Kurizki, “Control of decay into Non-Markov reservoirs by the quantum Zeno effect,” (preprint).

Luis, A.

A. Luis and J. Per̆ina, “Zeno effect in parametric down-conversion,” Phys. Rev. Lett. 76, 4340 (1996).
[Crossref] [PubMed]

Maddox, J.

J. Maddox, “Fuzzy-sets make fuzzy-logic,” Nature (London) 306, 111 (1983).

Misra, B.

B. Misra and E. C. G. Sudarshan, “The Zeno’s paradox in quantum theory,” J. Math. Phys. 18, 756 (1977).
[Crossref]

Mitsunaga, M.

I. A. Walmsley, M. Mitsunaga, and C. L. Tang, “Theory of quantum beats in optical transmission-correlation and pump-probe experiments for a general Raman configuration,” Phys. Rev. A 38, 4681 (1988).
[Crossref] [PubMed]

Oreg, J.

J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in multilevel systems,” Phys. Rev. A 29, 690 (1984).
[Crossref]

Per?ina, J.

A. Luis and J. Per̆ina, “Zeno effect in parametric down-conversion,” Phys. Rev. Lett. 76, 4340 (1996).
[Crossref] [PubMed]

Peres, A.

A. Peres, “Quantum limited detectors for weak classical signals,” Phys. Rev. D 39, 2943 (1989).
[Crossref]

Petrosky, T.

T. Petrosky, S. Tasaki, and I. Prigogine, “Quantum Zeno effect,” Phys. Lett. A 151, 109 (1990).
[Crossref]

Plenio, M. B.

M. B. Plenio, P. L. Knight, and R. C. Thompson, “Inhibition of spontaneous decay by continuous measurements - proposal for realizable experiment,” Opt. Commun. 123, 278 (1996).
[Crossref]

Pons, M. L.

J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72, 56 (1994).
[Crossref] [PubMed]

Prigogine, I.

T. Petrosky, S. Tasaki, and I. Prigogine, “Quantum Zeno effect,” Phys. Lett. A 151, 109 (1990).
[Crossref]

Radmore, P. M.

P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a 3-level system,” J. Phys. B 15, 561 (1982).
[Crossref]

Schenzle, A.

V. Frerichs and A. Schenzle, “Quantum Zeno effect without collapse of the wave packet,” Phys. Rev. A 44, 1962 (1991).
[Crossref] [PubMed]

Scully, M. O.

M. O. Scully, “From lasers and masers to phaseonium and phasers,” Phys. Rep. 219, 191 (1992).
[Crossref]

Sherman, B.

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneousand induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353 (1994).
[Crossref]

Stroud, C. R.

Sudarshan, E. C. G.

B. Misra and E. C. G. Sudarshan, “The Zeno’s paradox in quantum theory,” J. Math. Phys. 18, 756 (1977).
[Crossref]

Tang, C. L.

I. A. Walmsley, M. Mitsunaga, and C. L. Tang, “Theory of quantum beats in optical transmission-correlation and pump-probe experiments for a general Raman configuration,” Phys. Rev. A 38, 4681 (1988).
[Crossref] [PubMed]

Tasaki, S.

T. Petrosky, S. Tasaki, and I. Prigogine, “Quantum Zeno effect,” Phys. Lett. A 151, 109 (1990).
[Crossref]

Thompson, R. C.

M. B. Plenio, P. L. Knight, and R. C. Thompson, “Inhibition of spontaneous decay by continuous measurements - proposal for realizable experiment,” Opt. Commun. 123, 278 (1996).
[Crossref]

Walmsley, I. A.

I. A. Walmsley, M. Mitsunaga, and C. L. Tang, “Theory of quantum beats in optical transmission-correlation and pump-probe experiments for a general Raman configuration,” Phys. Rev. A 38, 4681 (1988).
[Crossref] [PubMed]

Whitley, R. M.

Yudson, V.

A. Kozhekin, G. Kurizki, and V. Yudson, “Resonant population transfer in three-level atom by a single photon: spontaneous emission control,” (preprint).

J. Math. Phys. (1)

B. Misra and E. C. G. Sudarshan, “The Zeno’s paradox in quantum theory,” J. Math. Phys. 18, 756 (1977).
[Crossref]

J. Mod. Opt. (1)

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneousand induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353 (1994).
[Crossref]

J. Phys. B (1)

P. M. Radmore and P. L. Knight, “Population trapping and dispersion in a 3-level system,” J. Phys. B 15, 561 (1982).
[Crossref]

Nature (London) (2)

J. Maddox, “Fuzzy-sets make fuzzy-logic,” Nature (London) 306, 111 (1983).

P. L. Knight, “The quantum Zeno effect,” Nature (London) 344, 493 (1990).
[Crossref]

Opt. Commun. (1)

M. B. Plenio, P. L. Knight, and R. C. Thompson, “Inhibition of spontaneous decay by continuous measurements - proposal for realizable experiment,” Opt. Commun. 123, 278 (1996).
[Crossref]

Opt. Lett. (1)

Phys. Lett. A (1)

T. Petrosky, S. Tasaki, and I. Prigogine, “Quantum Zeno effect,” Phys. Lett. A 151, 109 (1990).
[Crossref]

Phys. Rep. (2)

See O. Kocharovskaya, “Amplification and lasing without inversion,” Phys. Rep. 219, 175 (1992) and references therein.
[Crossref]

M. O. Scully, “From lasers and masers to phaseonium and phasers,” Phys. Rep. 219, 191 (1992).
[Crossref]

Phys. Rev. A (7)

W. M. Itano et al., “Quantum Zeno effect,” Phys. Rev. A 41, 2295 (1990).
[Crossref] [PubMed]

A. G. Kofman and G. Kurizki, “Quantum Zeno effect on atomic excitation decay in resonators,” Phys. Rev. A 54, R3750 (1996).
[Crossref] [PubMed]

E. Block and P. R. Berman, “Quantum Zeno effect and quantum Zeno paradox in atomic physics,” Phys. Rev. A 44, 1466 (1991).
[Crossref] [PubMed]

L. E. Ballentine, “Quantum Zeno effect - comment,” Phys. Rev. A 43, 5165 (1991).
[Crossref] [PubMed]

V. Frerichs and A. Schenzle, “Quantum Zeno effect without collapse of the wave packet,” Phys. Rev. A 44, 1962 (1991).
[Crossref] [PubMed]

I. A. Walmsley, M. Mitsunaga, and C. L. Tang, “Theory of quantum beats in optical transmission-correlation and pump-probe experiments for a general Raman configuration,” Phys. Rev. A 38, 4681 (1988).
[Crossref] [PubMed]

J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in multilevel systems,” Phys. Rev. A 29, 690 (1984).
[Crossref]

Phys. Rev. D (1)

A. Peres, “Quantum limited detectors for weak classical signals,” Phys. Rev. D 39, 2943 (1989).
[Crossref]

Phys. Rev. Lett. (4)

A. Luis and J. Per̆ina, “Zeno effect in parametric down-conversion,” Phys. Rev. Lett. 76, 4340 (1996).
[Crossref] [PubMed]

A. Imamoglu, J. E. Feld, and S. E. Harris, “Nonlinear optical processes using electrically induced transparency,” Phys. Rev. Lett. 661154 (1991).

J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an absorbing medium,” Phys. Rev. Lett. 72, 56 (1994).
[Crossref] [PubMed]

S. E. Harris, “Normal modes for electromagnetically induced transparency,” Phys. Rev. Lett. 72, 52 (1994).
[Crossref] [PubMed]

Other (3)

A.G. Kofman and G. Kurizki, “Control of decay into Non-Markov reservoirs by the quantum Zeno effect,” (preprint).

A. Kozhekin, G. Kurizki, and V. Yudson, “Resonant population transfer in three-level atom by a single photon: spontaneous emission control,” (preprint).

G. Harel, A. Kozhekin, and G. Kurizki, “State control by interfering interaction histories,” (preprint).

Supplementary Material (4)

» Media 1: MOV (149 KB)     
» Media 2: MOV (382 KB)     
» Media 3: MOV (351 KB)     
» Media 4: MOV (255 KB)     

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Figures (11)

Figure 1.
Figure 1. Cavity mode with Lorentzian lineshape

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Figure 2.
Figure 2. The level scheme

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Figure 3.
Figure 3. Movie A: Evolution of excited-state population W in two-level atom coupled to cavity mode with Lorentzian lineshape on resonance, (∆ = 0): red curve - uninterrupted decay in cavity with F = (1 - R)-2 = 104, L=15 cm, and f=0.02; green curve - interrupted evolution at intervals τ = 3 × 10-8 s, yellow dots denote the interruption moments. Here γb ≃ γf = 106 s-1; [Media 1]

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Figure 4.
Figure 4. Movie B: Idem, for detuning ∆ = 108s-1 and F = 105 [Media 2]

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Figure 5.
Figure 5. DOM with cutoff

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Figure 6.
Figure 6. Movie C: Idem, for two-level atom γf = 106 s-1) coupled to waveguide field, with coupling C 2/3 = 106 s-1 and sharp cutoff. Red curve - uninterrupted evolution at cuto frequency (∆ = 0); green curve - interrupted evolution at intervals τ = 10-8 s for ∆ = 0, yellow dots mark interruption moments. [Media 3]

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Figure 7.
Figure 7. Dependence of effective decay rate κs (eq.(14)), on dephasing (relaxation) spectrum F(∆) and field reservoir response with cutoff G(ω): (a) Lorentzian dephasing spectrum; (b) sinc-function spectrum (impulsive measurements).

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Figure 8.
Figure 8. (a) Scheme of location-dependent interference of decay channels in a cavity. Vertical line denotes a thin film, fin (out) denote incoming and outgoing photon, r(l) and 1(2) subscripts denote photon propagation to right (left) near ω l -green (ω 2-red). (b) Level scheme for atoms in film.

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Figure 9.
Figure 9. Movie D: Wavepacket conversion from ω 1 input (green envelope) to ω 2 output (red envelope) in cavity under condition (16). τ ~ 50γc−1 gives an error of ~ 10-4 in population transfer. Distance between mirror and atom is assumed to be much less than pulse envelope (shown not to scale), so that the steady state approximation holds for the field between atom and mirror (multiple reflections occur within the pulse propagation time, and are not resolvable on this time scale). [Media 4]

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Figure 10.
Figure 10. Scheme of CM preparation of superposed parallel field-atom evolutions in a cavity. Red arrows denote atomic momenta. Dash-dotted blue lines denote diffraction gratings.

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Figure 11.
Figure 11. Coherence term ρeg (t) of the atomic reduced density matrix as a function of evolution time t, showing decoherence of the atomic state with time. By the CIPE method (Eqs.(21)–(23)) we can recreate ρeg = 1 at any desired time t if the required CM is successful.

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Equations (23)

Equations on this page are rendered with MathJax. Learn more.

G ( ω ) = G s ( ω ) + G b ( ω ) .
α e ( τ ) ≃ 1 − ∫ 0 τ dt ( τ − t ) Φ s ( t ) e i Δ t − γ b τ 2 ,
Φ s ( t ) = ∫ 0 ∞ dω G s ( ω ) e − i ( ω − ω s ) t .
W ( t = nτ ) ≈ [ 2 Re α e ( τ ) − 1 ] n ≈ e − κt ,
κ = 2 Re [ 1 − α e ( τ ) ] τ .
κ = κ s + γ b ,
κ s = ( 2 τ ) Re ∫ 0 τ dt ( τ − t ) Φ s ( t ) e i Δ t
α e ( τ ) ≈ 1 − g s 2 Γ s − i Δ [ τ + e ( i Δ − Γ s ) τ − 1 Γ s − i Δ ] .
τ ≪ ( Γ s + ∣ Δ ∣ ) − 1 , g s − 1 .
κ = κ s + γ b , κ s = g s 2 τ .
G s ( ω ) = [ C ω − ω s ( ω − ω s + Γ s ) ] Θ ( ω − ω s ) ,
τ ≪ min { Γ s − 1 , ∣ Δ ∣ − 1 , C − 2 3 } .
κ s = ( 2 5 2 π 1 2 3 ) C τ 1 2 .
κ s = ∫ G s ( Δ + ω a ) F ( Δ ) d Δ ,
κ s ∼ 2 g s 2 Γ ϕ Ω 2 ,
R 1 e i θ 1 ( 1 + R 2 e i θ 2 ) ( 1 + r 2 e i ϕ 2 ) ( 1 + R 1 e i θ 1 ) ( 1 − r 2 e i ϕ 2 R 2 e i θ 2 ) = g 1 2 g 2 2
γ b − 1 ≫ τ ≫ γ c − 1
U ( t ) | ψ A + F ( i ) 〉 = B ( h ( t ) | e 〉 | 0 〉 + ∑ m = 0 M f m ( t ) | g 〉 | 1 m 〉 ) + C | g 〉 | 0 〉 ,
| ψ A + F ( f ) 〉 = ∑ j β j | p → j 〉 U ( t j ) | ψ A + F ( i ) 〉 .
| ψ A + F ( f ) 〉 = ∑ j α j U ( t j ) | ψ A + F ( i ) 〉
∑ j α j = 1
∑ j α j h ( t j ) = 1
∑ j α j f m ( t j ) = 0 m = 1 . . M .

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