Abstract

We present the theory of optical excitation of a two-level quantum system, using an interaction Hamiltonian that permits both adiabatic following and diabatic following.

©1999 Optical Society of America

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References

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  1. K.-A. Suominen, “Time Dependent Two-State Models and Wave Packet Dynamics,” Univ. of Helsinki Report Series in Theoretical Physics HU-TFT-IR-92-1 (1992).
  2. L.D. Landau, “Zur Theorie der Energieübertragung.II,” Phys. Z. Sowjet Union 2, 46–51 (1932).
  3. C. Zener, “Non-Adiabatic Crossing of Energy Levels,” Proc. Roy. Soc. Lond. A 137, 696–702 (1932).
  4. N.F. Ramsey, Molecular Beams (Oxford Univ. Press, England,1956).
  5. C.E. Carroll and F.T. Hioe, “Analytic solutions for three-state systems with overlapping pulses,” Phys. Rev. A 42, 1522–1531 (1990).
    [Crossref] [PubMed]
  6. L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications, Inc., New York,1987), Sec. 2.4.
  7. P. Güttinger, “Das Verhalten von Atomen im magnetischen Drehfeld,” Zeits. f. Phys. 73, 169–184 (1931).
    [Crossref]

1990 (1)

C.E. Carroll and F.T. Hioe, “Analytic solutions for three-state systems with overlapping pulses,” Phys. Rev. A 42, 1522–1531 (1990).
[Crossref] [PubMed]

1932 (2)

L.D. Landau, “Zur Theorie der Energieübertragung.II,” Phys. Z. Sowjet Union 2, 46–51 (1932).

C. Zener, “Non-Adiabatic Crossing of Energy Levels,” Proc. Roy. Soc. Lond. A 137, 696–702 (1932).

1931 (1)

P. Güttinger, “Das Verhalten von Atomen im magnetischen Drehfeld,” Zeits. f. Phys. 73, 169–184 (1931).
[Crossref]

Allen, L.

L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications, Inc., New York,1987), Sec. 2.4.

Carroll, C.E.

C.E. Carroll and F.T. Hioe, “Analytic solutions for three-state systems with overlapping pulses,” Phys. Rev. A 42, 1522–1531 (1990).
[Crossref] [PubMed]

Eberly, J.H.

L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications, Inc., New York,1987), Sec. 2.4.

Güttinger, P.

P. Güttinger, “Das Verhalten von Atomen im magnetischen Drehfeld,” Zeits. f. Phys. 73, 169–184 (1931).
[Crossref]

Hioe, F.T.

C.E. Carroll and F.T. Hioe, “Analytic solutions for three-state systems with overlapping pulses,” Phys. Rev. A 42, 1522–1531 (1990).
[Crossref] [PubMed]

Landau, L.D.

L.D. Landau, “Zur Theorie der Energieübertragung.II,” Phys. Z. Sowjet Union 2, 46–51 (1932).

Ramsey, N.F.

N.F. Ramsey, Molecular Beams (Oxford Univ. Press, England,1956).

Suominen, K.-A.

K.-A. Suominen, “Time Dependent Two-State Models and Wave Packet Dynamics,” Univ. of Helsinki Report Series in Theoretical Physics HU-TFT-IR-92-1 (1992).

Zener, C.

C. Zener, “Non-Adiabatic Crossing of Energy Levels,” Proc. Roy. Soc. Lond. A 137, 696–702 (1932).

Phys. Rev. A (1)

C.E. Carroll and F.T. Hioe, “Analytic solutions for three-state systems with overlapping pulses,” Phys. Rev. A 42, 1522–1531 (1990).
[Crossref] [PubMed]

Phys. Z. Sowjet Union (1)

L.D. Landau, “Zur Theorie der Energieübertragung.II,” Phys. Z. Sowjet Union 2, 46–51 (1932).

Proc. Roy. Soc. Lond. (1)

C. Zener, “Non-Adiabatic Crossing of Energy Levels,” Proc. Roy. Soc. Lond. A 137, 696–702 (1932).

Zeits. f. Phys. (1)

P. Güttinger, “Das Verhalten von Atomen im magnetischen Drehfeld,” Zeits. f. Phys. 73, 169–184 (1931).
[Crossref]

Other (3)

K.-A. Suominen, “Time Dependent Two-State Models and Wave Packet Dynamics,” Univ. of Helsinki Report Series in Theoretical Physics HU-TFT-IR-92-1 (1992).

N.F. Ramsey, Molecular Beams (Oxford Univ. Press, England,1956).

L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications, Inc., New York,1987), Sec. 2.4.

Supplementary Material (2)

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Figures (3)

Figure 1.
Figure 1. A plot of Pesc , given in (1), as a function of the rapidity parameter x, which is defined in (19). An expanded view of the approach to the singular oscillations near the adiabatic limit is given in the left graph.

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Figure 2.
Figure 2. A frame from a movie (352KB) showing adiabatic following of the Bloch vector as it precesses rapidly about Ω, which is rotating slowly with the relative rotation rate 1/40.

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Figure 3.
Figure 3. A frame from a movie (570KB) showing diabatic following of the Bloch vector as it achieves inversion and then full 2π rotation as a consequence of successive “kicks” by Ω.

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Equations (35)

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P esc = x 2 1 + x 2 sin ⁡ 2 ( π 2 1 + x 2 x 2 ) .
d S dt = Ω × S
Ω ≡ { r , 0 , Δ } ,
r = 2 dε ħ and Δ = ω 0 − ω L .
d Ω dt = A × Ω ,
S = ± ( Ω ( t ) − A ) T ad ,
( A 2 + Ω 2 ) T ad 2 = 1 ,
r = ± Ω sin ⁡ At , Δ = ± Ω cos ⁡ At ,
A = 2 ̂ A = { 0 , A , 0 } .
Ω ( t ) = A ̂ ( A ̂ · Ω 0 ) + ( A ̂ × Ω 0 ) sin ⁡ At
− A ̂ × ( A ̂ × Ω 0 ) cos ⁡ At ,
Ω = − 1 ̂ Ω sin ⁡ At − 3 ̂ Ω cos ⁡ At
= − Ω { sin ⁡ At , 0 , cos ⁡ At }
≡ Ω Ω ̂ ( t )
Ω ̂ = { − sin ⁡ At , 0 , − cos ⁡ At } ,
A ̂ = { 0,1,0 } ,
A ̂ × Ω ̂ = { − cos ⁡ At , 0 , sin ⁡ At } .
S ( t ) = α ( t ) A ̂ + β ( t ) Ω ̂ ( t )
+ γ ( t ) A ̂ × Ω ̂ ( t )
d 2 γ d t 2 + 1 T ad 2 γ = 0 .
α ( t ) = ΩA T ad 2 ( cos ⁡ ( t T ad ) − 1 ) ,
β ( t ) = 1 + A 2 T ad 2 ( cos ⁡ ( t T ad ) − 1 ) ,
γ ( t ) = − A T ad sin ⁡ ( t T ad ) .
S = A ̂ Ω A T ad 2 ( cos ⁡ t T ad − 1 ) + Ω ̂ ( 1 + A 2 T ad 2 ( cos ⁡ t T ad − 1 ) )
− ( A ̂ × Ω ̂ ) A T ad sin ⁡ t T ad ,
S = 1 ̂ { A T ad sin ⁡ t T ad cos ⁡ At − ( 1 + A 2 T ad 2 ( cos ⁡ t T ad − 1 ) ) sin ⁡ At }
+ 2 ̂ { ΩA T ad 2 ( cos ⁡ t T ad − 1 ) }
+ 3 ̂ { − A T ad sin ⁡ t T ad sin ⁡ At − ( 1 + A 2 T ad 2 ( cos ⁡ t T ad − 1 ) ) cos ⁡ At } .
x ≡ A Ω ≪ 1 ( highly adiabatic )
x ≫ 1 ( very nonadiabatic ) .
S 3 ( π ) = 1 − ( A T ad ) 2 ( 1 − cos ⁡ π A T ad ) .
P esc → 1 2 ( 1 − cos ⁡ π ) → 1 , ( x → ∞ , zero transfer ) ,
→ 1 2 x 2 ( 1 − cos ⁡ ( π x ) ) → 0 , ( x → 0 , complete transfer ) ,
S = 1 ̂ sin ⁡ ( t T ad − At ) + 2 ̂ Ω T ad ( cos ⁡ t T ad − 1 ) − 3 ̂ cos ⁡ ( t T ad − At ) .
( 1 T ad − A ) ≡ ( A 2 + Ω 2 − A ) ≈ Ω 2 2 A ≪ Ω ≪ A .

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