## Perfect following in the diabatic limit

Optics Express, Vol. 4, Issue 7, pp. 217-222 (1999)

http://dx.doi.org/10.1364/OE.4.000217

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### 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.

© Optical Society of America

*diabatic*following has not previously been examined. Here we present the dynamical solutions of a model simple enough to allow both of these two kinds of following to occur and to be explained within a common framework. All of this is done entirely analytically using only sines and cosines, although two short movies are included for help in visualizing the limiting cases.

*P*

_{esc}, then as the Landau-Zener (LZ) adiabaticity parameter ʌ becomes large:

*P*

_{esc}→

*e*

^{-πʌ}→ 0. [1] One says that the occupation probability of the initial state “follows” the adiabatic transformation of the ground state into the final excited state.

*diabatic*limit.

*P*

_{esc}is given by:

*x*is our rapidity parameter, to be defined below in (19). The super-slow limit

*x*→ 0 is the fully adiabatic limit in which 100% transfer occurs, i.e., no probability escapes from the adiabatically evolving initial state. The opposite limit for very large

*x*is the extremely rapid diabatic case in which the escape probability is unity, meaning that the system does not follow at all but remains in the initial bare state. A graph of the expression for

*P*

_{esc}is shown in Fig. 1.

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]

*ω*

_{0}being irradiated by a laser field with amplitude

*ε*and frequency

*ω*

_{L}. Spin-based magnetic analogies are obvious. We will start with the usual Bloch-vector equation for two-level evolution:

**S**is the Bloch vector and

**Ω**is the vector around which the Bloch vector precesses. In the ordinary rotating frame, having made the Rotating Wave Approximation (RWA) as usual (see, e.g., [6]), one finds that

**Ω**has the three components

**Ω**rotates uniformly, maintaining its length. This is guaranteed if

**Ω**itself obeys the same kind of equation as

**S**, namely

**A**is a new constant “axis” vector. Thus we must satisfy (4) at the same time as the primary equation (2). It is not difficult to verify that a special solution that obeys both equations, apparently first reported by Güttinger in 1931 [7

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

**A**and

**Ω**:

*T*

_{ad}is a constant with the dimensions of time and constrained by

**S**·

**S**= 1, or

*A*, the length of

**A**, is

**Ω**’s uniform rotation rate. When

*A*≪ Ω then

**Ω**rotates slowly compared to the precession of

**S**. Solution (5) for

**S**(

*t*) is sometimes referred to as the “spin-locking” solution.

**Ω**simply rotates. The specific time-dependences of its components can be chosen to be:

**Ω**

_{0}may be oriented in any direction in the

**1**–

**3**plane. Since

**A**is a constant vector, we can write the exact solution of (4) in terms of

**Ω**

_{0}and the unit vector

**Â**as

**Ω**

_{0}= -

**3̂**Ω = {0, 0, -Ω}, for simplicity and because it corresponds to excitation by a laser field that is turned on at

*t*= 0. Then the expression for

**Ω**(

*t*) reduces to

**S**will “follow”

**Ω**, we will consider more “natural” components of

**S**, those along three orthogonal (but time dependent) directions that include

**Ω̂**, defined by the three unit vectors:

**Ω̂**(

*t*) about the

**A**axis).

*α*,

*β*and

*γ*. We can then use the equation for

**S**to find the

*α*,

*β*,

*γ*equations:

*dα*/

*dt*=

*Ωγ*,

*dβ*/

*dt*=

*Aγ*, and

*dγ*/

*dt*= -

*Aβ*- Ω

*α*. Here the initial conditions are

*α*

_{0}=

*γ*

_{0}= 0, and

*β*

_{0}= 1 if the atom starts in its ground state.

*γ*separates from the others in second order:

**S**) demands

*α*

^{2}+

*β*

^{2}+

*γ*

^{2}= 1, which is easily confirmed.

*α*,

*β*,

*γ*solutions are substituted into (12) this gives the exact solution for the Bloch vector:

*A*≪ Ω, i.e.,

**Ω**changes slowly compared to the precession of

**S**around

**Ω**. This means that the “rapidity” parameter

*x*=

*A*/Ω is very small:

*x*≪ 1 the motion is easily visualized, and Movie #1 is given in Fig. 2 to show the Bloch vector faithfully following

**Ω**, as expected. In the movie,

*x*= 1/40, so the Bloch vector precesses 40 times around

**Ω**while

**Ω**makes a full revolution. To show the precession more clearly in the movie, a slightly different initial condition was chosen in which

**Ω**is too fast for

**S**to follow. We must calculate the probability that at the end of a

*π*rotation of

**Ω**from -

**3**to +

**3**, the Bloch vector has remained aligned with -

**3**, i.e., the atom has been left in its ground state. Thus we must evaluate

*At*=

*π*, and (18) gives:

*At*=

*π*point. In the adiabatic case, as Movie #1 showed, the Bloch vector simply continues to follow. The diabatic case is more interesting. In the limit

*A*≫ Ω we may take

*AT*

_{ad}≈ 1 and retain only terms of order Ω

*T*

_{ad}or higher in (18). Then we find the extremely simple expression:

**2**is practically negligible compared to the other components, we see that the Bloch vector rotates smoothly in the

**1**–

**3**plane with full amplitude, eventually even achieving full inversion. Just as in the adiabatic limit, the rate of rotation is very slow but now the rate, 1/

*T*

_{ad}-

*A*, is slow because

*A*is large, not small:

**ω**sweeps rapidly past

**S**on every revolution, but each kick it gives to

**S**is coherent with the next kick when

**Ω**encounters

**S**again on its next revolution. Therefore, even in the extreme diabatic case, it could be said that S follows

**Ω**, if only in the sense that it “remembersΩ

**Ω**and accumulates its influence from one kick to the next.

*opposite sense*from

**Ω**, making it even harder to visualize the nature of the kick-to-kick coherence. All of this is seen in Movie #2, given in Fig. 3. The movie parameters are

*AT*

_{ad}= 1 and Ω

*T*

_{ad}= 1/3, i.e.,

*x*is greater than unity, but is still far from the extreme diabatic limit, allowing the encounters of

**S**and

**Ω**to be seen as kicks, but still permitting full inversion.

## Acknowledgements

## References

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 |

3. | C. Zener, “Non-Adiabatic Crossing of Energy Levels,” Proc. Roy. Soc. Lond. |

4. | N.F. Ramsey, |

5. | C.E. Carroll and F.T. Hioe, “Analytic solutions for three-state systems with overlapping pulses,” Phys. Rev. A |

6. | L. Allen and J.H. Eberly, |

7. | P. Güttinger, “Das Verhalten von Atomen im magnetischen Drehfeld,” Zeits. f. Phys. |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(270.1670) Quantum optics : Coherent optical effects

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 25, 1999

Published: March 29, 1999

**Citation**

Denise Sawicki and J. Eberly, "Perfect following in the diabatic limit," Opt. Express **4**, 217-222 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-7-217

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### References

- 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).
- L.D. Landau, "Zur Theorie der Energieubertragung.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).
- N. F. Ramsey, Molecular Beams (Oxford Univ. Press, England, 1956).
- 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]
- L. Allen and J.H . Eberly, Optical Resonance and Two-Level Atoms (Dover Publications, Inc., New York, 1987), Sec. 2.4.
- P. Guttinger, "Das Verhalten von Atomen im magnetischen Drehfeld," Zeits. f. Phys. 73, 169-184 (1931). [CrossRef]

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