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Optimal control in laser-induced population transfer for two- and three-level quantum systems. (English) Zbl 1059.81195
Summary: We apply the techniques of control theory and of sub-Riemannian geometry to laser-induced population transfer in two- and three-level quantum systems. The aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences. Sub-Riemannian geometry and singular-Riemannian geometry provide a natural framework for this minimization, where the optimal control is expressed in terms of geodesics. We first show that in two-level systems the well-known technique of “\(\pi\)-pulse transfer” in the rotating wave approximation emerges naturally from this minimization. In three-level systems driven by two resonant fields, we also find the counterpart of the “\(\pi\)-pulse transfer”. This geometrical picture also allows one to analyze the population transfer by adiabatic passage.

81V80 Quantum optics
49N90 Applications of optimal control and differential games
53C17 Sub-Riemannian geometry
Full Text: DOI
[1] Vitanov, Annu. Rev. Phys. Chem. 52 pp 763– (2001)
[2] Drese, Eur. Phys. J. D 5 pp 119– (1999)
[3] Guérin, Phys. Rev. A 63 pp 031403– (2001)
[4] Yatsenko
[5] Korolkov, Adv. Chem. Phys. 101 pp 327– (1997)
[6] Holthaus, Phys. Rev. A 49 pp 1950– (1994)
[7] Daleh, Phys. Rev. A 37 pp 4950– (1988)
[8] Bartana, Chem. Phys. 267 pp 95– (2001)
[9] Band, J. Chem. Phys. 101 pp 7528– (1994)
[10] Malinovsky, Phys. Rev. A 56 pp 4929– (1997)
[11] Guérin, Phys. Rev. A 65 pp 023409– (2002)
[12] Khaneja, Phys. Rev. A 63 pp 032308– (2001)
[13] A. Isidori,Nonlinear Control Systems(Springer Verlag, New York, 1995). · Zbl 0878.93001
[14] B. W. Shore,The Theory of Coherent Atomic Excitation(Wiley, New York, 1990), Vol. 1, pp. 235 and 304.
[15] Guérin, Phys. Rev. A 56 pp 1458– (1997)
[16] A. Bellaiche, ”The tangent space in sub-Riemannian geometry,”Sub-Riemannian Geometry(Birkhuser, Basel, 1996), pp. 1–78.
[17] M. Gromov, ”Carnot-Carathodory spaces seen from within,” in Ref. 16, pp. 79–323. · Zbl 0864.53025
[18] R. Montgomery,A Tour of Sub-Riemannian Geometry, Mathematical Surveys and Monographs (American Mathematical Society, Providence, 2002).
[19] Agrachev, Acta Applicandae Mathematicae 57 pp 287– (1999)
[20] B. W. Shore,The Theory of Coherent Atomic Excitation(Wiley, New York, 1990), Vol. 2, p. 792.
[21] L. S. Pontryagin, V. Boltianski, R. Gamkrelidze and E. Mitchtchenko,The Mathematical Theory of Optimal Processes(Wiley, New York, 1961).
[22] V. Jurdjevic,Geometric Control Theory(Cambridge University Press, Cambridge, 1997). · Zbl 0940.93005
[23] Agrachev, J. Dyn. Control Syst. 2 pp 321– (1996)
[24] Chakir, J. Dyn. Control Syst. 2 pp 359– (1996)
[25] Carroll, Phys. Rev. A 42 pp 1522– (1990)
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