Quantum states of an oscillator with periodic time-dependent frequency under quasiresonant condition: Unperturbed evolution, perturbative effects, and anharmonic effects

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 59, Issue 5, 1999, Pages 3270-3279

  Degli Esposti Boschi, C ^a   Ferrari, L ^a,b  

^a INFM   (Italy)

^b UNIVERSITY OF BOLOGNA   (Italy)

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Indexed keywords

EID: 0142080817     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.59.3270     Document Type: Article

References (25)

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20. The (Formula presented) terms that appear in Eqs. (8) (and related following equations) are linear combinations of (Formula presented) and (Formula presented), just like the zeroth- and first-order terms, with coefficients of order (Formula presented). The square roots (Formula presented) in the trigonometric functions have not been expanded, since when (Formula presented), one has a direct connection with the quasiresonance condition (9).
21. It can be easily seen (see also Ref. 10) that the complete set of quasiresonant values is (Formula presented). We have selected the even ones just for the sake of simplicity, since the odd ones simply change the sign of (Formula presented).
22. See, for example, A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1970), Vol. 2, Chap. XVII.
23. The reason why the integral of (Formula presented) in (Formula presented) is exponentially decreasing in (Formula presented) is that the peak in Fig. 22 [whose height is proportional to (Formula presented)] has a width vanishing like (Formula presented), so that the integral contribution of each peak is proportional to (Formula presented). This can be seen from Eq. (12), on studying (Formula presented).
24. See, for example, J. J. Sakurai, Modern Quantum Mechanics, edited by San Fu Tuan, revised ed. (Addison Wesley, Reading, MA, 1994).
25. As shown in Ref. 4 and stressed in Sec. I, there is a “band” of values of (Formula presented), around each quasiresonant value, that yields an exponential increase of the mean energy, that is, a hyperbolic form of the functions depending on (Formula presented). The corresponding rates decrease from (Formula presented) [Eq. (8d)] to zero at the band edges. Outside the bands, the functions depending on (Formula presented) become trigonometric, and the effect of the frequency fluctuation is simply a periodic modulation of the mean energy.