Non-dispersive wave packets in periodically driven quantum systems

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498 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 the basis of the above claim. 26 From our point of view, which, as already stated above, attributes the non-dispersive character of the wave packet to a classical non-linear resonance, the accuracy of the harmonic approximation (which, anyway, always remains an approximation) is irrelevant for the existence of non-dispersive wave packets. The best proof is that [44] concludes, on the basis of the validity of the harmonic approximation, that non-dispersive wave packets should not exist for n0 60 in CP eld, in complete contradiction to numerically exact experiments showing their existence down to n0 = 15 [49]. On the other hand, it is an interesting question how good the harmonic approximation actually is in the pure CP case. The interested reader may nd a more quantitative discussion of this point in Section 7.2. 5. Other resonances 5.1. General considerations We have so far restricted our attention to non-dispersive wave packets anchored to the principal resonance of periodically driven Hamiltonian systems. In Section 3.1, we already saw that any harmonic of the unperturbed classical motion can dominate the harmonic expansion (57) of the classical Hamilton function, provided it is resonantly driven by the external perturbation, i.e., s − !t const; s¿0 integer : (225) This is the case when the sth harmonic of the classical internal frequency is resonant with the external driving !. As depends on the classical unperturbed action, the corresponding classical resonant action is de ned by (Is)= 9H0 ! (Is)= : (226) 9I s At this action, the period of the classical motion is s times the period of the external drive. Precisely, like in the s = 1 case (the principal resonance), for any integer s¿1; Floquet eigenstates of the driven system exist which are localized on the associated classical stability islands in phase space. The energy of these eigenstates can again be estimated through the semiclassical quantization of the secular dynamics. To do so, we start from Eqs. (57) and (58) and transform to slowly varying variables (the “rotating frame”) de ned by ˆ = − !t s ; Î = I; ˆPt = Pt + !I s : (227) 26 Note that the non-harmonic terms, being entirely due to the Coulomb eld, are not removed or decreased by the addition of a magnetic eld. They are just hidden by a larger harmonic term.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 499 The Floquet Hamiltonian in this rotating frame now reads ˆH = ˆP + H0(Î) − !Î s + +∞ Vm(Î) cos m ˆ m − s + !t s m=−∞ ; (228) which is periodic with period =sT with T =2 =!. Passing to the rotating frame apparently destroys the T -periodicity of the original Hamiltonian. This, however, is of little importance, the crucial point being to keep the periodicity sT of the internal motion. If we now impose the resonance condition (225), the major contribution to the sum in Eq. (228) will come from the slowly evolving resonant term m=s, while the other terms vanish upon averaging the fast variable t over one period , leading to the secular Hamilton function Hsec = ˆPt + H0(Î) − !Î s + Vs(Î) cos(s ˆ ) : (229) This averaging procedure eliminates the explicit time dependence of ˆH, and is tantamount to restricting the validity of Hsec to the description of those classical trajectories which comply with Eq. (225) and, hence, exhibit a periodicity with period . This will have an unambiguous signature in the quasi-energy spectrum, as we shall see further down. The structure of the secular Hamiltonian is simple and reminds us of the result for the principal s = 1 resonance, Eq. (64). However, due to the explicit appearance of the factor s¿1 in the argument of the cos term, a juxtaposition of s resonance islands close to the resonant action, Eq. (226), is created. It should be emphasized that these s resonance islands are actually s clones of the same island. Indeed, a trajectory trapped inside a resonance island will successively visit all the islands: after one period of the drive, is approximately increased by 2 =s, corresponding to a translation to the next island. At the center of the islands, there is a single, stable resonant trajectory whose period is exactly = sT . At lowest order in ; all the quantities of interest can be expanded in the vicinity of Is, exactly as for the principal resonance in Section 3.1.1. Vs is consistently evaluated at the resonant action. The pendulum approximation of the secular Hamiltonian then reads: Hpend = ˆPt + H0(Î s) − ! s Î s + 1 ′′ H 0 (Î s)(Î − Î s) 2 2 + Vs(Î s) cos s ˆ ; (230) with the centers of the islands located at Î = Î s = Is ; (231) ⎧ ⎪⎨ ⎪⎩ ˆ = k 2 s ˆ = k 2 s + s if Vs(Î s)H ′′ 0 (Î s) ¡ 0 ; if Vs(Î s)H ′′ 0 (Î s) ¿ 0 ; (232) where k is an integer running from 0 to s−1. For H ′′ 0 (Î s)—see Eq. (67)—positive, these are minima of the secular Hamiltonian, otherwise they are maxima. The extension of each resonance island is, as a direct generalization of the results of Section 3.1.1: ˆ = 2 s ; (233)
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Non-dispersive wave packets in periodically driven quantum systems

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