Non-dispersive wave packets in periodically driven quantum systems

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438 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 This semiclassical quantization scheme is expected to work provided the classical phase space velocity is su ciently large, see Section 2. This may fail close to the stable and unstable xed points where the velocity vanishes. Near the stable equilibrium point, the expansion of the Hamiltonian at second order leads to an approximate harmonic Hamiltonian with frequency !harm = | V1(Î 1)H ′′ 0 (Î 1)| : (80) In this harmonic approximation, the semiclassical quantization is known to be exact [7]. Thus, close to the stable equilibrium, the quasi-energy levels, labeled by the non-negative integer N , are given by the harmonic approximation. There are various cases which depend on the signs of H ′′ 0 (Î 1), V1(Î 1), and , with the general result given by EN;k = H0(Î 1) − !Î 1 + k + ˝! − sign(H 4 ′′ 0 (Î 1)) | V1(Î 1)|− N + 1 ˝!harm 2 : (81) For N =0; k=0, this gives a fairly accurate estimate of the energy of the non-dispersive wave packet with optimum localization. The EBK semiclassical scheme provides us also with some interesting information on the eigenstate. Indeed, the invariant tori considered here are tubes surrounding the resonant stable periodic orbit. They cover the [0; 2 ] range of the t variable but are well localized in the transverse (Î=I; ˆ = −!t) plane, with an approximately Gaussian phase space distribution. Hence, at any xed time t, the Floquet eigenstate will appear as a Gaussian distribution localized around the point (I = Î 1; = !t). As this point precisely de nes the resonant, stable periodic orbit, one expects the N = 0 state to be a Gaussian wave packet following the classical orbit. In the original (p; z) coordinates, the width of the wave packet will depend on the system under consideration through the change of variables (p; z) → (I; ), but the Gaussian character is expected to be approximately valid for both the phase space density and the con guration space wave function, as long as the change of variables is smooth. Let us note that low-N states may be considered as excitations of the N = 0 “ground” state. Such states has been termed “ otons” in [42] where their wave-packet character was, however, not considered. The number of eigenstates trapped within the resonance island—i.e., the number of non-dispersive wave packets—is easily evaluated in the semiclassical limit, as it is the maximum N with librational motion. It is roughly the area of the resonance island, Eq. (71), divided by 2 ˝: Number of trapped states 8 V1(Î 1) ˝ H ′′ 0 (Î : (82) 1) Near the unstable xed point—that is at the energy which separates librational and rotational motion—the semiclassical quantization fails because of the critical slowing down in its vicinity [93]. The corresponding quantum states—known as separatrix states [84]—are expected to be dominantly localized near the unstable xed point, simply because the classical motion there slows down, and the pendulum spends more time close to its upright position. This localization is once again of purely classical origin, but not perfect: some part of the wave function must be also localized along the separatrix, which autointersects at the hyperbolic xed point. Hence, the Floquet eigenstates
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 439 associated with the unstable xed points are not expected to form non-dispersive wave packets with optimum localization (see also Section 2.3). 3.1.4. The Mathieu approach The pendulum approximation, Eq. (66), for a resonantly driven system can also be found by a pure quantum description [39,42]. Let us consider a Floquet state of the system. Its spatial part can be expanded in the eigenbasis of the unperturbed Hamiltonian H0, H0| n〉 = En| n〉 ; (83) while the time-periodic wave function can be expanded in a Fourier series. One obtains | (t)〉 = cn;k exp(−ik!t)| n〉 ; (84) n;k where the coe cients cn;k are to be determined. The Schrodinger equation for the Floquet states (eigenstates of H), Eq. (75), with quasi-energies E and time dependence (73), reads cn;k(E + k˝! − En)= 2 〈 n|V | n+p〉(cn+p;k+1 + cn+p;k−1) : (85) p For = 0, the solutions of Eq. (85) are trivial: E = En − k˝!, which is nothing but the unperturbed energy spectrum modulo ˝!. 11 In the presence of a small perturbation, only quasi-degenerate states with values close to En − k˝! will be e ciently coupled. In the semiclassical limit, see Section 2, Eq. (40), the unperturbed eigenenergies En, labeled by a non-negative integer, are locally approximately spaced by ˝ , where is the frequency of the unperturbed classical motion. Close to resonance, ! (Eq. (65)), and thus: En − k˝! En+1 − (k +1)˝! En+2 − (k +2)˝! ::: ; (86) so that only states with the same value of n − k will be e ciently coupled. The rst approximation is thus to neglect the couplings which do not preserve n − k. This is just the quantum version of the secular approximation for the classical dynamics. Then, the set of Eqs. (85) can be rearranged in independent blocks, each subset being characterized by n − k. The various subsets are in fact identical, except for a shift in energy by an integer multiple of ˝!. This is nothing but the ˝!-periodicity of the Floquet spectrum already encountered in Sections 3.1.2 and 3.1.3. As a consequence, we can consider the n − k = 0 block alone. Consistently, since Eq. (86) is valid close to the center of the resonance only, one can expand the quantities of interest in the vicinity of the center of the resonance, and use semiclassical approximations for matrix elements of V . Let n0 denote the e ective, resonant quantum number such that, 11 Note that, as a consequence of the negative sign of the argument of the exponential factor in Eq. (84), the energy shift k˝! appears here with a negative sign in the expression for E—in contrast to semiclassical expressions alike Eq. (81), where we chose the more suggestive positive sign. Since k = −∞ ::: + ∞, both conventions are strictly equivalent.
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Non-dispersive wave packets in periodically driven quantum systems

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