Non-dispersive wave packets in periodically driven quantum systems

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460 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 16. Surface of section of the classical phase space of a 1D hydrogen atom driven by a linearly polarized microwave eld, for di erent values of the phase: !t = 0 (top left), !t = =2 (top right), !t = (bottom left), !t =3 =2 (bottom right). The action angle variables I; are de ned by Eq. (117). At this value of the eld amplitude, F0 =0:034, the principal resonance island (and a small remnant of the s = 2 resonance island) remain as the only regions of regular motion, in a globally chaotic phase space. |ψ(z)| 2 0.0015 0.001 0.0005 0 0.0015 0.001 0.0005 t=0 t=0.25 t=0.5 t=0.75 0 0 5000 0 5000 z (atomic units) Fig. 17. Wave-packet eigenstate anchored to the hyperbolic xed point of the principal resonance of the 1D hydrogen atom driven by a linearly polarized microwave eld, in con guration space, for the same phases of the driving eld as in Fig. 16. The wave function is partly localized, especially close to the outer turning point at t =0:5 × 2 =!, but the localization is far from being perfect. Comparison to Figs. 12, 13 and 16 shows that the state evolves in phase opposition with the stable, non-dispersive wave-packet, with signi cantly worse localization properties.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 461 Fig. 18. Husimi representation of the “hyperbolic” wave-packet eigenstate of Fig. 17, for the same phases (scales as in Fig. 16). Clearly, the quantum mechanical eigenstate of the atom in the eld follows the classical evolution. It is partly localized on top of the hyperbolic xed point, but also spreads along the separatrix con ning the principal resonance. The localization is more visible at t = 0 (top left), the spreading more visible at t =0:5 × 2 =! (bottom left). dynamics along the other degrees of freedom. In atomic systems, the Coulomb degeneracy makes it possible to gain a full understanding of this phenomenon. The starting point is similar to the 1D analysis in Section 3.3.1, that is the expression of the Floquet Hamiltonian—whose eigenstates are of interest—as a function of action-angle coordinates (I; ), (L; ), (M; ) introduced in Section 3.2. Using Eqs. (125) and (126) and the Fourier expansion, Eqs. (121)–(123), of the position operator, one obtains with H = Pt − 1 + F 2I 2 Xm(I)=I 2 J ′ m(me) 1 − M 2 L 2 +∞ m=−∞ [ − Xm cos cos(m − !t)+Ym sin sin(m − !t)] (152) ; m=0 ; m Ym(I)=I (153) 2 √ 1 − e2Jm(me) ; m=0 ; me (154) X0(I)=− 3 2 eI 2 ; (155) Y0(I)=0 (156)
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Non-dispersive wave packets in periodically driven quantum systems

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