Non-dispersive wave packets in periodically driven quantum systems

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464 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 19. Isovalue curves of the angular part 1, Eq. (160), of the secular Hamiltonian Hsec represented in the plane of the L0 = L=Î 1 and coordinates. The slow evolution of the Kepler ellipse of a Rydberg electron driven by a resonant, linearly polarized microwave eld, takes place along such isovalue curves. L0 = L=Î represents the total angular momentum (a circular trajectory in a plane containing the eld polarization axis has L0 = 1), and the [canonically conjugate] angle between the eld polarization axis and the major axis of the Kepler ellipse. The separatrix emanating from the unstable xed point (L0 =0; = 0) separates rotational and librational motion, both “centered” around their respective stable xed points (L0 =1, arbitrary) and (L0 =0, = =2). The former corresponds to a circular orbit centered around the nucleus. The latter represents a straight linear orbit perpendicular to the eld axis. The unstable xed point corresponds to linear motion along the polarization axis. However, this initially degenerate Kepler ellipse will slowly precess in the azimuthal plane. The equipotential curves shown here satisfy the quantization condition (163), for n0 = 21 and p =0:::20. At lowest order, the motion in the (L0; ) plane is independent of the microwave eld strength and of the resonant principal quantum number n0. Importantly, the loops of constant 1 are independent of the microwave amplitude F and scale simply with Î. Thus, the whole quantization in the (L; ) plane has to be done only once. With this prescription we can unravel the semiclassical structure of the quasienergy spectrum induced by the additional degree of freedom spanned by (L; ), as an amendment to the spectral structure of the one-dimensional model discussed in Section 3.3.1. Fig. 19 shows the equipotential curves of 1 in the (L; ) plane. For a comparison with quantal data, the equipotential lines plotted correspond to the quantized values of 1 for n0 =21. Using the well-known properties of the Bessel functions [95], it is easy to show that 1(L; ) has the following xed points: • (L = Î 1, arbitrary ). This corresponds to a Kepler ellipse with maximum angular momentum, i.e., a circular orbit in a plane containing the microwave polarization axis along ˆz. As such a circle corresponds to a degenerate family of elliptical orbits with arbitrary orientation of the major axis, is a dummy angle. This xed point corresponds to a global maximum of 1(L = Î 1)=Î 2 1=2, and is surrounded by “rotational” trajectories in the (L; ) plane. An alternative representation of the (L; ) motion on the unit sphere, spanned by L and the z and -components of the Runge–Lenz vector, contracts the line representing this orbit in Fig. 19 to an elliptic xed point [87]. • (L=0; = =2; 3 =2). This corresponds to a degenerate straight line trajectory perpendicular to the microwave eld. Because of the azimuthal symmetry around the electric eld axis, the two points
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 465 I > 25 24 23 22 21 20 19 (a) (b) (c) 18 −1.0 −0.5 0.0 0.5 −1.0 −0.5 0.0 0.5 −1.0 −0.5 0.0 0.5 1.0 (θ+δ 1)/π Fig. 20. Isovalue curves of the secular Hamiltonian Hsec, Eq. (159), generating the (Î; ˆ ) motion of a Rydberg electron in a resonant microwave eld. Î and ˆ correspond to the atomic principal quantum number, and to the polar angle of the electron on the Kepler ellipse, respectively. The scaled microwave amplitude is xed at F0 =0:03. Since the isovalues of Hsec depend on the transverse motion in (L; ) via the constant value of 1, Eq. (160), contours (—) are shown for three characteristic values of 1, corresponding to xed quantum numbers p =0; 10; 20, Eq. (163), of the angular motion for the n0 = 21 resonant manifold. Only the “ground state” orbit satisfying Eq. (78) with N = 0 is shown, together with the separatrix (- - -) between librational and rotational motion in the (Î; ˆ ) plane. The separatrix encloses the principal resonance island in phase space, see also Eqs. (66) and (71). Panel (a) corresponds to the orbit with L0 = L=n0 1 (rotational orbit, p = 20), panel (b) to the orbit close to the separatrix of the angular motion (p = 10), panel (c) to the librational orbit close to the stable xed point L0 =0, = =2. Note that the resonance island is smallest for librational, largest for rotational, and of intermediate size for separatrix modes of the angular motion. actually correspond to the same physics. The oscillating dipole clearly vanishes there, resulting in a global minimum of 1(L =0; = =2; 3 =2) = 0. This stable xed point is surrounded by “librational” trajectories in the (L; ) plane. • (L =0; =0; ). This corresponds to a degenerate, straight line trajectory along the microwave eld, i.e., the situation already considered in the 1D model of the atom. = 0 and correspond to the two orbits pointing up and down, which are of course equivalent. This is a saddle point of 1(L =0; =0; )=J ′ 2 1 (1)Î 1: Hence, it is an unstable equilibrium point. As an implication, in the real 3D world, the motion along the microwave axis, with the phase of the radial motion locked on the external drive, is angularly unstable (see also Section 4.1). This leads to a slow precession of the initially degenerate Kepler ellipse o the axis, and will manifest itself in the localization properties of the 3D analog of the non-dispersive wave packet displayed in Fig. 13. This motion takes place along the separatrix between librational and rotational motion. Once the quantized values of 1 (represented by the trajectories in Fig. 19) have been determined, we can quantize the (Î; ˆ ) motion with these values xed. Fig. 20 shows the equipotential lines of Hsec, for the three values of 1 corresponding to the p = 0, 10 and 20 states, see Eq. (163), of the n0 = 21 manifold. In each case, the contour for the lowest state N = 0 has been drawn, together with the separatrix between the librational and rotational (Î; ˆ ) modes. The separatrix determines the size of the principal resonance island for the di erent substates of the transverse motion. Note that the principal resonance is largest for the p = 20 state, localized closest to the stable circular orbit (hence associated with the maximum value of 1), whereas the smallest resonance island is obtained for the p = 0 state, localized in the vicinity of (though not precisely at) the straight line orbit perpendicular to the eld axis (minimum value of 1). For the latter orbit itself, the rst-order >
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Non-dispersive wave packets in periodically driven quantum systems

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