Non-dispersive wave packets in periodically driven quantum systems

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494 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 38. Temporal dynamics of the non-dispersive wave packet of a three-dimensional hydrogen atom exposed to a linearly polarized, resonant microwave eld, in the presence of a parallel static electric eld. F0 =0:03, Fs;0 =0:009, Fs;0=F0 =0:3, n0 = 60. Driving eld phases at the di erent stages of the wave-packet’s evolution: !t = 0 (top left), =2 (top right), 3 =4 (bottom left), (bottom right). It is well localized in the three dimensions of space and repeats its shape periodically (cf. Fig. 13 for the analogous dynamics in the restricted 1D model, where no additional static electric eld is needed). The nucleus is at the center of the plot which extends over ±8000 Bohr radii. The microwave polarization axis and the static eld are oriented along the vertical axis. papers [144,44,46,54,62,30,151]. The Hamiltonian of the system in the coordinate frame corotating with the CP eld reads (compare with Eq. (184), for the pure CP case) H = p2 x + p 2 y + p 2 z 2 − 1 r − (! − !c=2)Lz + Fx + !2 c 8 (x2 + y 2 ) ; (214) where !c is the cyclotron frequency. In atomic units, the cyclotron frequency !c = −qB=m equals the magnetic eld value. It can be both positive or negative, depending on the direction of the magnetic eld. 23 This additional parameter modi es the dynamical properties which characterize the equilibrium points, the analysis of which may be carried out alike the pure CP case treated in 23 A di erent convention is used (quantization axis de ned by the orientation of the magnetic eld) in many papers on this subject. It leads to unnecessarily complicated equations.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 495 Section 3.4. A detailed stability analysis can be found in [30,46,154] and we summarize here the main results only. Since changing the sign of F in Eq. (214) is equivalent to changing the sign of x from positive to negative, we only consider the equilibrium position at xeq ¿ 0 (compare Eqs. (185)–(188)). For non-vanishing magnetic eld, its position is given by !(! − !c)xeq − 1 x2 eq − F =0: (215) Rede ning the dimensionless parameter q (see Eq. (186)) via 1 q = !(! − !c)x3 eq ; (216) we obtain for the microwave amplitude F =[!(! − !c)] 2=3 (1 − q)=q 1=3 and (217) Eeq =[!(! − !c)] 1=3 (1 − 4q)=2q 2=3 for the equilibrium energy. (218) Harmonic expansion of Eq. (214) around the equilibrium point (xeq;yeq =0;zeq = 0) allows for a linear stability analysis in its vicinity. Alike the pure CP case, the z motion decouples from the motion in the (x; y) plane. For the latter, we recover the generic harmonic Hamiltonian discussed in Section 3.4, Eq. (191), provided we substitute ˜! = ! − !c=2 : (219) When expanded at second order around the equilibrium point, Hamiltonian (214) takes the standard form of a rotating anisotropic oscillator, Eq. (191), with ˜! replacing !, and with the stability parameters: a = 1 ˜! 2 2 ! c 4 b = 1 ˜! 2 2 ! c 4 2 − x3 eq 1 + x3 eq ; : (220) The regions of stability of the equilibrium point (xeq;yeq;zeq) are thus obtained from the domains of stability of the 2D rotating anisotropic oscillator, given by Eqs. (192) and (193). They are visualized in terms of the physical parameters F and !c, (using the standard scaled electric eld F0 = Fn 4 0 = F!−4=3 ) in Fig. 39, with the black region corresponding to Eq. (192), and the grey region to Eq. (193). Observe that the presence of the magnetic eld tends to enlarge the region of stability in parameter space; for !c = 0 (pure CP case, no magnetic eld) the stability region is quite tiny, in comparison to large values of |!c|: 24 On the other hand, the stability diagram 24 As long as we are interested in long-lived wave packets, the region of small F0 is of interest only. At higher F0 and for !c ¡!, the strong driving eld will ionize the atom rather fast—see Section 7.1. This makes the gray region F0 ¿ 0:1 of little practical interest.
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