Non-dispersive wave packets in periodically driven quantum systems

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470 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 25. Comparison of the numerically exact level dynamics (...) with the semiclassical prediction (—), for the n0 =22(N = 1) manifold of a 3D hydrogen atom exposed to a microwave eld with frequency ! =1=(21) 3 , resonant with the n0 = 21 manifold. For su ciently high F0, the quantum states originating at F0 = 0 from the unperturbed n0 =22 level are captured by the principal resonance island and then represent the rst excited state of the motion in the (Î; ˆ ) plane (i.e., N = 1 in Eq. (78)). Since the island’s size depends on the angular (L; ) motion (value of p in Eq. (163), see also Fig. 20), states with large p enter the resonance zone rst. For these, the agreement between quantum and semiclassical quasi-energies starts to be satisfactory at lower F0 values than for low-p states. Finally note that, as already mentioned at the end of Section 3.3.1, all wave-packet eigenstates have a nite decay rate which induces a slow, global reduction of the electronic density localized on the resonantly driven classical periodic orbit. However, the time-scale of this decay is of the order of thousands to millions of Kepler cycles, and therefore leaves our above conclusions unaffected. However, some very intriguing consequences of the non-vanishing continuum coupling will be discussed in Section 7.1. 3.4. Rydberg states in circularly polarized microwave elds As shown in the preceding section, the use of a linearly polarized microwave eld is not su cient to produce a non-dispersive wave packet fully localized in all three dimensions, due to the azimuthal symmetry around the microwave polarization axis. To get more exibility, one may consider the case of arbitrary polarization. It turns out that the results are especially simple in circular polarization. They are the subject of this section. In most experiments on microwave driven Rydberg atoms, linearly polarized (LP) microwaves have been used [132,133,135–137]. For circular polarization (CP), rst experiments were performed for alkali atoms in the late eighties [138,139], with hydrogen atoms following only recently [134]. The latter experiments also studied the general case of elliptic polarization (EP). While, at least theoretically, di erent frequency regimes were considered for CP microwaves (for a review, see [140])—we shall restrict our discussion here to resonant driving. Given a di erent microwave
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 471 Fig. 26. Same as Fig. 25, but for N = 2. The quantum states originate from the manifold n0 = 20. The resonance island in the (Î; ˆ ) coordinates is too small to support any N = 2 states for F0 ¡ 0:03, as seen from the negative slope of the quasienergy levels. The quantum states “cross” the separatrix as the amplitude is further increased, and successively enter the resonance zone, starting from the largest value of p in Eq. (163) (the resonance island size increases with p). Even for F0 ¿ 0:04; only a minority of substates of the n0 = 20 manifold is well represented by the resonant semiclassical dynamics (indicated by the solid lines). polarization, and thus a di erent form of the interaction Hamiltonian, Eq. (111), Kepler trajectories which are distinct from those considered in the LP case will be most e ciently locked on the external driving. Hence, in the sequal, we shall launch non-dispersive wave packets along periodic orbits which are distinct from those encountered above. Historically, the creation of non-dispersive wave packets in CP and LP microwave elds, respectively, has been considered quite independently. In particular, in the CP case, the notion of non-dispersive wave packets has been introduced [34] along quite di erent lines than the one adopted in this review. The original work, as well as subsequent studies of the CP situation [44,45,47,49,50,55,61,63] used the fact that, in this speci c case, the time dependence of the Hamiltonian may be removed by a unitary transformation to the rotating frame (see below). Thus, the stable periodic orbit at the center of the island turns into a stable equilibrium point in the rotating frame. This allows the expansion of the Hamiltonian into a Taylor series in the vicinity of the xed point, and in particular a standard harmonic treatment using normal modes. We shall review this line of reasoning in detail below. It is, however, instructive to rst discuss the very same system using the general resonance approach exposed in Section 3.1. 3.4.1. Hamiltonian With ˆz the propagation direction of the microwave, the electric eld rotates in the x–y plane, and Hamiltonian (106) takes the following explicit form: HCP = ˜p2 2 1 − + F{x cos(!t)+y sin(!t)} : (164) r
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Non-dispersive wave packets in periodically driven quantum systems

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