Non-dispersive wave packets in periodically driven quantum systems

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516 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 However, contrary to the situation for the driven hydrogen atom discussed in Sections 3.3.1 and 3.3.2, there is a fundamental di erence between the one dimensional model of the driven three-body Coulomb problem and the full 3D problem. For the one-electron system, we have seen that the classical Kepler ellipse performs a slow precession in the angular variables, though remains bounded and does not ionize. In contrast, if one permits deviations from collinearity in the driven frozen planet dynamics, it is found that the transverse direction is generally unstable and leads to rapid ionization. This transverse ionization is simply due to the fact that the external eld destroys the intricate electron electron correlation which creates the unperturbed frozen planet. Notwithstanding, it has been shown that the application of an additional, weak static electric eld allows to compensate for the transverse instability, and to establish a classically globally stable dynamical situation for the frozen planet. The transverse con nement through the static eld again justi es the collinear model, and rst quantum calculations performed for this restricted model show the existence of a wave packet associated with the principal resonance between the frozen planet orbit and the driving eld, which faithfully traces the classical trajectory at the period of the drive. As for driven one-electron systems, these non-dispersive two-electron wave packets exhibit life times of typically 10 6 driving eld periods. 33 Hence, there is strong evidence that a resonant external forcing allows for the creation of quantum eigenstates with a quasi-classical temporal evolution, even in the presence of strong two-particle correlations. 6.3. Non-dispersive wave-packets in isolated core excitation of multielectron atoms Another example of non-dispersive wave packets in a two-component atomic system has recently been proposed for two-electron atoms [174–176]. The scheme uses an isolated-core excitation in which one of the electrons is transfered to a Rydberg trajectory by a short laser pulse, forming an initially well-localized wave packet. A second source continuously drives a transition between two discrete states of the remaining atomic core. The latter induces Rabi oscillations (or a coherent superposition) between two Rydberg series to which the rst electron is excited. If the Rabi frequency (controlled by the continuous drive of the core) is matched with the Kepler frequency of the orbit of the outer electron, the autoionization rate of the latter may be strongly suppressed, provided the respective phases are also matched properly: if the electron approaches its inner turning radius (where the con guration-interaction between Rydberg electron and core— leading to autoionization—is strongest) while the core is in its ground state, autoionization becomes impossible since the con guration-interaction does not compensate for the ionization potential of the Rydberg electron. On the other hand, when the electron is far from the nucleus (and electron– electron interaction is weak), the core may be in its excited state, without ejecting the Rydberg electron. Consequently, autoionization is suppressed for the center of the Rydberg wave packet. During time evolution, however, the wave packet spreads, its head and its tail desynchronize with the Rabi evolution of the core, and eventually approach the region close to the nucleus (where 33 Again, in contrast to the driven one electron problem, nothing guarantees that the life times obtained for the 1D model carry over to the real 3D object. On the contrary, rst results on the bare 3D Coulomb problem [173] indicate a strong dependence of the lifetimes on the dimension of the accessible con guration space.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 517 con guration-interaction is most pronounced) when the core is not in its ground state. Then these parts of the wave packet autoionize, and the remaining Rydberg population is reshaped into a localized wave packet, since the spreading tails have been chopped o . Hence, these wave packets exhibit a rather rapid “melting” (on a time scale of at most some hundred Kepler periods)—to be compared to hundreds of thousands or even more Kepler cycles performed by non-dispersive wave packets in microwave-driven hydrogen atoms studied above (which also ionize, however very slowly, see Section 7.1). The present scenario is in some sense reminiscent of the one in Section 6.1, with a (quantum) two-level core replacing the rotating molecular core. As mentioned above, a two-level system alone can only exchange one quantum with the outer electron and thus cannot provide an exact phase locking mechanism for the highly excited Rydberg electron. However, the two-level core is here driven by an external electromagnetic eld and consequently gains an additional degree of freedom which can be used for the phase locking mechanism. The drawback is that this phase locking implies losses (through autoionization). Nevertheless, the quasi-classical evolution over ∼ 100 Kepler cycles is still quite impressive, and presumably stems from the relatively sharp con nement of e cient con guration-interaction within a spatial region close to the inner turning point of the Rydberg wave packet. 7. Characteristic properties of non-dispersive wave packets After presenting several examples of non-dispersive wave packets in the previous chapters, we now study their speci c properties in more detail. Especially, several important physical processes which may a ect the existence of wave packets have so far been hidden under the carpet [177]. The two most important ones, at least for driven atoms, are ionization and spontaneous emission, and they will be discussed in detail below. First, let us brie y discuss the general properties of wave-packet eigenstates under the variation of various parameters of the driven system (e.g., microwave amplitude and frequency, the strength of an external static eld, etc.). 7.1. Ionization rates and chaos-assisted tunneling Atoms driven by microwaves will eventually ionize. Therefore, the non-dispersive wave-packet states discussed up till now cannot be, rigorously speaking, discrete states, they are rather resonances [23] with some nite lifetimes. Importantly, as we shall discuss in detail below, these lifetimes may be extremely long, of the order of millions of microwave periods. In that sense, they are comparable to those of highly excited atomic Rydberg states, which also decay, by spontaneous emission, on time scales of few millions of classical periods. Even more importantly, the lifetimes of the non-dispersive wave packets are typically orders of magnitude larger than the lifetimes of other states in the Floquet spectrum: the wave packets are particularly resistant to ionization. This is due to the classical con nement of the electron inside the regular island. To ionize, the electron has no other option but to tunnel out of the classically con ning island, before gaining energy by di usive excitation [131]. The resonance island is strictly con ning only for a one-dimensional system. For multi-dimensional systems, the tori in the resonance islands are not fully isolating and a very slow classical di usion process might eventually lead to ionization. This, however, takes place
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