Non-dispersive wave packets in periodically driven quantum systems

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514 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 [163]. In [51], the following model Hamiltonian has been proposed: H = ˜p2 2 − 1 |˜r + ã(t)| ; (261) where ã(t) denotes the position of the center of the Coulomb eld w.r.t. the molecular center of mass, and is assumed to rotate in the x–y plane with constant frequency !: cos !t −sin !t ã(t)= ã: sin !t cos !t (262) In the rotating frame, one obtains the Hamiltonian [163] H = p2 x + p 2 y + p 2 z 2 − 1 r + a!2x − !Lz + a2 ! 2 2 ; (263) which, apart from the constant term a 2 ! 2 =2, is equivalent to the one describing an atom driven by a CP eld (compare Eq. (184)). Note that the role of the microwave amplitude (which can be arbitrarily tuned in the CP problem) is taken by a! 2 , i.e., a combination of molecular parameters, what, of course, restricts the experimental realization of non-dispersive wave packets in the molecule to properly selected molecular species [51]. With the help of the stability analysis outlined in Section 3.4.4, Eqs. (185)–(188), the equilibrium position xeq of the molecular Rydberg electron is easily estimated according to (assuming a small value of a, limited by the size of the molecular core) xeq (I=J) 2=3 (264) with I the molecular momentum of inertia, J = !I the rotational quantum number. To optimize the angular localization of the wave packet, it is necessary that xeq be su ciently large (from Section 3.2, xeq ∼ n 2 0 , where n0 is the electronic principal quantum number). Thus, for given I, J should be small. In [51], a hydrogen–tritium molecule is considered, which yields n0 18 for J =1. Note, however, that such reasoning is not justi ed. The e ective Hamiltonian (261) implies a classical description of the molecular rotation (much as the classical treatment of the periodic drive in Eq. (54), with a well-de ned phase) de ned by the position vector ã(t). For such an approach to be valid, J must be su ciently large. The molecular rotation plays the role of the microwave eld in the analogous CP problem, the number of rotational quanta is just equivalent to the average number of photons de ning the amplitude of the (classical) coherent state of the driving eld. Clearly, if J is too small, the e ect of an exchange of angular momentum between the Rydberg electron and the core on the quantum state of the core (and, hence, on a! 2 assumed to be constant in Eq. (263)) cannot be neglected and, therefore, precludes any semiclassical treatment, see also [68,92]. In other words, if J is too small, the number of rotational states of the core which are coupled via the interaction is too small to mimic a quasi-classical evolution as suggested by Eq. (262). Nontheless, this caveat does not completely rule out the existence of molecular non-dispersive wave packets, provided a fast rotation of a core with large momentum of inertia (to render xeq su ciently large, Eq. (264), such that the electronic wave packet gets localized far away from the molecular core) can be realized, as also suggested in [51].
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 515 6.2. Driven Helium in a frozen planet con guration In the previous examples of non-dispersive wave packets, the key point has been the generic appearance of a nonlinear resonance for periodically driven quantum systems whose unperturbed dynamics is integrable. A natural question to ask is whether the concept of non-dispersive wave packets can be generalized to systems which exhibit mixed regular–chaotic dynamics even in the absence of the external perturbation. In the atomic realm, such a situation is realized for the helium atom, where electron–electron interactions provide an additional source of non-linearity. The corresponding Hamiltonian writes in atomic units HHe = ˜p2 1 2 + ˜p2 2 2 − − 2 r1 2 1 + : (265) r2 |˜r1 −˜r2| As a matter of fact, the classical and quantum dynamics of the three-body Coulomb problem generated by Hamiltonian (265) has been a largely unexplored “terra incognita” until very recently [164], since the dimensionality of the phase space dynamics increases from e ectively two to e ectively eight dimensions when a second electron is added to the familiar Kepler problem. Furthermore, the exact quantum mechanical treatment of the helium atom remains a formidable task since the early days of quantum mechanics, and considerable advances could be achieved only very recently, with the advent of modern semiclassical and group theoretical methods [165–168]. Already the classical dynamics of this system exhibits a largely chaotic phase space structure, which typically leads to the rapid autoionization of the associated doubly excited quantum states of the atom. One of the major surprises in the analysis of the three-body Coulomb problem during the last decades has therefore been the discovery of a new, highly correlated and classically globally stable electronic con guration, the “frozen planet” [169,170]. The appeal of this con guration resides in its counterintuitive, asymmetric character where both electrons are located on the same side of the nucleus. Furthermore, this con guration turns out to be the most robust of all known doubly excited two-electron con gurations, in the sense that it occupies a large volume in phase space. Its stability is due to the strong coupling of the two electrons by the 1=|˜r1 − ˜r2| term in Hamiltonian (265), which enforces their highly correlated motion. The frozen planet is an ideal candidate to test the prevailance of the concept of non-dispersive wave packets in systems with intrinsically mixed dynamics. In a recent study [171,110,172,173] the response of this highly correlated two-electron con guration to a periodic force has been investigated from a classical and from a quantum mechanical point of view. The Hamiltonian for the driven problem writes, in the length gauge, H = HHe + F cos(!t)(z1 + z2) : (266) Guided by the experience on non-dispersive wave packets in one electron Rydberg states, the driving frequency ! was chosen near resonant with the natural frequency FP ≈ 0:3n −3 i of the frozen planet, where ni denotes the principal quantum number of the inner electron. It was found that, for a suitably chosen driving eld amplitude F, a non-linear resonance between the correlated electronic motion and the external drive can be induced in the classical dynamics, at least for the collinear frozen planet where the three particles (two electrons and the nucleus) are aligned along the polarization axis of the driving eld.
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Non-dispersive wave packets in periodically driven quantum systems

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