Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
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A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 535<br />
the energy spac<strong>in</strong>gs <strong>in</strong> the immediate vic<strong>in</strong>ity of the <strong>wave</strong>-packet state directly. Hence, micro<strong>wave</strong><br />
probe spectroscopy should be an extremely sensitive probe, s<strong>in</strong>ce it allows for the unambiguous<br />
identi cation of the <strong>wave</strong> packet via the characterization of its local spectral environment.<br />
Given the spectroscopic resolution which is nowadays available <strong>in</strong> the optical as well as <strong>in</strong> the<br />
micro<strong>wave</strong> doma<strong>in</strong>, the spectroscopic approach outl<strong>in</strong>ed above seems to be the method of choice for<br />
an unambiguous identi cation, and—where possible—for an e cient launch of non-<strong>dispersive</strong> <strong>wave</strong><br />
<strong>packets</strong> along a periodic orbit of the classical dynamics. What it requires, however, is a precise<br />
determ<strong>in</strong>ation of Floquet spectra from the accompany<strong>in</strong>g <strong>quantum</strong> calculation. Fortunately, both for<br />
hydrogen and for alkali atoms, the necessary theoretical <strong>quantum</strong> data may be obta<strong>in</strong>ed from already<br />
exist<strong>in</strong>g software [225,226].<br />
Although we elaborated <strong>in</strong> this review paper only the case of the hydrogen atom, the general<br />
concepts are also fruitful for non-hydrogenic atoms. Indeed, the major di erence between the Rydberg<br />
electron <strong>in</strong> a hydrogen atom and <strong>in</strong> a non-hydrogenic atom is the existence <strong>in</strong> the latter case of an<br />
ionic core which a ects the classical and <strong>quantum</strong> dynamics of the Rydberg electron. On the scale<br />
of a Rydberg atom, the ionic core is a extremely small object which will thus <strong>in</strong>duce a very<br />
local perturbation. As long as the Rydberg electron does not approach the ionic core, it behaves<br />
completely similarly <strong>in</strong> hydrogen or non-hydrogenic atoms. Thus, the properties of non-<strong>dispersive</strong><br />
<strong>wave</strong> <strong>packets</strong> trac<strong>in</strong>g circular or elliptical classical orbits are essentially <strong>in</strong>dependent of the ionic<br />
core, and the hydrogenic analysis holds. For orbits which come close to the nucleus, the ionic core<br />
may scatter the Rydberg electron. Thus, <strong>in</strong>stead of be<strong>in</strong>g <strong>in</strong>de nitely trapped on a torus <strong>in</strong>side a<br />
resonance island, it may happen that the Rydberg electron hops from a torus to another one when it<br />
gets close to the nucleus. This of course will a ect the long time classical and <strong>quantum</strong> dynamics.<br />
Nevertheless, it rema<strong>in</strong>s true that most of the time the classical dynamics—and consequently the<br />
phase lock<strong>in</strong>g phenomenon responsible for the existence of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong>—is identical<br />
to the hydrogenic dynamics. From the <strong>quantum</strong> po<strong>in</strong>t of view, the ionic core is responsible for the<br />
existence of non-zero <strong>quantum</strong> defects <strong>in</strong> the low angular momentum channels. The energy levels,<br />
mixed by the micro<strong>wave</strong> driv<strong>in</strong>g, will thus be signi cantly shifted from their hydrogenic positions.<br />
However, the structure of the energy levels—grouped <strong>in</strong> manifolds—will essentially survive, see<br />
[225]. It is likely that some non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> also exist <strong>in</strong> non-hydrogenic atomic species.<br />
8.3. Preparation through tailored pulses<br />
Another, <strong>in</strong>direct method for prepar<strong>in</strong>g non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> is also available. This will<br />
be the method of choice for <strong>wave</strong>-<strong>packets</strong> mov<strong>in</strong>g along classical orbits of large angular momentum<br />
(small eccentricity). Such states, obviously, are not accessible to a direct optical excitation from<br />
weakly excited, low angular momentum states. The same general scheme may be also applicable<br />
to high eccentricity <strong>wave</strong> <strong>packets</strong> although <strong>in</strong> that case we expect that the direct excitation may be<br />
more e cient and exible. The method to be discussed here consists of two stages. We rst prepare<br />
the atom <strong>in</strong> a well chosen and well de ned <strong>in</strong>itial highly excited state, and then turn the micro<strong>wave</strong><br />
eld on relatively slowly, from zero amplitude to its plateau value Fmax.<br />
A non-<strong>dispersive</strong> <strong>wave</strong> packet is a s<strong>in</strong>gle eigenstate |E〉 of the Floquet Hamiltonian describ<strong>in</strong>g<br />
the <strong>driven</strong> system at xed driv<strong>in</strong>g eld amplitude F. As shown <strong>in</strong> Sections 3.3.2, 3.4 and 7.3,<br />
the evolution of the quasi-energies of the <strong>driven</strong> atom with an external control parameter like the<br />
driv<strong>in</strong>g eld amplitude is rather complicated. It re ects the dramatic transformation of the structure