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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444 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
call this situation “optimal resonance”. From Eq. (101), it is associated with an <strong>in</strong>teger value of n0.<br />
On the contrary, a half-<strong>in</strong>teger value of n0 corresponds to = 1 and the least optimal case.<br />
3.2. Rydberg states <strong>in</strong> external elds<br />
3.2.1. Rydberg atoms<br />
In order to construct non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong>, a <strong>quantum</strong> system subject to periodic driv<strong>in</strong>g<br />
with classically non-l<strong>in</strong>ear dynamics is needed. The latter requirement rules out the harmonic<br />
oscillator, and all its variants. The simplest periodic driv<strong>in</strong>g is certa<strong>in</strong>ly provided by an externally<br />
applied, monochromatic electromagnetic eld. Extremely stable, tunable and well controlled sources<br />
exist over a wide range of frequencies.<br />
Furthermore, <strong>in</strong>coherent processes which destroy the phase coherence of the <strong>quantum</strong> <strong>wave</strong> function<br />
have to be m<strong>in</strong>imized. Otherwise, they will spoil the localization properties of the non-<strong>dispersive</strong><br />
<strong>wave</strong> <strong>packets</strong> and—<strong>in</strong> the worst case—destroy them completely. Therefore, the characteristic time<br />
scales of the <strong>in</strong>coherent processes should be at least much longer than the period of the driv<strong>in</strong>g<br />
eld. In this respect, atomic electrons appear as very good candidates, s<strong>in</strong>ce—given suitable experimental<br />
conditions—atoms can be considered as practically isolated from the external world, with<br />
spontaneous emission of photons as the only <strong>in</strong>coherent process. Spontaneous emission is usually a<br />
very slow mechanism, especially for highly excited states: the spontaneous lifetime of typical atomic<br />
states is at least four or ve orders of magnitude longer than the classical Kepler period (typically<br />
nanoseconds vs. femtoseconds for weakly excited states [22]).<br />
The Coulomb <strong>in</strong>teraction between the nucleus and the electrons is highly non-l<strong>in</strong>ear, which is<br />
very favorable. The e ciency of the coupl<strong>in</strong>g with an external electromagnetic eld is known to<br />
<strong>in</strong>crease rapidly with the degree of excitation of the atom [22]. As we have seen <strong>in</strong> the preceed<strong>in</strong>g<br />
Sections 3.1.1 and 3.1.2, non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> are the <strong>quantum</strong> mechanical counterparts of<br />
non-l<strong>in</strong>ear resonances <strong>in</strong> <strong>periodically</strong> <strong>driven</strong> Hamiltonian <strong>systems</strong>, where the period of the drive<br />
matches some <strong>in</strong>tr<strong>in</strong>sic time-scale of the unperturbed Hamiltonian dynamics. Due to the immediate<br />
correspondence between the classical Kepler problem and the hydrogen atom, the relevant time-scale<br />
<strong>in</strong> this simplest atomic system is the unperturbed classical Kepler period, which—compare Eq. (40)—<br />
co<strong>in</strong>cides with the <strong>in</strong>verse level spac<strong>in</strong>g between neighbour<strong>in</strong>g eigenstates of the unperturbed atom,<br />
for large <strong>quantum</strong> numbers. Hence, the driv<strong>in</strong>g eld frequency has to be chosen resonant with an<br />
atomic transition <strong>in</strong> the Rydberg regime, typically around the pr<strong>in</strong>cipal <strong>quantum</strong> number n0 = 60.<br />
This is the micro<strong>wave</strong> doma<strong>in</strong>, where excellent sources exist. Thus, we believe that atomic Rydberg<br />
states are very well suited for the experimental preparation of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> 13 .<br />
In most cases, the energy scale <strong>in</strong>volved <strong>in</strong> the dynamics of Rydberg electrons is so small that<br />
the <strong>in</strong>ner electrons of the ionic core can be considered as frozen and ignored. Thus, we will consider<br />
ma<strong>in</strong>ly the hydrogen atom as the simplest prototype. Multi-electron e ects are discussed <strong>in</strong><br />
Section 6.2. Alternative <strong>systems</strong> for observ<strong>in</strong>g non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> are considered <strong>in</strong><br />
Section 5.2.<br />
13 Note, however, that this is a speci c choice. Any <strong>driven</strong> <strong>quantum</strong> system with a su ciently high density of states and<br />
mixed regular–chaotic classical dynamics will exhibit non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong>. S<strong>in</strong>ce a mixed phase space structure<br />
is the generic scenario for dynamical <strong>systems</strong>, non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> are expected to be a completely general and<br />
ubiquitous phenomenon.