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) 409–547 429<br />
the equilibrium po<strong>in</strong>t than further away from it. The <strong>quantum</strong> eigenfunctions have the same property:<br />
the probability density is large near the equilibrium po<strong>in</strong>t. This trivial enhancement of the probability<br />
density is already well known for a one-dimensional system where the WKB <strong>wave</strong> function, Eq.<br />
(37), diverges when the momentum tends to zero. The localization e ect near an unstable po<strong>in</strong>t is<br />
just the <strong>quantum</strong> manifestation of the critical slow<strong>in</strong>g down of the classical particle [60].<br />
3. <strong>Non</strong>-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> and their realization <strong>in</strong> various atomic <strong>systems</strong><br />
3.1. General model—non-l<strong>in</strong>ear resonances<br />
In this section, we present the general theory of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong>. As expla<strong>in</strong>ed <strong>in</strong><br />
Section 1.4, the basic <strong>in</strong>gredients for build<strong>in</strong>g a non-<strong>dispersive</strong> <strong>wave</strong> packet are a non-l<strong>in</strong>ear dynamical<br />
system and an external periodic driv<strong>in</strong>g which is resonant with an <strong>in</strong>ternal frequency of<br />
the dynamical system. We present here a very general theory start<strong>in</strong>g out from classical mechanics<br />
which provides us with the most suggestive approach to non-l<strong>in</strong>ear resonances. In a second step, we<br />
choose a pure <strong>quantum</strong> approach giv<strong>in</strong>g essentially the same physics.<br />
We use a one-dimensional model, which displays all the <strong>in</strong>terest<strong>in</strong>g features of non-l<strong>in</strong>ear resonances.<br />
While the direct l<strong>in</strong>k between classical non-l<strong>in</strong>ear resonances, the correspond<strong>in</strong>g Floquet<br />
states, and non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> has been identi ed only recently [43,32,33,49,64] some<br />
aspects of the developments presented below may be found <strong>in</strong> earlier studies [39,85,40,42].<br />
Several complications not <strong>in</strong>cluded <strong>in</strong> the simple one-dimensional model are important features of<br />
“real <strong>systems</strong>”. They are discussed at a later stage <strong>in</strong> this paper:<br />
• the e ect of additional degrees of freedom, <strong>in</strong> Sections 3.3.2–3.5;<br />
• higher non-l<strong>in</strong>ear resonances (where the driv<strong>in</strong>g frequency is a multiple of the <strong>in</strong>ternal frequency),<br />
<strong>in</strong> Section 5;<br />
• an unbounded phase space, lead<strong>in</strong>g to the decay of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> (as “open <strong>quantum</strong><br />
<strong>systems</strong>”), <strong>in</strong> Section 7.1;<br />
• sources of “decoherence”, such as spontaneous emission of atomic <strong>wave</strong> <strong>packets</strong>, <strong>in</strong> Section 7.2;<br />
• deviations from temporal periodicity, <strong>in</strong> Section 8.3.<br />
In particular cases, an apparently simpler approach is also possible (such as the use of the rotat<strong>in</strong>g<br />
frame for a Rydberg atom exposed to a circularly polarized electromagnetic eld, see Section 3.4).<br />
Despite all its advantages, it may be quite speci c and too restricted to reveal non-l<strong>in</strong>ear resonances<br />
as the actual cause of the phenomenon. Here, we seek the most general description.<br />
3.1.1. Classical dynamics<br />
Let us start from a time-<strong>in</strong>dependent, bounded, one-dimensional system described by the Hamilton<br />
function H0(p; z). S<strong>in</strong>ce energy is conserved, the motion is con ned to a one-dimensional manifold<br />
<strong>in</strong> two-dimensional phase space. Except for energies which de ne a xed po<strong>in</strong>t of the Hamiltonian<br />
dynamics (such that 9H0=9z = 0 and 9H0=9p = 0; these xed po<strong>in</strong>ts generically only exist at some<br />
isolated values of energy, for example at E = 0 for the harmonic oscillator), the motion is periodic<br />
<strong>in</strong> time, and the phase space trajectory is a simple closed loop.