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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496 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
F 0<br />
4<br />
2<br />
0<br />
-2<br />
-4<br />
-2 -1 0 1 2<br />
ωc /ω<br />
Fig. 39. (Shaded) Regions of stability of the equilibrium po<strong>in</strong>t (xeq;yeq;zeq) for circularly polarized driv<strong>in</strong>g (amplitude F,<br />
frequency !) of a hydrogen atom, <strong>in</strong> the presence of a magnetic eld (correspond<strong>in</strong>g cyclotron frequency !c). The black<br />
and gray regions correspond to the two regions of stability, described by Eqs. (192) and (193), respectively.<br />
does not provide us any detailed <strong>in</strong>formation on the actual size of the resonance island surround<strong>in</strong>g<br />
the equilibrium po<strong>in</strong>t. However, it is precisely the size of the resonance island which is crucial<br />
for anchor<strong>in</strong>g non-<strong>dispersive</strong> <strong>quantum</strong> <strong>wave</strong> <strong>packets</strong> close to the classical equilibrium po<strong>in</strong>t (see<br />
Section 3.1.3).<br />
An alternative approach to characterize the stability properties of the classical motion near (xeq;yeq;<br />
zeq) has been advertized <strong>in</strong> [30,44,46,54,62,144,151]: the concept of zero velocity surfaces (ZVS).<br />
In order to construct a ZVS, the Hamilton function is expressed <strong>in</strong> terms of velocities rather than<br />
canonical momenta. For the harmonic Hamiltonian (191), the calculation yields<br />
H = v2 x + v2 y !2<br />
+<br />
2 2 [(a − 1)x2 +(b− 1)y 2 ] : (221)<br />
Thus, the “k<strong>in</strong>etic energy” becomes a positive function of velocities, and one can de ne the ZVS as<br />
S = H − v2 x + v 2 y<br />
2<br />
; (222)<br />
the generalization of an e ective potential for <strong>in</strong>teractions which mix position and momentum coord<strong>in</strong>ates.<br />
Note that, when the velocities co<strong>in</strong>cide with the canonical momenta, S is noth<strong>in</strong>g but<br />
the potential energy surface. We prefer to denote it S <strong>in</strong>stead of V , to stress the di erence. As<br />
discussed <strong>in</strong> detail <strong>in</strong> [30], a ZVS may be used to locate the equilibrium po<strong>in</strong>ts. However, their<br />
stability properties are not obvious (contrary to the potential surface, where m<strong>in</strong>ima de ne stable<br />
xed po<strong>in</strong>ts, while maxima and saddle po<strong>in</strong>ts are unstable). For a ZVS, saddles are also unstable,<br />
but maxima may either be stable or unstable. For example, the rst stability region, Eq. (192), of<br />
the rotat<strong>in</strong>g 2D anistropic Hamiltonian is associated with a stable m<strong>in</strong>imum of the ZVS. The second<br />
region of stability, Eq. (193), corresponds to a; b 6 1 and thus to a maximum of the ZVS. However,<br />
the ZVS does not show any qualitative change whether (a − b) 2 +8(a + b) is positive or negative,