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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458 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
z (atomic units)<br />
(a)<br />
8000<br />
6000<br />
4000<br />
2000<br />
0<br />
0 1 2 3 4<br />
Time (micro<strong>wave</strong> periods)<br />
∆z ∆p (h unit)<br />
(b)<br />
100<br />
10<br />
1<br />
Heisenberg limit<br />
0 1 2 3 4<br />
Time (micro<strong>wave</strong> periods)<br />
Fig. 14. (a) Dashed l<strong>in</strong>e: Classical temporal evolution of the position of the Rydberg electron resonantly <strong>driven</strong> by a<br />
l<strong>in</strong>early polarized micro<strong>wave</strong> eld, <strong>in</strong> the one-dimensional model, Eq. (137), of the hydrogen atom. (n0 = 60; scaled eld<br />
amplitude F0 = Fn 4 0 =0:053). Thick l<strong>in</strong>e: Expectation value 〈z〉 for the non-<strong>dispersive</strong> <strong>wave</strong> packet shown <strong>in</strong> Fig. 13, as a<br />
function of time. It follows the classical trajectory remarkably well (except for collisions with the nucleus). Dotted l<strong>in</strong>e:<br />
The position uncerta<strong>in</strong>ty z = 〈z 2 〉−〈z〉 2 of the <strong>wave</strong> packet. z be<strong>in</strong>g much smaller than 〈z〉 (except near collisions<br />
with the nucleus) highlights the e cient localization of the <strong>wave</strong> packet. (b) Uncerta<strong>in</strong>ty product z p of the <strong>wave</strong><br />
packet. The <strong>periodically</strong> repeat<strong>in</strong>g maxima of this quantity <strong>in</strong>dicate the collision of the electron with the atomic nucleus.<br />
Note that the m<strong>in</strong>imum uncerta<strong>in</strong>ty at the outer turn<strong>in</strong>g po<strong>in</strong>t of the <strong>wave</strong> packet is very close to the Heisenberg limit<br />
˝=2. Although the <strong>wave</strong> packet is never a m<strong>in</strong>imal one, it is nevertheless well localized and an excellent approximation<br />
of a classical particle.<br />
is unstable, the <strong>quantum</strong> eigenstate cannot rema<strong>in</strong> fully localized—some probability has to ow<br />
away along the unstable manifold of the classical ow <strong>in</strong> the vic<strong>in</strong>ity of the hyperbolic xed po<strong>in</strong>t.<br />
Consequently, such an eigenstate is partially localized along the separatrix between librational and<br />
rotational motion. For an illustration, rst consider Fig. 16, which shows classical surfaces of section<br />
of the <strong>driven</strong> (1D) hydrogen atom, at F0 =0:034, aga<strong>in</strong> for di erent phases !t. Comparison with<br />
Fig. 12 shows a larger elliptic island at this slightly lower eld amplitude, as well as remnants<br />
of the s = 2 resonance island at slightly larger actions I=n0 1:2 :::1:3. The time evolution of<br />
the electronic density of the eigenstate localized near the hyperbolic xed po<strong>in</strong>t is displayed <strong>in</strong><br />
Fig. 17, for di erent phases of the driv<strong>in</strong>g eld. Clearly, as compared to Fig. 13, the <strong>wave</strong> packet<br />
moves <strong>in</strong> phase opposition to the driv<strong>in</strong>g eld, and displays slightly irregular localization properties.<br />
Accord<strong>in</strong>gly, the Husimi representation <strong>in</strong> Fig. 18 exhibits reasonably good localization on top of<br />
the hyperbolic po<strong>in</strong>t at phase !t = , but the electronic probability spreads signi cantly along the<br />
separatrix layer at phase, as visible at !t =0.<br />
In the above discussion of the localization properties of the <strong>wave</strong>-packet eigenstate we represented<br />
the <strong>wave</strong> function <strong>in</strong> the I– phase space of classically bounded motion (i.e., classical motion with<br />
negative energy). However, as we shall see <strong>in</strong> more detail <strong>in</strong> Section 7.1, the micro<strong>wave</strong> driv<strong>in</strong>g<br />
actually <strong>in</strong>duces a non-vanish<strong>in</strong>g overlap of all Floquet eigenstates [120,121], and, hence, of the<br />
<strong>wave</strong>-packet eigenstates, with the atomic cont<strong>in</strong>uum. It su ces to say here that the associated -<br />
nite decay rates <strong>in</strong>duce nite life times of approx. 10 6 unperturbed Kepler orbits for the <strong>quantum</strong><br />
objects considered <strong>in</strong> this section, and are therefore irrelevant on the present level of our discussion.<br />
In Figs. 13, 15, 17 and 18, a nite decay rate would manifest as a slow reduction of the