Non-dispersive wave packets in periodically driven quantum systems

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