Non-dispersive wave packets in periodically driven quantum systems

→ ;
→ ;
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 451
→ : (130)
It is therefore useful to introduce the “scaled” total angular momentum and its component along the
z-axis, by choosing
= I 2 ; (131)
and doing so leads to
L0 = L
I ;
M0 = M
I
: (132)
The eccentricity of the classical ellipse then reads
e = 1 − L2 0 ; (133)
and only depends—as it should—on scaled quantities. Similarly, the Euler angles describing the
orientation of the ellipse are scaled quantities, by virtue of Eq. (130).
When dealing with non-dispersive wave packets, it will be useful to scale the amplitude and the
frequency of the external eld with respect of the action Î 1 of the resonant orbit. With the above
choice of , Eq. (131), the scaling relation (129) for ! de nes the scaled frequency
!0 = !I 3 ; (134)
which turns into !0 = Î 3
1 with the resonance condition, Eq. (65), and enforces
Î 1 = ! −1=3 ; (135)
by virtue of Eq. (119). Correspondingly, the scaled external eld is de ned as
F0 = FI 4 ; (136)
which, with Eq. (135), turns into F0 =F! −4=3 at resonance. Hence, except for a global multiplicative
factor I −2 , the Hamiltonian of a hydrogen atom in an external eld depends only on scaled quantities.
Finally, note that the quantum dynamics is not invariant with respect to the above scaling transformations.
Indeed, the Planck constant ˝ xes an absolute scale for the various action variables. Thus,
the spectrum of the Floquet Hamiltonian will not be scale invariant, while the underlying classical
phase space structure is. This latter feature will be used to identify in the quantum spectrum the
remarkable features we are interested in.
3.3. Rydberg states in linearly polarized microwave elds
We are now ready to consider speci c examples of non-dispersive wave packets. We consider
rst the simplest, one-dimensional, driven hydrogen atom, as de ned by Eqs. (113) and (114).