Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 421
At longer times, the wave packet continues to alternate between collapses and revivals. In Fig. 4, we
show the temporal evolution of the product z p. It is initially close to the Heisenberg limit (minimum
value) ˝=2, and oscillates at the frequency of the classical motion with a global increase. When
the wave packet has completely spread, the uncertainty product is roughly constant, with apparently
erratic uctuations. At the revival time, the uncertainty undergoes again rather orderly oscillations
of a relatively large magnitude reaching, at minima, values close to ˝. That is a manifestation of its
partial relocalization.
For the three-dimensional hydrogen atom, the energy spectrum is exactly the same as in one
dimension. This implies that the temporal dynamics is built from exactly the same frequencies;
thus, the 3D dynamics is essentially the same as the 1D dynamics. 5 Indeed, collapses and revivals
of the wave packet were also observed, under various experimental conditions, in the laboratory
[5,11,25,26]. Fig. 5 shows the evolution of a minimum uncertainty wave packet of the 3D atom,
initially localized on a circular Kepler orbit of the electron. It is built as a linear combination
of circular hydrogenic states (i.e., states with maximum angular and magnetic quantum numbers
L = M = n − 1), using the same Gaussian distribution of the coe cients as in Fig. 3. As expected,
the wave packet spreads along the circular trajectory (but not transversally to it) and eventually
re-establishes its initial shape after Trevival. Fig. 6 shows the corresponding evolution of a swarm of
classical particles, for the same initial phase space density. The spreading of the classical distribution
and of the quantum wave packet proceeds very similarly, whereas the revival is completely absent
in the classical evolution, which once more illustrates its purely quantum origin.
Finally, let us notice that collapse and revival of a 3D wave packet depend on the principal
quantum number n0 only—see Eqs. (34) and (35)—and are independent of other parameters which
characterize the classical motion, such as the eccentricity and the orientation of the classical elliptical
trajectory. This establishes that a 3D wave packet with low average angular momentum (and,
a fortiori, a 1D wave packet as shown in Fig. 3)—which deeply explores the non-linearity of the
Coulomb force—does not disperse faster than a circular wave packet which essentially feels a constant
force. Hence, arguments on the non-linear character of the interaction should be used with
some caution.
There have been several experimental realizations of electronic wave packets in atoms [4,5,11,25,
27,28], either along the pure radial coordinate or even along angular coordinates too. However, all
these wave packets dispersed rather quickly.
1.4. How to overcome dispersion
Soon after the discovery of quantum mechanics, the spreading of wave packets was realized
and attempts were made to overcome it [2]. From Eq. (13), it is however clear that this is only
possible if the populated energy levels are equally spaced. In practice, this condition is only met for
the harmonic oscillator (or simple tops and rotors). In any other system, the anharmonicity of the
energy ladder will induce dispersion. Hence, the situation seems hopeless.
5 In a generic, multidimensional, integrable system, there are several di erent classical frequencies along the various
degrees of freedom. Hence, only partial revivals of the wave packet at various times are observed. The 3D hydrogen
atom is not generic, because the three frequencies are degenerate, which opens the possibility of a complete revival,
simultaneously along all three coordinates.