Non-dispersive wave packets in periodically driven quantum systems

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carries over to quantum mechanics, and allows the creation of non-dispersive wave packets, as will
be explained in detail in Section 3.1.2.
1.5. The interest of non-dispersive wave packets
Schrodinger dreamt of the possibility of building quantum wave packets following classical trajectories
[2]. He succeeded for the harmonic oscillator, but failed for other systems [31]. It was
then believed that wave packets must spread if the system is non-linear, and this is correct for
time-independent systems. However, this is not true in general, and we have seen in the previous
section that clever use of the non-linearity may, on the contrary, stabilize a wave packet and preserve
it from spreading. Such non-dispersive wave packets are thus a realization of Schrodinger’s
dream.
One has to emphasize strongly that they are not some variant of the coherent states of the harmonic
oscillator. They are of intrinsically completely di erent origin. Paradoxically, they exist only if there
is some non-linearity, i.e., some unharmonicity, in the classical system. They have some resemblance
with classical solitons which are localized solutions of a non-linear equation that propagate without
spreading. However, they are not solitons, as they are solutions of the linear Schrodinger equation.
They are simply new objects.
Non-dispersive wave packets in atomic systems were identi ed for the hydrogen atom exposed to
a linearly polarized [32,33] and circularly polarized [34] microwave elds quite independently and
using di erent physical pictures. The former approach associated the wave packets with single Floquet
states localized in the vicinity of the periodic orbit corresponding to atom-microwave non-linear
resonance. The latter treatment relied on the fact that a transformation to a frame corotating with the
microwave eld removes the explicit time-dependence of the Hamiltonian for the circular polarization
(see Section 3.4). The states localized near the equilibria of the rotating system were baptized
“Trojan wave packets” to stress the analogy of the stability mechanism with Trojan asteroids. Such
an approach is, however, restricted to a narrow class of systems where the time-dependence can be
removed and lacks the identi cation of the non-linear resonance as the relevant mechanism. We thus
prefer to use in this review the more general term “non-dispersive wave packets” noting also that
in several other papers “non-spreading wave packets” appear equally often.
Apart from their possible practical applications (for example, for the purpose of quantum control
of atomic or molecular fragmentation processes [28], or for information storage [35–38] in
a con ned volume of (phase) space for long times), they show the fruitful character of classical
non-linear dynamics. Indeed, here the non-linearity is not a nuisance to be minimized, but
rather the essential ingredient. From complex non-linear dynamics, a simple object is born. The
existence of such non-dispersive wave packets is extremely di cult to understand (let alone to
predict) from quantum mechanics and the Schrodinger equation alone. The classical non-linear
dynamics point of view is by far more illuminating and predictive. It is the classical mechanics
inside which led Berman and Zaslavsky [39] to the pioneering discussion of states associated
with the classical resonance island, the subsequent studies [40–42] further identi ed such states
for driven one-dimensional systems using the Mathieu approach without, however, discussing the
wave-packet aspects of the states. The best proof of the importance of the classical mechanics
inside is that the non-dispersive wave packets could have been discovered for a very long
time (immediately after the formulation of the Schrodinger equation), but were actually identi ed