Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 443
of such states is given by Eq. (82) which can be rewritten, using Eq. (100), as
Number of trapped states 4 |q| : (103)
At the center of the island (states with small
a ( ; q) reads [95]
and=or large |q|), an approximate expression for
a ( ; q) ≈−2|q| +4 + 1
|q|
:
2
(104)
It does not depend on (what physically means in our case that the states deeply inside the resonance
island are insensitive to the boundary condition). When inserted in Eq. (102), it yields exactly the
energy levels of Eq. (78), with = N . Hence, the Mathieu approach agrees with the harmonic
approximation within the resonance island, for su ciently large islands.
For a ( ; q) ¿ 2q, the pendulum undergoes a rotational, unbounded motion, which again can be
quantized using WKB. For small |q|, a ( ; q) ≈ 4([( +1)=2]− =2) 2 (where [ ] stands for the integer
part), see [95]. With this in Eq. (102), one recovers the known Floquet spectrum, in the limit of
a vanishingly weak perturbation. Note, however, that in this rotational mode, the eigenstates are
sensitive to the boundary conditions (and the a ( ; q) values depend on ). This is essential for the
correct → 0 limit.
Around a ( ; q) =2q, the pendulum is close to the separatrix between librational and rotational
motion: the period of the classical motion tends to in nity (critical slowing down). That explains
the locally enhanced density of states apparent in Fig. 9(a).
In Fig. 9(b), we also plot the semiclassical WKB prediction for the quantized a (q) values.
Obviously, the agreement with the exact “quantum” Mathieu result is very good, even for weakly
excited states, except in the vicinity of the separatrix. This is not unexpected because the semiclassical
approximation is known to break down near the unstable xed point, see Sections 2 and 3.1.3 above.
The Mathieu equation yields accurate predictions for properties of non-dispersive wave packets in
periodically driven quantum systems. Indeed, in the range a ( ; q) ∈ [ − 2q; 2q], the classical motion
is trapped inside the resonance island, and the corresponding quantum eigenstates are expected to
be non-dispersive wave packets. In particular, the lowest state in the resonance island, associated
with the ground state of the pendulum = 0, corresponds to the semiclassical eigenstate N = 0, see
Eq. (78), and represents the non-dispersive wave packet with the best localization properties. As
can be seen in Fig. 9(b), the semiclassical quantization for this state is in excellent agreement with
the exact Mathieu result. This signi es that Eq. (81) can be used for quantitative predictions of the
quasi-energy of this eigenstate.
As already mentioned, the characteristic exponent does not play a major role inside the resonance
island, as the eigenvalues a ( ; q) there depends very little on . However, it is an important parameter
outside the resonance, close to the separatrix especially at small q. Indeed, at q = 0, the minimum
eigenvalue is obtained for =0 : a0( =0;q= 0) = 0. This implies that, even for very small q,
the ground state enters most rapidly the resonance island. On the opposite, the worst case is =1
where the lowest eigenvalue is doubly degenerate: a0( =1;q=0)=a1( =1;q=0)=1. As we are
interested in the ground state of the pendulum (the one with maximum localization), the situation
for = 0 is preferable: not only the state enters rapidly the resonance island, but it is also separated
from the other states by an energy gap and is thus more robust versus any perturbation. We will