Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 483
by the coupling to the closest Floquet state [43,145], typically much less resistant against ionization,
as we shall discuss in detail in Section 7.1).
Thus, strictly speaking, when we speak of non-dispersive wave packets as speci c Floquet states,
we really have in mind a generic situation, far from any avoided crossing. In particular, the examples
of wave packet states shown in the gures above correspond to such situations. The observed
accuracy of the semiclassical approximation has important practical consequences: in order to obtain
the “exact” wave packets numerically, we do not need many eigenvalues for a given set of
parameters. Using the Lanczos algorithm for the partial diagonalization of a matrix, it is enough to
extract few (say ve) eigenvalues only, centered on the semiclassical prediction. The accuracy of the
latter (a fraction of the mean level spacing) is su cient for a clear identi cation of the appropriate
quantum eigenvalue. Actually, in a real diagonalization of the Floquet Hamiltonian, we identify the
wave-packet states by both their vicinity to the semiclassical prediction for the energy, and the large
(modulus of the) slope of the level w.r.t. changes of F0, induced by its large dipole moment in
the rotating frame (by virtue of the Hellman–Feynman theorem [18]). The latter criterion is actually
also very useful for the identi cation of excited wave packets in a numerically exact spectrum. The
state |1; 3〉 (in the harmonic approximation) presented in Fig. 28 is precisely the excited wave packet
which appears in Fig. 30 as a “line” with a negative slope, meeting the |0; 0〉 state in a broad avoided
crossing around F0 0:036. From that gure, it is apparent that the harmonic prediction for the
energy is not excellent for the |1; 3〉 state. On the other hand, the slope of the Floquet state almost
matches the slope given by the harmonic approximation, which con rms that the exact wave-function
is still well approximated by its harmonic counterpart. Note that also from the experimental point
of view it is important to get accurate and simple semiclassical estimates of the energies of the
non-dispersive wave packets, since it may help in their preparation and unambiguous identi cation.
For a more detailed discussion, see Section 8.2.
It is interesting to compare the accuracy of the harmonic approximation to the pendulum description
outlined previously in Section 3.4.2. The latter results from lowest order perturbation theory in
F. Taking the small F limit we get !+;!z → !, and !− → ! √ 3F0, for the harmonic modes, see
Eqs. (194)–(196). The latter result coincides—as it should—with the pendulum prediction, Eqs. (80)
and (173). Similarly, the energy of the stable equilibrium point, Eq. (187), becomes at rst-order in
F:
Eeq = − 3
2n2 + n
0
2 0F + O(F 2 ) ; (200)
which coincides with the energy of the center of the resonance island, see Section 3.4.2. Thus, the
prediction of the resonance analysis agrees with the harmonic approximation in the rotating frame.
For a more accurate estimate of the validity of both approaches, we have calculated—for the 2D
model of the atom, but similar conclusions are reached in 3D—the energy di erence between the
exact quantum result and the prediction using a semiclassical quantization of the secular motion in
the (M; ) plane together with the Mathieu method in the (Î; ˆ ) plane on the one side, and the
prediction of the harmonic approximation around the xed point, Eq. (198), on the other side. In
Fig. 31, we compare the results coming from both approaches. As expected, the semiclassical approach
based on the Mathieu equation is clearly superior for very small microwave amplitudes, as
it is “exact” at rst-order in F. On the other hand, for the harmonic approximation to work well,
the island around the xed point has to be su ciently large. Since the size of the island increases