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