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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538 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
Fig. 61. Snapshots of the electronic density for a two-dimensional hydrogen atom exposed to a circularly polarized<br />
micro<strong>wave</strong> eld with <strong>in</strong>creas<strong>in</strong>g amplitude. The micro<strong>wave</strong> eld amplitude is switched on accord<strong>in</strong>g to Eq. (284), with<br />
maximum scaled eld F0;max =0:03 and Tswitch = 400 × 2 =!, where ! is resonant with n0 = 60 (frequency ! =1=(60:5) 3 ).<br />
The evolution of the <strong>in</strong>itial circular state n = M = 60 is numerically computed by solv<strong>in</strong>g the time-dependent Schrod<strong>in</strong>ger<br />
equation <strong>in</strong> a convenient Sturmian basis. Top left—t = 0 (<strong>in</strong>itial circular state); top middle—t = 20 micro<strong>wave</strong> periods;<br />
top right—t = 60 periods, bottom left—t = 100 periods; bottom middle—the nal state, t = 400 periods; bottom right—the<br />
non-<strong>dispersive</strong> <strong>wave</strong> packet (exact Floquet eigenstate): it is almost <strong>in</strong>dist<strong>in</strong>guishable from the previous <strong>wave</strong> function,<br />
what proves that the excitation process e ciently and almost exclusively populates the state of <strong>in</strong>terest. The box extends<br />
over 10 000 Bohr radii <strong>in</strong> both directions, with the nucleus at the center. The micro<strong>wave</strong> eld is along the horizontal axis,<br />
po<strong>in</strong>t<strong>in</strong>g to the right.<br />
we nd<br />
n++n−<br />
(−3− )<br />
2<br />
Funharmonic ∼ n 0 : (281)<br />
S<strong>in</strong>ce we want to switch the eld to a maximum value Fmax ∼ n −4<br />
0 , the Landau–Zener formula<br />
yields<br />
∼ Fmax<br />
E F<br />
unharmonic ∼ n n−+n++1<br />
0<br />
(282)<br />
∼ n n−+n+−2<br />
0 micro<strong>wave</strong> periods (283)<br />
for the scal<strong>in</strong>g behavior of the timescale which guarantees diabatic switch<strong>in</strong>g through avoided cross<strong>in</strong>gs<br />
of the <strong>wave</strong>-packet eigenstate with excited states of the elliptic island. Let us stress that this is<br />
only a very rough estimate of the switch<strong>in</strong>g time, some numerical factors (not necessarily close to<br />
unity) are not taken <strong>in</strong>to account.<br />
The above predictions can be checked, e.g., by a numerical <strong>in</strong>tegration of the time-dependent<br />
Schrod<strong>in</strong>ger equation for a micro<strong>wave</strong>-<strong>driven</strong> atom, tak<strong>in</strong>g <strong>in</strong>to account the time-dependent amplitude<br />
of the eld. An exemplary calculation on the two-dimensional model atom (see Section 3.4.3) can<br />
be found <strong>in</strong> [61], for CP driv<strong>in</strong>g. Fig. 61 shows the evolution of the electronic density of the atomic