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
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
482 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
Fig. 30. Spectrum of the two-dimensional hydrogen atom <strong>in</strong> a circularly polarized micro<strong>wave</strong> eld of frequency<br />
! =1=(60:5) 3 , as a function of the scaled micro<strong>wave</strong> amplitude F0 = F! −4=3 . In order to test the accuracy of the<br />
harmonic prediction, we subtract the semiclassical energy for the ground state |0; 0〉 <strong>wave</strong> packet, Eq. (198), from the<br />
result of the exact numerical diagonalization, and rescale the energy axis <strong>in</strong> units of the mean level spac<strong>in</strong>g. The almost<br />
horizontal l<strong>in</strong>e slightly above zero corresponds to the non-<strong>dispersive</strong> ground state <strong>wave</strong> packet, which typically undergoes<br />
small avoided cross<strong>in</strong>gs with other Floquet states. A relatively large avoided cross<strong>in</strong>g occurs when two <strong>wave</strong> packet-like<br />
states meet, as here happens around F0 0:036. The other (“collid<strong>in</strong>g”) state is the excited <strong>wave</strong> packet |1; 3〉. The<br />
dashed l<strong>in</strong>e <strong>in</strong>dicates the harmonic prediction, Eq. (198), for this state, which is obviously less accurate. Note, however,<br />
that the slope is correctly predicted.<br />
states are excited by one <strong>quantum</strong> <strong>in</strong> either of the normal modes !±;z, and are therefore signi cantly<br />
more extended <strong>in</strong> space. Aga<strong>in</strong>, as already mentioned <strong>in</strong> Sections 3.3.1 and 3.3.2, these <strong>wave</strong>-packet<br />
eigenstates have nite, but extremely long life-times (several thousands to millions of Kepler orbits),<br />
due to the eld-<strong>in</strong>duced ionization. For a detailed discussion of their decay properties see Section 7.1.<br />
As already demonstrated <strong>in</strong> the LP case (see Figs. 21 and 22), the semiclassical prediction for<br />
the energies of the non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> is usually excellent. In order to stress the (small)<br />
di erences, we plot <strong>in</strong> Fig. 30 a part of the Floquet spectrum of the two-dimensional model atom,<br />
i.e., quasi-energy levels versus the (scaled) micro<strong>wave</strong> amplitude, after subtraction of the prediction<br />
of the harmonic approximation around the stable xed po<strong>in</strong>t, Eq. (198), for the ground state <strong>wave</strong><br />
packet |0; 0〉. The result is shown <strong>in</strong> units of the mean level spac<strong>in</strong>g, estimated 20 to be roughly 2=n 4 0 .<br />
If the harmonic approximation was exact, the ground state <strong>wave</strong> packet would be represented by a<br />
horizontal l<strong>in</strong>e at zero. The actual result is not very far from that, which proves that the semiclassical<br />
method predicts the correct energy with an accuracy mostly better than the mean level spac<strong>in</strong>g. The<br />
other states of the system appear as energy levels which rapidly evolve with F0, and which exhibit<br />
extremely small avoided cross<strong>in</strong>gs—hence extremely small coupl<strong>in</strong>gs—with the <strong>wave</strong> packet. In the<br />
vic<strong>in</strong>ity of such avoided cross<strong>in</strong>gs, the energy levels are perturbed, the diabatic <strong>wave</strong> functions mix<br />
(the <strong>wave</strong>-packet eigenstates get distorted), and, typically, the lifetime of the state decreases (<strong>in</strong>duced<br />
20 This estimate follows from the local energy splitt<strong>in</strong>g, ∼ n −3<br />
0 , divided by the number n0=2 of photons needed to ionize<br />
the <strong>in</strong>itial atomic state by a resonant driv<strong>in</strong>g eld.