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) 409–547 507<br />
Fig. 44. Left: Po<strong>in</strong>care surface of section of the one-dimensional hydrogen atom under l<strong>in</strong>early polarized driv<strong>in</strong>g,<br />
Eq. (137), for resonant driv<strong>in</strong>g at twice the Kepler frequency. The scaled eld strength is F0 =0:03, and the phase<br />
is xed at !t =0: The s = 2 resonance islands are apparent, embedded <strong>in</strong> the chaotic sea, and separated from the s =1<br />
resonance by <strong>in</strong>variant tori. Right: Husimi representation [145] of a Floquet eigenstate (for n0 = 60) anchored to the s =2<br />
resonance displayed on the left.<br />
with s—compare Eqs. (120) and (121). Consider rst the simpler 1D model of the atom. We discuss<br />
the s = 2 case only, s<strong>in</strong>ce similar conclusions can be obta<strong>in</strong>ed for higher s values. The left panel <strong>in</strong><br />
Fig. 44 shows the classical phase space structure (Po<strong>in</strong>care surface of section) for F0 =0:03, with the<br />
s = 2 resonance completely embedded <strong>in</strong> the chaotic sea, and well separated from the much larger<br />
pr<strong>in</strong>cipal resonance island. From our experience with the pr<strong>in</strong>cipal resonance, and from the general<br />
considerations on s:1 resonances above, we expect to nd Floquet eigenstates which are localized on<br />
this classical phase space structure and mimic the temporal evolution of the correspond<strong>in</strong>g classical<br />
trajectories. Indeed, the right plot <strong>in</strong> Fig. 44 displays a Floquet eigenstate obta<strong>in</strong>ed by “exact”<br />
numerical diagonalization, which precisely exhibits the desired properties.<br />
Aga<strong>in</strong>, this observation has its direct counterpart <strong>in</strong> the realistic 3D atom, where the 2:1 resonance<br />
allows for the construction of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> along elliptic trajectories, as we shall<br />
demonstrate now. We proceed as for the s = 1 case (Section 3.3.2): the secular Hamiltonian is<br />
obta<strong>in</strong>ed by averag<strong>in</strong>g the full Hamiltonian, Eq. (152), after transformation to the “rotat<strong>in</strong>g frame”,<br />
Eq. (228), over one period = sT of the resonantly <strong>driven</strong> classical trajectory:<br />
Hsec = ˆPt − 1<br />
<br />
!Î<br />
− + F 1 − 2<br />
2Î s<br />
M 2<br />
L 2 [ − Xs(Î) cos cos ˆ + Ys(Î) s<strong>in</strong> s<strong>in</strong> ˆ ] ; (257)<br />
where Xs and Ys are given by Eqs. (153) and (154). This can be condensed <strong>in</strong>to<br />
with<br />
Hsec = ˆPt − 1 !Î<br />
− 2<br />
2Î s + F s cos(s ˆ + s) (258)<br />
<br />
s(Î; L; ):= 1 −<br />
tan s(L; ):= Ys<br />
Xs<br />
M 2<br />
L2 <br />
X 2<br />
s cos2 + Y 2<br />
s s<strong>in</strong> 2 ; (259)<br />
tan = Js(se) √ 1 − e 2<br />
eJ ′ s(se)<br />
tan : (260)