Non-dispersive wave packets in periodically driven quantum systems

530 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
where is a parameter. In our case, for example, the microwave amplitude may be tuned, leading
to V = x and = F. The interesting quantities are then the eigenvalues Ei( ) and the eigenfunctions
| i( )〉 of Eq. (275). Di erentiating the Schrodinger equation with respect to , one shows (with
some algebra) [15] that the behavior of Ei( ) with may be viewed as the motion of N ctitious
classical particles (where N is the dimension of the Hilbert space) with positions Ei and momenta
pi = Vii = 〈 i|V | i〉, governed by the Hamiltonian
Hcl =
N
p 2 i
2
+ 1
2
N
N
|Lij| 2
; (276)
(Ej − Ei) 2
i=1
i=1 j=1;j=i
where Lij =(Ei − Ej)〈 i|V | j〉 are additional independent variables obeying the general Poisson
brackets for angular momenta. The resulting dynamics, although non-linear, is integrable [15].
Let us now consider the parametric motion of some eigenstate |n+;n−〉, for example of the ground
state wave packet |0; 0〉. Its coupling to other states is quite weak—because of its localization in
a well de ned region of phase space—and the corresponding Lij are consequently very small. If
we rst suppose that the wave-packet state is well isolated (in energy) from other wave packets
(i.e., states with low values of n+;n−), the ctitious particle associated with |0; 0〉 basically ignores
the other particles and propagates freely at constant velocity. It preserves its properties across the
successive interactions with neighboring states, in particular its shape: in that sense, it is a solitonic
solution of the equations of motion generated by Hamiltonian (276).
Suppose that, in the vicinity of some F values, another wave-packet state (with low n+;n− quantum
numbers) becomes quasi-degenerate with |0; 0〉. In the harmonic approximation, see Section
3.4.4, the two states are completely uncoupled; it implies that the corresponding Lij vanishes and
the two levels cross. The coupling between the two solitons stems from the di erence between the
exact Hamiltonian and its harmonic approximation, i.e., from third order or higher terms, beyond the
harmonic approximation. Other |n+;n−〉 states having di erent slopes w.r.t. F induce “solitonic collisions”
at some other values of F. To illustrate the e ect, part of the spectrum of the two-dimensional
hydrogen atom in a CP microwave is shown in Fig. 59, as a function of the scaled microwave amplitude
F0. For the sake of clarity, the energy of the ground state wave packet |0; 0〉 calculated in
the harmonic approximation, Eq. (174) is substracted, such that it appears as an almost horizontal
line. Around F0 =0:023, it is crossed by another solitonic solution, corresponding to the |1; 4〉 wave
packet, 36 what represents the collision of two solitons. Since this avoided crossing is narrow and
well isolated from other avoided crossings, the wave functions before and after the crossing preserve
their shape and character, as typical for an isolated two-level system. This may be further veri ed by
wave-function plots before and after the collision (see [63] for more details). The avoided crossings
become larger (compare Fig. 30) with increasing F0. In fact, as mentioned in Section 3.4.4, we
have numerically veri ed that the solitonic character of the ground state wave packet practically
disappears at the 1:2 resonance, close to F0 0:065 [63]. For larger F0, while one may still nd
nicely localized wave packets for isolated values of F0, the increased size of the avoided crossings
makes it di cult to follow the wave packet when sweeping F0. For such strong elds, the
ionization rate of wave-packet states becomes appreciable, comparable to the level spacing between
36 Similarly to the wave packet |1; 3〉 discussed in Fig. 30, the semiclassical harmonic prediction for the energy of |1; 4〉
is not satisfactory. However, the slope of the energy level is well reproduced as a function of F0.