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