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) 409–547 437<br />
To summarize, the construction of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> <strong>in</strong> a time-periodic system is equivalent<br />
to nd<strong>in</strong>g localized Floquet eigenstates. The existence of such states is far from obvious, as the<br />
Floquet spectrum is usually very complex, composed of quasi-bound states, resonances and cont<strong>in</strong>ua.<br />
This is why a semiclassical analysis can be very helpful <strong>in</strong> nd<strong>in</strong>g these objects.<br />
3.1.3. Semiclassical approximation<br />
Deal<strong>in</strong>g with highly excited states, a semiclassical approximation can be used to determ<strong>in</strong>e quasienergies<br />
and Floquet eigenstates [80]. If the driv<strong>in</strong>g perturbation is su ciently weak, we have shown<br />
<strong>in</strong> Section 3.1.1 that the classical dynamics close to a non-l<strong>in</strong>ear resonance is essentially regular and<br />
accurately described by the pendulum Hamiltonian (66). It describes a system with two degrees of<br />
freedom (along t and ˆ , with their conjugate momenta ˆPt and Î, respectively) which is essentially<br />
regular. For semiclassical quantization, we may then use the standard EBK rules, Eq. (45), <strong>in</strong>troduced<br />
<strong>in</strong> Section 2.<br />
The momentum ˆPt is a constant of the motion, and the isovalue curves of ˆPt; Î; ˆ ly<strong>in</strong>g on the<br />
<strong>in</strong>variant tori can be used for the EBK quantization scheme. Along such a curve, t evolves from 0 to<br />
2 =!, with ˆ = − !t kept constant. Thus, itself is changed by 2 , what implies that the Maslov<br />
<strong>in</strong>dex of the unperturbed (I; ) motion has to be <strong>in</strong>cluded, lead<strong>in</strong>g to the follow<strong>in</strong>g quantization<br />
condition for ˆPt,<br />
1<br />
2<br />
T<br />
0<br />
ˆPt dt = ˆPtT<br />
2 =<br />
<br />
k +<br />
4<br />
˝ (76)<br />
with <strong>in</strong>teger k. S<strong>in</strong>ce T =2 =! is just the period of the resonant driv<strong>in</strong>g, we get the quantized values<br />
of ˆPt :<br />
<br />
ˆPt = k + ˝! ; (77)<br />
4<br />
which are equally spaced by ˝!. Thus, we recover semiclassically the !-periodicity of the Floquet<br />
spectrum.<br />
For the motion <strong>in</strong> the (Î; ˆ ) plane, we can use the isocontour l<strong>in</strong>es of the pendulum Hamiltonian<br />
Hpend, Eq. (66), as closed paths, keep<strong>in</strong>g ˆPt and t constant. Depend<strong>in</strong>g on the nature of the pendulum<br />
motion (librational or rotational), the topology of the closed paths is di erent, lead<strong>in</strong>g to dist<strong>in</strong>ct<br />
expressions:<br />
• For trapped librational motion, <strong>in</strong>side the resonance island, the path is isomorphic to a circle <strong>in</strong><br />
the (Î; ˆ ) plane, with a Maslov <strong>in</strong>dex equal to two. The quantization condition is<br />
<br />
1<br />
Î d<br />
2<br />
ˆ <br />
= N + 1<br />
<br />
˝ (librational motion)<br />
2<br />
(78)<br />
with N a non-negative <strong>in</strong>teger. Of special <strong>in</strong>terest is the “fundamental” state, N =0, which exhibits<br />
maximum localization with<strong>in</strong> the resonance island and is therefore expected to represent the optimal<br />
non-<strong>dispersive</strong> <strong>wave</strong> packet.<br />
• For unbounded rotational motion, outside the resonance island, the path <strong>in</strong>cludes a 2 phase change<br />
for and acquires the Maslov <strong>in</strong>dex of the unperturbed motion:<br />
2 1<br />
Î d<br />
2<br />
ˆ <br />
= N + ˝ (rotational motion) :<br />
4<br />
(79)<br />
0