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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434 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
s<strong>in</strong>ce the “k<strong>in</strong>etic energy” is negative—and the maximum of Hpend is now a stable equilibrium<br />
po<strong>in</strong>t, as the reader may easily check by standard l<strong>in</strong>ear stability analysis <strong>in</strong> the vic<strong>in</strong>ity of the xed<br />
po<strong>in</strong>t. Thus, <strong>in</strong> compact form, if V1(Î 1)H ′′<br />
0 (Î 1) is positive, ˆ = is a stable equilibrium po<strong>in</strong>t, while<br />
ˆ = 0 is unstable. If V1(Î 1)H ′′<br />
0 (Î 1) is negative, the stable and unstable po<strong>in</strong>ts are <strong>in</strong>terchanged.<br />
There are two qualitatively di erent types of motion:<br />
• Close to the stable equilibrium po<strong>in</strong>t of the pendulum, ˆ oscillates <strong>periodically</strong>, with an amplitude<br />
smaller than . This is the “librational motion” of the pendulum <strong>in</strong>side the resonance island.<br />
Any trajectory started with<strong>in</strong> this region of phase space (the shaded area <strong>in</strong> Fig. 8) exhibits<br />
librational motion. It should be realized that the resonance island con nes the motion to nite<br />
<strong>in</strong>tervals <strong>in</strong> Î and ˆ , and thereby strongly a ects all trajectories with action close to the resonant<br />
action. Accord<strong>in</strong>g to Eqs. (68) and (69), the resonance island is associated with the energy range<br />
[H0(Î 1) − !Î 1 −| V1(Î 1)|;H0(Î 1) − !Î 1 + | V1(Î 1)|].<br />
• For any <strong>in</strong>itial energy outside that energy range the pendulum has su cient k<strong>in</strong>etic energy to<br />
rotate. This is the “rotational motion” of the pendulum outside the resonance island, where ˆ is an<br />
unbounded and monotonous function of time. Far from the center of the island, the motion occurs<br />
at almost constant unperturbed action Î, with an almost constant angular velocity <strong>in</strong> ˆ , tend<strong>in</strong>g to<br />
the unperturbed motion. This illustrates that the e ect of the perturbation is important for <strong>in</strong>itial<br />
conditions close to the resonance island, but negligible for non-resonant trajectories.<br />
The size of the resonance island can be simply estimated from Eq. (66) and Fig. 8. The extension<br />
<strong>in</strong> ˆ is 2 , its width <strong>in</strong> Î (which depends on ˆ )is<br />
<br />
<br />
<br />
Î =4<br />
V1(Î 1)<br />
<br />
H ′′<br />
0 (Î <br />
<br />
<br />
; (70)<br />
1) <br />
and the total area [3]<br />
<br />
<br />
<br />
A( )=16<br />
V1(Î 1)<br />
<br />
H ′′<br />
0 (Î <br />
<br />
<br />
: (71)<br />
1) <br />
The dependence of A( ) on | | implies that even a small perturbation may <strong>in</strong>duce signi cant<br />
changes <strong>in</strong> the phase space structure, provided the perturbation is resonant.<br />
The above picture is valid <strong>in</strong> the rotat<strong>in</strong>g frame de ned by Eqs. (60)–(62). If we go back to<br />
the orig<strong>in</strong>al action-angle coord<strong>in</strong>ates (I; ), the stable (resp. unstable) xed po<strong>in</strong>t of the secular<br />
Hamiltonian is mapped on a stable (resp. unstable) periodic orbit whose period is exactly equal<br />
to the period of the driv<strong>in</strong>g perturbation, as a consequence of Eq. (65). Any trajectory started <strong>in</strong><br />
the vic<strong>in</strong>ity of the stable periodic orbit will correspond to an <strong>in</strong>itial po<strong>in</strong>t close to the xed po<strong>in</strong>t<br />
<strong>in</strong> the rotat<strong>in</strong>g frame, and thus will rema<strong>in</strong> trapped with<strong>in</strong> the resonance island. In the orig<strong>in</strong>al<br />
coord<strong>in</strong>ate frame, it will appear as a trajectory evolv<strong>in</strong>g close to the stable periodic orbit forever.<br />
In particular, the di erence <strong>in</strong> between the stable periodic orbit and any orbit trapped <strong>in</strong> the<br />
resonance island rema<strong>in</strong>s bounded with<strong>in</strong> (− ; + ), for arbitrarily long times. This means that the<br />
phase of any trapped trajectory cannot drift with respect to the phase of the periodic orbit. As the<br />
latter evolves at the driv<strong>in</strong>g frequency, we reach the conclusion that the phase of any trajectory