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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504 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
Fig. 40. A non-<strong>dispersive</strong> s = 1 <strong>wave</strong> packet of the gravitational bouncer, Eq. (249). The <strong>quantum</strong> number of the resonant<br />
state is chosen as n0 = 1000, to match typical experimental dimensions [158]. The left column shows the time evolution<br />
of the <strong>wave</strong> packet for !t =0; =2; ;3 =2 (from top to bottom). The right column shows the correspond<strong>in</strong>g phase space<br />
(Husimi, see Eq. (249)) representation (z-axis horizontal as <strong>in</strong> the left column, momentum p on the vertical axis). The<br />
parameters are ! 0:1487, =0:025. The periodic, non<strong>dispersive</strong> dynamics of the <strong>wave</strong> packet bounc<strong>in</strong>g o the mirror<br />
<strong>in</strong> the gravitational eld is apparent.<br />
while the full, time-dependent Hamiltonian reads<br />
(3 I)2=3 I<br />
H = +<br />
2<br />
2=3<br />
∞<br />
2 (3I)2=3<br />
s<strong>in</strong>(!t) − s<strong>in</strong>(!t)<br />
(3 ) 1=3 4=3<br />
Thus, the resonant action (226) is given by<br />
Î s =<br />
2 s 3<br />
3! 3<br />
with the associated strength of the e ective coupl<strong>in</strong>g<br />
n=1<br />
cos(n )<br />
n 2 : (253)<br />
(254)<br />
Vs = (3I)2=3<br />
: (255)<br />
s2 4=3<br />
Us<strong>in</strong>g the framework of Sections 3.1.1, 3.1.2 (for s = 1), and 5.1 (for s¿1), the reader may easily<br />
compute the various properties of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> <strong>in</strong> this system. 29 An example for<br />
s=1 is presented <strong>in</strong> Fig. 40, for the resonant pr<strong>in</strong>cipal <strong>quantum</strong> number n0=1000, (i.e., Î 1=1000:75)<br />
where both, the (time-periodic) probability densities <strong>in</strong> con guration and phase space are shown.<br />
Note that such high n0 values (or even higher) correspond to typical experimental fall<strong>in</strong>g heights<br />
(around 0:1 mm for n0 = 1000) <strong>in</strong> experiments on cold atoms [158]. Therefore, the creation of<br />
an atomic <strong>wave</strong> packet <strong>in</strong> such an experiment would allow to store the atom <strong>in</strong> a quasi-classical<br />
29 There is, however, a tricky po<strong>in</strong>t: the Maslov <strong>in</strong>dex <strong>in</strong> this system is 3, with a contribution 1 com<strong>in</strong>g from the outer<br />
turn<strong>in</strong>g po<strong>in</strong>t, and 2 from z = 0, s<strong>in</strong>ce the oscillat<strong>in</strong>g plane acts as a hard wall. Hence, the relation between the pr<strong>in</strong>cipal<br />
<strong>quantum</strong> number and the action is I1 = n +3=4, see Eq. (38).