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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422 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
∆z ∆p (hbar unit)<br />
100<br />
10<br />
1<br />
Heisenberg limit<br />
Revival<br />
0 10 20<br />
Time (Kepler periods)<br />
Fig. 4. Time evolution of the uncerta<strong>in</strong>ty product z p (<strong>in</strong> units of ˝) of the <strong>wave</strong> packet shown <strong>in</strong> Fig. 3. Start<strong>in</strong>g<br />
out from m<strong>in</strong>imum uncerta<strong>in</strong>ty, z p ˝=2 (the Heisenberg limit, Eq. (1)), the <strong>wave</strong> packet exhibits some transient<br />
spread<strong>in</strong>g on the time-scale of a Kepler period Trecurrence, thus re ect<strong>in</strong>g the classical motion (compare top left of Fig. 3),<br />
collapses on a time-scale of few Kepler cycles (manifest <strong>in</strong> the damp<strong>in</strong>g of the oscillations of z p dur<strong>in</strong>g the rst ve<br />
classical periods), shows a fractional revival around t 10 × Trecurrence, and a full revival at t 20 × Trecurrence. Note that,<br />
nontheless, even at the full revival the contrast of the oscillations of the uncerta<strong>in</strong>ty product is reduced as compared to<br />
the <strong>in</strong>itial stage of the evolution, as a consequence of higher-order corrections which are neglected <strong>in</strong> Eq. (15).<br />
Fig. 5. Time evolution of a <strong>wave</strong> packet launched along a circular Kepler trajectory, with the same Gaussian weights cn,<br />
Eq. (11), as employed for the one-dimensional example displayed <strong>in</strong> Fig. 3, i.e., centered around the pr<strong>in</strong>cipal <strong>quantum</strong><br />
number n0 = 60. S<strong>in</strong>ce the relative phases accumulated dur<strong>in</strong>g the time evolution only depend on n0—see Eq. (13)—we<br />
observe precisely the same behavior as <strong>in</strong> the one-dimensional case: classical propagation at short times (top), followed<br />
by spread<strong>in</strong>g and collapse (middle), and revival (bottom). The snapshots of the <strong>wave</strong> function are taken at times (<strong>in</strong> units<br />
of Trecurrence) t = 0 (top left), t =0:5 (top right), t = 1 (middle left), t =12:5 (middle right), and t =19:45 (bottom). The<br />
cube size is 10 000 Bohr radii, centered on the nucleus (marked with a cross). The radius of the circular <strong>wave</strong>-packet<br />
trajectory equals approx. 3600 Bohr radii.<br />
Surpris<strong>in</strong>gly, it is classical mechanics which provides us with a possible solution. Indeed, as<br />
discussed above, a <strong>quantum</strong> <strong>wave</strong> packet spreads exactly as the correspond<strong>in</strong>g swarm of classical<br />
particles. Hence, dispersion can be overcome if all classical trajectories behave similarly <strong>in</strong>