Non-dispersive wave packets in periodically driven quantum systems

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422 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 ∆z ∆p (hbar unit) 100 10 1 Heisenberg limit Revival 0 10 20 Time (Kepler periods) Fig. 4. Time evolution of the uncertainty product z p (in units of ˝) of the wave packet shown in Fig. 3. Starting out from minimum uncertainty, z p ˝=2 (the Heisenberg limit, Eq. (1)), the wave packet exhibits some transient spreading on the time-scale of a Kepler period Trecurrence, thus re ecting the classical motion (compare top left of Fig. 3), collapses on a time-scale of few Kepler cycles (manifest in the damping of the oscillations of z p during the rst ve classical periods), shows a fractional revival around t 10 × Trecurrence, and a full revival at t 20 × Trecurrence. Note that, nontheless, even at the full revival the contrast of the oscillations of the uncertainty product is reduced as compared to the initial stage of the evolution, as a consequence of higher-order corrections which are neglected in Eq. (15). Fig. 5. Time evolution of a wave packet launched along a circular Kepler trajectory, with the same Gaussian weights cn, Eq. (11), as employed for the one-dimensional example displayed in Fig. 3, i.e., centered around the principal quantum number n0 = 60. Since the relative phases accumulated during the time evolution only depend on n0—see Eq. (13)—we observe precisely the same behavior as in the one-dimensional case: classical propagation at short times (top), followed by spreading and collapse (middle), and revival (bottom). The snapshots of the wave function are taken at times (in units 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 cube size is 10 000 Bohr radii, centered on the nucleus (marked with a cross). The radius of the circular wave-packet trajectory equals approx. 3600 Bohr radii. Surprisingly, it is classical mechanics which provides us with a possible solution. Indeed, as discussed above, a quantum wave packet spreads exactly as the corresponding swarm of classical particles. Hence, dispersion can be overcome if all classical trajectories behave similarly in
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 423 Fig. 6. Classical time evolution of a Gaussian (in spherical coordinates) phase space density tted to the minimum uncertainty wave packet of Fig. 5 at time t = 0 (left). As time evolves, the classical phase space density spreads along the circular Kepler orbit (t =12:5, right), but exhibits no revival. Hence, wave-packet spreading is of purely classical origin, only the revival is a quantum feature. The cube size is 10 000 Bohr radii, centered on the nucleus (marked with a cross). The radius of the circular wave-packet trajectory equals approx. 3600 Bohr radii. the long time limit. In other words, if an initial volume of phase space remains well localized under time evolution, it is reasonable to expect that a wave packet built on this initial volume will not spread either. The simplest example is to consider a stable xed point: by de nition [3], every initial condition in its vicinity will forever remain close to it. The corresponding wave packet indeed does not spread at long times ::: though this is of limited interest, as it is simply at rest. Another possibility is to use a set of classical trajectories which all exhibit the same periodic motion, with the same period for all trajectories. This condition, however, is too restrictive, since it leads us back to the harmonic oscillator. Though, we can slightly relax this constraint by allowing classical trajectories which are not strictly periodic but quasi-periodic and staying forever in the vicinity of a well-de ned periodic orbit: a wave packet built on such orbits should evolve along the classical periodic orbit while keeping a nite dispersion around it. It happens that there is a simple possibility to generate such classical trajectories locked on a periodic orbit, which is to drive the system by an external periodic driving. The general theory of non-linear dynamical systems (described in Section 3.1.1) [3,29] shows that when a non-linear system (the internal frequency of which depends on the initial conditions) is subject to an external periodic driving, a phase locking phenomenon—known as a non-linear resonance—takes place. For initial conditions where the internal frequency is close to the driving frequency (quasi-resonant trajectories), the e ect of the coupling is to force the motion towards the external frequency. In other words, trajectories which, in the absence of the coupling, would oscillate at a frequency slightly lower than the driving are pushed forward by the non-linear coupling, and trajectories with slightly larger frequency are pulled backward. In a certain region of phase space—termed “non-linear resonance island”—all trajectories are trapped, and locked on the external driving. At the center of the resonance island, there is a stable periodic orbit which precisely evolves with the driving frequency. If the driving is a small perturbation, this periodic orbit is just the periodic orbit which, in the
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Non-dispersive wave packets in periodically driven quantum systems

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