Non-dispersive wave packets in periodically driven quantum systems

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424 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 7. The initially Gaussian distributed swarm of classical particles, shown in Fig. 6, after evolution in the presence of the Coulomb eld of the nucleus, with a resonant, circularly polarized microwave eld in the plane of the circular Kepler trajectory added. The non-linear resonance between the unperturbed Kepler motion and the driving eld locks the phase of the particles on the phase of the driving eld. As opposed to the free (classical) evolution depicted in Fig. 6, the classical distribution does not exhibit dispersion along the orbit, except for few particles launched from the tail of the initial Gaussian distribution, which are not trapped by the principal resonance island. absence of driving, has exactly the driving frequency. All the trajectories in the resonance island are winding around the central orbit with their phases locked on the external driving. The crucial point for our purposes is that the resonance island occupies a nite volume of phase space, i.e., it traps all trajectories in a window of internal frequencies centered around the driving frequency. The size of this frequency window increases with the amplitude of the system-driving coupling, and, as we shall see in Section 3.1, can be made large enough to support wave-packet eigenstates of the corresponding quantum system. The classical trapping mechanism is illustrated in Fig. 7 which shows a swarm of classical particles launched along a circular Kepler orbit of a three-dimensional hydrogen atom exposed to a resonant, circularly polarized microwave eld: the e ect of the microwave eld is to lock the particles in the vicinity of a circular trajectory. Note that also the phase along the classical circular trajectory is locked: the particles are grouped in the direction of the microwave eld and follow its circular motion without any drift. There is a striking di erence with the situation shown previously in Fig. 6, where the cloud of particles rapidly spreads in the absence of the microwave eld (the same swarm of initial conditions is used in the two gures). In Fig. 7, there are few particles (about 10%) in the swarm which are not phase locked with the microwave eld. This is due to the nite subvolume of phase space which is e ectively phase locked. Particles in the tail of the initial Gaussian distribution may not be trapped [30]. Although the microwave eld applied in Fig. 7 amounts to less than 5% of the Coulomb eld along the classical trajectory, it is su cient to synchronize the classical motion. The same phenomenon
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 425 carries over to quantum mechanics, and allows the creation of non-dispersive wave packets, as will be explained in detail in Section 3.1.2. 1.5. The interest of non-dispersive wave packets Schrodinger dreamt of the possibility of building quantum wave packets following classical trajectories [2]. He succeeded for the harmonic oscillator, but failed for other systems [31]. It was then believed that wave packets must spread if the system is non-linear, and this is correct for time-independent systems. However, this is not true in general, and we have seen in the previous section that clever use of the non-linearity may, on the contrary, stabilize a wave packet and preserve it from spreading. Such non-dispersive wave packets are thus a realization of Schrodinger’s dream. One has to emphasize strongly that they are not some variant of the coherent states of the harmonic oscillator. They are of intrinsically completely di erent origin. Paradoxically, they exist only if there is some non-linearity, i.e., some unharmonicity, in the classical system. They have some resemblance with classical solitons which are localized solutions of a non-linear equation that propagate without spreading. However, they are not solitons, as they are solutions of the linear Schrodinger equation. They are simply new objects. Non-dispersive wave packets in atomic systems were identi ed for the hydrogen atom exposed to a linearly polarized [32,33] and circularly polarized [34] microwave elds quite independently and using di erent physical pictures. The former approach associated the wave packets with single Floquet states localized in the vicinity of the periodic orbit corresponding to atom-microwave non-linear resonance. The latter treatment relied on the fact that a transformation to a frame corotating with the microwave eld removes the explicit time-dependence of the Hamiltonian for the circular polarization (see Section 3.4). The states localized near the equilibria of the rotating system were baptized “Trojan wave packets” to stress the analogy of the stability mechanism with Trojan asteroids. Such an approach is, however, restricted to a narrow class of systems where the time-dependence can be removed and lacks the identi cation of the non-linear resonance as the relevant mechanism. We thus prefer to use in this review the more general term “non-dispersive wave packets” noting also that in several other papers “non-spreading wave packets” appear equally often. Apart from their possible practical applications (for example, for the purpose of quantum control of atomic or molecular fragmentation processes [28], or for information storage [35–38] in a con ned volume of (phase) space for long times), they show the fruitful character of classical non-linear dynamics. Indeed, here the non-linearity is not a nuisance to be minimized, but rather the essential ingredient. From complex non-linear dynamics, a simple object is born. The existence of such non-dispersive wave packets is extremely di cult to understand (let alone to predict) from quantum mechanics and the Schrodinger equation alone. The classical non-linear dynamics point of view is by far more illuminating and predictive. It is the classical mechanics inside which led Berman and Zaslavsky [39] to the pioneering discussion of states associated with the classical resonance island, the subsequent studies [40–42] further identi ed such states for driven one-dimensional systems using the Mathieu approach without, however, discussing the wave-packet aspects of the states. The best proof of the importance of the classical mechanics inside is that the non-dispersive wave packets could have been discovered for a very long time (immediately after the formulation of the Schrodinger equation), but were actually identi ed
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Non-dispersive wave packets in periodically driven quantum systems

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