Non-dispersive wave packets in periodically driven quantum systems

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456 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 12. Surface of section of the classical phase space of a 1D hydrogen atom, driven by a linearly polarized microwave eld of amplitude F0 =0:053, for di erent values of the phase: !t = 0 (top left), !t = =2 (top right), !t = (bottom left), !t =3 =2 (bottom right). At this driving strength, the principal resonance remains as the only region of regular motion of appreciable size, in a globally chaotic phase space. The action-angle variables I; are de ned by Eq. (117), according to which a collision with the nucleus occurs at =0. Fig. 12 shows a typical Poincare surface of section of the classical dynamics of the driven Rydberg electron, at di erent values of the phase of the driving eld. The eld amplitude is chosen su ciently high to induce largely chaotic dynamics, with the principal resonance as the only remnant of regular motion occupying an appreciable volume of phase space. The gure clearly illustrates the temporal evolution of the elliptic island with the phase of the driving eld, i.e., the locking of the electronic motion on the external driving. The classical stability island follows the dynamics of the unperturbed electron along the resonant trajectory. The distance from the nucleus is parametrized by the variable , see Eq. (117). At !t = , the classical electron hits the nucleus (at =0), its velocity diverges and changes sign discontinuously. This explains the distortion of the resonance island as it approaches =0; . Quantum mechanically, we expect a non-dispersive wave-packet eigenstate to be localized within the resonance island. The semiclassical prediction of its quasi-energy, Eq. (149), facilitates to identify the non-dispersive wave packet within the exact Floquet spectrum, after numerical diagonalization of the Floquet Hamiltonian (75). The wave-packet’s con guration space representation is shown in Fig. 13, for the same phases of the eld as in the plots of the classical dynamics in Fig. 12. Clearly, the wave packet is very well localized at the outer turning point of the Kepler electron at phase !t = 0 of the driving eld, and is re ected o the nucleus half a period of the driving eld later. On re ection, the electronic density exhibits some interference structure, as well as some transient spreading. This is a signature of the quantum mechanical uncertainty in the angle : part of the wave function, which still approaches the Coulomb singularity, interferes with the other part already
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 457 |ψ(z)| 2 0.0005 0 0.0005 t=0,1 t=0.25 t=0.5 t=0.75 0 0 5000 0 5000 z (atomic units) Fig. 13. Con guration space representation of the electronic density of the non-dispersive wave-packet eigenstate of a 1D hydrogen atom, driven by a linearly polarized microwave eld, for the same eld amplitude and phases as in Fig. 12. The eigenstate is centered on the principal resonance of classical phase space, at action (principal quantum number) n0 = 60. In con guration space, the wave packet is localized at the outer turning point (with zero average velocity) at time t =0, then propagates towards the nucleus which it hits at t=T =0:5 (where T =2 =! is the microwave period). Afterwards, it propagates outward to the apocenter which is reached at time t=T = 1. After one period, the wave packet recovers exactly its initial shape, and will therefore propagate along the classical trajectory forever, without spreading. The wave packet has approximately Gaussian shape (with time-dependent width) except at t=T 0:5. At this instant, the head of the wave packet, which already has been re ected o the nucleus, interferes with its tail, producing interference fringes. re ected o the nucleus. The transient spreading is equally manifest in the temporal evolution of the uncertainty product z p itself, which is plotted in Fig. 14. Apart from this singularity at !t = , the wave packet is approximately Gaussian at any time, with a time-dependent width (compare !t=0 and =2). To complete the analogy between classical and quantum motion, we nally calculate the Husimi distribution—the phase space representation of the wave packet eigenstate de ned in Section 1.2, Eq. (30)—in order to obtain a direct comparison between classical and quantum dynamics in phase space. Fig. 15 shows the resulting phase space picture, again for di erent phases of the driving eld. The association of the quantum mechanical time evolution with the classical resonance island (see Fig. 12) is unambiguous. The transient spreading at the collision with the nucleus (!t = )isdue to the divergence of the classical velocity upon re ection. As discussed in Section 3.1.1 and visible in Fig. 10, there is an hyperbolic xed point (i.e., an unstable equilibrium point) at (Î =n0; ˆ =0): it corresponds to the unstable equilibrium position of the pendulum when it points “upwards”. For the driven system, it corresponds to an unstable periodic orbit resonant with the driving frequency: it is somewhat similar to the stable orbit supporting the non-dispersive wave packets, except that is shifted in time by half a period. As discussed in Section 2.3, the classical motion slows down at the hyperbolic xed point (the time to reach the unstable equilibrium point with zero velocity diverges [93]), and the eigenfunction must exhibit a maximum of the electronic density at this position. In addition, due to the periodicity of the drive, the corresponding (“hyperbolic”) wave-packet eigenstate necessarily follows the dynamics of a classical particle which evolves along the unstable periodic orbit. However, because the orbit
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Non-dispersive wave packets in periodically driven quantum systems

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