Non-dispersive wave packets in periodically driven quantum systems

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528 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 linearly polarized wave can be decomposed into two circularly polarized waves, and the extreme rotational state (Fig. 24) is a coherent superposition of two circular wave-packets—each locked on one circularly polarized component—moving in the opposite sense. The decay of each component can then be obtained from the preceding discussion. A completely di erent picture emerges for other wave packets. Their electronic densities averaged over one period are concentrated along either the (extreme librational) or z-axes and do not vanish close to the nucleus (Fig. 23). They have non-negligible dipole elements with Floquet states built on low lying atomic states (i.e., states practically una ected by the driving eld). Because of the cubic power dependence of the decay rate, Eq. (271), on the energy of the emitted photon, these will dominate the spontaneous emission. Thus, in contrast to the CP case, spontaneous decay will lead to the destruction of the wave packet. A quantitative analysis con rms this qualitative picture. To this end, a general master equation formalism can be developed [71], which allows to treat the ionization process induced by the driving eld exactly, while the spontaneous emission is treated perturbatively, as in the preceding section. Applied to the one-dimensional model of the atom (Section 3.3.1), with the density of eld modes of the real three-dimensional world, it is possible to approximately model the behavior of the separatrix states of the three-dimensional atom. A non-dispersive wave packet may decay either by ionization or by spontaneous emission, the total decay rate being the sum of the two rates [71]. Like in the CP case, the decay rate to the atomic continuum decreases exponentially with n0 (since it is essentially a tunneling process, see Section 7.1, Eq. (267)) while the spontaneous decay rate depends algebraically on n0 [22]. The wave packet is a coherent superposition of atomic states with principal quantum number close to n0 = ! −1=3 , and the dipole matrix element between an atomic state n and a weakly excited state scales as n −3=2 [22]. Since the energy of the emitted photon is of order one (in atomic units), Eq. (271) shows 3 3 that the spontaneous emission rate should decrease like =n0 . The numerical results, presented in Fig. 58, fully con rm this 1=n3 0 prediction. However, the spontaneous decay of real, 3D wave packets with near 1D localization properties (see Fig. 23, middle column, and Fig. 38) is certainly slower. Indeed, these states are combinations of atomic states with various total angular momenta L; among them, only the low-L values decay rapidly to weakly excited states, the higher L components being coupled only to higher excited states. In other words, they are dominantly composed by extremal parabolic Rydberg states, which have well-known decay properties [22]. Altogether, their decay rate is decreased by a factor of the order of n0, yielding a n −4 0 law instead of n −3 0 . On the other hand, since the ionization process is dominated by tunneling in the direction of the microwave polarization axis [43,67,87,194], the ionization rate in 3D remains globally comparable to the ionization rate in 1D, for the wave packet launched along straight line orbits. This remains true even if the generic uctuations of the ionization rate (see Section 7.1) may induce locally (in some control parameter) large deviations between individual 3D and 1D decay rates. 35 Therefore, the transition from dominant ionization to dominant spontaneous decay will shift to slightly higher values of n0 in 3D. As in the CP case studied above, this cross-over may be moved to smaller values of n0 by reducing the ionization rate, i.e., by decreasing F0. 35 A similar behavior is observed in circular polarization for the 2D and 3D non-dispersive wave packets: they exhibit comparable ionization rates, but distinct uctuations [50].
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 529 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 ionisation rate radiative rate total rate 10 30 50 70 90 110 130 principal quantum number n 0 Fig. 58. Comparison of the spontaneous decay rate, the ionization rate, and their sum, for a non-dispersive wave packet in a linearly polarized microwave eld, as a function of the principal quantum number n0. Microwave amplitude F0 =0:04442, decay rates in atomic units. The full decay rate exhibits a cross-over from a dominantly coherent (ionization) to a dominantly incoherent (spontaneous emission) regime. The uctuations of the rate present in the coherent regime are suppressed in the incoherent regime. The data presented here are obtained by an exact numerical calculation on the one-dimensional model of the atom [71], see Section 3.3.1. 7.3. Non-dispersive wave packet as a soliton The non-dispersive character of the wave packets discussed in this review brings to mind solitons, i.e., solutions of non-linear wave equations that propagate without deformation: the non-linearity is there essential to overcome the spreading of the solution. The non-dispersive wave packets discussed by us are, on the other hand, solutions of the linear Schrodinger wave equation, and it is not some non-linearity of the wave equation which protects them from spreading, but rather the periodic driving. Thus, at rst glance, there seems to be no link between both phenomena. This is not fully correct. One may conceive non-dispersive wave packets as solitonic solutions of particular nonlinear equations, propagating not in time, but in parameter space [63]. The evolution of energy levels in such a space, called “parametric level dynamics”, has been extensively studied (see [15,195] for reviews), both for time-independent and for periodically time-dependent systems. In the latter case, the energy levels are the quasi-energies of the Floquet Hamiltonian (see Section 3.1.2). For the sake of simplicity, we consider here the two-dimensional hydrogen atom exposed to a circularly polarized microwave (Section 3.4.3), where the explicit time dependence can be removed by transforming to the rotating frame (see Section 3.4.4), but completely similar results are obtained for the Floquet Hamiltonian of any periodically time-dependent system. The Hamiltonian, given by Eq. (184), H = ˜p2 1 − 2 r + Fx − !Lz ; (274) may be thought of as an example of a generic system of the form H( )=H0 + V ; (275)
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Non-dispersive wave packets in periodically driven quantum systems

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