Non-dispersive wave packets in periodically driven quantum systems

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520 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 |Shift| Distribution 10 0 10 -1 10 -2 10 -2 10 -1 10 0 10 1 10 -3 (a) |AC Stark Shift| (b) Fig. 52. The distribution of energy shifts (a) and ionization widths (b) for the non-dispersive wave packet of a two-dimensional hydrogen atom in a circularly polarized eld, obtained by numerical diagonalization of the Hamiltonian (large bins), compared to the random matrix model (small bins). Both distributions are shown on a double logarithmic scale to better visualize the behavior over a large range of shift and width values. Since the energy shift may be positive or negative, we show the distribution of its modulus. The random matrix model ts very well the numerical results, with both distributions showing regions of algebraic behavior followed by an exponential cut-o . were used for a single t [65]. This allowed us to study the dependence of the parameters = ; = on n0 and F0. The dependence on n0 is shown in Fig. 53. Clearly, the tunneling rate = decreases exponentially with n0. Since n0 is the inverse of the e ective Planck constant in our problem (see the discussion in Section 3.3.1 and Eq. (147)), this shows that = ˙ exp − S ˝e ; (267) where S, corresponding to some e ective imaginary action [188], is found to be given for our speci c choice of parameters by S 0:06 ± 0:01 (as tted from the plot). Such an exponential dependence is a hallmark of a tunneling process, thus con rming that the wave packets are strongly localized in the island and communicate with the outside world via tunneling. The n0 dependence of the dimensionless chaotic ionization rate = is very di erent: it shows a slow, algebraic increase with n0. A simple analysis based on a Kepler map [131] description would yield a linear increase with n0, whereas our data seem to suggest a quadratic function of n0. This discrepancy is not very surprising, bearing in mind the simplicity of the Kepler map approach. Similarly, we may study, for xed n0, the dependence of = and = on F0, i.e., on the microwave eld strength. Such studies, performed for both linear and circular polarizations, have indicated that, not very surprisingly, the chaotic ionization rate = increases rather smoothly with the microwave amplitude F0. On the other hand, the tunneling rate = shows pronounced non-monotonic variations with F0, see Fig. 54. This unexpected behavior can nontheless be explained [65]. The bumps in = occur at microwave eld strengths where secondary nonlinear resonances emerge within the resonance island in classical phase space. For circular polarization, this corresponds to some resonance between two eigenfrequencies !+ and !− (see Section 3.4 and Figs. 30 and 59) of the dynamics in the classical resonance island. Such resonances strongly perturb the classical dynamics and necessarily a ect the quantum transport from the island. Let us stress nally that, even for rather strong microwave elds (say F0=0:05), where most of the other Floquet states have lifetimes of few tens or hundreds of microwave periods, and irrespective of the polarization of the driving eld or of the dimension of the accessible con guration space (1D, 2D or 3D), the lifetime (modulo uctuations) of a non-dispersive wave packet is typically of the Width Distribution 10 1 10 -1 10 -3 10 -5 10 -7 10 -5 10 -3 10 -1 Width 10 1 10 3
γ /∆ σ/∆ 10 -1 10 -2 10 -3 0.2 0.1 (a) (b) 0.0 20 40 60 80 100 Principal Quantum Number n 0 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 521 10 -1 10 -2 10 -3 10 -1 10 -3 10 -5 10 -7 0.02 0.04 0.06 Fig. 53. E ective tunneling rate = of the wave packet (a), as a function of the e ective quantum number n0 = ! −1=3 (the inverse of the e ective Planck constant), for xed classical dynamics, F0 =0:0426. Note the exponential decrease for su ciently high n0 (the vertical scale is logarithmic). The corresponding e ective chaotic ionization rate = (b) smoothly increases with n0, approximately as n 2 0. Fig. 54. The tunneling rate = (panel (a)) and the chaotic ionization rate = (panel (b)), as a function of the scaled microwave amplitude F0, for wave-packet eigenstates of a two-dimensional hydrogen atom in a circularly polarized microwave eld. Observe the oscillatory behavior of the tunneling rate. The bumps are due to secondary nonlinear resonances in the classical dynamics of the system. order of 10 5 Kepler periods, for n0 60. This may be used for their possible experimental detection, see Section 8.4. 7.2. Radiative properties So far, we have considered the interaction of the atom with the coherent driving eld only. However, this is not the full story. Since the driving eld couples excited atomic states, it remains to be seen to which extent spontaneous emission (or, more precisely, the coupling to other, initially unoccupied modes of the electromagnetic eld) a ects the wave packet properties. This is very important, since the non-dispersive wave packets are supposed to be long living objects, and spontaneous emission obviously limits their lifetime. Furthermore, we have here an example of decoherence e ects due to interaction with the environment. More generally, the interaction of non-dispersive wave packets with an additional weak external electromagnetic eld may provide a useful tool to probe their properties. In particular, their localization within the resonance island implies that an external probe will couple them e ciently only to neighboring states within the island. In turn, that should make their experimental characterization easy and unambiguous. Of course, external drive (microwave eld) and probe must not be treated on the same footing. One should rather consider the atom dressed γ/∆ σ/∆ F 0 (b) (a)
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Non-dispersive wave packets in periodically driven quantum systems

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