Non-dispersive wave packets in periodically driven quantum systems

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526 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 SQUARED DIPOLE 1.0 0.5 0.0 ω−ω + ω ω+ ω− 0.0 0.5 1.0 1.5 2.0 Fig. 56. Square of the dipole matrix element (scaled w.r.t. n0, i.e., divided by n 2 0) connecting the |0; 0; 0〉 non-dispersive wave packet of a three-dimensional hydrogen atom in a circularly polarized microwave eld with other Floquet states, as a function of the energy di erence between the two states. The stick spectrum is the exact result obtained from a numerical diagonalization with ! =1=60 3 , corresponding to a principal quantum number n0 = 60, and scaled amplitude F0 =0:04446; in natural units, the microwave frequency !=2 is 30:48 GHz, and the microwave amplitude 17:6 V=cm. The crosses represent the analytic prediction within the harmonic approximation [68]. There are three dominant lines ( + polarized) discussed in the text, other transitions (as well as transitions with − or polarizations) are negligible, what proves the validity of the harmonic approximation. If a weak probe eld (in the microwave domain) is applied to the system in addition to the driving eld, its absorption spectrum should therefore show the three dominant lines, allowing an unambiguous characterization of the non-dispersive wave packet. and the semiclassical energies observed in Figs. 30 and 59). This is not completely surprising as the energy levels themselves are well reproduced by this harmonic approximation, see Section 3.4.4. However, the photoabsorption spectrum probes the wave functions themselves (through the overlaps) which are well known to be much more sensitive than the energy levels. The good agreement for both the energy spectrum and the matrix elements is a clear-cut proof of the reliability of the harmonic approximation for physically accessible principal quantum numbers, say n0 ¡ 100; in fact, it is good down to n0 30, and the non-dispersive wave packet exists even for lower n0 values (e.g. n0 = 15 in [49]) although the harmonic approximation is not too good at such low quantum numbers. There were repeated claims in the literature [30,44,46,54,62,144] that the stability island as well as the e ective potential are necessarily unharmonic in the vicinity of the equilibrium point, and that the unharmonic terms will destroy the stability of the non-dispersive wave packets. The present results prove that these claims are doubly wrong: rstly, as explained in Section 3.4.4, harmonicity is not a requirement for non-dispersive wave-packets to exist (the only condition is the existence of a su ciently large resonance island); secondly, the harmonic approximation is clearly a very good approximation even for moderate values of n0. Multiplication by the free space density of states transforms Fig. 56 in Fig. 57, which shows that the corresponding spontaneous decay rates are very low, of the order of 100 Hz at most. They are few orders of magnitude smaller than the ionization rates and thus may be di cult to observe. With increasing n0, the spontaneous rate decreases algebraically while the ionization rate decreases exponentially, see Section 7.2.4. Thus for large n0, the spontaneous emission may be the dominant ωp
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 527 Γ [s -1 ] 20 10 0 ω−ω + ω ω +ω − ω+2ω − 0 10 20 30 40 50 60 ωp /2π [GHz] Fig. 57. Same as Fig. 56, but for the decay rates along the di erent transitions and in natural units. The density of modes of the electromagnetic eld completely kills the transition at frequency !−!+, invisible in the gure. One can see a small line at frequency approximately ! +2!− (see arrow), an indication of a weak breakdown of the harmonic approximation. process. For F0 0:05 the cross-over may be expected around n0 = 200. However, for smaller F0, the ionization rate decreases considerably, and for F0 0:03 both rates become comparable around n0 = 60. Still, a rate of few tens of photons (or electrons in the case of ionization) per second may be quite hard to observe experimentally. To summarize, resonance uorescence of non-dispersive wave packets in circularly polarized microwave occurs only in the microwave range (close to the driving frequency). In particular, the elastic component (dominant in the semiclassical limit) does not destroy the wave packet, the wave packet merely converts the microwave photon into a photon emitted with the same polarization, but in a di erent direction. Let us stress that we assumed the free space density of modes in this discussion. Since the microwave eld may be also supplied to the atom by putting the latter in a microwave cavity, it should be interesting to investigate how the density of modes in such a cavity a ects the spontaneous emission rate either by increasing or decreasing it (see [191] for a review) or, for special cavities (waveguides), even invalidates the concept of a decay rate [192,193]. 7.2.4. Linearly polarized microwave Let us now discuss the spontaneous emission of non-dispersive wave packets driven by a linearly polarized microwave eld. The situation becomes complicated since we should consider di erent wave-packets corresponding to (see Fig. 23) extreme liberational states (p = 0, located perpendicularly to the polarization axis), separatrix states elongated along the polarization axis, and extreme rotational (maximal p, doughnut shaped) states of the resonantly driven manifold. Clearly, all these wave-packet states have di erent localization properties and spontaneous emission will couple them to di erent nal states. No systematic analysis of the e ect has been presented until now, only results based on the simpli ed one-dimensional model are available [71]. Those are of relevance for the spontaneous emission of the separatrix based wave packet and are reviewed below. Quantitatively we may, however, expect that the spontaneous emission properties of the extreme rotational wave packet will resemble those of the non-dispersive wave packet in circular polarization. Indeed, the
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Non-dispersive wave packets in periodically driven quantum systems

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