Non-dispersive wave packets in periodically driven quantum systems

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540 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 EXCITATION PROBABILITY 1.0 0.8 0.6 0.4 0.2 0.0 59 60 n0 61 Fig. 63. Overlap between the wave-function obtained at the end of the microwave turn-on and the exact target state representing the non-dispersive wave packet as a function of n0, obtained for the two-dimensional hydrogen atom (circles). Fmax is as in Fig. 61, and the switching time is Tswitch = 250 microwave periods. The initial state corresponds to a circular n = 60 state of a 2D hydrogen atom. For a given initial state | 0〉 of the atom, only the resonance condition de ning the driving eld frequency is to some extent restrictive, as depicted in Fig. 63. It is crucial that the initially excited eld-free state is adiabatically connected (through the Mathieu equation) to the ground state wave packet. The best choice is “optimal resonance”, but the adiabaticity is preserved if n0 is changed by less than one-half, see Section 3.1.2. This corresponds to a relative change of ! of the order of 3=2n0. Given the spectral resolution of presently available microwave generators, the de nition of the frequency with an accuracy of less than 1% is not a limitation. The exact numerical calculation displayed in Fig. 63 fully con rms that e cient excitation is possible as long as n0 = ! −1=3 − 1=2 matches the e ective principal quantum number of the initially excited eld-free state within a margin of ±1=2 (in the range [59.5,60.5]). In conclusion, the preparation of non-dispersive wave packets by excitation of a Rydberg state followed by careful switching of the microwave eld can be considered as an e cient method, provided a clean experimental preparation of the atomic initial state can be achieved. Furthermore, the boundaries—Eqs. (278) and (283)—imposed on the timescale for the switching process leave a su cient exibility for the experimentalist to e ciently prepare the wave packet. A nal word is in place on the homogeneity of the driving eld amplitude experienced by the atoms in the “ at top region” of the interaction, i.e., after the switching from the eld-free state into the wave packet state at F(t)=Fmax. In any laboratory experiment, a slow drift of the amplitude will be unavoidable over the interaction volume. Hence, slightly di erent non-dispersive wave packets will coexist at various spatial positions. Since the ionization rate of non-dispersive wave packets is rather sensitive with respect to detailed values of the parameters, see Section 7.1, this should manifest itself by a deviation of the time dependence of the ionization yield from purely exponential decay. 8.4. Life time measurements Given the above, rather e cient experimental schemes for the population of non-dispersive wavepacket eigenstates—either via direct optical Floquet absorption or through an appropriate switching
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 541 procedure—we still need some means to prove that we really did populate the wave packet. As a matter of fact, to provide unambiguous experimental evidence, one has to test various characteristic properties of the wave packet, so as to exclude accidental coincidences. A natural way is Floquet spectroscopy (see Section 8.2), i.e. probing the structure of the Floquet quasi-energy levels, in either the optical or the microwave regime (via absorption, stimulated emission, Raman spectroscopy etc.). Another possibility is to explore unique properties of wave packet Floquet states. For example, as discussed in Section 7.1, these states exhibit extremely small ionization rates. Hence, an experimentally accessible quantity to identify these states is the time dependence of their survival probability, i.e., of the probability not to ionize during an interaction time t. It is given by [43,97] P(t)= |c | 2 exp(− t) ; (285) where the c denote the expansion coe cients of the initial eld-free state in the Floquet basis, at a given value of microwave amplitude F. If the selective population of the wave packet is successful, only one Floquet state contributes to P(t), and the decay of the population to the atomic continuum should manifest in its exponential decrease, as opposed to a multiexponential decrease in the case of a broad distribution of the c over the Floquet states [43,145,206–208,220]. Of course, the distinction between an exponential and an algebraic decay law requires the variability of the experimental interaction time over more than one order of magnitude. This is a nontrivial task in experiments on atomic Rydberg states of hydrogen, since the typical velocities of the atomic beam are of the order of 1000 m=s. That signi cantly restricts the interval on which the interaction time may be changed, taking into account the typical size (in the cm-range) of the atom– eld interaction region [133,228]. However, the feasibility of such measurements has already been demonstrated in microwave experiments on rubidium Rydberg states, where the interaction time has been scanned from approx. 100 to approx. 100 000 eld cycles, i.e., over three orders of magnitude [220,221]. Note that, whereas the dynamics of the driven Rydberg electron along a Kepler ellipse of large eccentricity will certainly be a ected by the presence of a non-hydrogenic core, non-dispersive wave packets as the ones discussed in Sections 3.4 and 3.5 can certainly be launched along circular trajectories, since the Rydberg electron of the rubidium atom essentially experiences a Coulomb eld on such a circular orbit. To use the character of the decay as a means to identify the wave packet, the microwave eld amplitude should be su ciently large to guarantee that other Floquet states localized in the chaotic sea (see Section 7.1) decay rapidly. Otherwise, the observation of a mono-exponential decay simply suggests that we succeeded in populating a single Floquet state—not necessarily a wave packet [220]. The appropriate choice of the driving eld amplitude F, such that appreciable ionization is achieved for the longer experimentally accessible interaction times, should therefore allow for the experimental identi cation of the mono-exponential decay from the wave packet to the atomic continuum, but also—by varying F—of the variations of the decay rate with F, which is predicted to uctuate wildly over several orders of magnitude, see Section 7.1. Note, however, that this requires an excellent homogeneity of the microwave eld (e.g., provided by a high-quality microwave cavity), as the uctuations take place over rather small intervals of F.
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Non-dispersive wave packets in periodically driven quantum systems

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