Non-dispersive wave packets in periodically driven quantum systems

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490 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 4. Manipulating the wave packets We have shown in the previous sections that non-dispersive wave packets are genuine solutions of the Floquet eigenvalue problem, Eq. (75), under resonant driving, for arbitrary polarization of the driving eld. The semiclassical approximation used to guide our exact numerical approach directly demonstrates the localization of the electronic density in well de ned regions of phase space, which protect the atom against ionization induced by the external eld (see, however, Section 7.1). We have also seen that classical phase space does not only undergo structural changes under changes of the driving eld amplitude (Figs. 10, 12 and 16), but also under changes of the driving eld ellipticity (Fig. 33). Therefore, the creation of non-dispersive wave packets can be conceived as an easy and e cient means of quantum control, which allows the manipulation and the controlled transfer of quantum population across phase space. In particular, one may imagine the creation of a wave packet moving along the polarization axis of a linearly polarized microwave eld. A subsequent, smooth change through elliptical to nally circular polarization allows to transfer the electron to a circular orbit. Adding additional static elds to Hamiltonian (75) provides us with yet another handle to control the orientation and shape of highly excited Rydberg trajectories, and, hence, to manipulate the localization properties of non-dispersive wave-packet eigenstates in con guration and phase space. The key point is that trapping inside the non-linear resonance island is a robust mechanism which protects the non-dispersive wave packet very e ciently from imperfections. This allows to adjust the wave-packet’s properties at will, just by adiabatically changing the properties of the island itself. Moreover, when the strength of the external perturbation increases, chaos generically invades a large part of classical phase space, but the resonance islands most often survives. The reason is that the phase locking phenomenon introduces various time scales in the system, which have di erent orders of magnitude. That makes the system quasi-integrable (for example through some adiabatic approximation a la Born–Oppenheimer) and—locally—more resistant to chaos. Hereafter, we discuss two possible alternatives of manipulating the wave packets. One is realized by adding a static electric eld to the LP microwave drive [72]. Alternatively, the addition of a static magnetic eld to CP driving enhances the region of classical stability, and extends the range of applicability of the harmonic approximation [30,44,46,54,62,144,151]. 4.1. Rydberg states in linearly polarized microwave and static electric elds Let us rst consider a Rydberg electron driven by a resonant, linearly polarized microwave, in the presence of a static electric eld. We already realized (see the discussion in Section 3.3.2) that the classical 3D motion of the driven Rydberg electron is angularly unstable in a LP microwave eld. It turns out, however, that a stabilization of the angular motion is possible by the addition of a static electric eld Fs parallel to the microwave polarization axis [72,152,153]. The corresponding Hamiltonian reads H = p2 x + p 2 y + p 2 z 2 − 1 r + Fz cos !t + Fsz ; (207)
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 491 which we examine in the vicinity of the s=1 resonance. As in Section 3.3.2, the angular momentum projection M on the z-axis remains a constant of motion, and we shall assume M =0 in the following. Compared to the situation of a pure microwave eld, there is an additional time scale, directly related to the static eld. Indeed, in the presence of a perturbative static eld alone, it is known that the Coulomb degeneracy of the hydrogenic energy levels (in L) is lifted. The resulting eigenstates are combinations of the n0 substates of the n0 manifold (for 0 6 L 6 n0 − 1). 22 The associated energy levels are equally spaced by a quantity proportional to Fs (3n0Fs in atomic units). Classically, the trajectories are no longer closed but rather Kepler ellipses which slowly librate around the static eld axis, periodically changing their shapes, at a (small) frequency 3n0Fs!Kepler =1=n3 0 . Thus, the new time scale associated with the static electric eld is of the order of 1=Fs;0 Kepler periods, where Fs;0 = Fsn 4 0 (208) is the scaled static eld. This is to be compared to the time scales 1=F0 and 1= √ F0, which characterize the secular time evolution in the (L; ) and (Î; ˆ ) coordinates, respectively (see discussion in Section 3.3.2), in the presence of the microwave eld alone. To achieve con nement of the electronic trajectory in the close vicinity of the eld polarization axis, we need 1=F0 1=Fs;0, with both, Fs and F small enough to be treated at rst-order. If we now consider the s = 1 resonance, we deduce the secular Hamiltonian by keeping only the term which does not vanish after averaging over one Kepler period. For the microwave eld, this term was already identi ed in Eq. (159). For the static eld, only the static Fourier component of the atomic dipole, Eqs. (155) and (156), has a non-vanishing average over one period. Altogether, this nally leads to Hsec = ˆPt − 1 2Î 2 − !Î + FsX0(Î;L) cos + F 1(Î; L; )cos ( ˆ + 1) : (209) Since the last two terms of this Hamiltonian depend di erently on ˆ , it is no more possible, as it was in the pure LP case (see Section 3.3.2), to perform the quantization of the slow LP motion rst. Only the secular approximation [18] which consists in quantizing rst the fast variables (Î; ˆ ), and subsequently the slow variables (L; ), remains an option for the general treatment. However, since we are essentially interested in the wave-packet eigenstate with optimal localization properties, we shall focus on the ground state within a su ciently large resonance island induced by a microwave eld of an appropriate strength. This motivates the harmonic expansion of the secular Hamiltonian around the stable xed point at Î = Î 1 = ! −1=3 ; ˆ = − 1 : (210) with the characteristic frequency, see Eq. (80): 3F 1(Î 1;L; ) (Î 1L; )= : (211) Î 2 1 22 These states are called “parabolic” states, since the eigenfunctions are separable in parabolic coordinates [7].
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Non-dispersive wave packets in periodically driven quantum systems

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