Non-dispersive wave packets in periodically driven quantum systems

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536 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 of classical phase space, manifesting in an abundance of avoided crossings of various sizes in the level dynamics. Still, as exempli ed in Figs. 9, 30 and 59, the wave-packet states may be followed rather easily under changes of F (parametrized by t, during the switching of the pulse) in the level dynamics, in agreement with their “solitonic” character (see Section 7.3). Nontheless, the very same gures illustrate clearly that the targeted wave-packet state undergoes many avoided crossings as the microwave amplitude is swept. To remain in a single eigenstate, the avoided crossings should be passed either adiabatically or diabatically, with a branching ratio at an individual crossing being described by the well known Landau–Zener scenario [7,227]. Consequently, if we want to populate an individual wave-packet eigenstate from a eld-free atomic state | 0〉, we need some knowledge of the energy level dynamics. Then it is possible to identify those eld-free states which are connected to the wave packet via adiabatic and=or diabatic transitions in the network of energy levels, and subsequently to design F(t) such as to transfer population from | 0〉 to |E〉 most e ciently. A precise experimental preparation of | 0〉 is the prerequisite of any such approach. When the driving eld is increased from zero, the major modi cation in the classical phase space is the emergence of the resonance island (see Figs. 22, 25 and 26). Quantum mechanically, the states with initial principal quantum number close to n0 = ! −1=3 will enter progressively inside the resonance island. For a one-dimensional system, the Mathieu equation, discussed in Section 3.1.4, fully describes the evolution of the energy levels in this regime. As shown for example in Fig. 9, the non-dispersive wave packet with the best localization, i.e., N = 0, is—in this simple situation— adiabatically connected to the eld-free state with principal quantum number closest to n0, i.e. the eigenstate = 0 of the Mathieu equation. When the Mathieu parameter q, Eq. (100), is of the order of unity, the state of interest is trapped in the resonance island, which happens at eld amplitudes given by Eq. (150) for the one-dimensional atom, and for linear polarization of the microwave eld. A similar scaling is expected for other polarizations, too. In the interval F 6 Ftrapping, the eld has to be increased slowly enough such as to avoid losses from the ground state to the excited states of the Mathieu equation, at an energy separation of the order of n −4 0 . The most favorable situation is then the case of “optimal” resonance (see Section 3.1.2), when n0 is an integer, the situation in Fig. 9. The wave-packet state is always separated from the other states by an energy gap comparable to its value at F = 0, i.e., of the order of 3=(2n4 0 ). The situation is less favorable if n0 is not an integer, because the energy gap between the wave packet of interest and the other states is smaller when F → 0. The worst case is met when n0 is half-integer: the free states n0 +1=2 and n0 − 1=2 are quasi-degenerate, and selective excitation of a single wave packet is thus more di cult. The appropriate time scale for switching on the eld is given by the inverse of the energy splitting, i.e., for “optimal” resonance trapping ∼ n 4 0 = n0 × 2 =! (278) or n0 driving eld periods. Once trapped in the resonance island, the coupling to states localized outside the island will be residual—mediated by quantum mechanical tunneling, see Section 7.1—and the size of the avoided crossings between the trapped and the untrapped states is exponentially small. After adiabatic switching into the resonance island on a time-scale of n0 Kepler orbits, we now have to switch diabatically from Ftrapping to some nal F value, in order to avoid adiabatic losses from the wave packet into other states while passing through the avoided crossings.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 537 The preceding discussion is based on a one-dimensional model and the Mathieu equation. Taking into account the other “transverse” degrees of freedom is not too di cult. Indeed, as noticed in Sections 3.3.2 and 3.4, the various time-scales of the problem are well separated. The transverse motion is slow and can be adiabatically separated from the fast motion in (Î; ˆ ). Instead of getting a single set of energy levels, one gets a family of sets, the various families being essentially uncoupled. An example for the 3D atom in a linearly polarized microwave eld is shown in Fig. 22. It follows that the estimate for the trapping eld and the switching time are essentially the same as for 1D systems. Inside the resonance island, the situation becomes slightly more complicated, because there is not a single frequency for the secular motion, but several frequencies along the transverse degrees of freedom. For example, in CP, it has been shown that there are three eigenfrequencies, Eqs. (194) and (195), in the harmonic approximation—see Section 3.4.4. This results in a large number of excited energy levels which may have avoided crossings with the “ground state”, i.e., the non-dispersive wave packet we want to prepare. As shown in Section 7.3, most of these avoided crossings are extremely small and can be easily crossed diabatically. However, some of them are rather large, especially when there is an internal resonance between two eigenfrequencies. Examples are given in Figs. 30 and 59, where !+ =3!− and 4!−, respectively. These avoided crossings are large and dangerous, because the states involved lie inside the resonance island, which thus loses its protective character. They are mainly due to the unharmonic character of the Coulomb potential. Their size may be qualitatively analyzed as we do below on the CP example, expecting similar sizes of the avoided crossings for any polarization. The unharmonic corrections to the harmonic approximation around the stable xed point xe;pe in the center of the non-linear resonance—as outlined in Section 3.4—are due to the higher-order terms (−1) j ˜x j =j!x j+1 e in the Taylor series of the Coulomb potential, where ˜x; ˜y are excursions from the equilibrium position. ˜x and ˜y can be expressed as linear combinations of a † ± and a± operators [68] giving ˜x; ˜y ∼ ! −1=2 ∼ n 3=2 0 . Furthermore, the equilibrium distance from the nucleus scales as the size of the atom, xe ∼ n2 0 , and, therefore, ˜x j =x j+1 e ∼ n (−2−j=2) 0 : (279) A state |n+;n−〉 is obtained by the excitation of N+ quanta in the !+-mode and of N− quanta in the !−-mode, respectively, i.e., by the application of the operator product (a † +) n+ (a † − )n− on the wave-packet state |0; 0〉, which—by virtue of Eq. (279)—will be subject to an unharmonic correction scaling like Eunharmonic ∼ n (−2−(n++n−)=2) 0 : (280) Hence, the size of the avoided crossings between the wave-packet eigenstate and excited states of the local potential around the stable xed point decreases with the number of quanta in the excited modes. In addition, we can determine the width Funharmonic of such avoided crossings in the driving eld amplitude F, by di erentiation of energy (198) of |n+;n−〉 with respect to F. Then, the di erence between the energies of two eigenstates localized in the resonance island is found to scale like Fn0. De ning Funharmonic by the requirement that Fn0 be of the order of Eunharmonic,
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Non-dispersive wave packets in periodically driven quantum systems

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