Non-dispersive wave packets in periodically driven quantum systems

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482 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 30. Spectrum of the two-dimensional hydrogen atom in a circularly polarized microwave eld of frequency ! =1=(60:5) 3 , as a function of the scaled microwave amplitude F0 = F! −4=3 . In order to test the accuracy of the harmonic prediction, we subtract the semiclassical energy for the ground state |0; 0〉 wave packet, Eq. (198), from the result of the exact numerical diagonalization, and rescale the energy axis in units of the mean level spacing. The almost horizontal line slightly above zero corresponds to the non-dispersive ground state wave packet, which typically undergoes small avoided crossings with other Floquet states. A relatively large avoided crossing occurs when two wave packet-like states meet, as here happens around F0 0:036. The other (“colliding”) state is the excited wave packet |1; 3〉. The dashed line indicates the harmonic prediction, Eq. (198), for this state, which is obviously less accurate. Note, however, that the slope is correctly predicted. states are excited by one quantum in either of the normal modes !±;z, and are therefore signi cantly more extended in space. Again, as already mentioned in Sections 3.3.1 and 3.3.2, these wave-packet eigenstates have nite, but extremely long life-times (several thousands to millions of Kepler orbits), due to the eld-induced ionization. For a detailed discussion of their decay properties see Section 7.1. As already demonstrated in the LP case (see Figs. 21 and 22), the semiclassical prediction for the energies of the non-dispersive wave packets is usually excellent. In order to stress the (small) di erences, we plot in Fig. 30 a part of the Floquet spectrum of the two-dimensional model atom, i.e., quasi-energy levels versus the (scaled) microwave amplitude, after subtraction of the prediction of the harmonic approximation around the stable xed point, Eq. (198), for the ground state wave packet |0; 0〉. The result is shown in units of the mean level spacing, estimated 20 to be roughly 2=n 4 0 . If the harmonic approximation was exact, the ground state wave packet would be represented by a horizontal line at zero. The actual result is not very far from that, which proves that the semiclassical method predicts the correct energy with an accuracy mostly better than the mean level spacing. The other states of the system appear as energy levels which rapidly evolve with F0, and which exhibit extremely small avoided crossings—hence extremely small couplings—with the wave packet. In the vicinity of such avoided crossings, the energy levels are perturbed, the diabatic wave functions mix (the wave-packet eigenstates get distorted), and, typically, the lifetime of the state decreases (induced 20 This estimate follows from the local energy splitting, ∼ n −3 0 , divided by the number n0=2 of photons needed to ionize the initial atomic state by a resonant driving eld.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 483 by the coupling to the closest Floquet state [43,145], typically much less resistant against ionization, as we shall discuss in detail in Section 7.1). Thus, strictly speaking, when we speak of non-dispersive wave packets as speci c Floquet states, we really have in mind a generic situation, far from any avoided crossing. In particular, the examples of wave packet states shown in the gures above correspond to such situations. The observed accuracy of the semiclassical approximation has important practical consequences: in order to obtain the “exact” wave packets numerically, we do not need many eigenvalues for a given set of parameters. Using the Lanczos algorithm for the partial diagonalization of a matrix, it is enough to extract few (say ve) eigenvalues only, centered on the semiclassical prediction. The accuracy of the latter (a fraction of the mean level spacing) is su cient for a clear identi cation of the appropriate quantum eigenvalue. Actually, in a real diagonalization of the Floquet Hamiltonian, we identify the wave-packet states by both their vicinity to the semiclassical prediction for the energy, and the large (modulus of the) slope of the level w.r.t. changes of F0, induced by its large dipole moment in the rotating frame (by virtue of the Hellman–Feynman theorem [18]). The latter criterion is actually also very useful for the identi cation of excited wave packets in a numerically exact spectrum. The state |1; 3〉 (in the harmonic approximation) presented in Fig. 28 is precisely the excited wave packet which appears in Fig. 30 as a “line” with a negative slope, meeting the |0; 0〉 state in a broad avoided crossing around F0 0:036. From that gure, it is apparent that the harmonic prediction for the energy is not excellent for the |1; 3〉 state. On the other hand, the slope of the Floquet state almost matches the slope given by the harmonic approximation, which con rms that the exact wave-function is still well approximated by its harmonic counterpart. Note that also from the experimental point of view it is important to get accurate and simple semiclassical estimates of the energies of the non-dispersive wave packets, since it may help in their preparation and unambiguous identi cation. For a more detailed discussion, see Section 8.2. It is interesting to compare the accuracy of the harmonic approximation to the pendulum description outlined previously in Section 3.4.2. The latter results from lowest order perturbation theory in F. Taking the small F limit we get !+;!z → !, and !− → ! √ 3F0, for the harmonic modes, see Eqs. (194)–(196). The latter result coincides—as it should—with the pendulum prediction, Eqs. (80) and (173). Similarly, the energy of the stable equilibrium point, Eq. (187), becomes at rst-order in F: Eeq = − 3 2n2 + n 0 2 0F + O(F 2 ) ; (200) which coincides with the energy of the center of the resonance island, see Section 3.4.2. Thus, the prediction of the resonance analysis agrees with the harmonic approximation in the rotating frame. For a more accurate estimate of the validity of both approaches, we have calculated—for the 2D model of the atom, but similar conclusions are reached in 3D—the energy di erence between the exact quantum result and the prediction using a semiclassical quantization of the secular motion in the (M; ) plane together with the Mathieu method in the (Î; ˆ ) plane on the one side, and the prediction of the harmonic approximation around the xed point, Eq. (198), on the other side. In Fig. 31, we compare the results coming from both approaches. As expected, the semiclassical approach based on the Mathieu equation is clearly superior for very small microwave amplitudes, as it is “exact” at rst-order in F. On the other hand, for the harmonic approximation to work well, the island around the xed point has to be su ciently large. Since the size of the island increases
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Non-dispersive wave packets in periodically driven quantum systems

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