Non-dispersive wave packets in periodically driven quantum systems

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538 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 61. Snapshots of the electronic density for a two-dimensional hydrogen atom exposed to a circularly polarized microwave eld with increasing amplitude. The microwave eld amplitude is switched on according to Eq. (284), with maximum scaled eld F0;max =0:03 and Tswitch = 400 × 2 =!, where ! is resonant with n0 = 60 (frequency ! =1=(60:5) 3 ). The evolution of the initial circular state n = M = 60 is numerically computed by solving the time-dependent Schrodinger equation in a convenient Sturmian basis. Top left—t = 0 (initial circular state); top middle—t = 20 microwave periods; top right—t = 60 periods, bottom left—t = 100 periods; bottom middle—the nal state, t = 400 periods; bottom right—the non-dispersive wave packet (exact Floquet eigenstate): it is almost indistinguishable from the previous wave function, what proves that the excitation process e ciently and almost exclusively populates the state of interest. The box extends over 10 000 Bohr radii in both directions, with the nucleus at the center. The microwave eld is along the horizontal axis, pointing to the right. we nd n++n− (−3− ) 2 Funharmonic ∼ n 0 : (281) Since we want to switch the eld to a maximum value Fmax ∼ n −4 0 , the Landau–Zener formula yields ∼ Fmax E F unharmonic ∼ n n−+n++1 0 (282) ∼ n n−+n+−2 0 microwave periods (283) for the scaling behavior of the timescale which guarantees diabatic switching through avoided crossings of the wave-packet eigenstate with excited states of the elliptic island. Let us stress that this is only a very rough estimate of the switching time, some numerical factors (not necessarily close to unity) are not taken into account. The above predictions can be checked, e.g., by a numerical integration of the time-dependent Schrodinger equation for a microwave-driven atom, taking into account the time-dependent amplitude of the eld. An exemplary calculation on the two-dimensional model atom (see Section 3.4.3) can be found in [61], for CP driving. Fig. 61 shows the evolution of the electronic density of the atomic
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 539 EXCITATION PROBABILITY 1.0 0.8 0.6 0.4 0.2 0.0 0 500 1000 Tswitch 1500 Fig. 62. 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 the switching time Tswitch, for the two-dimensional hydrogen atom (circles). The lled squares indicate the results obtained for a fully three-dimensional atom. Fmax, !, and n0 as in Fig. 61. wave function (initially prepared in the circular Rydberg state n = M = 60) during the rising part of the driving eld envelope, modeled by F(t)=Fmax sin 2 t : (284) 2Tswitch The driving eld frequency was chosen according to the resonance condition with the n0 =60 state, with a maximum scaled amplitude F0;max =0:03. Inspection of Fig. 59 shows that, for this value of Fmax, the crossing between the wave-packet eigenstate and the state |n+ =1;n− =4〉 has to be passed diabatically after adiabatic trapping within the principal resonance. By virtue of the above estimations of the adiabatic and the diabatic time-scales, the switching time (measured in driving eld cycles) has to be chosen such that n0 ¡Tswitch ¡n 3 0 (in microwave periods). Clearly, the pulse populates the desired wave packet once the driving eld amplitude reaches its maximum value. More quantitatively, the overlap of the nal state after propagation of the time-dependent Schrodinger equation with the wave-packet eigenstate of the driven atom in the eld (bottom-right panel) amounts to 94%. Since losses of atomic population due to ionization are negligible on the time scales considered in the gure, 6% of the initial atomic population is lost during the switching process. The same calculation, done for the realistic three-dimensional atom with n0 = 60 gives the same result, proving that the z direction (which is neglected in the 2D model) is essentially irrelevant in this problem. Fig. 62 shows the e ciency of the proposed switching scheme as a function of the switching time Tswitch, expressed in units of microwave periods. Observe that too long switching times tend to be less e ective, since the avoided crossings passed during the switching stage are not traversed diabatically. The rough estimate, Eq. (283), overestimates the maximum switching time by one order of magnitude. On the other hand, too short switching times do not allow the wave packet to localize inside the resonance island. However, a wide range of switching times remains where good e ciency is achieved.
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Non-dispersive wave packets in periodically driven quantum systems

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