Non-dispersive wave packets in periodically driven quantum systems

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530 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 where is a parameter. In our case, for example, the microwave amplitude may be tuned, leading to V = x and = F. The interesting quantities are then the eigenvalues Ei( ) and the eigenfunctions | i( )〉 of Eq. (275). Di erentiating the Schrodinger equation with respect to , one shows (with some algebra) [15] that the behavior of Ei( ) with may be viewed as the motion of N ctitious classical particles (where N is the dimension of the Hilbert space) with positions Ei and momenta pi = Vii = 〈 i|V | i〉, governed by the Hamiltonian Hcl = N p 2 i 2 + 1 2 N N |Lij| 2 ; (276) (Ej − Ei) 2 i=1 i=1 j=1;j=i where Lij =(Ei − Ej)〈 i|V | j〉 are additional independent variables obeying the general Poisson brackets for angular momenta. The resulting dynamics, although non-linear, is integrable [15]. Let us now consider the parametric motion of some eigenstate |n+;n−〉, for example of the ground state wave packet |0; 0〉. Its coupling to other states is quite weak—because of its localization in a well de ned region of phase space—and the corresponding Lij are consequently very small. If we rst suppose that the wave-packet state is well isolated (in energy) from other wave packets (i.e., states with low values of n+;n−), the ctitious particle associated with |0; 0〉 basically ignores the other particles and propagates freely at constant velocity. It preserves its properties across the successive interactions with neighboring states, in particular its shape: in that sense, it is a solitonic solution of the equations of motion generated by Hamiltonian (276). Suppose that, in the vicinity of some F values, another wave-packet state (with low n+;n− quantum numbers) becomes quasi-degenerate with |0; 0〉. In the harmonic approximation, see Section 3.4.4, the two states are completely uncoupled; it implies that the corresponding Lij vanishes and the two levels cross. The coupling between the two solitons stems from the di erence between the exact Hamiltonian and its harmonic approximation, i.e., from third order or higher terms, beyond the harmonic approximation. Other |n+;n−〉 states having di erent slopes w.r.t. F induce “solitonic collisions” at some other values of F. To illustrate the e ect, part of the spectrum of the two-dimensional hydrogen atom in a CP microwave is shown in Fig. 59, as a function of the scaled microwave amplitude F0. For the sake of clarity, the energy of the ground state wave packet |0; 0〉 calculated in the harmonic approximation, Eq. (174) is substracted, such that it appears as an almost horizontal line. Around F0 =0:023, it is crossed by another solitonic solution, corresponding to the |1; 4〉 wave packet, 36 what represents the collision of two solitons. Since this avoided crossing is narrow and well isolated from other avoided crossings, the wave functions before and after the crossing preserve their shape and character, as typical for an isolated two-level system. This may be further veri ed by wave-function plots before and after the collision (see [63] for more details). The avoided crossings become larger (compare Fig. 30) with increasing F0. In fact, as mentioned in Section 3.4.4, we have numerically veri ed that the solitonic character of the ground state wave packet practically disappears at the 1:2 resonance, close to F0 0:065 [63]. For larger F0, while one may still nd nicely localized wave packets for isolated values of F0, the increased size of the avoided crossings makes it di cult to follow the wave packet when sweeping F0. For such strong elds, the ionization rate of wave-packet states becomes appreciable, comparable to the level spacing between 36 Similarly to the wave packet |1; 3〉 discussed in Fig. 30, the semiclassical harmonic prediction for the energy of |1; 4〉 is not satisfactory. However, the slope of the energy level is well reproduced as a function of F0.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 531 RESCALED ENERGY 3 2 1 0 -1 -2 -3 0.018 0.02 0.022 F0 0.024 0.026 Fig. 59. The quasi-energy spectrum of a two-dimensional hydrogen atom in a circularly polarized microwave eld, as a function of the scaled microwave amplitude F0 (for n0 = 60). In order to emphasize the dynamics of wave-packet states, the semiclassical prediction, Eq. (174) for the ground state wave-packet energy is substracted from the numerically calculated energies. Consequently, the ground state wave packet |0; 0〉 is represented by the almost horizontal line. The dashed line represents the semiclassical prediction for the |1; 4〉 wave packet. Although it is rather far from the exact result, the slope of the energy level is well reproduced. The size of the avoided crossing between the “solitonic” levels |0; 0〉 and |1; 4〉 is a direct measure of the failure of the harmonic approximation. consecutive states and the simple solitonic model breaks down. In order to understand the variations of the (complex) energies of the resonances with F, a slightly more complicated model—level dynamics in the complex plane—should be used [15]. 8. Experimental preparation and detection of non-dispersive wave packets In the preceding sections, we have given an extensive theoretical description of the characteristic properties of non-dispersive wave packets in driven Rydberg systems. We have seen that these surprisingly robust “quantum particles” are ubiquitous in the interaction of electromagnetic radiation with matter. However, any theoretical analysis needs to be confronted with reality, and we have to deal with the question of creating and identifying non-dispersive wave packets in a laboratory experiment. In our opinion, none of the currently operational experiments on the interaction of Rydberg atoms with microwave elds allows for an unambiguous identi cation of non-dispersive wave packets, although some of them [133] certainly have already populated such states. In the following, we shall therefore start out with a brief description of the typical approach of state-of-the-art experiments, and subsequently extend on various alternatives to create and to probe non-dispersive wave packets in a real experiment. We do not aim at a comprehensive review on the interaction of Rydberg atoms with microwave elds, but rather refer to [43,32,69,133,196] for a detailed treatment of various aspects of this intricate problem. Here, we strictly focus on issues pertinent to our speci c purpose.
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