Non-dispersive wave packets in periodically driven quantum systems

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458 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 z (atomic units) (a) 8000 6000 4000 2000 0 0 1 2 3 4 Time (microwave periods) ∆z ∆p (h unit) (b) 100 10 1 Heisenberg limit 0 1 2 3 4 Time (microwave periods) Fig. 14. (a) Dashed line: Classical temporal evolution of the position of the Rydberg electron resonantly driven by a linearly polarized microwave eld, in the one-dimensional model, Eq. (137), of the hydrogen atom. (n0 = 60; scaled eld amplitude F0 = Fn 4 0 =0:053). Thick line: Expectation value 〈z〉 for the non-dispersive wave packet shown in Fig. 13, as a function of time. It follows the classical trajectory remarkably well (except for collisions with the nucleus). Dotted line: The position uncertainty z = 〈z 2 〉−〈z〉 2 of the wave packet. z being much smaller than 〈z〉 (except near collisions with the nucleus) highlights the e cient localization of the wave packet. (b) Uncertainty product z p of the wave packet. The periodically repeating maxima of this quantity indicate the collision of the electron with the atomic nucleus. Note that the minimum uncertainty at the outer turning point of the wave packet is very close to the Heisenberg limit ˝=2. Although the wave packet is never a minimal one, it is nevertheless well localized and an excellent approximation of a classical particle. is unstable, the quantum eigenstate cannot remain fully localized—some probability has to ow away along the unstable manifold of the classical ow in the vicinity of the hyperbolic xed point. Consequently, such an eigenstate is partially localized along the separatrix between librational and rotational motion. For an illustration, rst consider Fig. 16, which shows classical surfaces of section of the driven (1D) hydrogen atom, at F0 =0:034, again for di erent phases !t. Comparison with Fig. 12 shows a larger elliptic island at this slightly lower eld amplitude, as well as remnants of the s = 2 resonance island at slightly larger actions I=n0 1:2 :::1:3. The time evolution of the electronic density of the eigenstate localized near the hyperbolic xed point is displayed in Fig. 17, for di erent phases of the driving eld. Clearly, as compared to Fig. 13, the wave packet moves in phase opposition to the driving eld, and displays slightly irregular localization properties. Accordingly, the Husimi representation in Fig. 18 exhibits reasonably good localization on top of the hyperbolic point at phase !t = , but the electronic probability spreads signi cantly along the separatrix layer at phase, as visible at !t =0. In the above discussion of the localization properties of the wave-packet eigenstate we represented the wave function in the I– phase space of classically bounded motion (i.e., classical motion with negative energy). However, as we shall see in more detail in Section 7.1, the microwave driving actually induces a non-vanishing overlap of all Floquet eigenstates [120,121], and, hence, of the wave-packet eigenstates, with the atomic continuum. It su ces to say here that the associated - nite decay rates induce nite life times of approx. 10 6 unperturbed Kepler orbits for the quantum objects considered in this section, and are therefore irrelevant on the present level of our discussion. In Figs. 13, 15, 17 and 18, a nite decay rate would manifest as a slow reduction of the
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 459 Fig. 15. Husimi representation of the wave-packet eigenstate of Fig. 13 in classical phase space, for the same phases !t and scales (0 6 6 2 ; 30 6 I 6 90) as employed for the classical surface of section in Fig. 12. Clearly, the quantum mechanical eigenstate of the atom in the eld follows the classical evolution without dispersion, except for its transient spreading when re ected o the nucleus (at !t = , bottom left), due to the divergence of the classical velocity at that position. electronic density, without a ecting its shape or localization properties, after 10 6 classical Kepler periods. 3.3.2. Realistic three-dimensional atom Extending our previous analysis to the three-dimensional hydrogen atom driven by a linearly polarized microwave eld, we essentially expand the accessible phase space. Since the Hamiltonian HLP = ˜p2 1 − + Fz cos(!t) (151) 2 r is invariant under rotations around the eld polarization axis, the projection of the angular momentum is a conserved quantity and gives rise to a good quantum number M. Hence, only two dimensions of con guration space are left, which, together with the explicit, periodic time dependence, span a ve-dimensional phase space. In the 1D situation described previously, the key ingredient for the existence of non-dispersive wave packets was the phase locking of the internal degree of freedom on the external drive. In the 3D situation, there remains one single drive, but there are several internal degrees of freedom. In the generic case, not all internal degrees of freedom can be simultaneously locked on the external drive, and one can expect only partial phase locking, i.e., only partially localized wave packets. The non-trivial task is to understand how the phase locking of one degree of freedom modi es the
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Non-dispersive wave packets in periodically driven quantum systems

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