Non-dispersive wave packets in periodically driven quantum systems

More documents

Recommendations

Info

486 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 an unstable periodic orbit embedded in a chaotic sea—which disappears in the semiclassical limit [81,147]. 3.5. Rydberg states in elliptically polarized microwave elds The origin of non-dispersive wave packets being their localization inside the resonance island (locking the frequency of the electronic motion onto the external drive) suggests that such wave packets are quite robust and should exist not only for CP and LP, but also for arbitrary elliptical polarization (EP). The possible existence of non-dispersive wave packets for EP was mentioned in [30,148], using the classical “pulsating SOS” approach. The method, however, did not allow for quantitative predictions, and was restricted to elliptic polarizations very close to the CP case. However, the robustness of such wave packets for arbitrary EP is obvious once the localization mechanism inside the resonance island is well understood [73,69]. Let us consider an elliptically polarized driving eld of constant amplitude. With the ellipticity parameter ∈ [0; 1], V = F(x cos !t + ysin !t) ; (201) establishes a continuous transition between linear ( = 0) and circular ( = 1) polarization treated in the two preceding sections. 21 This general case is slightly more complicated than both limiting cases LP and CP. For LP microwaves (see Section 3.3.2), the conservation of the angular momentum projection onto the polarization axis, M, makes the dynamics e ectively two-dimensional. For the CP case, transformation (184) to the frame rotating with the microwave frequency removes the explicit time dependence (see Section 3.4). None of these simpli cations is possible in the general EP case, and the problem is truly three dimensional and time-dependent. To illustrate the transition from LP to CP via EP, the two-dimensional model of the atom is su - cient, and we shall restrain our subsequent treatment to this computationally less involved case. The classical resonance analysis for EP microwave ionization has been described in detail in [149,150]. It follows closely the lines described in detail in Sections 3.3 and 3.4. By expanding the perturbation, Eq. (201), in the action-angle coordinates (I; M; ; 3.2.4), one obtains the following secular Hamiltonian: ) of the two-dimensional atom (see Section Hsec = ˆPt − 1 2Î 2 − !Î + F[V1(Î; M; ; ) cos ˆ − U1(Î; M; ; ) sin ˆ ] ; (202) with V1(Î; M; ; ) = cos (X1 + Y1) ; U1(Î; M; ; ) = sin (Y1 + X1) : (203) 21 Note, however, that = 0 de nes a linearly polarized eld along the x-axis, i.e., in the plane of elliptical polarization for ¿0. In Section 3.3, the polarization vector was chosen along the z-axis. The physics is of course the same, but the algebraic expressions are slightly di erent, requiring a rotation by an angle = =2 around the y-axis.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 487 This can be nally rewritten as Hsec = ˆPt − 1 2Î 2 − !Î + F 1(Î; M; ; ) cos( ˆ + 1) : (204) Both, 1 and 1, depend on the shape and orientation of the electronic elliptical trajectory, as well as on , and are given by and 1(Î; M; ; )= tan 1(M; ; )= U1 V2 1 + U2 1 ; (205) V1 ; (206) which is once more the familiar form of a system with a resonance island in the (Î; ˆ ) plane. The expressions obtained are in fact very similar to the ones we obtained for the three-dimensional atom exposed to a circularly polarized microwave eld, Eqs. (169)–(171), in Section 3.4.2. This is actually not surprising: the relevant parameter for the transverse dynamics is the magnitude of the atomic dipole oscillating with the driving eld, i.e., the scalar product of the oscillating atomic dipole with the polarization vector. The latter can be seen either as the projection of the oscillating atomic dipole onto the polarization plane or as the projection of the polarization vector on the plane of the atomic trajectory. If one considers a three-dimensional hydrogen atom in a circularly polarized eld, the projection of the polarization vector onto the plane of the atomic trajectory is elliptically polarized with ellipticity = L=M. This is another method to rediscover Hamiltonian (204) from Hamiltonian (169). To obtain a semiclassical estimation of the energies of the non-dispersive wave packets we proceed precisely in the same way as for LP and CP. Since the radial motion in (Î; ˆ ) is much faster than in the transverse=angular degree of freedom de ned by (M; ), we rst quantize the e ective perturbation 1(Î; M; ; ) driving the angular motion. Fig. 33 shows 1=Î 2 , as a function of M0 = M=Î and , for two di erent values of the driving eld ellipticity . Note that 1 becomes more symmetric as → 0, since this limit de nes the LP case, where the dynamics cannot depend on the rotational sense of the electronic motion around the nucleus. The four extrema of 1 de ne the possible wave-packet eigenstates. Whereas the minima at = =2; 3 =2 correspond to elliptic orbits of intermediate eccentricity 0 ¡e¡1 perpendicular to the driving eld major axis, the ( -independent) maxima at M0 = ±1 de ne circular orbits which co- or contra-rotate with the driving eld. In the limit → 0, the M0= maxima are associated with the same value of 1. Hence, the actual Floquet eigenstates appear as tunneling doublets in the Floquet spectrum, each member of the doublet being a superposition of the co- and contra-rotating wave packets. The quantization of the fast motion in the (Î; ˆ ) plane is similar to the one already performed in the LP and CP cases. As we already observed (see Fig. 20), the size of the resonance island in the (Î; ˆ ) plane is proportional to √ 1. Correspondingly, also the localization properties of the wave packet along the classical trajectory improve with increasing 1. We therefore conclude from Fig. 33 that the eigenstates corresponding to the minima of 1 cannot be expected to exhibit strong longitudinal localization, whereas the eigenstates localized along the circular orbits at the maxima of 1 can.
Page 1 and 2:
Abstract Physics Reports 368 (2002)
Page 3 and 4:
A. Buchleitner et al. / Physics Rep
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28: A. Buchleitner et al. / Physics Rep
Page 43 and 44: → ; → ; A. Buchleitner et al. /
Page 77: A. Buchleitner et al. / Physics Rep
Page 97 and 98: |ψ (z)| 2 |ψ (z)| 2 0.3 0.2 0.1 0
Page 113 and 114: γ /∆ σ/∆ 10 -1 10 -2 10 -3 0.
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139:
show all

Non-dispersive wave packets in periodically driven quantum systems

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?