Non-dispersive wave packets in periodically driven quantum systems

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452 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 For a real 3D atom, this corresponds to driving the electron initially prepared in a one-dimensional eccentricity one orbit along the polarization axis of the eld. In fact, it turns out that the one dimensionality of the dynamics is not stable under the external driving: the Kepler ellipse (with orientation xed in con guration space by the Runge–Lenz vector) slowly precesses o the eld polarization axis (see Sections 3.3.2 and 4.1). Thus, the 1D presentation which follows has mostly pedagogical value—being closest to the general case discussed later. However, a one-dimensional model allows to grasp essential features of the driven atomic dynamics and provides the simplest example for the creation of non-dispersing wave packets by a near-resonant microwave eld. A subsequent section will describe the dynamics of the real 3D atom under linearly polarized driving, and amend on the aws and drawbacks of the one-dimensional model. 3.3.1. One-dimensional model From Eqs. (113) and (114), the Hamiltonian of the driven 1D atom reads H = p2 2 1 − + Fz cos(!t); z¿0 : (137) z This has precisely the general form, Eq. (54), and we can therefore easily derive explicit expressions for the secular Hamiltonian subject to the semiclassical quantization conditions, Eqs. (76), (78) and (79), as well as for the quantum mechanical eigenenergies, Eq. (102), in the pendulum approximation. With the Fourier expansion, Eq. (54), and identifying and V (p; z) in Eq. (120) with F and z in Eq. (137), respectively, the Fourier coe cients in Eq. (55) take the explicit form V0 = 3 2 I 2 ; Vm = −I 2 J ′ m(m) ; m=0 : (138) m The resonant action—which de nes the position of the resonance island in Fig. 8—is given by Eq. (135). In a quantum description, the resonant action coincides with the resonant principal quantum number: n0 = Î 1 = ! −1=3 ; (139) since the Maslov index vanishes in 1D, see Section 3.2, and ˝ ≡ 1 in atomic units. The resonant coupling is then given by V1 = −Î 2 J ′ 1(1) (140) and the secular Hamiltonian, Eq. (64), reads Hsec = ˆPt − 1 2Î 2 − !Î − J ′ 1(1)Î 2 F cos ˆ : (141) This Hamiltonian has the standard form of a secular Hamiltonian with a resonance island centered around Î = Î 1 = ! −1=3 = n0; ˆ = ; (142) sustaining librational motion within its boundary. Those energy values of Hsec which de ne contour lines (see Fig. 8) such that the contour integrals, Eqs. (78) and (79), lead to non-negative integer values of N , are the semiclassical quasienergies
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 453 of the 1D hydrogen atom under external driving. The non-dispersive wave-packet eigenstate of this model atom in the electromagnetic eld is represented by the ground state N =0 of Hsec, localized (in phase space) near the center of the resonance island. A detailed comparison of the semiclassical energies to the exact quantum solution of our problem will be provided in the next subsection, where we treat the three-dimensional atom in the eld. There it will turn out that the spectrum of the 1D model is actually neatly embedded in the spectrum of the real 3D atom. In the immediate vicinity of the resonance island, the secular Hamiltonian can be further simpli ed, leading to the pendulum approximation, see Section 3.1.1 and Eq. (66). The second derivative of the unperturbed Hamiltonian with respect to the action is H ′′ 0 = − 3 n 4 0 ; (143) and the pendulum Hamiltonian reads Hpend = ˆPt − 3 2n2 − J 0 ′ 1(1)n 2 0F cos ˆ − 3 2n4 (Î − n0) 0 2 : (144) Remember that n0 is the resonant action, not necessarily an integer. As we are interested in states deeply inside the resonance island, we can employ the harmonic approximation around the stable xed point (Î = n0; ˆ = ), and nally obtain the semiclassical energies of the non-dispersive wave packets: EN;k = k! − 3 2n2 + J 0 ′ 1(1)n 2 0F − where, in agreement with Eq. (80) √ F !harm = 3J ′ 1 (1) n0 = ! 3J ′ 1 (1)F0 N + 1 !harm ; (145) 2 (146) is the classical librational frequency in the resonance island. The quantum number k re ects the global ! periodicity of the Floquet spectrum, as a consequence of Eq. (77). As already noted in Section 3.2.5, the semiclassical quantization breaks the scaling of the classical dynamics. Nonetheless, the semiclassical energy levels can be written in terms of the scaled parameters introduced above, by virtue of Eqs. (136) and (142): EN;k=0 = 1 n2 − 0 3 2 + J ′ N +1=2 1(1)F0 − 3J n0 ′ 1 (1)F0 : (147) Note that the term (N +1=2)=n0 highlights the role of 1=n0 as an e ective Planck constant. As discussed in Section 3.1.4, the fully “quantum” quasi-energies of the resonantly driven atom can be obtained using the very same pendulum approximation of the system, together with the solutions of the Mathieu equation. In our case, the characteristic exponent in the Mathieu equation is given by Eq. (101) and the Mathieu parameter is, according to Eqs. (100), (140) and (143): q = 4 3 Fn6 0J ′ 1(1) = 4 3 F0n 2 0J ′ 1(1) : (148)
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Non-dispersive wave packets in periodically driven quantum systems

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