Non-dispersive wave packets in periodically driven quantum systems

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