Non-dispersive wave packets in periodically driven quantum systems

474 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
and the quasi-energy levels are
EN;k = k! − 3
2n2 + n
0
2 0F −
N + 1
!harm : (174)
2
Note that, by construction, −3=2n2 0 + Fn20 is nothing but the energy at the center of the resonance
island, i.e., the energy of the resonant circular orbit. For very small F, the resonance island shrinks
and may support only a small number of states, or even no state at all. In this regime, the harmonic
approximation, Eq. (81), breaks down. Alternatively, one can apply a quantum treatment of the
pendulum motion in the (Î; ˆ ) plane, as explained in Section 3.1.4 and discussed in Section 3.3.1
for the 1D model of the atom exposed to a linearly polarized microwave. The analysis—essentially
identical to the one in Section 3.3.1—yields the following expression for the energy levels:
Ek;N = k! − 3
2n2 −
0
3aN ( ; q)
8n4 0
; (175)
where aN ( ; q) are the Mathieu eigenvalues (compare with Eq. (99) for the general case), with
and
q = 4
3 Fn6 0
(176)
= −2n0 (mod 2) (177)
the characteristic exponent.
These expressions are valid for the states localized close to the resonant circular orbit. For the
other states, the calculation is essentially identical, the only amendment being the use of the values
of 1 following from the quantization of the secular motion, instead of the maximum value n 2 0 .
Finally, as the center of the resonance island corresponds to a circular trajectory in the (x; y) plane,
the Floquet states associated with the non-dispersive wave packets will be essentially composed of
combinations of circular states |n; L = M = n − 1〉, with coe cients described by the solutions of
the Mathieu equation, as explained in Section 3.1.4. This Mathieu formalism has been rediscovered
in this particular CP situation via complicated approximations on the exact Schrodinger equation in
[55]. We believe that the standard resonance analysis using the pendulum approximation leads, at
the same time, to simpler calculations, and to a much more transparent physical picture.
3.4.3. The two-dimensional model
We shall now discuss the simpli ed 2D model of the CP problem, which amounts to restricting
the motion to the (x; y) plane, but retains almost all the features of the full 3D problem. Instead
of the six-dimensional phase space spanned by the action-angle variables (I; ); (L; ); (M; ), one
is left with a four-dimensional submanifold with coordinates (I; ); (M;
secular Hamiltonian then reads (cf. Eqs. (169), (170)):
); see Section 3.2.4. The
Hsec = ˆPt − 1
2Î 2 − !Î + FV1(Î;M) cos( ˆ + ) (178)