Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 475
with (see Eq. (166))
V1(Î;M)=Î 2
J ′ √
1 − e2 1(e) + sign(M) J1(e) ;
e
where (as in Eq. (124))
(179)
e =
M 2
1 −
Î 2 : (180)
However, the Maslov index for the (Î; ˆ ) motion is di erent. Indeed, the energy spectrum of the 2D
atom is given by Eq. (116). Thus, quantized values of the action are half-integer multiples of ˝. The
relation between the resonant action Î 1 = ! −1=3 and the corresponding principal quantum number
now reads (with ˝ = 1):
n0 = Î 1 − 1:
(181)
2
As explained in Section 3.1.4, the optimal case for the preparation of non-dispersive wave packets—
where the states are the most deeply bound inside the resonance island—is for integer values of n0,
i.e., frequencies (cf. with Eqs. (65), (119))
1
! =
: (182)
(n0 +1=2) 3
For the energy levels of the non-dispersive wave packets, this also implies that the characteristic
exponents in the Mathieu equation—see Section 3.1.4—are shifted by one unit:
= −2n0 (mod 2) = −2Î 1 + 1 (mod 2): (183)
3.4.4. Transformation to the rotating frame
The resonance analysis developed above is restricted to rst order in the amplitude F of the
external drive. Extensions to higher orders are possible, but tedious. For CP, an alternative approach
is possible, which allows higher orders to be included quite easily. It is applicable to CP only and
thus lacks the generality of the resonance approach we used so far. Still, it is rather simple and
deserves an analysis.
In CP, one may remove the time dependence of Hamiltonian (164) by a transformation to the
non-inertial frame rotating with the external frequency !. The unitary fransformation U =exp(i!Lzt)
leads to [141,142]
Hrot = UHCP U † 9U † ˜p2 1
+iU = −
9t 2 r + Fx − !Lz : (184)
Classically, such an operation corresponds to a time-dependent rotation of the coordinate frame
spanned by x = x cos !t + y sin !t, y = y cos !t − x sin !t (and dropping the bar hereafter). 17
17 Passing to the rotating frame implies a change of to = − !t in Eq. (165). That is de nitely di erent from the
change → ˆ = − !t, Eq. (60), used in the resonance analysis. Both transformations are unfortunately known under the
same name of “passing to the rotating frame”. This is quite confusing, but one has to live with it. Along the resonantly
driven circular orbit we are considering here, it happens that the azimuthal angle and the polar angle actually coincide.
It follows that the two approaches are equivalent in the vicinity of this orbit.