Non-dispersive wave packets in periodically driven quantum systems

502 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
by j = n − sk. Because of the overall ! perodicity of the spectrum, changing j by s units (i.e.,
shifting all k-values by 1) is irrelevant, so that it is enough to consider the s independent ladders
0 6 j 6 s−1. Furthermore, in analogy to Section 3.1.4, Eq. (91), we can replace the coupling matrix
elements in Eq. (228) by the resonantly driven Fourier coe cients of the classical motion,
〈 n|V | n+s〉 〈 n−s|V | n〉 Vs(Î s) : (242)
With these approximations, and the shorthand notation r = n − n0, Eq. (85) takes the form of s
independent sets of coupled equations, identi ed by the integer j: 28
dr = Vs[dr+s + dr−s] ; (243)
2
where
E − En0 + n0 − j
˝! −
s
˝2r2 ′′
H 0 (Î s)
2
dr = cn0+r;(n0+r−j)=s ; (244)
as a generalization of the notation in Eq. (93). Again, r is not necessarily an integer, but the various
r values involved in Eq. (243) are equal modulo s. Precisely as in the case of the principal resonance,
Eq. (243) can be mapped on its dual space expression, via Eq. (94):
f( )=Ef( ) (245)
− ˝2 ′′
H 0 (Î s)
2 d2
d 2 + En0 − n0 − j
˝! + Vs cos(s )
s
and identi ed with the Mathieu equation (97) through
s =2v;
a =
8
˝ 2 s 2 H ′′
0 (Î s)
q = 4 Vs(Î s)
s 2 ˝ 2 H ′′
0 (Î s)
E − En0 +(n0 − j)˝ !
s
; j =0;:::;s− 1 ;
: (246)
The quasienergies associated with the s resonance in the pendulum approximation then follow immediately
as
E ;j = En0 − (n0 − j)˝ !
s + ˝2s2 8
H ′′
0 (Î s)a ( ; q) ; (247)
where the index j runs from 0 to s − 1, and labels the eigenvalues of the Mathieu equation
[95]. Again, the boundary condition for the solution of the Mathieu equation is incorporated via the
characteristic exponent, which reads
= −2 n0 − j
(mod 2);
s
j =0;:::s− 1 : (248)
The structure of this quasi-energy spectrum apparently displays the expected ˝!=s periodicity. However,
the characteristic exponent —and consequently the a ( ; q) eigenvalues—depend on j, what
28
For s = 1, this equation reduces of course to Eq. (92). We here use n0 instead of Î s; the two quantities di er only by
the Maslov index, Eq. (241).