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