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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440 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
with Eqs. (40) and (65),<br />
<br />
dEn <br />
<br />
dn = ˝! : (87)<br />
n=n0<br />
Note that, by this de nition, n0 is not necessarily an <strong>in</strong>teger. In the semiclassical limit, where n0<br />
is large, the WKB approximation connects n0 to the center of the classical resonance island, see<br />
Eq. (39),<br />
n0 + 4 = Î 1<br />
˝<br />
; (88)<br />
En0 = H0(Î 1) ; (89)<br />
with be<strong>in</strong>g the Maslov <strong>in</strong>dex along the resonant trajectory. Furthermore, for n close to n0, we can<br />
expand the unperturbed energy at second order <strong>in</strong> (n − n0),<br />
En En0 +(n− n0)˝! + 1 d<br />
2<br />
2En dn2 <br />
<br />
(n − n0) 2 ; (90)<br />
n0<br />
where the second derivative d2En=dn 2 is directly related to the classical quantity H ′′<br />
0 , see Eq. (67),<br />
with<strong>in</strong> the semiclassical WKB approximation, Eq. (39). Similarly, the matrix elements of V are<br />
related to the classical Fourier components of the potential [7], Eq. (55),<br />
〈 n|V | n+1〉 〈 n+1|V | n+2〉 V1(Î 1) ; (91)<br />
evaluated at the center Î = Î 1—see Eq. (65)—of the resonance zone.<br />
With these <strong>in</strong>gredients and r = n − n0, Eq. (85) is transformed <strong>in</strong> the follow<strong>in</strong>g set of approximate<br />
equations:<br />
<br />
<br />
E − H0(Î 1)+! Î 1 − ˝<br />
<br />
−<br />
4<br />
˝2 ′′<br />
H 0 (Î 1)r<br />
2 2<br />
<br />
dr = V1(dr+1 + dr−1) ; (92)<br />
where<br />
dr ≡ cn0+r;n0+r : (93)<br />
Note that, because of Eq. (87), the r values are not necessarily <strong>in</strong>tegers, but all have the same<br />
fractional part.<br />
The tridiagonal set of coupled equations (92) can be rewritten as a di erential equation. Indeed,<br />
if one <strong>in</strong>troduces the follow<strong>in</strong>g function associated with the Fourier components dr:<br />
f( )= <br />
exp(ir )dr ; (94)<br />
Eq. (92) can be written as<br />
<br />
− ˝2<br />
2<br />
r<br />
H ′′<br />
0 (Î 1) d2<br />
d 2 + H0(Î 1) − !<br />
<br />
Î 1 − ˝<br />
<br />
+ V1 cos<br />
4<br />
<br />
f( )=Ef( ) ; (95)<br />
which is noth<strong>in</strong>g but the <strong>quantum</strong> version of the pendulum Hamiltonian, Eq. (66). Thus, the present<br />
calculation is just the purely <strong>quantum</strong> description of the non-l<strong>in</strong>ear resonance phenomenon. The