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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A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 453<br />
of the 1D hydrogen atom under external driv<strong>in</strong>g. The non-<strong>dispersive</strong> <strong>wave</strong>-packet eigenstate of this<br />
model atom <strong>in</strong> the electromagnetic eld is represented by the ground state N =0 of Hsec, localized<br />
(<strong>in</strong> phase space) near the center of the resonance island. A detailed comparison of the semiclassical<br />
energies to the exact <strong>quantum</strong> solution of our problem will be provided <strong>in</strong> the next subsection, where<br />
we treat the three-dimensional atom <strong>in</strong> the eld. There it will turn out that the spectrum of the 1D<br />
model is actually neatly embedded <strong>in</strong> the spectrum of the real 3D atom.<br />
In the immediate vic<strong>in</strong>ity of the resonance island, the secular Hamiltonian can be further simpli ed,<br />
lead<strong>in</strong>g to the pendulum approximation, see Section 3.1.1 and Eq. (66). The second derivative of<br />
the unperturbed Hamiltonian with respect to the action is<br />
H ′′<br />
0 = − 3<br />
n 4 0<br />
; (143)<br />
and the pendulum Hamiltonian reads<br />
Hpend = ˆPt − 3<br />
2n2 − J<br />
0<br />
′ 1(1)n 2 0F cos ˆ − 3<br />
2n4 (Î − n0)<br />
0<br />
2 : (144)<br />
Remember that n0 is the resonant action, not necessarily an <strong>in</strong>teger. As we are <strong>in</strong>terested <strong>in</strong> states<br />
deeply <strong>in</strong>side the resonance island, we can employ the harmonic approximation around the stable<br />
xed po<strong>in</strong>t (Î = n0; ˆ = ), and nally obta<strong>in</strong> the semiclassical energies of the non-<strong>dispersive</strong> <strong>wave</strong><br />
<strong>packets</strong>:<br />
EN;k = k! − 3<br />
2n2 + J<br />
0<br />
′ 1(1)n 2 0F −<br />
where, <strong>in</strong> agreement with Eq. (80)<br />
√<br />
F<br />
<br />
!harm = 3J ′ 1 (1)<br />
n0<br />
<br />
= ! 3J ′ 1 (1)F0<br />
<br />
N + 1<br />
<br />
!harm ; (145)<br />
2<br />
(146)<br />
is the classical librational frequency <strong>in</strong> the resonance island. The <strong>quantum</strong> number k re ects the<br />
global ! periodicity of the Floquet spectrum, as a consequence of Eq. (77).<br />
As already noted <strong>in</strong> Section 3.2.5, the semiclassical quantization breaks the scal<strong>in</strong>g of the classical<br />
dynamics. <strong>Non</strong>etheless, the semiclassical energy levels can be written <strong>in</strong> terms of the scaled<br />
parameters <strong>in</strong>troduced above, by virtue of Eqs. (136) and (142):<br />
EN;k=0 = 1<br />
n2 <br />
−<br />
0<br />
3<br />
2 + J ′ <br />
N +1=2<br />
1(1)F0 − 3J<br />
n0<br />
′ 1 (1)F0<br />
<br />
: (147)<br />
Note that the term (N +1=2)=n0 highlights the role of 1=n0 as an e ective Planck constant.<br />
As discussed <strong>in</strong> Section 3.1.4, the fully “<strong>quantum</strong>” quasi-energies of the resonantly <strong>driven</strong> atom<br />
can be obta<strong>in</strong>ed us<strong>in</strong>g the very same pendulum approximation of the system, together with the<br />
solutions of the Mathieu equation. In our case, the characteristic exponent <strong>in</strong> the Mathieu equation<br />
is given by Eq. (101) and the Mathieu parameter is, accord<strong>in</strong>g to Eqs. (100), (140) and (143):<br />
q = 4<br />
3 Fn6 0J ′ 1(1) = 4<br />
3 F0n 2 0J ′ 1(1) : (148)