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
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 491<br />
which we exam<strong>in</strong>e <strong>in</strong> the vic<strong>in</strong>ity of the s=1 resonance. As <strong>in</strong> Section 3.3.2, the angular momentum<br />
projection M on the z-axis rema<strong>in</strong>s a constant of motion, and we shall assume M =0 <strong>in</strong> the follow<strong>in</strong>g.<br />
Compared to the situation of a pure micro<strong>wave</strong> eld, there is an additional time scale, directly related<br />
to the static eld. Indeed, <strong>in</strong> the presence of a perturbative static eld alone, it is known that the<br />
Coulomb degeneracy of the hydrogenic energy levels (<strong>in</strong> L) is lifted. The result<strong>in</strong>g eigenstates are<br />
comb<strong>in</strong>ations of the n0 substates of the n0 manifold (for 0 6 L 6 n0 − 1). 22 The associated energy<br />
levels are equally spaced by a quantity proportional to Fs (3n0Fs <strong>in</strong> atomic units). Classically, the<br />
trajectories are no longer closed but rather Kepler ellipses which slowly librate around the static<br />
eld axis, <strong>periodically</strong> chang<strong>in</strong>g their shapes, at a (small) frequency 3n0Fs!Kepler =1=n3 0 . Thus,<br />
the new time scale associated with the static electric eld is of the order of 1=Fs;0 Kepler periods,<br />
where<br />
Fs;0 = Fsn 4 0<br />
(208)<br />
is the scaled static eld. This is to be compared to the time scales 1=F0 and 1= √ F0, which characterize<br />
the secular time evolution <strong>in</strong> the (L; ) and (Î; ˆ ) coord<strong>in</strong>ates, respectively (see discussion <strong>in</strong> Section<br />
3.3.2), <strong>in</strong> the presence of the micro<strong>wave</strong> eld alone. To achieve con nement of the electronic<br />
trajectory <strong>in</strong> the close vic<strong>in</strong>ity of the eld polarization axis, we need 1=F0 1=Fs;0, with both, Fs<br />
and F small enough to be treated at rst-order.<br />
If we now consider the s = 1 resonance, we deduce the secular Hamiltonian by keep<strong>in</strong>g only the<br />
term which does not vanish after averag<strong>in</strong>g over one Kepler period. For the micro<strong>wave</strong> eld, this<br />
term was already identi ed <strong>in</strong> Eq. (159). For the static eld, only the static Fourier component of<br />
the atomic dipole, Eqs. (155) and (156), has a non-vanish<strong>in</strong>g average over one period. Altogether,<br />
this nally leads to<br />
Hsec = ˆPt − 1<br />
2Î 2 − !Î + FsX0(Î;L) cos + F 1(Î; L; )cos ( ˆ + 1) : (209)<br />
S<strong>in</strong>ce the last two terms of this Hamiltonian depend di erently on ˆ , it is no more possible, as it<br />
was <strong>in</strong> the pure LP case (see Section 3.3.2), to perform the quantization of the slow LP motion rst.<br />
Only the secular approximation [18] which consists <strong>in</strong> quantiz<strong>in</strong>g rst the fast variables (Î; ˆ ), and<br />
subsequently the slow variables (L; ), rema<strong>in</strong>s an option for the general treatment. However, s<strong>in</strong>ce<br />
we are essentially <strong>in</strong>terested <strong>in</strong> the <strong>wave</strong>-packet eigenstate with optimal localization properties, we<br />
shall focus on the ground state with<strong>in</strong> a su ciently large resonance island <strong>in</strong>duced by a micro<strong>wave</strong><br />
eld of an appropriate strength. This motivates the harmonic expansion of the secular Hamiltonian<br />
around the stable xed po<strong>in</strong>t at<br />
Î = Î 1 = ! −1=3 ; ˆ = − 1 : (210)<br />
with the characteristic frequency, see Eq. (80):<br />
<br />
3F 1(Î 1;L; )<br />
(Î 1L; )=<br />
: (211)<br />
Î 2<br />
1<br />
22 These states are called “parabolic” states, s<strong>in</strong>ce the eigenfunctions are separable <strong>in</strong> parabolic coord<strong>in</strong>ates [7].