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) 409–547 529<br />
10 -16<br />
10 -15<br />
10 -14<br />
10 -13<br />
10 -12<br />
10 -11<br />
10 -10<br />
10 -9<br />
10 -8<br />
10 -7<br />
10 -6<br />
ionisation rate<br />
radiative rate<br />
total rate<br />
10 30 50 70 90 110 130<br />
pr<strong>in</strong>cipal <strong>quantum</strong> number n 0<br />
Fig. 58. Comparison of the spontaneous decay rate, the ionization rate, and their sum, for a non-<strong>dispersive</strong> <strong>wave</strong> packet <strong>in</strong> a<br />
l<strong>in</strong>early polarized micro<strong>wave</strong> eld, as a function of the pr<strong>in</strong>cipal <strong>quantum</strong> number n0. Micro<strong>wave</strong> amplitude F0 =0:04442,<br />
decay rates <strong>in</strong> atomic units. The full decay rate exhibits a cross-over from a dom<strong>in</strong>antly coherent (ionization) to a<br />
dom<strong>in</strong>antly <strong>in</strong>coherent (spontaneous emission) regime. The uctuations of the rate present <strong>in</strong> the coherent regime are<br />
suppressed <strong>in</strong> the <strong>in</strong>coherent regime. The data presented here are obta<strong>in</strong>ed by an exact numerical calculation on the<br />
one-dimensional model of the atom [71], see Section 3.3.1.<br />
7.3. <strong>Non</strong>-<strong>dispersive</strong> <strong>wave</strong> packet as a soliton<br />
The non-<strong>dispersive</strong> character of the <strong>wave</strong> <strong>packets</strong> discussed <strong>in</strong> this review br<strong>in</strong>gs to m<strong>in</strong>d solitons,<br />
i.e., solutions of non-l<strong>in</strong>ear <strong>wave</strong> equations that propagate without deformation: the non-l<strong>in</strong>earity is<br />
there essential to overcome the spread<strong>in</strong>g of the solution. The non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> discussed<br />
by us are, on the other hand, solutions of the l<strong>in</strong>ear Schrod<strong>in</strong>ger <strong>wave</strong> equation, and it is not some<br />
non-l<strong>in</strong>earity of the <strong>wave</strong> equation which protects them from spread<strong>in</strong>g, but rather the periodic<br />
driv<strong>in</strong>g. Thus, at rst glance, there seems to be no l<strong>in</strong>k between both phenomena. This is not fully<br />
correct. One may conceive non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> as solitonic solutions of particular nonl<strong>in</strong>ear<br />
equations, propagat<strong>in</strong>g not <strong>in</strong> time, but <strong>in</strong> parameter space [63]. The evolution of energy levels <strong>in</strong><br />
such a space, called “parametric level dynamics”, has been extensively studied (see [15,195] for<br />
reviews), both for time-<strong>in</strong>dependent and for <strong>periodically</strong> time-dependent <strong>systems</strong>. In the latter case,<br />
the energy levels are the quasi-energies of the Floquet Hamiltonian (see Section 3.1.2).<br />
For the sake of simplicity, we consider here the two-dimensional hydrogen atom exposed to a<br />
circularly polarized micro<strong>wave</strong> (Section 3.4.3), where the explicit time dependence can be removed<br />
by transform<strong>in</strong>g to the rotat<strong>in</strong>g frame (see Section 3.4.4), but completely similar results are obta<strong>in</strong>ed<br />
for the Floquet Hamiltonian of any <strong>periodically</strong> time-dependent system. The Hamiltonian, given by<br />
Eq. (184),<br />
H = ˜p2 1<br />
−<br />
2 r + Fx − !Lz ; (274)<br />
may be thought of as an example of a generic system of the form<br />
H( )=H0 + V ; (275)