Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 525
becomes dominant, since its intensity scales as ! 5=3 , while the intensities of the other two components
are proportional to ! 2 , i.e., are typically weaker by a factor n0 = ! −1=3 . This implies that the
non-dispersive wave packet decays exclusively (in the harmonic approximation) to its immediate
neighbor states, emitting a photon with frequency in the microwave range, comparable to the driving
frequency. Direct decay to the atomic |n =1;L= M =0〉 ground state or to weakly excited states
of the system is forbidden by the selection rules of the dipole operator. This is easily understood:
the CP non-dispersive ground state wave packet |0; 0; 0〉 is built essentially from states with large
angular momentum (of the order of n0), and as it can lose only one unit of angular momentum per
spontaneous emission event, it can decay only to similar states. When the harmonic approximation
breaks down, additional lines may appear, but, for the same reason, in the microwave range only.
Another important observation is that the inelastic component at ! + !− is by far stronger than
the one at ! − !+. This is entirely due to the cubic power of the transition frequency entering the
expression for the rate (271). Note the sign di erence, due to the sign di erence between ± modes
in the harmonic Hamiltonian, Eq. (197).
In the semiclassical limit ! = n −3
0 → 0, the decay is dominated by the elastic component, and the
total decay rate is [68]
= 2 3 ! 5=3 q −2=3
3
; (272)
when, multiplied by the energy ! of the spontaneous photon, gives the energy loss due to spontaneous
emission:
dE
dt = 2 3 ! 8=3q−2=3 =
3
2 3 ! 4 |xeq| 2
;
3
(273)
where we used Eq. (186). This is nothing but the result obtained from classical electrodynamics [190]
for a point charge moving on a circular orbit of radius |xeq| with frequency !. Since the charge loses
energy, it cannot survive on a circular orbit and would eventually fall onto the nucleus following a
spiral trajectory. This model stimulated Bohr’s original formulation of quantum mechanics. Let us
notice that the non-dispersive wave packet is the rst physical realization of the Bohr model. There
is no net loss of energy since, in our case, the electron is driven by the microwave eld and an
emission at frequency ! occurs in fact as an elastic scattering of a microwave photon. Thus the
non-dispersive wave packet is a cure of the long-lasting Bohr paradox.
Fig. 56 shows the square of the dipole matrix elements connecting the non-dispersive wave packet,
for n0 = 60 (i.e., microwave frequency ! =1=603 ) and scaled microwave eld F0 =0:04446, to other
Floquet states with lower energy. These are the results of an exact numerical diagonalization of the
full Floquet Hamiltonian. They are presented as a stick spectrum because the widths of the important
lines are very narrow on the scale of the gure, which is given as a function of the energy di erence
between the initial and the nal state, that is the frequency of the scattered photon. Thus, this gure
shows the lines that could be observed when recording the photoabsorption of a weak microwave
probe eld. As expected, there is a dominant line at the frequency ! of the microwave, and two other
lines at frequencies !−!− and !+!+ with comparable intensities, while all other lines are at least
10 times weaker. This means that the harmonic approximation works here very well; its predictions,
indicated by the crosses in the gure, are in good quantitative agreement with the exact result (apart
from tiny shifts recognizable in the gure, which correspond to the mismatch between the exact