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) 409–547 523<br />
eld. This is thus some k<strong>in</strong>d of spontaneous emission or rather resonance uorescence of the atom<br />
under coherent driv<strong>in</strong>g. It can be seen as spontaneous emission of the dressed atom, where an <strong>in</strong>itial<br />
Floquet state decays spontaneously to another Floquet state with a lower quasi-energy, the energy<br />
di erence be<strong>in</strong>g carried by the spontaneous photon. As an immediate consequence, the spectrum of<br />
the emitted photons is composed of the resonance frequencies of the Floquet system, the same that<br />
are <strong>in</strong>volved <strong>in</strong> Eq. (270). The decay rate along a transition depends on the dipole matrix element<br />
connect<strong>in</strong>g the <strong>in</strong>itial and the nal states, but also on the density of modes for the emitted photons.<br />
If we consider, for simplicity, the case of free atoms, one obta<strong>in</strong>s<br />
if = 4 3 (Ei − Ef) 3<br />
|〈Ef|T|Ei〉|<br />
3<br />
2 ; (271)<br />
where Ei − Ef is the positive energy di erence between the <strong>in</strong>itial and nal Floquet states. As the<br />
matrix element of the dipole operator T is <strong>in</strong>volved, clearly the localization properties of the Floquet<br />
states will be of primordial importance for the spontaneous emission process.<br />
The total decay rate (<strong>in</strong>verse of the lifetime) of a state |Ei〉 is obta<strong>in</strong>ed by summ<strong>in</strong>g the partial rates<br />
if connect<strong>in</strong>g the <strong>in</strong>itial state to all states with lower energy. It is not straightforward to determ<strong>in</strong>e<br />
which Floquet states contribute most to the decay rate—the two factors <strong>in</strong> Eq. (271) compete: while<br />
|〈Ef|T|Ei〉| 2 tends to favor states localized close to the <strong>in</strong>itial state (maximum overlap), the factor<br />
(Ei − Ef) 3 (due to the density of modes <strong>in</strong> free space) favors transitions to much less excited states.<br />
Which factor w<strong>in</strong>s depends on the polarization of the driv<strong>in</strong>g eld.<br />
7.2.3. Circular polarization<br />
Consider rst a circularly polarized micro<strong>wave</strong> eld. A rst analysis of spontaneous emission<br />
has been given <strong>in</strong> [59], where the rotat<strong>in</strong>g frame (see Section 3.4) approach was used. The <strong>driven</strong><br />
problem becomes then time-<strong>in</strong>dependent, and the analysis of spontaneous emission appears to be<br />
simple. This is, however, mislead<strong>in</strong>g, and it is quite easy to omit some transitions with considerable<br />
rate. The full and correct analysis, both <strong>in</strong> the rotat<strong>in</strong>g and <strong>in</strong> the standard frame [68], discusses this<br />
problem extensively. The reader should consult the orig<strong>in</strong>al papers for details.<br />
A crucial po<strong>in</strong>t is to realize that the Floquet spectrum of the Hamiltonian <strong>in</strong> CP splits <strong>in</strong>to separate<br />
blocks, all of them be<strong>in</strong>g identical, except for a shift by an <strong>in</strong>teger multiple of the driv<strong>in</strong>g frequency<br />
!. Each block corresponds to a xed <strong>quantum</strong> number = k + M where k labels the photon<br />
block (Fourier component) <strong>in</strong> the Floquet approach, while M is the azimutal <strong>quantum</strong> number. This<br />
merely signi es that the absorption of a driv<strong>in</strong>g photon of circular polarization + <strong>in</strong>creases M<br />
by one unit. In other words, is noth<strong>in</strong>g but the total angular momentum (along the direction of<br />
propagation of the micro<strong>wave</strong> eld) of the entire system compris<strong>in</strong>g the atom and the driv<strong>in</strong>g eld.<br />
The separate blocks are coupled by spontaneous emission. S<strong>in</strong>ce, aga<strong>in</strong>, the spontaneously emitted<br />
photon carries one <strong>quantum</strong> of angular momentum, spontaneous emission couples states with<strong>in</strong> the<br />
same -block (for polarization of the emitted photon w.r.t. the z-axis, which leaves M <strong>in</strong>variant)<br />
or <strong>in</strong> neighbor<strong>in</strong>g -blocks ′ = ± 1 (see Fig. 55). + polarization of the emitted photon gives<br />
rise to higher frequency photons s<strong>in</strong>ce—for the same <strong>in</strong>itial and nal Floquet states—the energy<br />
di erence <strong>in</strong> the + channel is larger by ˝! than <strong>in</strong> the channel (and by 2˝! than <strong>in</strong> the −<br />
channel), as immediately observed <strong>in</strong> Fig. 55. As the emission rate, Eq. (271), changes with the<br />
cubic power of the energy di erence, spontaneous photons with + polarization are expected to be<br />
dom<strong>in</strong>ant.