Non-dispersive wave packets in periodically driven quantum systems

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522 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 by the external drive as a strongly coupled system, or use the Floquet picture described above, and treat the additional mode(s) of the probe (environment) as a perturbation. We rst start with the simplest situation, where a single mode of the environment is taken into account. 7.2.1. Interaction of a non-dispersive wave packet with a monochromatic probe eld Let us rst consider the addition of a monochromatic probe eld of frequency !p. The situation is very similar to the probing of a time-independent system by a weak monochromatic eld, with the only di erence that the Floquet Hamiltonian replaces the usual time-independent Hamiltonian. Thus, the weak probe eld may induce a transition between two Floquet states if it is resonant with this transition, i.e., if ˝!p is equal to the quasi-energy di erence between the two Floquet states. According to Fermi’s Golden Rule, the transition probability is proportional to the square of the matrix element coupling the initial Floquet state |Ei〉 to the nal one |Ef〉. Using the Fourier representation of Floquet states, |E(t)〉 = exp(−ik!t)|E k 〉 ; (268) k and averaging over one driving eld cycle 2 =! we get 〈Ef|T|Ei〉 = ∞ 〈E k f|T|E k i 〉 : (269) k=−∞ T denotes the transition operator, usually some component of the dipole operator depending on the polarization of the probe beam. If the quasi-energy levels are not bound states but resonances with nite lifetime (for example, because of multiphoton transition amplitudes to the continuum) this approach is easily extended [18], yielding the following expression for the photoabsorption cross-section of the probe eld at frequency !p: 4 !p (!p)= Im c |〈Ef|T|Ei〉| 2 f 1 Ef − Ei − !p + 1 Ef − Ei + !p ; (270) where is the ne structure constant, and where Ef and Ei are the complex energies of the initial and nal Floquet states. The sum extends over all the Floquet states of the system. Although the Floquet energy spectrum is itself !-periodic (see Section 3.1.2), this is not the case for the photoabsorption cross-section. Indeed, the Floquet states at energies Ef and Ef+! have the same Fourier components, but shifted by one unit in k, resulting in di erent matrix elements. The photoabsorption spectrum is thus composed of series of lines separated by ! with unequal intensities. When the driving is weak, each Floquet state has a dominant Fourier component. The series then appears as a dominant peak accompanied by side bands shifted in energy by an integer multiple of the driving frequency !. In the language of the scattering theory [18,23,24], these side bands can be seen as the scattering of the probe photon assisted by one or several photons of the drive. In any case, the Floquet formalism is well suited, since it contains this weak driving regime as a limiting case, as well as the strong driving regime needed to generate a non-dispersive wave packet. 7.2.2. Spontaneous emission from a non-dispersive wave-packet We now address the situation where no probe eld is added to the microwave eld. Still, photons of the driving eld can be scattered in the (initially empty) remaining modes of the electromagnetic
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 523 eld. This is thus some kind of spontaneous emission or rather resonance uorescence of the atom under coherent driving. It can be seen as spontaneous emission of the dressed atom, where an initial Floquet state decays spontaneously to another Floquet state with a lower quasi-energy, the energy di erence being carried by the spontaneous photon. As an immediate consequence, the spectrum of the emitted photons is composed of the resonance frequencies of the Floquet system, the same that are involved in Eq. (270). The decay rate along a transition depends on the dipole matrix element connecting the initial and the nal states, but also on the density of modes for the emitted photons. If we consider, for simplicity, the case of free atoms, one obtains if = 4 3 (Ei − Ef) 3 |〈Ef|T|Ei〉| 3 2 ; (271) where Ei − Ef is the positive energy di erence between the initial and nal Floquet states. As the matrix element of the dipole operator T is involved, clearly the localization properties of the Floquet states will be of primordial importance for the spontaneous emission process. The total decay rate (inverse of the lifetime) of a state |Ei〉 is obtained by summing the partial rates if connecting the initial state to all states with lower energy. It is not straightforward to determine which Floquet states contribute most to the decay rate—the two factors in Eq. (271) compete: while |〈Ef|T|Ei〉| 2 tends to favor states localized close to the initial state (maximum overlap), the factor (Ei − Ef) 3 (due to the density of modes in free space) favors transitions to much less excited states. Which factor wins depends on the polarization of the driving eld. 7.2.3. Circular polarization Consider rst a circularly polarized microwave eld. A rst analysis of spontaneous emission has been given in [59], where the rotating frame (see Section 3.4) approach was used. The driven problem becomes then time-independent, and the analysis of spontaneous emission appears to be simple. This is, however, misleading, and it is quite easy to omit some transitions with considerable rate. The full and correct analysis, both in the rotating and in the standard frame [68], discusses this problem extensively. The reader should consult the original papers for details. A crucial point is to realize that the Floquet spectrum of the Hamiltonian in CP splits into separate blocks, all of them being identical, except for a shift by an integer multiple of the driving frequency !. Each block corresponds to a xed quantum number = k + M where k labels the photon block (Fourier component) in the Floquet approach, while M is the azimutal quantum number. This merely signi es that the absorption of a driving photon of circular polarization + increases M by one unit. In other words, is nothing but the total angular momentum (along the direction of propagation of the microwave eld) of the entire system comprising the atom and the driving eld. The separate blocks are coupled by spontaneous emission. Since, again, the spontaneously emitted photon carries one quantum of angular momentum, spontaneous emission couples states within the same -block (for polarization of the emitted photon w.r.t. the z-axis, which leaves M invariant) or in neighboring -blocks ′ = ± 1 (see Fig. 55). + polarization of the emitted photon gives rise to higher frequency photons since—for the same initial and nal Floquet states—the energy di erence in the + channel is larger by ˝! than in the channel (and by 2˝! than in the − channel), as immediately observed in Fig. 55. As the emission rate, Eq. (271), changes with the cubic power of the energy di erence, spontaneous photons with + polarization are expected to be dominant.
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Non-dispersive wave packets in periodically driven quantum systems

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