Floquet resonance

Feshbach-type resonances [51], also known as Fano resonances [52] and Floquet resonances [22] depending on the system studied, are formed in a different manner. We encounter this type of metastable states whenever a bound system is coupled to an external continuum. In the same spirit as before, one can define a reference Hamiltonian in which the closed channel containing the bound states is uncoupled from the open channel through which the asymptote can be reached. When the coupling is introduced, the previously bound state decays into the continuum of the open channel. The distinction from shape-type resonances, described above, is that the resonance state decays into a different channel of the reference Hamiltonian. 

This dependence of the H+ KE on the XUV-IR delay in this case of the longer, 35 fs FWHM, IR pulse can be understood in terms of the adiabatic-ity of the Floquet dynamics underlying the dissociation processes, and the way that the IR intensity affects both the preparation and the propagation of the Floquet components of the wavepackets. More precisely, the IR probe pulse projects the various vibrational components of the wavepacket onto Floquet resonances, whose widths vary with the intensity of the IR pulse. We recall that these resonances are of two types Shape resonances supported by the lower adiabatic potential defined at the one-photon crossing between the dressed (g, n), (u, n ) channels and leading to efficient dissociation through the BS mechanism, or Feshbach resonances, vibrationally trapped in the upper adiabatic potential well. 

The primary Floquet resonance condition is, approximately, 0 e) so that... 

For the driven atom, we developed an accurate approach without any adjustable parameter, and with no other approximation than the confinement of the accessible configuration space to two dimensions. This method was successfully applied for the study of the near resonantly driven frozen planet configuration. Floquet states were found that are well localized in the associated phase space and propagate along near-... 

The energy levels of a molecule placed in an off-resonant microwave field can be calculafed by diagonalizing fhe mafrix of fhe Floquef Hamiltonian in the basis of direct products y) ), where y) represents in the eigenstates of the molecule in the absence of the field and ) - fhe Fourier componenfs in Eq. (8.21). The states k) are equivalent to photon number states in the alternative formalism using the quantum representation of the field [11, 15, 26], The eigensfales of the Floquet Hamiltonian are the coherent superpositions... 

Let us examine more closely what occurs on the right-hand boundary of the 1/1 resonance horn [Fig. 9(a)]. In a sequence of one-parameter bifurcation diagrams with respect to oj/co0, each taken at a successively higher forcing amplitude FA, we observe that as FA increases, the bifurcation point to a torus changes. The point of exit of the Floquet multipliers of the periodic... 

E. This double -1 point is yet another codimension-two bifurcation, which will be discussed in detail later. Another period 1 Hopf curve extends from point F through points G and H. F is another double -1 point and, as one moves away from F along the Hopf curve, the angle at which the complex multipliers leave the unit circle decreases from it. The points G and H correspond to angles jt and ixr respectively and are hard resonances of the Hopf bifurcation because the Floquet multipliers leave the unit circle at third and fourth roots of unity, respectively. Points G and H are both important codimension-two bifurcation points and will be discussed in detail in the next section. The Hopf curves described above are for period 1 fixed points. Subharmonic solutions (fixed points of period greater than one) can also bifurcate to tori via Hopf bifurcations. Such a curve exists for period 2 and extends from point E to K, where it terminates on a period 2 saddle-node curve. The angle at which the complex Floquet multipliers leave the unit circle approaches zero at either point of the curve. 

FIGURE 8 (a) Detail of the tip of the 3/1 resonance horn illustrating typical way in which period 3 resonance horns close around a point with Floquet multipliers at the third root of unity (point F). (fa) and (c) The saddle-node pairings change from section AA to section BB and the unstable manifolds of the period 3 saddles no longer make up a phase locked torus. (d) A one-parameter vertical cut through the third root of unity point. The three saddles coalesce with the period one focus that is undergoing the Hopf bifurcation. 

Fig. 15.10 Cross sections as a function of rf field strength for the first four orders of sideband resonances of the K 29s + K 27d radiative collisions in a 4 MHz rf field, (a) The zero-photon resonant collision cross section, (b) the +1 sideband resonance, (c) the —2 sideband resonance, and (d) the +3 sideband resonance. The solid line shows the experimental data, the bold line indicates the prediction the Floquet theory, and the dashed fine is the result of numerical integration of the transition probability (from ref. 17).

At the resonance w(t) = A(x), the adiabatic potentials i.e. the eigenvalues of (5.9) show avoided crossing and the population splits into the two adiabatic Floquet states. In the case of quadratically chirped pulses, the instantaneous frequency meets the resonance condition twice and near-complete excitation can be achieved due to the constructive interference. The nonadi-abatic transition matrix Ujj for the two-level problem of (5.9) is given by the ZN theory [33] as... 

There are also important advances concerning the theoretical description of Rydberg atoms in strong radiation fields. Buchleitner et d. (1995) report on fully fiedged three-dimensional computations of the microwave ionization problem. They use the method of complex rotation discussed in Section 10.4.1, adapted to the computation of the resonances of the Floquet operator. The computed ionization probabilities are in good overall agreement with existing experimental data. 

Using Rydberg atoms and microwave fields it has been possible to observe virtually all one electron strong field phenomena. The attraction of these experiments is that they can be more controlled than most laser experiments, with the result that more quantitative information can be extracted. The insights gained from these experiments can be profitably transferred to optical experiments. To demonstrate the latter point we demonstrate that apparently non-resonant microwave ionization, in fact, occurs by resonant transitions through intermediate states. These experiments demonstrated clearly the power of Floquet analysis of such processes, and the ideas were subsequently applied to the analogous problem of laser multiphoton ionization. 

Rydberg atoms and microwave fields constitute an ideal system for the study of atom-strong field effects, and they have been used to explore the entire range of one electron phenomena [5]. Here we focus on an illustrative example, which has a clear parallel in laser experiments, a series of experiments which show that apparently non-resonant microwave ionization of nonhydronic atoms proceeds via a sequence of resonant microwave multiphoton transitions and that this process can be understood quantitatively using a Floquet, or dressed state approach. 

In the following section the experimental approach is briefly described. The initial observations of microwave ionization and the completely non-resonant picture initially used to describe it are then presented. Then microwave multiphoton transitions in a two level system analogous to the rate limiting step of microwave ionization are described both experimentally and theoretically. Experiments on this two level system with well controlled pulses of microwaves to show the applicability of an adiabatic Floquet theory to pulses are then described. We finally return to microwave ionization to see evidence for the resonant nature of the process. 

Although extending the Landau-Zener description to many cycles accurately describes the evolution from the non-resonant interaction with a single cycle to the resonant interaction with many cycles, this approach is not very convenient to use nor does it easily lead to any analytic predictions. Instead we use a Floquet approach [20-22]. We assume that we have two states, the (n -b 2)s state and the (n, 3) Stark state. The effective Hamiltonian,... 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

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