Floquet Level Crossings

The propagator given in Eq. 16 becomes periodic in time only when the two characteristic frequencies, uir and Wc are commensurate. Then Floquet energy level crossings occur (see Figs. 16 and 17) under the condition [52]  

In that case a new characteristic frequency can be defined that is equal to 

This new characteristic frequency enables the BMFT representation to be reduced to a single mode Floquet (SMFT) representation. An effective Hamiltonian can be 

At any Floquet energy level crossing condition all states n, k with the same Floquet state energy + k ujc = lujt obey 

Replacing nn+kv by I and combining all the elements with equal I for all possible integers 

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The schematic of Floquet energy levels is shown in Fig. 16. For certain characteristic frequencies these energy differences can become zero and a Floquet energy level crossing occurs ... 

Fig. 17 The Floquet energy levels Enk = nour+kuc as a function of the spinning frequency ujr for a constant ouc value for cOr < ojc- Level crossings are indicated by symbols open circle for 5,1 fllled circle for 9,2 open square for 4,1 fllled square for 7,2 open triangle for 3,1, filled triangle for 5,2 open diamond for 2,1 filled diamond for 3,2 and inverted triangle for 1,1 ...

With the constraints of Eq. 47 we can, in general, use perturbation theory to find the approximate eigenvalues oiT-Lp. In the coming sections we will do so by using van Vleck perturbation theory, but only after discussing Floquet energy level crossings. 

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... 

The crossing conditions of the Floquet energy levels were defined before as... 

Fig. 8.12 Amplitudes and frequencies of the field which lead to CDT. The/w// lines represent the parameters obtained from the two-level model of Eq. (8.53) and the crosses represent the parameters obtained with the effective Floquet Hamiltonian of Eq. (8.36). The parameters given by the first, second, third and fourth crossing for each frequency are plotted in blue, red, green and magenta, respectively. Figure adapted from Ref. [2]...

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