State, Floquet

The first stoand of Ham s argument [11] is that V(c()) supports continuous bands of Floquet states, with wave functions of the form... 

Here, pj denotes the nonadiabatic transition probability for one passage of the avoided crossing Xt, and / are the dynamical phases due to the nonadiabatic transition atX, is the kth adiabatic Floquet state, Xq = i and X3 = 2- The transition amplimde Eq. (152) can be explicitly expressed as... 

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 states an) are the Floquet states and is the Floquet Hamiltonian. The diagonal matrix elements of Hp in the Floquet basis are... 

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... 

Fig. 13a-d The number operators a N and b, and the ladder operators c and d F[ in the manifold of Floquet states nk) with n) the Fourier states corresponding to the spinning frequency ujr and fc) the Fourier states corresponding to the characteristic RF frequency ujc... 

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

The last expression is the expectation value calculated with the semiclassical model with initial phase 0O. We thus conclude that, if one considers only observables of the molecule, the Floquet evolution with a coherent state in the initial condition is equivalent to the semiclassical model. We remark that a somewhat related construction, linking the evolution from cavity dressed states directly to the semiclassical model (i.e., without the intermediate level of Floquet states as we do here), was established in Ref. 16. 

The models we have discussed so far correspond to continuous (CW) lasers with a fixed sharp frequency and constant intensity. They can be easily adapted to the case of pulsed lasers that have slowly varying envelopes. They can furthermore have a chirped frequency—that is, a frequency that changes slowly with time. For periodic (or quasi-periodic) semiclassical Hamiltonians, the Floquet states are the stationary states of the problem. Processes controlled by chirped laser pulses include additional time-dependent parameters (the pulse... 

Since the phase of the instantaneous Floquet states is not uniquely specified at each r, one can always choose the geometrical phase as zero by requiring that at each time r ( / Vr / )jr =0. However, if one follows a closed loop in parameter space the eigenvector at the end of the loop will differ from the initial one by a phase. This phase, which depends only on the geometry of the loop but not on the speed, is the geometrical Berry phase [44]. If only one parameter is varied, the closed-loop geometrical phase is 0. If two parameters are varied, it can be 0 or n. If more than two parameters are varied, it can take any value. 

More than one Floquet state can be involved in the dynamics—for example, if the initial condition is a linear combination of the instantaneous eigenvectors. These Floquet states span a subspace ff, and the adiabatic transport can be formulated in terms of eigenvectors ... 

We label these two continuous branches by the instantaneous Floquet states v and Y ] The two eigenvalues 7.1 can be deduced from an effective local dressed Hamiltonian... 

This half-scrap could be generalized for a /i-multi photon process (n > 2) in a multilevel system, with the use of the full quasi-energies and Floquet states (calculated numerically). 

The last term on the right-hand side of Eq. (C6) characterizes the nonadiabatic couplings between the instantaneous Floquet states (off-diagonal terms, of the form m) and the Berry phase (real... 

A number of basic properties of the Floquet states (f)) can be infered easily from Eqs. (14 and 19). First, if E, is a quasi-energy, i.e., an eigenvalue of Hp associated with an eigenstate 4> ,(f)>, then... 

Second, the properties of Floquet eigenstates are such as to produce a very simple stroboscopic way of following the motion of a general wavepacket of the time-dependent laser-driven system Consider the evolution operator W(f -I- T,t) between times f and t + T. Starting at time t from a Floquet state... 

At best, the local electronic Floquet eigenvectors can be taken as a new basis, an adiabatic one, on which the total molecular Floquet states can be expanded. The channel amplitudes in this basis are coupled by kinetic non-adiabatic couplings, arising from the commutator between and... 

Care must be exercised to distinguish the concept of adiabatic Floquet dynamics introduced here, which refers to an adiabatic time-evolution, or to the slow variations of the Floquet basis with time, from the concept of adiabatic representation defined in the previous section, which refers to the slow variations of the electronic Hamiltonian (Floquet or not) with respect to nuclear motions (i.e., the noncommutativity of the electronic Hamiltonian Hei and the nuclear KE operator Tjv). Where confusion is possible and to be avoided, we shall refer to this concept of adiabaticity related to the BO approximation as the R-adiabaticity, while adiabaticity in actual time evolution will be termed t-adiabaticity. Non-adiabatic effects in time evolution are due to a fast variation of the (Floquet) Hamiltonian with time, causing Floquet states to change rapidly in time, to the extent that in going from one time slice to another, a resonance may be projected onto many new resonances as well as diffusion (continuum) states [40], and the Floquet analysis breaks down completely. We will see in Section 5 how one can take advantage of such effects to image nuclear motions by an ultrafast pump-probe process. 

The sum is taken over all the discrete vibrational levels if of state g>. Vr (f) is the component of the wavepacket on the g channel evolved up to time t from the field-free vibrational state v > prepared at time f = 0. Note that Pbound(y if) actually represents the total bound state population at any time after tj, since no further decay is then possible, the laser being turned off at such a time. It is clear that Eq. (71) gives a useful approximation for the result of a full time-dependent wavepacket evolution, [Eq. (73)], only if the assumption of an adiabatic transport of Floquet states is valid. 

S. Guerin, F. Monti, J.M. Dupont, H. Jauslin, On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models, J. Phys. A Math. Gen. 30 (1997) 7193. 

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