Floquet eigenstates

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... 

On the other hand, there are many instances when the rotating wave approximation cannot be used. For example, in order to find the energy levels of a molecule placed in a strong microwave field, it is necessary to diagonalize a large piece of the full Floquet matrix involving multiple n-states and multiple eigenstates of Hq, as discussed in Section 8.3.4. 

In the presence of the field, the molecular states are coherent superpositions of the states AM ) A ). In principle, the basis set must include an infinite number of states A ). However, the Floquet Hamiltonian matrix is block-diagonal and the diagonal matrix elements of the Floquet matrix separate in values SiS k-k increases. This suggests that it may be possible to include in the basis set a finite number of states from - max to max seek convergence with respect to In other words, the eigenstates of the Floquet Hamiltonian... 

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... 

Complex rotation can be usefully applied also to the case of the interaction of an atom with a time-dependent perturbation. With the Floquet formalism by Shirley [41], it was shown that, for a time-periodic field, the dressed states of the combined atom-field system can be characterized non-perturbatively by the eigenstates of a time-independent, infinite-dimensional matrix. The combination of the Floquet approach with complex rotation, proposed by Chu, Reinhardt, and coworkers [37, 42, 43], permits to account for the field-induced coupling to the continuum in an efficient way. As in the time-independent case, this results in complex eigenvalues (this time to the Floquet Hamiltonian matrix) and again the imaginary parts give the transition rate to the continuum. This combination has since then been successfully used to examine various strong field phenomena a review can be found in Ref. [44]. 

From the discussion above, this coincides with the expectation value (A)(t) calculated with the evolution in the Floquet picture of an initial condition of the photon field that is a photon number eigenstate eM (with arbitrary k) that is, (.A) t) = J(ft 0o) A 4)(f Go)) - according to Eq. (46). We have seen on... 

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... 

Similarly to the result obtained for QRs, these unitary DSs are just the spatial transformations dictated by the symmetry of the nanotube potential and compensated by the appropriate translation in time. It is interesting to examine the quantum numbers associated with the DSs Rjv and Poo- Note that the Floquet states 0 (r, t) are eigenstates of P and as well. Recall, that for QRs we have = I and, therefore, the eigenvalues of Pjv are the Mh order roots of - 1. The situation is more intricate for nanotubes in circularly polarized fields, where we find P P = I. Owing to the foim of the interaction term, equation (28), and the periodicity embedded in P, it is natural to transform from z and t to another set of orthogonal coordinates o)t — Icqz and cjt + koz)/2. Afterwards, it is possible to rewrite a Floquet state as... 

T, it may undergo rapid time evolution over many adiabatic states due to the interaction with the external field. Such behavior of the Floquet state is sometimes described as a field-dressed state. Its importance in analysis in the field-induced dynamics is analogous to that of the energy eigenstate under a time-independent Hamiltonian. We will further show that even a small non-periodic term can be incorporated in this framework in an analogous manner as the nonadiabatic transitions in the field-free dynamics. 

Presumably the most straightforward approach to chemical dynamics in intense laser fields is to use the time-independent or time-dependent adiabatic states [352], which are the eigenstates of field-free or field-dependent Hamiltonian at given time points respectively, and solve the Schrodinger equation in a stepwise manner. However, when the laser field is approximately periodic, one can also use a set of field-dressed periodic states as an expansion basis. The set of quasi-static states in a periodic Hamiltonian is derived by a Floquet type analysis and is often referred to as the Floquet states [370]. Provided that the laser field is approximately periodic, advantages of using the latter basis set include (1) analysis and interpretation of the electron dynamics is clearer since the Floquet state population often vary slowly with the timescale of the pulse envelope and each Floquet state is characterized as a field-dressed quasi-stationary state, (2) under some moderate conditions, the nuclear dynamics can be approximated by mixed quantum-classical (MQC) nonadiabatic dynamics on the field-dressed PES. The latter point not only provides a powerful clue for interpretation of nuclear dynamics but also implies possible MQC formulation of intense field molecular dynamics. 

The Floquet theorem, when apphed to the quantum mechanics [370], implies the stationarity of Floquet states imder a perfectly periodic Hamiltonian. We define the electronic Floquet operator as 7ft = Hf — ihdt and the Floquet states as its periodic eigenstates which satisfy Tlt x t)) = A] A(f))- The above mentioned stationarity states that the solution of time-dependent Schrodinger equation ] t) can be expanded as... 

Substituting the parameterized Hamiltonian HsUH ) into Eq. (8.25), the Floquet-type operator becomes Hs TZ ) = Hs Tlt ) - ih- , which is now a true Floquet operator in the sense that it has the exact periodicity in the variable s. It thus follows that the eigenstates of this Floquet operator Ffs( 7 -f ) are also periodic in the variable s. We define parameterized Floquet states... 

Using the fact that the Floquet Hamiltonian is time-independent, the solution of the TDSE in T L Eq. (6.57) can be directly expressed as a function of the eigenstates and eigenvalues obtained through... 

This last equation shows that, in the adiabatic limit, the dynamics is restricted to the subspace of K. containing the initial wavefimction (. to)- Specifically, if the sys-temis at timer = fo in the Floquet instantaneous eigenstate l>jf(0, fo) = rf(to)),... 

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