Floquet formalism

S.-I. Chu, D.A. Telnov, Beyond the Floquet theorem Generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser fields, Phys. Rep. 390 (2004) 1. 

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]. 

A. Floquet Formalism from the Semiclassical Point of View... 

B. Floquet Formalism from Quantized Cavity Dressed States... 

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... 

We will discuss the Floquet approach from two different points of view. In the first one, discussed in Section II.A, the Floquet formalism is just a mathematically convenient tool that allows us to transform the Schrodinger equation with a time-dependent Hamiltonian into an equivalent equation with a time-independent Hamiltonian. This new equation is defined on an enlarged Hilbert space. The time dependence has been substituted by the introduction of one auxiliary dynamical variable for each laser frequency. The second point of... 

We will establish a precise relation between dressed states in a cavity and the Floquet formalism. We show that the Floquet Hamiltonian K can be obtained exactly from the dressed Hamiltonian in a cavity in the limit of infinite cavity volume and large number of photons K represents the Hamiltonian of the molecule interacting in free space with a field containing a large number of photons. We establish the physical interpretation of the operator... 

From the formulation of the Floquet formalism given above, we can establish the precise connection between the dynamics in the enlarged space C/f defined by the Floquet Hamiltonian K, and the one defined by the semiclassical Hamiltonian in with a classical description of the electric field ... 

Multiphoton processes taking place in atoms in strong laser fields can be investigated by the non-Hermitian Floquet formalism (69-71,12). This time-independent theory is based on the equivalence of the time-dependent Schrodin-ger description to a time-independent field-dressed-atom picture, under assumption of monochromaticity, periodicity and adiabaticity (69,72). Implementation of complex coordinates within the Floquet formalism allows direct determination of the complex energy associated with the decaying state. The... 

The methods axe applied to multielectron atoms represented by a one-electron potential (73-76). Ben-Tal, Moiseyev and coworkers also developed an approach combining complex-coordinates with the Floquet formalism and applied it to model Hamiltonians (77-82). They also considered non-periodic time-dependent Hamiltonians (81). 

As seen above, laser assisted and controlled photofragmentation dynamics can conceptually be viewed in two different ways. The time-dependent viewpoint offers a realistic time-resolved dynamical picture of the basic processes that are driven by an intense, short laser pulse. For pulses characterized by a long duration (as compared to the timescales of the dynamics), the laser field can be considered periodic, allowing the (quasi-) complete elimination of the time variable through the Floquet formalism, giving rise to a time-independent viewpoint. This formalism not only offers a useful and important interpretative tool in terms of the stationary field... 

We present the numerical methodology for solving Eqs. (39), resulting from the application of the Floquet formalism to the semiclassical Hamiltonian of the molecule plus field system. For this purpose, we present Eqs. (39) in the generic form of a system of a finite number of closed-coupled equations... 

T.T. Nguyen-Dang, F. Chateauneuf, O. Atabek, X. He, Time-resolved dynamics (f two-channel molecular systems in cw laser fields Wave-packet construction in the Floquet formalism, Phys. Rev. A 51 (1995) 1387. 

In each case, we first studied the laser driven dynamics of the system in the framework of the Floquet formalism, described in Sect. 6.5 of Chap. 6, which provides a geometrical interpretation of the laser driven dynamics and its dependence on the frequency and amplitude of the laser field, through the analysis of the eigenvalues of the Floquet operator, called quasienergies. Various effective models were used for that purpose. This analysis allowed us to explain the shape of the relevant quasienergy curves as a function of the laser parameters, and to obtain the parameters of the laser field that induce the CDT. We then used the MCTDH method to solve the TDSE for the molecule in interaction with the laser field and compare these results with those obtained from the effective Hamiltonian described in Sect. 8.2.3 above. 

---