Floquet matrix

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... 

Quantum dynamical calculations on the IRMPE/D of 614, 615 O3. Quantum interference effects and discussion of the possibility of mode-selective excitation and reaction Confirmation that OCS does not undergo IRMPD at 616 high laser fluences (ca. 250 J cm ). Laser-induced dielectric breakdown in OCS, OCS-He, and OCS-Ar does lead to dissociation, giving CO + S Ab initio study of SO2 IRMPE using the most proba- 617 ble path approximation to select the most important paths within the semiclassical Floquet matrix. Conclude that collisionless MPD of SO2 will not occur at laser field strengths <20 GW cm ... 

In this section, we discuss briefly the generalized Floqnet formnlation of TDDFT [28,60-64]. It can be applied to the nonperturbative stndy of mnltiphoton processes of many-electron atoms and molecules in intense periodic or qnasi-periodic (multicolor) time-dependent fields, allowing the transformation of time-dependent Kohn-Sham equations to an equivalent time-independent generalized Floquet matrix eigenvalue problems. 

It was shown that the three methods - multistep method, perturbation expansion and diagonalization of the Floquet matrix - are numerically equivalent in treating the time evolution of spin systems, though not equally efficient for the different cases. By treating the sample spinning in terms of an angular momentum operator, a physical interpretation of the Floquet description of MAS experiments in solids was presented by Boender et al The sample rotation was treated as a continuous irradiation on the spin system by the phonons from the RF field accordingly, the Floquet Hamiltonian consisted of the spin-phonon interaction. 

As a concrete illustration of the Floquet band structure for a threefold barrier, Section 3.4 of Child [50] contains an explicit analytical form for the matrix u ... 

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

The sums over n in the Fourier expansions (8.16) and (8.17) extend to infinity. Therefore, the matrix of the Floquet Hamiltonian in Eq. (8.20) is infinitely large. 

If the basis of molecular states is restricted only to two states a and / , the matrix of the Floquet Hamiltonian has the following form ... 

The size of the matrix as it operates on the perturbation vector is directly related to the eigenvalues of J (or of B). The eigenvalues of J are known as the Floquet multipliers fit the eigenvalues of B are the Floquet exponents / ,. In general the former are easier to evaluate, although we should identify the parameter p2 introduced in chapter 5 with the Hopf bifurcation formula as a Floquet exponent for the emerging limit cycle (then P2 < 0 implies stability, P2 > 0 gives instability, and P2 = 0 corresponds to a bifurcation between these two cases). 

The differential equation for M in (7) is non-antonomous and involves evaluation of the jacobian of the forced-model equations at the current value of the trajectory jc(x0, p, t) for each time step so that it must be integrated simultaneously with the system equations. Upon convergence on a fixed point, the matrix M becomes the monodromy matrix whose eigenvalues are those of the jacobian of the stroboscopic map evaluated at the fixed point and are called the Floquet multipliers of the periodic solution. 

There are many forms in which the Floquet approach can be more rigorously implemented.13-16 One which has been used often is the infinite matrix approach of Shirley.14 It corresponds roughly to the infinite set of sidebands. An alternative more compact approach has been described by Sambe15 and Christiansen-Dalsgaard.16... 

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]. 

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... 

As customary, the exponential of a matrix means the sum of the matrix series corresponding to the exponential function. The eigenvalues of i T) =e are called the Floquet multipliers. The eigenvalues of B are called the Floquet exponents. (There is some delicacy about the uniqueness of B which we will ignore because it is not relevant to our use.) Usually it is not possible to compute the Floquet exponents or multipliers. However, for low-dimensional systems of the kind we will investigate, there is a general theorem about the determinant of a fundamental matrix which is helpful. Let 4>(0 be a fundamental matrix for (4.1) with i (0) = I. Then... 

Proof. The quantity under the integral sign in the definition of A in (5.2) is the trace of the Jacobian matrix for the system (5.1) evaluated along the periodic orbit. Theorem 4.2 then applies. A periodic orbit for an autonomous system has one Floquet multiplier equal to 1. Since there are only two multipliers and one of them is 1, is the remaining one. The periodic orbit is asymptotically orbitally stable because, in view of Lemma 5.1, A<0. ... 

This system is periodic and therefore the Floquet theory described in Section 4, Chapter 3, applies. Let 4>(/) be the fundamental matrix solution of (2.2). The Floquet multipliers of (2.2) are the eigenvalues of 4>(w) if /i is a Floquet multiplier and /i = e" then A is called a Floquet exponent. Only the real part of a Floquet exponent is uniquely defined. 

The choice of this form for the initial density matrix is in line with the above definitions of the Fourier and Floquet space, and maintains the form of all operators in Floquet space as in Eq. 28. 

The zero-order contribution to Hef / is thus just equal to the diagonal term Ho, and depends on the matrix elements H k as in Eq. 56. We will discuss briefly which bimodal Floquet elements Hnk can contribute to the diagonal block Hq in the single-mode representation. According to Eq. 55, in order to obtain I = 0 the integer k must be equal to -nn/v. There is always a possibility that n = 0 and k = 0 and hence Hqo will always contribute to Ho-... 

This Floquet Hamiltonian can be further simplified by diagonalizing the constant matrix with T = j... 

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