Floquet eigenvalue

In the quasiperiodic case of two or several incommensurate frequencies the Floquet eigenvalues cover the real line densely, and the overlap between Brillouin zones is much more intricate. 

This structure of the eigenvectors and eigenvalues of Floquet Hamiltonians can be understood by considering an alternative interpretation of the Floquet eigenvalue problem. 

Substituting equation (7) into the time-dependent Schrodinger equation one obtains the Floquet eigenvalue problem... 

Figure 3. Floquet band structure for a threefold cyclic barrier (a) in the plane wave case after using Eq. (A.l 1) to fold the band onto the interval —I < and (b) in the presence of a threefold potential barrier. Open circles in case (b) mark the eigenvalues at = 0, 1, consistent with periodic boundary conditions. Closed circles mark those at consistent with sign-changing...

According to the Floquet theorem [Arnold 1978], this equation has a pair of linearly-independent solutions of the form x(z,t) = u(z, t)e p( 2nizt/p), where the function u is -periodic. The solution becomes periodic at integer z = +n, so that the eigenvalues e we need are = ( + n). To find the infinite product of the we employ the analytical properties of the function e z). It has two simple zeros in the complex plane such that... 

This is an eigenvalue equation for the Floquet exponents and the coefficients To see this more clearly, let us introduce the basis of states an), where the Greek letter labels the molecular states and the Roman letter the Fourier components. Then, Eq. (8.19) can be written as... 

When we come to look at the stability of the limit cycle which is born at the Hopf bifurcation point, we shall meet a quantity known as the Floquet multiplier , conventionally denoted p2, which plays a role similar to that played for the stationary state by the eigenvalues and k2. If / 2 is negative, the limit cycle will be stable and should correspond to observable oscillations if P2 is positive the limit cycle will be unstable. 

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). 

Upon convergence, the eigenvalues of dF/dx (the characteristic or Floquet multipliers FMt) are independent of the particular point on the limit cycle (i.e. the particular Poincare section or anchor equation used). One of them, FMn, is constrained to be unity (Iooss and Joseph, 1980) and this may be used as a numerical check of the computed periodic trajectory the remaining FMs determine the stability of the periodic orbit, which is stable if and only if they lie in the unit circle in the complex plane ( FM, < 1,1 i = n - 1). The multiplier with the largest absolute value is usually called the principal FM (PFM). When (as a parameter varies) the PFM crosses the unit circle, the periodic orbit loses stability and a bifurcation occurs. 

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. 

The Floquet multipliers determine the stability and character of the fixed point (or limit cycle) much in the same way as the eigenvalues of the jacobian... 

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... 

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. 

Let the Floquet multipliers (eigenvalues of (7 )) of the variational equation be l,Pi,P2,. ..,p i, where the terms are listed according to multiplicity and the first one corresponds to the eigenvector e. Finally, recall from the fundamental theory of ordinary differential equations [H2, chap. 1, thm. 3.3] that... 

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. 

This effective Hamiltonian is again not unique but can be chosen such that its eigenvalue differences are smaller than l/2o t. Maricq [100-102] and others [14, 103] have demonstrated that the Magnus expansion of the effective Hamiltonian in AHT and the van Vleck transformation approach of the Floquet Hamiltonian are equivalent. At the time points krt the Floquet solution for the propagator in Eq. 24 has the form... 

The eigenvectors of K are the same ones as those for K, and the eigenvalues just have to be multiplied by Ha. The difference with the preceding discussion is that here Ho and V are both of order e. Thus we take as the unperturbed Floquet Hamiltonian just... 

As we have stated, the Floquet Hamiltonian (113) has no terms that are resonant if we take small enough e, and the iteration of the KAM procedure converges. However, if we take e large enough, we encounter new resonances that are not present at zero or small fields that is, they are not related to degeneracies of the unperturbed eigenvalues of Kq that lead to the zero-field resonances we have discussed in the previous subsection. These new resonances are related to degeneracies of the new effective unperturbed operator K 0(e), which appear at some specific finite values of e. These are the dynamical resonances. 

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... 

The Structure of Eigenvectors and Eigenvalues of Floquet Hamiltonians—The Concept of Dressed Hamiltonian... 

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