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Journal of Mathematical Physics

The Floquet analysis and noninteger higher harmonics generation

José M. Cerveró, Juan D Lejarreta

Published Apr 1, 1999• 7 citations
Publisher

Abstract

We consider here the exact solution of a nonrelativistic quantum system composed of a two-level atom interacting with a laser with arbitrary large frequency and intensity. We use the analogy of this system and a ℏ/2-spin particle interacting with a time-dependent magnetic field. A systematic use of the dynamical symmetry underlying the physical system is made. Actually the Hamiltonian is a Hermitian element of the SU(2) Lie Algebra. The exact Temporal Evolution Operator in terms of a Generalized Displacement Operator of the group is constructed. It is possible to develop a nonperturbative method that allows us to solve exactly the model for any value of the relevant frequencies (Rabi, Laser, and Atom Frequencies) and in so doing the usual Rotating Wave and Small Interaction approximations are unnecessary. The properties of periodicity of this model and the phenomenon of harmonic generation are considered by using Floquet Analysis. We find that in addition to the so far well-known spectrum composed by odd harmonics, this model generates another type of noninteger harmonics whose frequencies and amplitudes are determined for any value of the relevant parameters of the system.

DOI
10.1063/1.532831

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We consider here the exact solution of a nonrelativistic quantum system composed of a two-level atom interacting with a laser with arbitrary large frequency and intensity. We use the analogy of this system and a ℏ/2-spin particle interacting with a time-dependent magnetic field. A systematic use of the dynamical symmetry underlying the physical system is made. Actually the Hamiltonian is a Hermitian element of the SU(2) Lie Algebra. The exact Temporal Evolution Operator in terms of a Generalized Displacement Operator of the group is constructed. It is possible to develop a nonperturbative method that allows us to solve exactly the model for any value of the relevant frequencies (Rabi, Laser, and Atom Frequencies) and in so doing the usual Rotating Wave and Small Interaction approximations are unnecessary. The properties of periodicity of this model and the phenomenon of harmonic generation are considered by using Floquet Analysis. We find that in addition to the so far well-known spectrum composed by odd harmonics, this model generates another type of noninteger harmonics whose frequencies and amplitudes are determined for any value of the relevant parameters of the system.
The extended quantum Rabi models make a significant contribution to understanding of the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, the anisotropic Rabi model and the asymmetric Rabi model, into the rotating frame and regard them as periodically driven quantum systems. The analytical solutions of the quasienergy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the nonstroboscopic dynamics of physical observables such as atomic inversion, transverse magnetization, and atom-field correlation. For the anisotropic Rabi model, the quasienergy fits well with the numerical results even when the rotating-wave coupling is in the deep strong-coupling regime $g\ensuremath{\simeq}\ensuremath{\hbar}\ensuremath{\omega}$ if the counter-rotating term is small enough compared to the driving frequency. Avoided level crossing may occur for quasienergy with the same parity when the positive branch spectrum lines for the total excitation number NNN cross the negative branch lines for N+2N+2N+2, while the high-frequency expansion fails to predict this due to the conservation of the total excitation number. Furthermore, we present analytical and numerical studies of the long-time evolution of population and conclude that the analytical method is credible for the population dynamics. For the asymmetric Rabi model, we find that the external bias field which breaks the parity symmetry of the total excitation number tends to cluster the upper and lower branches into two bundles, and the detuning induced gap in the first temporal Brillouin zone shows a quadratic dependence on the bias. Fourier analysis is applied to extract the frequency composition and two-frequency driving behavior is revealed. Varying the bias strength will change the time-averaged value of the oscillation, which shows how the bias competes with the detuning and atom-field coupling in the driving dynamics. Both models prove that treating the Hamiltonian in the rotating frame by Floquet theory gives an alternative tool in the study of interaction between atom and light.
In a previous paper, we have already considered the system composed by a two level atom interacting with a coherent external electromagnetic field. No application whatsoever has been made of the rotating wave approximation. Being specially interested in the problem of higher harmonic generations for the case of very intense laser fields, we have developed in this letter a much more efficient way to obtain these solutions as well as to carry out some calculations in a range in which the parameters take extreme values. Also the formalism allows us now to provide analytic expressions in the WKB regime for the electric dipole moment and the population inversion. The spectrum can be decomposed in periodic and non-periodic contributions. Only the latter depends upon the Floquet exponent and can be responsible of the main complexities of the observed Rabi revivals and the hyper-Raman shift.
The spectra produced by high-order harmonic generation (HHG) typically exhibit well-defined peaks at odd integers times the laser frequency. However, in recent investigations of HHG from correlated materials, spectra exhibit signals at noninteger harmonics which do not conform to the well-known symmetry-based selection rules for HHG spectra. Here, we use the Fermi-Hubbard model to study HHG from a linear chain of atoms. This model allows us to study both the correlated and uncorrelated phases through a specification of the amount of onsite electron-electron repulsion. The presence of signal at noninteger harmonics can be interpreted as originating from the population of multiple Floquet states. We show how this coupling to different Floquet states depends on the characteristics of the driving pulse and the strength of the electron-electron interaction in the system.
The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.
In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively. We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies , which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.
We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold-atom experiments.