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arXiv:1804.01805 · Zeitschrift für Naturforschung A

The Floquet Theory of the Two-Level System Revisited

Heinz–Jürgen Schmidt

Published Jun 14, 2018• 16 citations
arXivHTMLPDF

Abstract

Abstract In this article, we reconsider the periodically driven two-level system especially the Rabi problem with linear polarisation. The Floquet theory of this problem can be reduced to its classical limit, i.e. to the investigation of periodic solutions of the classical Hamiltonian equations of motion in the Bloch sphere. The quasienergy is essentially the action integral over one period and the resonance condition due to Shirley is shown to be equivalent to the vanishing of the time average of a certain component of the classical solution. This geometrical approach is applied to obtain analytical approximations to physical quantities of the Rabi problem with linear polarisation as well as asymptotic formulas for various limit cases.

DOI
10.1515/zna-2018-0211

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Abstract In this article, we reconsider the periodically driven two-level system especially the Rabi problem with linear polarisation. The Floquet theory of this problem can be reduced to its classical limit, i.e. to the investigation of periodic solutions of the classical Hamiltonian equations of motion in the Bloch sphere. The quasienergy is essentially the action integral over one period and the resonance condition due to Shirley is shown to be equivalent to the vanishing of the time average of a certain component of the classical solution. This geometrical approach is applied to obtain analytical approximations to physical quantities of the Rabi problem with linear polarisation as well as asymptotic formulas for various limit cases.
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We develop a low-frequency perturbation theory in the extended Floquet Hilbert space of a periodically driven quantum systems, which puts the high- and low-frequency approximations to the Floquet theory on the same footing. It captures adiabatic perturbation theories recently discussed in the literature as well as diabatic deviation due to Floquet resonances. For illustration, we apply our Floquet perturbation theory to a driven two-level system as in the Schwinger–Rabi and the Landau-Zener–Stückelberg–Majorana models. We reproduce some known expressions for transition probabilities in a simple and systematic way and clarify and extend their regime of applicability. We then apply the theory to a periodically-driven system of fermions on the lattice and obtain the spectral properties and the low-frequency dynamics of the system.
We develop a low-frequency perturbation theory in the extended Floquet Hilbert space of a periodically driven quantum systems, which puts the high- and low-frequency approximations to the Floquet theory on the same footing. It captures adiabatic perturbation theories recently discussed in the literature as well as diabatic deviation due to Floquet resonances. For illustration, we apply our Floquet perturbation theory to a driven two-level system as in the Schwinger-Rabi and the Landau-Zener-Stuckelberg-Majorana models. Were produce some known expressions for transition probabilities in a simple and systematic way and clarify and extend their regime of applicability. We then apply the theory to a periodically-driven system of fermions on the lattice and obtain the spectral properties and the low-frequency dynamics of the system.
Rabi oscillations and Floquet states are likely the most familiar concepts associated with a periodically time-varying Hamiltonian. Here, we present an exactly solvable model of a two-level system coupled to both a continuum and a classical field that varies sinusoidally with time, which sheds light on the relationship between the two problems. For a field of the rotating-wave-approximation form, results show that the dynamics of the two-level system can be mapped exactly onto that for a static field, if one shifts the energy separation between the two levels by an amount equal to ℏω, where ω is the frequency of the field and ℏ is Planck's constant. This correspondence allows one to view Rabi oscillations and Floquet states from the simpler perspective of their time-independent-problem equivalents. The comparison between the rigorous results and those from perturbation theory helps clarify some of the difficulties underlying textbook proofs of Fermi's golden rule, and the discussions on quantum decay and linear response theory.
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.
We present an analytical and numerical Floquet treatment of a driven polar two-level quantum system characterized by both longitudinal and transverse coupling to a periodic field. Analytically, we derive a dressed-frame effective Hamiltonian up to first order in the inverse driving frequency, incorporating the longitudinal coupling nonperturbatively. This yields closed expressions for the effective transverse coupling strength and the effective detuning, both of which are modified by the presence of the longitudinal interaction. In the nonpolar limit, these expressions recover the usual near-resonant Rabi coupling and the Bloch-Siegert shift. As a second main result, we develop a numerical flow-equation framework that yields a time-independent effective Hamiltonian across a broad range of transverse and longitudinal coupling strengths. This dual framework is relevant for a variety of platforms, including driven polar quantum systems, optical lattices, superconducting circuits, and solids subject to surface acoustic waves.