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Physical Review Letters

Strong-Coupling Theory of Periodically Driven Two-Level Systems

Ying Wu, Xiaoxue Yang

Published Jan 5, 2007• 212 citations
Publisher

Abstract

We present a simple but highly efficient iterating approach for strong-coupling periodically driven two-level systems. The obtained explicit approximating analytical solution reproduces accurately the exact numerical solution in the strong-coupling regime for a wide frequency range including resonance, far-off resonance, harmonic, and subharmonic cases. Our theory is suitable for single- and multi-period periodic driving and for the periodic driving with a few-cycle pulse as well, and it gives a general formula for calculating the strong-field ac Stark effect in such diverse situations.

DOI
10.1103/physrevlett.98.013601

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We present a simple but highly efficient iterating approach for strong-coupling periodically driven two-level systems. The obtained explicit approximating analytical solution reproduces accurately the exact numerical solution in the strong-coupling regime for a wide frequency range including resonance, far-off resonance, harmonic, and subharmonic cases. Our theory is suitable for single- and multi-period periodic driving and for the periodic driving with a few-cycle pulse as well, and it gives a general formula for calculating the strong-field ac Stark effect in such diverse situations.
We show that a simple nonperturbative two-level model of an atom driven by a very strong periodic field, results in a rich picture of very high harmonic generation and related phenomena. It reproduces experimentally observed plateau, yields for the first time simple analytic formulas for the plateau cutoff frequency, critical driving intensity, and saturation, and predicts intensity-induced multi-resonances. One of the most fascinating phenomenon discovered recently in nonlinear interaction of light with atoms and ions, is very high-order (up to 135) odd harmonic generation (HHG) by intense (∼ 1013 W/cm2 and higher) optical laser radiation in rare gasses and some ions [1], The spectra of generated harmonics drasticly deviate from the perturbation theory predictions [2], In particular, intensity of harmonics, falling monotonically with their orders only up to a certain point, levels off forming a so-called “plateau", and falls monotonically again beyond it. Generally, harmonte generation depends on phase-matching conditions and nonlinear response of individual atoms. It has recently become clear however [3,4] that the major features of HHG, in particular the plateau, result mainly from general properties of atomic nonlinear response. The most direct and apparently successful way so far to approach the problem theoretically has been numerical simulation of the Schrödinger [2,5-7] (including an empiric rule [6]) for many-electron atoms using Hartry-Slater approximation. This approach requires, however, tremendous amount of calculations and involves many processes, making it difficult to gain simple insights. An interesting simplified model [8] based on 3-D delta-potential with a single (ground) level [9] produces results in integration form. The idea of retaining a single energy scale (ionization energy) brings one close to an even simpler system: a two-level model atom. A two-level model of HHG [10], however, due to various complications introduced into it, failed to generate simple results, whereas an analytic solution [11] holds for a virtually degenerate two-level model only (see below) which is unapplicable to the experimental conditions [1-4,7,9]; besides, no relaxation was considered in [10,11].