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Physical Review A

Extreme nonlinear optics of two-level systems

Thorsten Tritschler, Oliver D. Mücke, Martin Wegener

Published Sep 12, 2003• 53 citations
Publisher

Abstract

For Rabi frequencies comparable to, or even larger than, the transition frequency of a two-level system, the regime of extreme nonlinear optics is reached. Here, we give an overview of the radiated light intensity as a function of carrier frequency of light, transition frequency, Rabi frequency, spectrometer frequency, as well as of the shape and duration of the exciting optical pulses. The graphical representations reveal an amazing complexity and beauty of the nonlinear optical response. Analytical results within the ``square-wave approximation'' qualitatively reproduce many of the intricate features of the exact numerical calculations.

DOI
10.1103/physreva.68.033404

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For Rabi frequencies comparable to, or even larger than, the transition frequency of a two-level system, the regime of extreme nonlinear optics is reached. Here, we give an overview of the radiated light intensity as a function of carrier frequency of light, transition frequency, Rabi frequency, spectrometer frequency, as well as of the shape and duration of the exciting optical pulses. The graphical representations reveal an amazing complexity and beauty of the nonlinear optical response. Analytical results within the ``square-wave approximation'' qualitatively reproduce many of the intricate features of the exact numerical calculations.
In an approach based on the nonlinear Riccati-type differential equation for the ratio of the level population amplitudes, rather than on the optical Bloch equations, we describe the response of a two-level system to a few--cycle light pulse of intensity from the regime of extreme nonlinear optics, i.e., when the Rabi and transition frequencies are comparable.Emphasis is put on the dependence of the spectra of the scattered light on the carrier--envelope phase, duration and strength of both resonant and off-resonant pulses.
We formulate the problem of a two-level system in a linearly polarized laser field in terms of a nonlinear Riccati-type differential equation and solve the equation analytically in time intervals much shorter than half the optical period.The analytical solutions for subsequent intervals are then stuck together in an iterative procedure to cover the whole scale time of the laser pulse.Very good quality of the iterative method is shown by recovering with it a number of subtle effects met in earlier numerically calculated photon--emission spectra from model molecular ions, double quantum wells, atoms, and semiconductors.The method is used to describe novel, carrier-envelope offset phase effects in the region of extreme nonlinear optics, i.e., when two--level systems are exposed to pulses of only a few cycles in duration and strength ensuring the Rabi frequency to approach the laser light frequence.
We calculate analytically the error introduced by the rotating-wave approximation (RWA) to the description of a two-level-atom response to a resonant sufficiently intense femtosecond optical pulse of a few-cycle duration and any envelope. The analytically calculated non-RWA refinements to the RWA response are of the lowest order, i.e., linear in the small ratio of the Rabi to optical frequencies. One type of such refinements is small-amplitude rapid oscillations at nearly twice the optical frequency superimposed on the standard RWA population-inversion evolution. The other refinement is an extension of the spectrum of the induced atomic dipole moment in which the third harmonic of the transition frequency appears. The spectrum of this harmonic has an internal structure dependent on both strength and duration of the pulse. The analytically found non-RWA corrections are in qualitative agreement with the recent results of different numerical approaches by other authors and a recent experiment.