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Physical Review Letters

Breakdown of the Area Theorem: Carrier-Wave Rabi Flopping of Femtosecond Optical Pulses

Steven C. Hughes

Published Oct 19, 1998• 236 citations
Publisher

Abstract

By solving Maxwell's curl equations coupled to a two-level atom, a theoretical study of carrier-wave Rabi flopping of femtosecond optical pulses of only several carrier-cycles time duration is reported. For pulse areas of $2\ensuremath{\pi}$, the usual self-induced transparency regime is essentially recovered. However, for larger pulse areas, carrier-wave Rabi flopping occurs that manifests in local carrier reshaping and subsequently to the production of significantly higher spectral components on the propagating pulse. These new features are not predicted by employing the area theorem or a slowly-varying-envelope approximation for the amplitude and phase terms---which is the usual approach.

DOI
10.1103/physrevlett.81.3363

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By solving Maxwell's curl equations coupled to a two-level atom, a theoretical study of carrier-wave Rabi flopping of femtosecond optical pulses of only several carrier-cycles time duration is reported. For pulse areas of 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ, the usual self-induced transparency regime is essentially recovered. However, for larger pulse areas, carrier-wave Rabi flopping occurs that manifests in local carrier reshaping and subsequently to the production of significantly higher spectral components on the propagating pulse. These new features are not predicted by employing the area theorem or a slowly-varying-envelope approximation for the amplitude and phase terms---which is the usual approach.
Summary form only. The development of high-intensity ultrashort optical pulses has lead to a number of fascinating nonlinear optical propagation studies including Rabi flopping, self-induced transparency, and photon echo, to name but a few. Common to the description of these effects, is that their propagation phenomena can be described quite adequately by employing the appropriate coupled matter-Maxwell equations within the slowly-varying-envelope approximation (SVEA), i.e., the envelopes of the electromagnetic field and polarization are assumed to vary little over an optical period and wavelength. However, for several-cycle optical pulses this approximation becomes seriously questionable.
By comparing the numerical solutions of Maxwell-Bloch equations beyond and within the slowly-varying envelope approximation and the rotating-wave approximation for the propagation of a few-cycle pulse laser in a resonant two-level atom medium, we found that both the Rabi flopping and the refractive index, and subsequently the carrier and the propagation velocity of the few-cycle pulse, are closely connected with the time-derivative behaviour of the electric field. This is because the Rabi flopping is such that the soliton pulse splits during propagation and that a shorter pulse propagates faster than a broader one.
We investigate the propagation of attosecond pulse in a dense two-level medium by using an iterative predictor-corrector finite-difference time-domain method. We find when attosecond pulse is considered, that the standard area theorem will break down even for small area pulses: ideal self-induced transparency cannot occur even for a $2\ensuremath{\pi}$ pulse, while the pulses whose areas are not integer multiples of $2\ensuremath{\pi}$, such as $1.8\ensuremath{\pi}$ and $2.2\ensuremath{\pi}$ pulses, cannot evolve to $2\ensuremath{\pi}$ pulses as predicted by the standard area theorem. Significantly higher spectra components can occur on all these small area propagating pulses due to strong carrier reshaping. Furthermore, these higher spectral components dependent sensitively on the pulse area: the larger the pulse area is, the more evident are these higher spectral components.
Summary form only given. In our recent experiments on the model semiconductor GaAs using 5 fs pulses, the Rabi period became as short as 3 fs, which is comparable to the period of light of 2.9 fs at the GaAs band edge. Hence, the medium's dynamics significantly distort the optical polarization within one optical cycle, which can be measured by a characteristic shape of the harmonics of the emitted light. In this paper we summarize our efforts to obtain a detailed understanding of these experiments by simulations based on a finite difference time-domain algorithm, solving the coupled Maxwell-Bloch equations in one dimension without using either the rotating wave approximation or the slowly varying envelope approximation. We consider not only transitions close to the band edge but also transitions up to 5.8 eV photon energy in a realistic fashion, i.e. comprising the known E/sub 1/ and E/sub 2/ resonances in GaAs which are due to band structure effects.
We solve the problem of ultrashort pulse propagation in a two-level medium beyond the rotating-wave (RWA) and slowly varying-envelope approximations. The method of solution is based on the Maxwell-Bloch equations represented in the form that allows one to switch between RWA and general (non-RWA) cases in the framework of a single numerical algorithm. Using this method, the effect of a subcycle pulse (containing less than a single period of field oscillations) on the two-level medium was analyzed. It is shown that for such short pulses, the clear breakdown of the area theorem occurs for the pulses of large enough area. Moreover, deviations from the area theorem appear to be strongly dependent on the pulse shape that cannot be observed for longer few-cycle pulses.
Summary form only given. Illuminating a semiconductor with a constant light intensity can lead to a periodic oscillation of the inversion, a phenomenon which is known as Rabi flopping. Using pulsed excitation, Rabi flopping has been observed on semiconductors and exhibited periods in the range from 100 fs to 1 ps. Hughes (Phys. Rev. Lett. vol. 81, p. 3363, 1998) theoretically investigated the question of what happens when the light intensity becomes so high that the period of one Rabi oscillation becomes comparable with that of only one cycle of light, 2.7 fs for the band edge of GaAs, and predicted a new phenomenon: carrier-wave Rabi flopping. During a carrier-wave Rabi oscillation, the optical polarization becomes strongly distorted, which can lead to a double-peak structure (the Fourier-transform of the Rabi oscillation) in the third-harmonic (3/spl omega/) spectra. This characteristic shape is quite different from the well-known off-resonant third-harmonic generation. Importantly, resonant effects contain information on the dynamics of electronic resonances-potentially on the timescale of a single optical cycle. It is clear that, under these conditions, the usual envelope approximation breaks down, i.e. both the rotating wave approximation (RWA) as well as the slowly varying envelope approximation (SVEA) must not be used. This paper investigates carrier-wave Rabi flopping experimentally.