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Optics Letters

Multiphoton resonances in the modulated fluorescence from an inhomogeneously broadened medium

Wilhelmus M. Ruyten

Published May 15, 1989• 12 citations
Publisher

Abstract

We consider the interaction of a fully amplitude-modulated, narrow-band laser with a strongly inhomogeneously broadened two-level medium and show that harmonics of the fluorescence of susceptibility, unlike the time-averaged fluorescence, retain the characteristic multiphoton resonance structure that is found in the absence of inhomogeneous broadening. This behavior correlates with the existence of double branches of the corresponding Bloch-Siegert resonance curves. For strong fields, the depth of modulation and the phase shift of the lowest fluorescence harmonic may deviate significantly from the results for weak, broadband excitation.

DOI
10.1364/ol.14.000506

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We consider the interaction of a fully amplitude-modulated, narrow-band laser with a strongly inhomogeneously broadened two-level medium and show that harmonics of the fluorescence of susceptibility, unlike the time-averaged fluorescence, retain the characteristic multiphoton resonance structure that is found in the absence of inhomogeneous broadening. This behavior correlates with the existence of double branches of the corresponding Bloch-Siegert resonance curves. For strong fields, the depth of modulation and the phase shift of the lowest fluorescence harmonic may deviate significantly from the results for weak, broadband excitation.
We present a detailed theoretical treatment of the response of a two-level atom to a fully-amplitude-modulated, narrow-band optical field. Specifically, we calculate the resonance behavior of modulated fluorescence and susceptibility components from the scalar continued-fraction solution of the optical Bloch equations. In the limit of vanishing damping the resonances are generally doubly branched: One set of branches reproduces the resonances of the time-averaged fluorescence, while the second set is different for each harmonic component. However, each branch is uniquely associated with an odd multiple quantum resonance. In analogy with magnetic resonance work, we derive analytical expressions for the generalized Bloch-Siegert shifts of the additional resonances. We discuss also the possibility of experimental verification of the theory.
The problem of a two-level atom interacting with modulated or multifrequency fields has been the subject of much work in magnetic resonance experiments. More recently, experiments dealing with optical interactions have been reported.1,2 Unlike the former, these have been performed in beam experiments to eliminate the effect of Doppler broadening. Here we calculate that Rabi structures, in particular, multiphoton resonances, should be observable even in a gas cell and/or for a broadband source. However, this is true only for fluorescence and absorption at the modulation frequency or harmonics thereof. By contrast, the time-averaged fluorescence, which is the quantity usually studied, becomes structureless on integration over the Doppler profile. The result, which is not consistent with a rate equation analysis, can be qualitatively understood from the resonance behavior of the system. In particular, earlier studies on generalized Bloch-Siegert shifts are extended to harmonic components of the interaction.
We study the fluorescence spectrum of a strongly driven two-level system (TLS) with modulated transition frequency, which is a bichromatically driven TLS and has multiple resonance frequencies. We are aiming to provide a reliable description of the fluorescence in a regime that is difficult to tackle with perturbation theory and the rotating-wave approximation (RWA), and illustrate the spectral features of the fluorescence under off- and multiphoton-resonance conditions. To go beyond the RWA, we use a semianalytical counter-rotating-hybridized rotating-wave method that combines a unitary transformation and Floquet theory to calculate the two-mode Floquet states and quasienergies for the bichromatically driven TLS. We then solve the master equation accounting for the spontaneous decay in the bases of the two-mode Floquet states, and derive a physically transparent fluorescence spectrum. In comparison with the numerically exact spectrum from the generalized Floquet-Liouville approach, the present spectrum is found to be applicable in a wide range of the parameters where the RWA and the secular approximation may break down. We find that the counter-rotating (CR) terms of the transverse field omitted in the RWA have non-negligible contributions to the spectrum under certain conditions. Particularly, at the multiphoton resonance the width of which is comparable with the Bloch-Siegert shift, the RWA and non-RWA spectra markedly differ from each other because of the CR-induced shift. We also analyze the symmetry of the spectrum in terms of the transition matrix elements between the two-mode Floquet states. We show that the strict symmetry of the spectrum cannot be expected without the RWA but the almost symmetric spectrum can be obtained at the single-photon resonance that takes the Bloch-Siegert shift into account if the driving is moderately strong and at the multiphoton resonance with a sufficiently weak transverse field.
The interaction of a strong bichromatic field with a two-level atom has been the subject of numerous experimental and theoretical investigations. We performed an experimental study of the Interaction of an intense 100% amplitude-modulated (AM) field with an ensemble of homogeneously broadened two-level atoms. An optically pumped sodium atomic beam is used as the atomic medium. The field is created by interfering two intense frequency-stabilized laser fields. The frequency of one field is held fixed while another field of equal amplitude is scanned in frequency. The beat frequency produced by interfering the two laser fields is equal to twice the modulation frequency of a 100% AM field. We monitor the time-averaged fluorescent intensity of the atoms as the frequency separation between the two fields is varied. A similar experiment was done in the past in the rf regime.1 For the first time, we report the observation of multiple, well-resolved resonances in the absorption spectrum of the field in the optical regime. We present data and theory for several values of detuning between the fixed field and atomic resonance. The experimental results are in good agreement with theory.
The dynamical behavior of a two-level system in an intense frequency-modulated field is obtained without recourse to the weak-modulation assumption. Solutions to the optical Bloch equations are expressed in terms of continued fractions involving 2\ifmmode\times\else\texttimes\fi{}2 matrices. Such matrix-continued fractions are numerically evaluated to obtain a wide variety of absorption and fluorescence spectra in frequency-modulated fields. Results in various special cases involving combinations of weak fields, weak modulation, strong modulation, and strong fields are given. Analytical results for weak fields but with an arbitrary degree of modulation are also presented. The spectra exhibit additional resonances for strong modulation. The main resonance shifts and broadens considerably due to strong modulation and saturation effects.