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Annual Meeting Optical Society of America

Rabi structures in a Doppler broadened gas?

Wilhelmus R. Ruyten

Published Jan 1, 1988• 0 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1988.my6

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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 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.
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.
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 nonlinear absorption or gain characteristics of an optical-frequency, Doppler-broadened atomic resonance involving levels with closely spaced structure are analyzed. The level structures are assumed to be resolved with respect to their natural widths, but not necessarily with respect to the Doppler width at the optical transition. The radiation field consists of two closely spaced monochromatic frequencies lying within the Doppler width of the resonance. This type of radiation field may be obtained, for example, from a laser operating in two of its Fabry-Perot resonator modes. It is shown that, because of saturation of level populations and double-quantum Raman transitions between levels, appreciable nonlinear coupling takes place between the two fields. This coupling shows a resonance behavior when the frequency separation of the two applied fields becomes equal to the frequency splitting between two of the components which form either level structure. The width of this resonance is determined entirely by the natural widths of these two level components and not by the Doppler width of the optical transition or the natural width of other level components. When such a resonance occurs, the over-all gain or attenuation characteristics of the atomic resonance change drastically. In practice the frequency spacing of the fields may be kept constant while the level structures are tuned, e.g., magnetically. The effects are analyzed for cases of running-wave and standing-wave radiation fields. The use of this effect in precise determination of level structures as well as mode or transition coupling of a gas laser is discussed. Portions of the analysis are applicable to resonances with more general forms of inhomogeneous broadening.
In the framework of the nonsecular perturbation theory based on the Bogoliubov averaging method, the coherent dynamics of multiphoton Raman transitions in a two-level spin system driven by an amplitude-modulated microwave field is studied. Closed-form expressions for the Rabi frequencies of these transitions are obtained beyond the rotating-wave approximation for the low-frequency driving component. It is shown that spin states dressed by the high-frequency component of the driving field are shifted due to the Bloch-Siegert-like effect caused by antiresonant interactions with the strong low-frequency driving. We predict that with increasing the order of the Raman transition, the Rabi frequency decreases and the contribution of the Bloch-Siegert shift to this frequency becomes dominant. It is found that the amplitude and phase of the Rabi oscillations strongly depend on the initial phase of the low-frequency field as well as on detuning from multiphoton resonance. The recent experimental data for the second- and third-order Raman transitions observed for the nitrogen-vacancy center in diamond (Z. Shu et al., arXiv:1804.10492) are well described in the frame of our approach. Our results may provide new possibilities for coherent control of quantum systems.
The resonance behavior of magnetic and optical two-level systems under the influence of strong modulated fields is studied in detail by solving the Bloch equations in the limit of vanishing damping, namely, in terms of matrix continued fractions. Thus both analytical and numerical results are obtained for the positions of the various multiple quantum resonances in the time-averaged as well as modulated interaction components. Specifically, the role of the orientation of the optical pump beam in magnetic-resonance configurations is investigated, and it is shown that there are several classes of resonances for which the behavior of the system is independent of the orientation of the pump beam. These are the geometries in which the static and oscillating magnetic fields are either orthogonal or parallel, the geometries leading to Haroche-like resonances (which, however, play a special role in modulated interaction components also), and a previously unknown class, labeled saddle-point resonances. However, other than for these special classes, the behavior of the system depends generally on the orientation of the optical pump beam, unlike as implied in some previous work on this problem. As in the recently studied case of a fully amplitude-modulated optical interaction [W. M. Ruyten, Phys. Rev. A 40, 1447 (1989)], it is found that the resonances in modulated interaction components are generally doubly branched, with one branch duplicating the resonances found in the time-averaged component. A new feature found for not-fully-amplitude-modulated and phase-modulated, off-resonance optical excitation (or magnetic resonance with nonparallel or nonorthogonal static and oscillating magnetic fields) is the occurrence of an extra type of resonance in modulated components of the interaction that cannot, like all other resonances of the system, be classified as power-shifted multiple quantum transitions.