A model for description of the shapes of the coherent population trapping (CPT) resonances at different geometries of excitation and observation is developed. The numerical calculations, based on the irreducible tensor operator formalism, take into account the experimental geometry, the velocity distribution of the atoms, the Gaussian distribution of the laser beam intensity and the high rank polarization moments (HRPM). The results for different laser beam diameters, aperture of the detection, position of the photodetector etc. are compared with the experimental data. A method for description of the experimental shapes is proposed.
The shape of the coherent-population-trapping ͑CPT͒ resonances was investigated theoretically and experimentally at different laser powers. The CPT resonances were observed in fluorescence on the degenerate two-level system of the ͑F =2→ F f =1͒ transition of the 87 Rb D 1 line by means of a Hanle effect configuration in an uncoated vacuum cell. Numerical simulations based on the density matrix formalism, which take into account the high-rank polarization moment ͑HRPMs͒ influence and the velocity distribution of the atoms, were used to calculate the shape of the nonlinear magnetic resonances. The comparison of the theoretical and experimental shapes of the CPT resonances demonstrated that the HRPMs influence the shape at all laser excitation powers, and this influence can be used to explain some peculiarities at the center of the CPT resonance shape.There has been increasing interest in the investigation of the coherent-population-trapping ͑CPT͒ resonances, prepared and registered in different ways, because of many applications in high-resolution spectroscopy, magnetometry, lasing without inversion, laser cooling, ultraslow group velocity propagation of light, etc. ͓1-3͔. In many applications to ensure a reliable operation, a good knowledge of the internal and external factors influencing the resonance shape is required ͓4͔.The CPT resonance shape has been studied experimentally and theoretically in many works ͑see ͓1-6͔, and references therein͒. Most of the experimental investigations ͓1,2,7-11͔ were performed at low laser power, the shape of the resonances was practically Lorentzian, and the dependence of the resonance width on the laser power density was linear up to a few mW/ cm 2 . However, more precise investigations of the shape of the resonances showed that the theoretical shape could not reproduce the measured one ͓4,12-16͔, and in some works ͓4,12,14,16͔ theoretical dependences were proposed describing better the shape of the resonances for the particular experimental conditions. At a high laser power, the influence of the multiphoton interactions increases. The coherences obtained, described in density matrix representation by the components MM Ј , are related in irreducible representation to the polarization moments q k of rank k ͓k =0,1, ... ,2F , q = M − MЈ, where F is the total angular momentum of the lower level͔. For the case of F = 2, the maximum rank of the polarization moment is k = 4, the so-called hexadecapole moment.The manifestation of the hexadecapole moments in laserstimulated polarized fluorescence intensity was observed and described in terms of the iteration approach more than three decades ago ͓17͔. Recently, the influence of the transverse components ͑k =4, q = ±4͒ of the Rb lower-level hexadecapole moments on the transverse ͑k =2, q = ±2͒ upper-level quadrupole components was discussed ͓18͔. The influence of the high-rank polarization moments ͑HRPMs͒ on the nonlinear magneto-optical rotation resonances and their potential applications in nonlinear optics and in magnetometry were st...
In the present work we consider theoretically the influence of an arbitrary oriented additional magnetic field on the main properties of the Coherent Population Trapping (CPT) resonances in fluorescence and/or in absorption. The exact numerical solutions are obtained for the tensor components p (p -f, p) which describe the population and the longitudinal alignment of the resonant (f) levels with quantum numbers Ff 0 and Ff = 1, exited with a single frequency laser field from the ground (p) state with F 1 and F =2. Numerical simulations was used to obtain the width and the amplitude dependences of the resonances under different parameters and parasitic fields. The results are in qualitative agreement with resent experimental investigations.
Numerical simulations are used to obtain the amplitudes and the widths of the nonlinear resonances related to the different ranks of the tensor components k q in dependence on the atomic system parameters, and the laser field power (Rabi parameter -d.E/ ). The main attention is paid to the effect of the velocity distribution of the atoms on the hexadecapole (k = 4) components and their influence on the fluorescence and/or absorption signals. A detailed examination of our basic equations allows us to directly find out a relation between different rank components for a chosen transition and to clarify their role in the observed resonances for a given geometry.
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