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Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS)

Analytical expressions for parameters of the dark resonance in a vacuum vapour cell

Denis V. Brazhnikov, Moustafa Abdel Hafiz, Grégoire Coget, Rodolphe Boudot

Published Jul 1, 2017• 0 citations
Publisher

Abstract

In spite of great number of papers devoted to theoretical and experimental study of the coherent population trapping phenomenon (CPT), still there is a lack of analytical results for the properties of the CPT-caused dark resonance in some principal cases. We will present the results of our study for single and double A schemes of atomic energy levels, which are of great importance for various applications of CPT (atom clocks, magnetometers, etc.). Analytical expressions for width and amplitude of the dark resonances have been obtained, which are valid for wide range of light wave intensities. The results concern closed as well as open A schemes. We will also discuss the difference between the two A schemes and state the condition when a double scheme can be effectively treated as a single A scheme. The analytical approximate formulas obtained have been compared with exact numerical solutions based on the optical Bloch equations as well as experimental.

DOI
10.1109/fcs.2017.8088974

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In spite of great number of papers devoted to theoretical and experimental study of the coherent population trapping phenomenon (CPT), still there is a lack of analytical results for the properties of the CPT-caused dark resonance in some principal cases. We will present the results of our study for single and double A schemes of atomic energy levels, which are of great importance for various applications of CPT (atom clocks, magnetometers, etc.). Analytical expressions for width and amplitude of the dark resonances have been obtained, which are valid for wide range of light wave intensities. The results concern closed as well as open A schemes. We will also discuss the difference between the two A schemes and state the condition when a double scheme can be effectively treated as a single A scheme. The analytical approximate formulas obtained have been compared with exact numerical solutions based on the optical Bloch equations as well as experimental.
This paper reports a dedicated theoretical and experimental study on the properties (signal amplitude and linewidth) of coherent population trapping resonances detected in vacuum vapor cells. Results are presented for conventional single-lambda schemes of atomic energy levels but also for double-lambda schemes, now widely used in various applications including atomic clocks and magnetometers. Approximate compact analytical expressions, valid for a wide range of light-wave intensities, i.e., beyond the low intensities or pump-probe regime, have been obtained. Analytical results are found to be in excellent agreement with exact numerical solutions based on the optical Bloch equations. Experimental results, obtained in a Cs vapor microfabricated cell, are reported and found to be in correct agreement with theoretical expressions.
We present a comparative study of the light-shifts (ac Stark shift) in a Rb vapour cell using two possible schemes for Rb atomic clocks: double resonance (DR) and coherent population trapping (CPT). For both schemes, the same wall-coated cell in a compact atomic resonator was used. The light-shift resulting from a monochromatic (DR) or a non-monochromatic (CPT) optical excitation was measured as a function of the laser intensity and the laser frequency and compared with existing theoretical results.
We study the peculiarities of the coherent population trapping (CPT) dark resonance in atomic vapor confined in a finite-size cell. Taking into account the scattering of atoms on anti-relaxation coating of the walls, we analyze the line shape of the CPT resonance in 133 C s depending on various parameters, such as the width of laser radiation spectrum, the type of pumping, and the magnitude of static magnetic field for both lin || lin and l i n ⊥ l i n polarization schemes. We show that for different parameters, the dependence of the line shape on the size of a cell is different, i.e., different factors affect the CPT resonance nonadditively. Thus, in the case when CPT resonance is excited by a narrowband laser, the shape of the resonance weakly depends on the size of a cell due to the laser-induced narrowing effect. We also show that the distortion of the line shape of the CPT resonance caused by nonresonant hyperfine sublevels of atomic excited states is modified with the size of a cell. Our calculation reveals the important role of accounting for the hyperfine and Zeeman structure of atomic levels. We have shown that the contrast of the CPT resonance differs about 5 times from the prediction of the simple three-level Λ model. The obtained results can be used for the optimization of the quantum frequency standards based on the CPT effect. In the final part of the paper, on the basis of the shot noise approximation we estimate the short-term stability of such standards.
Coherent population trapping (CPT) with losses towards off-resonant levels is analyzed on the basis of the coupled/noncoupled states. We derive that at a long interaction time the noncoupled state is depopulated at a rate that is decreasing for decreasing Raman detuning. This behavior produces a narrowing of the dark resonance with the interaction time. The asymptotic evolution of the atomic system is completely characterized by the effective width ${\ensuremath{\Gamma}}_{\mathrm{NC}}$ of the noncoupled state. The analytic expression for this width evidences a quadratic dependence on the Raman detuning. This dependence implies that the width of the dark resonance decreases asymptotically with the inverse of the square root of the interaction time. This narrowing law, independent of the interaction scheme, has a general validity for CPT in open systems. Our analysis points out the differences between CPT with losses and CPT in closed systems.
This work is concerned with the line profiles, amplitudes and widths of the coherent population trapping (CPT) resonances due to low frequency Zeeman coherences, in the Hanle configuration, applied to 87Rb atoms in the vacuum cell. Detailed calculations of the fluorescence were done with two different transverse intensity profiles of the excitation laser tuned to the open Fg = 2 → Fe = 1 transition. Comparison with experiment produces good agreement for the laser intensity dependence of CPT line-widths in the wide range of the laser intensity, when the laser transverse beam profile is Gaussian. Results for both step-like and Gaussian profile of the laser beam show asymptotic 1/√d dependence (d is the laser beam diameter) of CPT line-widths, for large range of the laser intensity.
We both numerically and analytically investigate on the light shift in Ramsey-CPT resonance for compact atomic clocks. The numerical calculation of the light shift is based on a density matrix analysis in Λ-type three state model. And the analytical method leads to an estimation equation of light shift in the Raman-Ramsey scheme. The estimation equation expresses the relationship between light shift and all pulse parameters. The results show that the estimation equation was reproduced well as the numerical results.