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Journal of Applied Physics

Frequency locking: A distinctive feature of the coherent population trapping

Eugene A. Tsygankov, Dmitry S. Chuchelov, Maria I. Vaskovskaya, Vitali V. Vassiliev, Sergei A. Zibrov, Vladimir L Velichansky

Published Mar 12, 2025• 4 citations
Publisher

Abstract

We investigate the use of phase modulation to generate an error signal for frequency stabilization to a reference atomic transition. The study focuses on the high-frequency modulation regime. Our theoretical analysis shows that, in the case of coherent population trapping, the maximal slope of the error signal remains constant as the modulation frequency increases, whereas it steadily decreases for other types of resonances. Experimental results with 87Rb atoms confirm this behavior under comparison with the double radio-optical resonance.

DOI
10.1063/5.0252868

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We investigate the use of phase modulation to generate an error signal for frequency stabilization to a reference atomic transition. The study focuses on the high-frequency modulation regime. Our theoretical analysis shows that, in the case of coherent population trapping, the maximal slope of the error signal remains constant as the modulation frequency increases, whereas it steadily decreases for other types of resonances. Experimental results with 87Rb atoms confirm this behavior under comparison with the double radio-optical resonance.
We study the case where phase modulation of the harmonic signal is used to obtain the error signal for the frequency stabilization to a reference atomic transition. High-frequency modulation, or analog of the Pound-Drever-Hall regime, is considered. We demonstrate that for coherent population trapping, the maximal error-signal slope retains at a certain level with growth in the modulation frequency, while for other types of resonances it drops steadily. The investigation of the low-frequency modulation regime reveals the stationarity effect. We show that in this case, the maximal steepness of the error signal does not depend on the modulation frequency and is reached at a fixed value of the frequency deviation.
A method of detecting Coherent Population Trapping resonance by phase modulation (PM) was studied to expand the frequency locking range (LR). We calculated error signals by density matrix analysis using an eigenvector algorithm and verified the calculated result using an 87Rb vapor cell and a Rb-D1 vertical cavity surface-emitting laser. By comparing the error signal for PM with that for conventional frequency modulation (FM), it was found that the LR with PM is one order of magnitude wider than that with FM without degrading the frequency stability.
We study the coherent population trapping resonance induced by the radiation of a microwave-current-modulated diode laser. The situation is considered when a harmonic phase modulation of the microwave field is used to generate the error signal for the frequency stabilization of atomic clocks. We demonstrate that in this case the resonance acquires a multipeak structure. If it is asymmetric and the components of the multipeak structure are not fully resolved, then the resonance frequency nonlinearly depends on the laser field intensity. The peaks become separated at an increase in the frequency of the phase modulation, providing a linear dependence of the frequency on the intensity of the laser field. This theoretical prediction is confirmed experimentally. Other advantages of the high-frequency modulation regime are also discussed.
A method of dynamic continuous-wave spectroscopy of coherent population trapping (CPT) resonances using phase modulation of the jump type is developed. The time evolution of the spectroscopic signal is investigated. A method for the formation of an error signal for frequency stabilization is proposed. We show that our approach has a reduced sensitivity to the line-shape asymmetry of the CPT resonance. The experimental results are in good qualitative agreement with theoretical predictions based on a mathematical model of a three-level $\mathrm{\ensuremath{\Lambda}}$ system in a bichromatic field. This method can be used in atomic frequency standards (including chip-scale atomic clocks).
We theoretically investigate the dynamic regime of coherent population trapping (CPT) in the presence of frequency modulation (FM). We have formulated the criteria for quasi-stationary (adiabatic) and dynamic (non-adiabatic) responses of atomic system driven by this FM. Using the density matrix formalism for Λ system, the error signal is exactly calculated and optimized. It is shown that the optimal FM parameters correspond to the dynamic regime of atomic-field interaction, which significantly differs from conventional description of CPT resonances in the frame of quasi-stationary approach (under small modulation frequency). Obtained theoretical results are in good qualitative agreement with different experiments. Also we have found CPT-analogue of Pound-Driver-Hall regime of frequency stabilization.
We present the research of dynamic continuous-wave spectroscopy of the coherent population trapping resonance at the phase-jump modulation. Λ system of levels supplemented by a nonabsorbing state and bichromatic optical field, whose spectral components have different intensities, are considered. We demonstrate that the asymmetry leads to an additional nonlinear shift of the error-signal frequency under unisotropic relaxation of the ground-state density-matrix elements. We also investigate the conventional approach where the frequency difference of the optical field components is harmonically modulated to obtain the error signal. Comparison demonstrates that in the high-frequency modulation regime the corresponding frequency shift is more linear than at the phase-jump modulation for nonshort integration times.