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arXiv:2606.23908 · Journal of Applied Physics

Analysis of the frequency shift in coherent population trapping resonance's dynamic continuous-wave spectroscopy at the phase-jump modulation and its comparison with the conventional approach

E. D. Chivilis, E. A. Tsygankov

physics.atom-phquant-phSubmitted Jun 22, 2026• Announced Jun 24, 2026• 0 citations
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Abstract

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.

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Comment on “Light Shifts of Coherent Population Trapping Resonances in a Periodically Modulated Laser Field” (JETP Letters 122, 139 (2025))

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In this comment, I demonstrate that an addition to the light shift of the coherent population trapping resonance’s frequency, shown in [V. I. Yudin, M. Yu. Basalaev, A. V. Taichenachev, O. N. Prudnikov, M. D. Radchenko, and V. G. Palchikov, JETP Lett. 122, 144 (2025)], is not due to a new physical effect but is consequence of the asymmetry. To do this, the following model is considered. The phase of a monochromatic optical field is harmonically modulated at the frequency corresponding to the interval between ground-state levels of the Λ-scheme. The carrier is tuned to the low-frequency transition, while the first-order sideband is resonant to the high-frequency transition. I present curves demonstrating the opposite asymmetry of the resonance for the cases of the shift deviating from the light one in different directions.

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The appearance of the population difference in the ground state of the $$\Lambda $$ -system due to the spontaneous decay of the upper level into lower ones is considered. The resulting asymmetry of the coherent population trapping resonance is shown to lead to a frequency shift of the quadrature signal, which nonlinearly depends on the optical field intensity. In this case, this contribution is not suppressed due to the use of high modulation frequencies compared to the relaxation rate of ground state polarization (analogue of the Pound–Drever–Hall technique). A significant smallness of this shift when using a binary mixture of noble buffer gases is discussed.

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We investigate the coherent population trapping resonance induced by a polychromatic optical field with an asymmetric spectrum, i.e., whose sidebands equidistant from the carrier have unequal powers. A situation is considered where a modulation is used to provide the so-called in-phase and quadrature signals in the optical field's transmission, which are used for stabilization of the local oscillator's frequency in chip-scale atomic clocks. In a general case, the frequencies of the signals nonlinearly depend on the optical field's intensity due to the asymmetry of the resonance. In this work, we demonstrate a) that this effect stems from a multi-peak structure of the resonance; b) a linear dependence of the frequency on the optical field's intensity at high modulation values; c) that the regime of the resolved structure has more advantages than suppression of the frequency shift due to the spectrum asymmetry.

arXiv:2311.17229•Nov 28, 2023•0 citations
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.
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 study the field shift of coherent population trapping (CPT) resonance excited by a bichromatic field in an open Λ system with account for the Gaussian profile of the laser radiation intensity. Two methods for error signal formation are considered: the harmonic frequency modulation and the step phase modulation (phase jumps). It is shown that the spatial inhomogeneity of the light beam leads to an essentially nonlinear dependence of the error signal shift on the laser radiation intensity. We propose an approach for the linearization of this dependence, which is important for the development of methods for suppressing the field shift in atomic clocks based on CPT resonances.
In this comment, I demonstrate that an addition to the light shift of the coherent population trapping resonance’s frequency, shown in [V. I. Yudin, M. Yu. Basalaev, A. V. Taichenachev, O. N. Prudnikov, M. D. Radchenko, and V. G. Palchikov, JETP Lett. 122, 144 (2025)], is not due to a new physical effect but is consequence of the asymmetry. To do this, the following model is considered. The phase of a monochromatic optical field is harmonically modulated at the frequency corresponding to the interval between ground-state levels of the Λ-scheme. The carrier is tuned to the low-frequency transition, while the first-order sideband is resonant to the high-frequency transition. I present curves demonstrating the opposite asymmetry of the resonance for the cases of the shift deviating from the light one in different directions.
The appearance of the population difference in the ground state of the Λ\Lambda Λ -system due to the spontaneous decay of the upper level into lower ones is considered. The resulting asymmetry of the coherent population trapping resonance is shown to lead to a frequency shift of the quadrature signal, which nonlinearly depends on the optical field intensity. In this case, this contribution is not suppressed due to the use of high modulation frequencies compared to the relaxation rate of ground state polarization (analogue of the Pound–Drever–Hall technique). A significant smallness of this shift when using a binary mixture of noble buffer gases is discussed.
We have found that the error-signal frequency corresponding to the coherent population trapping resonance can be displaced from that of "0−00-00−0" transition unperturbed by the optical field, although the frequency is not sensitive to changes in its intensity. We consider the double ΛΛΛ-system of levels interacting with the asymmetric polychromatic optical field to demonstrate that this effect arises due to intensity nonlinearity of the error-signal frequency shift. The experiment with 87^{87}87Rb atoms in Ar-N2_22​ buffer gas atomic cell shows how the displacement value depends on different parameters. The possible influence of the effect on the clocks' frequency stability and reproducibility are discussed.
We investigate the coherent population trapping resonance induced by a polychromatic optical field with an asymmetric spectrum, i.e., whose sidebands equidistant from the carrier have unequal powers. A situation is considered where a modulation is used to provide the so-called in-phase and quadrature signals in the optical field's transmission, which are used for stabilization of the local oscillator's frequency in chip-scale atomic clocks. In a general case, the frequencies of the signals nonlinearly depend on the optical field's intensity due to the asymmetry of the resonance. In this work, we demonstrate a) that this effect stems from a multi-peak structure of the resonance; b) a linear dependence of the frequency on the optical field's intensity at high modulation values; c) that the regime of the resolved structure has more advantages than suppression of the frequency shift due to the spectrum asymmetry.